1

On some mathematical connections between the Cyclic Universe, Inflationary Universe, p-adic Inflation, p-adic cosmology and various sectors of Number Theory Michele Nardelli 2,1 1 Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy Abstract This paper is a review, a thesis, of some interesting results that has been obtained in various researches concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmology. In the Section 1 we have described some equations concerning cosmic evolution in a Cyclic Universe. In the Section 2, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In the Section 3, we have described some equations concerning the generating Ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N = 1 five-dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In the Section 4, we have described some equations regarding the “null energy condition” violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with 1>w . In the Section 5, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In the Section 6, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In the Section 7, we have showed various and interesting mathematical connections between some equations concerning the p-adic Inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have showed the mathematical connections with various sectors of Number Theory, principally the Ramanujan’s modular equations, the Aurea Ratio and the Fibonacci’s numbers. Dedicated to the memory of Professor Anatolii Alexeevich Karatsuba (1937-2008), mathematical genius, whose original researches are for me always source of new great inspirations…

2

1. On some equations concerning cosmic evolution in a Cyclic Universe. [1] [16] The action for a scalar field coupled to gravity and a set of fluids iρ in a homogeneous, flat Universe, with line element ( )( )22222 xddNads r

+−= ττ is

( ) ( )( )∫ ⎥⎦

⎤⎢⎣

⎡+Σ−⎟

⎠⎞

⎜⎝⎛ +−= − φρβφτ VaaNaaNxdd ii

4422213 '21'3S . (1.1)

We use τ to represent conformal time and primes to represent derivatives with respect to τ . N is the lapse function. The background solution for the scalar field is denoted ( )τφ , and ( )φV is the scalar potential. The equations of motion for gravity, the matter and scalar field φ are straightforwardly derived by varying (1.1) with respect to a , N and φ , after which N may be set equal to unity. Expressed in terms of proper time t , the Einstein equations are

⎟⎠⎞

⎜⎝⎛ +++= MRVGH ρβρβφπ 4422

21

38 & , (1.2)

⎟⎠⎞

⎜⎝⎛ ++−−= MRVG

aa ρβρβφπ 442

21

38 &&&

, (1.3)

where a dot is a proper time derivative. With regard the trajectory in the ( )10 ,aa -plane, the Friedmann constraint reads

( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ −+=− 0

221

20

421

20 16

134'' φρβ Vaaaaa . (1.3b)

Now we solve the equations of motion immediately before and after the bounce. Before the bounce there is a little radiation present since it has been exponentially diluted in the preceding quintessence-dominated accelerating phase. Furthermore, the potential ( )φV becomes

negligible as φ runs off to minus infinity. The Friedmann constraint reads 22

'61' φ=⎟

⎠⎞

⎜⎝⎛

aa , and the

scalar field equation, ( ) 0''2 =φa , where primes denote conformal time derivatives. The general solution is

( )( )τφ inAH5ln23

= , ( )τφ inAHAAea 56/ == ,

( )( )τλλ inAHAa 51

0−+= , ( )( )τλλ inAHAa 5

11

−−= , (1.4)

3

where 6/∞≡ φλ e . We choose 0=τ to be the time when a vanishes so that 0<τ before collision. A is an integration constant which could be set to unity by rescaling space-time coordinates but it is convenient not to do so. The Hubble constants as defined in terms of the brane scale factors are

200 /' aa and 2

11 /' aa which at 0=τ take the values ( )inH53−+ λ and ( )inH5

3−− λ respectively.

Re-expressing the scalar field as a function of proper time ∫= τadt , we obtain

( ) ⎟⎠⎞

⎜⎝⎛= tinH52

3ln32φ . (1.5)

The integration constant ( ) 05 <inH has a natural physical interpretation as a measure of the contraction rate of the extra-dimension. We remember that when the brane separation is small, one can use the usual formula for Kaluza-Klein theory,

232

224

32

25 dyedseds φφ

+= − , (1.5b) where 2

4ds is the four-dimensional line element, y is the fifth spatial coordinate which runs from zero to L , and L is a parameter with the dimension of length. Thence, we have that:

φ

φ

φ 23

5

32

5

55 3

2 edt

ed

LdtdLH &=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

≡≡ , (1.6)

where φ

32

5 LeL ≡ is the proper length of the extra dimension, L is a parameter with dimensions of length, and 5t is the proper time in the five-dimensional metric,

dtedaedt φφτ 6

161

5−− =≡ , (1.7)

with t being FRW proper time. Notice that a shift ∞φ can always be compensated for by a rescaling of L . As the extra dimension shrinks to zero, 5H tends to a constant, ( )inH5 . Immediately after the bounce, scalar kinetic energy dominates and 5H remains nearly constant. The kinetic energy of the scalar field scales as 6−a and radiation scales as 4−a , so the former dominates at small a . It is convenient to re-scale a so that it is unity at scalar kinetic energy-radiation equality, rt , and denote the corresponding Hubble constant rH . The Friedmann constraint in eq. (1.3b) then reads

( ) ( )222 121' −+= aHa r , (1.8)

and the solution is

4

( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=23

31

32

535

2

2ln23

τ

τφ

r

r

H

HoutH , ττ rr HHa 221 22 += . (1.9)

The brane scale factors are

( ) ( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=+≡ −−− τλτλλλ φφ outHHHAeeaa r

r 32

531

61

1

23

6/6/10 2

21 ,

( ) ( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=+−≡ −−− τλτλλλ φφ outHHHAeeaa r

r 32

531

161

23

6/6/11 2

21 . (1.10)

Here the constant ( )( )31

561

/2 outHHA r= has been defined so that we match 0a and 1a to the incoming solution given in (1.4). As for the incoming solution, we can compute the Hubble

constants on the two branes after collision. They are ( ) 31

532

135

53 2 HHoutH r

−−− +± λλ on the positive

and negative tension branes respectively. For 532

5

2 HHr−< λ , the case of relatively little radiation

production, immediately after collision 0a is expanding but 1a is contracting. Whereas for

532

5

2 HHr−> λ , both brane scale factors expand after collision. If no scalar potential ( )φV were

present, the scalar field would continue to obey the solution (1.9), converging to

( )⎟⎟⎠

⎞⎜⎜⎝

⎛=

rC H

outH525

2ln32φ . (1.11)

This value is actually larger than ∞φ for 25

35 2−< λHHr , the case of weak production of radiation.

However, the presence of the potential ( )φV alters the expression (1.11) for the final resting value of the scalar field. As φ crosses the potential well travelling in the positive direction, 5H is reduced

to a renormalized value ( ) ( )outHoutH 55ˆ < , so that the final resting value of the scalar field can be

smaller than ∞φ . If this is the case, then 1a never crosses zero, instead reversing to expansion shortly after radiation dominance. If radiation dominance occurs well after φ has crossed the potential well, eq. (1.11) provides a reasonable estimate for the final resting value, if we use the corrected value ( )outH5

ˆ . The dependence of (1.11) is simply understood: while the Universe is

kinetic energy dominated, a grows at 31

t and φ increases logarithmically with time. However,

when the Universe becomes radiation dominated and 21

ta ∝ , Hubble damping increases and φ converges to the finite limit above. With regard the eqs. (1.6-1.11), we note the following connections with number theory:

5

( ) ( ) 171,3174802104,3322 7/267/1633/5 =Φ+Φ≅== − ; ( ) ( ) 826,2828427125,282 7/147/132/3 =Φ+Φ≅== − ; ( ) ( ) 3168,0314980262,02 7/407/203/5 =Φ+Φ≅= −−− ; ( ) ( ) 1231,1122462048,122 7/437/166/1 =Φ+Φ≅== − ; ( ) ( ) 6553,5656854249,5322 7/377/252/5 =Φ+Φ≅== − . Note that, 8244832 +=×= , where 8 and 24 are the “modes” that correspond to the physical vibrations of a superstring and the physical vibrations of the bosonic strings.

Here, we have used the following expression: ( ) 7/nΦ , with ...618033987,12

15=

+=Φ that is

the Aurea ratio, n is a natural number and 7 are the compactified dimensions of the M-Theory. Using the following potential ( ) ( ) ( )φφ φ FeVV c−−= 10 , (1.12) we consider the motion of φ back and forth across the potential well. V may be accurately approximated by φceV −− 0 . For this pure exponential potential, there is a simple scaling solution

( ) ptta = , ( )2031

tppeVV c −

−=−= − φ , 22c

p = , (1.13)

which is an expanding or contracting Universe solution according to whether t is positive or negative. A the end of the expanding phase of the cyclic scenario, there is a period of accelerated expansion which makes the Universe empty, homogeneous and flat, followed by φ rolling down the potential ( )φV into the well. After φ has rolled sufficiently and the scale factor has begun to contract, the Universe accurately follows the above scaling solution down the well until φ encounters the potential minimum. Let us consider the behaviour of φ under small shifts in the contracting phase. In the background scalar field equation and the Friedmann equation, we set

δφφφ += B and HHH B δ+= , where Bφ and BH are the background quantities given from (1.13). To linear order in δφ , one obtains

031312 =

−−

++ δφφδφδ

tp

tp &&& , (1.14)

with two linearly independent solutions, 1−≈ tδφ and pt 31− , where 1<<p . In the contracting phase, the former solution grows as t tends to zero. However, this solution is simply an infinitesimal shift in the time to the Big Crunch: φδφ &∝ . We next the incoming and outgoing collision velocity, which we have parameterized as ( )inH5 and

( )outH5 . Within the scaling solution (1.13), we can calculate the value of incoming velocity by treating the prefactor of the potential ( )φF in eq. (1.12) as a Heaviside function which is unity for

minφφ > and zero for minφφ < , where minφ is the value of φ at the minimum of the potential. We compute the velocity of the field as it approaches minφ and use energy conservation at the jump in V to infer the velocity after minφ is crossed. In the scaling solution, the total energy as φ

6

approaches minφ from the right is 2

22 3

21

tpV =+φ& , and this must equal the total energy 2

21φ&

evaluated for φ just to the left of minφ . Hence, we find that ( )ppVtp 31/6/6 min −==φ& at the minimum and, according to eq. (1.6),

( )2

23

21

min5

618

min

−−−≈

c

eVc

inHφ

. (1.15)

Note that from the eq. (1.15), we obtain:

( ) 2

223

min2

25 61

)(8][min

−−−≈

ceV

cinH

φ

,

where the number 8 is connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

At the bounce, this solution is matched to an expanding solution with ( ) ( ) ( ) 01 55 >+−= inHoutH χ , (1.16) where χ is a small parameter which arises because of the inelasticity of the collision. We shall simply assume a small positive χ is given, and follow the evolution forwards in time. Since χ is small, the outgoing solution is very nearly the time reverse of the incoming solution as φ starts back across the potential well after the bounce: the scaling solution is given in (1.13), but with t positive. We can treat χ as a perturbation and use the solution in eq. (1.14) discussed above,

1−≈ tδφ and pt 31− . One can straightforwardly compute the perturbation in 5Hδ in this growing mode by matching at minφ as before. One finds 2

55 /12 cHH Bχδ = where BH5 is the background

value, at the minimum. Beyond this point, 5Hδ grows as φ

23

/6 et c ∝ for large c , whereas in the

background scaling solution 5H decays with φ as φ⎟⎟⎠

⎞⎜⎜⎝

⎛− 2/

23 c

e . The departure occurs when the scalar field has attained the value

7

χ

φφ12

ln2 2

minc

cDep += , min

2

212 Vc

V ⎟⎠⎞

⎜⎝⎛≤

χ . (1.17)

As φ passes beyond Depφ the kinetic energy overwhelms the negative potential and the field passes onto the plateau 0V with 5H nearly constant and equal to

( ) ( )inHcoutHc

5

62

5 12ˆ

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

χχ , (1.18)

until the radiation, matter and vacuum energy become significant and 5H is then damped away to zero. Note that we can rewrite the eq. (1.18) as follow:

( )ccoutH

62

5 12ˆ

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

χχ

2

23

21

min

618

min

−−−×

c

eVc

φ

. (1.18b)

Also this equation is related with the number 8, i.e. with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

,

and with the number 12 (12 = 24 / 2) that is related to the physical vibrations of the bosonic strings by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

The time spent to the left of the potential well ( )minφφ < is essentially identical in the incoming and outgoing stages for 1<<χ , namely

min

min23 Vct ≈ . (1.19)

8

For the outgoing solution, when φ has left the scaling solution but before radiation domination, the definition eq. (1.6) may be integrated to give the time since the Big Bang at each value of φ ,

( )( ) ( )outH

eHeddt

5

23

5

23

ˆ32

ˆ32

φφ

φφ

φφφ ≈== ∫ ∫& . (1.20)

Also this equation can be rewritten as follow:

( )( )

φφ

φφ

φφφ 2

3

5

23

32

ˆ32 e

Heddt ≈== ∫ ∫&

cc6

2

12/ ⎟⎟

⎠

⎞⎜⎜⎝

⎛χ

χ2

23

21

min

618

min

−−−×

c

eVc

φ

. (1.20b)

The time in eq. (1.20) is a microphysical scale. The corresponding formula for the time before the Big Crunch is very different. In the scaling solution (1.13) one has for large c

( )( ) ( )

min2

2/2/

min

minmin 62 tc

ec

eV

tcc φφφφ

φ−−

−=−= . (1.21)

The large exponential factor makes the time to the Big Crunch far longer than the time from the Big Bang, for each value of φ . This effect is due to the increase in 5H after the bounce, which, in turn, is due to the positive value of χ . As the scalar field passes beyond the potential well, it runs onto the positive plateau 0V . The value of ( )outH5 is nearly cancelled in the passage across the potential

well, and is reduced to 5H given in eq. (1.18). Once radiation domination begins, the field quickly converges to the large t (Hubble-damped) limit of eq. (1.9), namely

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= rC HoutH /ˆ2ln

32

525

φ , (1.22)

where rH is the Hubble radius at kinetic-radiation equality. Also the eq. (1.22) can be rewritten as follow

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

− r

c

C Hc

eVc

c /61

812

2ln32

2

23

21

min

62

25 minφ

χχφ . (1.22b)

The dependence is obvious: the asymptotic value of φ depends on the ratio of ( )outH5

ˆ to rH .

Increasing ( )outH5ˆ pushes φ further, likewise lowering rH delays radiation domination allowing

the logarithmic growth of φ in the kinetic energy dominated phase to continue for longer. The solution of the scalar field equation is, after expanding eq. (1.9) for large τ , converting to proper time ( )∫= ττ dat and matching,

( ) ∫−−≈t

r Vdtaata

H0 ,

3333

φφ& , (1.23)

9

where as above we define ( )ta to be unity at kinetic-radiation equal density. We have that φ may reach its maximal value maxφ and turn around during the radiation, matter or quintessence dominated epoch. For example, maxφ is reached in the radiation era, if, from eq. (1.23),

( ) 11052

,

51

4max <⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛≈ C

m

r

m VV

tt

tt φ

φ

, (1.24)

where mt is the time of matter domination. For turn around in the matter era, we require

( ) 3010331

,

61

4 ≤⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛≤× −

Cm

r

VV

tt φ

φ

. (1.25)

Finally, if the field runs to very large Cφ , so that ( ) Cc

C ceVV φφ φ −≈/, is exponentially small, then φ

only turns around in the quintessence-dominated era. For our scenario to be viable, we require there to be a substantial epoch of vacuum energy domination (inflation) before the next Big Crunch. The number of e-foldings eN of inflation is given by usual slow-roll formula,

∫ ≈= 2, c

eVVdN

Cc

e

φ

φ

φ , (1.26)

for our model potential. For example, if we demand that the number of baryons per Hubble radius be diluted to below unity before the next contraction, which is certainly over-kill in guaranteeing that the cyclic solution is an attractor, we set 803 10≥eNe , or 60≥eN . This is easily fulfilled if Cφ is of order unity Planck units. Hence, the eq. (1.26) can be rewritten as follow:

602,

≥≈= ∫ ce

VVdN

Cc

e

φ

φ

φ . (1.26b)

With regard the eqs. (1.24-1.25, 1.26b) we have the following mathematical connections with the Aurea ratio: ( ) ( ) 0941,29059693843,003444185,297/417/49 =+=Φ+Φ − ; ( ) ( ) 95894,0381966011,0576974982,07/147/8 =+=Φ+Φ −− ;

( ) 4914670835,07/33,10 =Φ − ; ( ) 437054,291804914670835,0arcsin =⋅π

;

( ) 8715438560,07/2 =Φ − ; ( ) 361456,291808715438560,0arccos =⋅π

;

( ) ( ) 135048,6010060654,3103444185,297/507/49 =+=Φ+Φ ;

( ) 4914670835,07/33,10 =Φ − ; ( ) 562946,601804914670835,0arccos =⋅π

;

10

( ) 8715438560,07/2 =Φ − ; ( ) 638544,601808715438560,0arcsin =⋅π

.

From the formulae given above we can also calculate the maximal value Cφ in the cyclic solution: for large c and for min

1ttr−>> χ , it is

⎟⎟⎠

⎞⎜⎜⎝

⎛≈−

minmin ln

32

ttr

C χφφ , (1.27)

where we used rr tH ≈−1 , the beginning of the radiation-dominated epoch. From eq. (1.27) we obtain

22

3

0

min2

min

1 cer

VVNc

tt

⎟⎟⎠

⎞⎜⎜⎝

⎛≈χ

. (1.28)

This equation provides a lower bound on rt . The extreme case is to take 1min ≈V . Then using

1200 10−≈V , 10≈c , 60≈eN , we find 2510−≈rt seconds. In this case the maximum temperature

of the Universe is 1010≈ GeV. This is not very different to what one finds in simple inflationary models. We have shown that a cyclic universe solution exists provided we are allowed to pass through the Einstein-frame singularity according to the matching conditions, eqs. (1.15) and (1.16). Specifically, we assumed that ( ) ( ) ( )inHoutH 55 1 χ+−= where χ is a non-negative constant, corresponding to branes whose relative speed after collision is greater than or equal to the relative speed before collision. Our argument showed that, for each 0≥χ , there is a unique value of

( )outH5 that is perfectly cyclic. Now we show that an increase in velocity is perfectly compatible with energy and momentum conservation in a collision between a positive and negative tension brane, provided a greater density of radiation is generated on the negative tension brane. We shall assume that all other extra dimensions and moduli are fixed, and the bulk space-time between the branes settles down to a static state after the collision. We shall take the densities of radiation on the branes after collision as being given. By imposing Israel matching in both initial and final states, as well as conservation of total energy and momentum, we shall be able to completely fix the state of the outgoing branes and in particular the expansion rate of the extra dimension ( )outH5 , in terms of ( )inH5 . The initial state of empty branes with tensions T and T− , and with corresponding velocities 0<+v and 0>−v obeys

22 11 −+ −=− vTvT ; 22 11 −+ −

−−

=v

Tv

TEtot ; 22 11 −

−

+

+

−−

−=

vTv

vTvPtot . (1.29)

The first equation follows from Israel matching on the two branes as the approach, and equating the kinks in the brane scale factors. The second and third equations are the definitions of the total

11

energy and momentum. The three equations (1.29) imply that the incoming, empty state has −+ −= vv , 0=totE and that the total momentum is

( )

( )( )0

411 2

5

5 <−

=inLH

inTLHPtot , (1.30)

where we identify −+ − vv with the contraction speed of the fifth dimension, ( )inLH5 . For the eq. (1.15), we can rewrite the eq. (1.30) also as follow:

0

618

411

161

82

2

23

21

min

2

23

21

min

min

min

<

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−=

−

−

c

eVc

L

c

eVc

TLPtot

φ

φ

. (1.30b)

The corresponding equations for the outgoing state are easily obtained, by replacing T with

++ ≡+ TT ρ for the positive tension brane, and T− with −− −≡+− TT ρ for the negative tension brane, assuming the densities of radiation produced at the collision on each brane, +ρ and −ρ respectively, are given from a microphysical calculation, and are both positive. Writing ( ) ( )±± = θtanhoutv , where ±θ are the associated rapidities, one obtains two solutions

( )( )221

21sinh −+

−

−+ −+−= TTPP

T tottotθ ; ( )( )221

21sinh −+

−

+− −−±= TTPP

T tottotθ , (1.31)

where ++ +≡ ρTT , −− −≡ ρTT with +ρ and −ρ the densities of radiation on the positive and negative tension branes respectively, after collision. Both +ρ and −ρ are assumed to be positive. In the first solution, with signs (– +), the velocities of the positive and negative tension branes are the same after the collision as they were before it. In the second, with signs (– –), the positive tension brane continues in the negative y direction but the negative tension brane is also moving in the negative y direction. The corresponding values for ( )outv± and V are

( ) ( )( ) ( )2222222

221

2 −+−

−+

−+−

+

−+++

−+−=

TTPTTP

TTPPoutv

tottot

tottot , ( ) ( )( ) ( )2222222

221

2 −+−

−+

−+−

−

−+++

−−±=

TTPTTP

TTPPoutv

tottot

tottot ,

( ) ( )( ) ( ) ( )2212222

2222222

/2

−+−

−+−+

−+−

−+

−+−+

−+++−=

TTPTTTTPTTPTTP

Vtottot

tottot , or ( ) ( )

( )221

2222222 2

−+−

−+−

−+

++

−+++−=

TTPPTTPTTP

Vtottot

tottot , (1.32)

where the first solution for V holds for the (– +) case, and the second for the (– –) case. We are interested in the relative speed of the branes in the outgoing state, since that gives the expansion rate of the extra dimension, ( ) ( ) ( )outLHoutvoutv 5=+− −+ , compared to their relative speed

( )inLHv 52 −=− + in the incoming state. We find in the (– +) solution,

12

( )( )

( ) ( )( ) ( )2222222

22

5

5

24

2−+

−−++

−+

−+++

+=

−=

TTPTTPTP

voutvoutv

inHoutH

tottot

tot , (1.33)

and in the (– –) solution

( )( )

( )( ) ( )2222222

22

2

22

5

5

24

−+−

−+

−+

−+++

+−=

TTPTTPTP

PTT

inHoutH

tottot

tot

tot

. (1.34)

with totP given by (1.30) in both cases. We note that we can rewrite the relation above mentioned, i.e. ( ) ( ) ( )outLHoutvoutv 5=+− −+ also as follow:

( ) ( ) ( )outLHoutvoutv 5=+− −+ ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−+−=

−2

23

21

min

6181

min

c

eVc

Lφ

χ . (1.34b)

At this point we need to consider how the densities of radiation +ρ and −ρ depend on the relative speed of approach of the branes. At very low speeds, ( ) 15 <<inLH , one expects the outer brane collision to be nearly adiabatic and an exponentially small amount of radiation to be produced. The (– +) solution has the speeds of both branes nearly equal before and after collision: we assume that it is this solution, rather than the (– –) solution which is realised in this low velocity limit. As

( )inLH5 is increased, we expect +ρ and −ρ to grow. Now, if we consider +ρ and −ρ to be both TPtot <<<< , then the second term in the denominator dominates. If more radiation is produced on

the negative tension brane, +− > ρρ , then

( )( ) ( ) ( )

⎥⎦⎤

⎢⎣⎡ −+≈+≡ +−

TinHoutH

211

5

5 ρρχ (1.34c)

and so χ is small and positive. This is the condition necessary to obtain cyclic behaviour. Conceivably, the brane tension can change from T to tTT −=' at collision. Then, we obtain

( ) ( )⎥⎦⎤

⎢⎣⎡ +−+≈+ +−

Tt

2211 ρρχ . (1.34d)

For the (– +) solution, we can straightforwardly determine an upper limit for

( ) ( ) ( )χ+≡ 1/ 55 inHoutH . Consider, for example, the case there the brane tension in unchanged at

collision, 0=t . The expression in (1.33) gives ( ) ( )inHoutH 55 / as a function of +T , −T and totP . It

is greatest, at fixed −T and totP , when TT =+ , its smallest value. For 22 TPtot < , it is maximized for

222totPTT −=− , and equal to ( )2

2

41

TPtot+ when equality holds. For 2TPtot ≥ , it is maximized when

0=−T , its smallest value, and 22 2TPtot = , when it is equal to 154700538,134= . This is more than

13

enough for us to obtain the small values of χ needed to make the cyclic scenario work. A reduction in brane tension at collisions 0>t further increases the maximal value of the ratio. To obtain cyclic behaviour, we need χ to be constant from bounce to bounce. That is, compared to the tension before collision, the fractional change in tension and the fractional production of radiation must be constant.

We note that for 154700538,134= , we have the following mathematical connections with the

Aurea ratio:

154700538,134= ( ) ( ) 1539499,1220384833,0933565132,07/227/1 =+=Φ+Φ= −− ;

154700538,134= ( ) ( ) 156677713,15386437257,06180339887,07/97/7 =+=Φ+Φ= −− .

2. On some equations concerning cosmological perturbations in a Big Crunch/Big Bang space-time and M-Theory model of a Big Crunch/Big Bang transition. [2] [3] [4] [5] [16]

We consider a positive or negative tension brane with cosmological symmetry but which moves through the five-dimensional bulk. The motion through the warped bulk induces expansion or contraction of the scale factor on the brane. The scale factor on the brane obeys a “modified Friedmann” equation,

4265

35

2

3631

±±

±±± +−+±=

bbK

MLMH Cρρ , (2.1)

where ±ρ is the density (not including the tension) of matter or radiation confined to the brane, ±b is the brane scale factor, and ±H is the induced Hubble constant on the positive (negative) tension brane. Choosing conformal time on each brane, and neglecting the 2ρ terms equations (2.1) become

C+−+= ++++24

35

2

31' Kbb

LMb ρ , C+−−= −−−−

2435

2

31' Kbb

LMb ρ . (2.2)

where prime denotes conformal time derivative. The corresponding acceleration equations for +''b and −''b , from which C disappears, are derived by differentiating equations (2.2) and using ( ) ( )dbPbbd 334 −= ρρ with P being the pressure of matter or radiation on the brines. We now

show that these two equations can be derived from a single action provided we equate the conformal times on each brane. Consider the action ( ) ( )[ ]∫ −−+−

−++++

− −−+−−−= 4222422235

3 '3'3 bKbbNLbKbbNLMxdtNd ρρS , (2.3) where N is a lapse function introduced to make the action time reparameterization invariant. Varying with respect to ±b and then setting 1=N gives the correct acceleration equations for +''b and −''b following from (2.2). These equations are equivalent to (2.2) up to two integration constants.

14

We rewrite the action (2.3) in terms of a four-dimensional effective scale factor a and a scalar field φ , defined by

⎟⎠⎞

⎜⎝⎛=+ 6

cosh φab , ⎟⎠⎞

⎜⎝⎛−=− 6

sinh φab .

Clearly, a and φ transform as a scale factor and as a scalar field under rescalings of the spatial coordinates xr . To interpret φ more physically, note that for static branes the bulk space-time is perfect Anti-de Sitter space with line element ( )22/22 xddtedY LY r

+−+ . The separation between the branes is given by

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

+

6

cothlnlnφ

LaaLd ,

so d tends from zero to infinity as φ tends from minus infinity to zero. In terms of a and φ , the action (2.3) becomes

( ) maKaaLMxdtd SS +⎥⎦⎤

⎢⎣⎡ +−−= ∫ 22223

53

213 φ&& , (2.4)

which is recognized as the action for Einstein gravity with line element ( )( )ji

ij dxdxdtta γ+− 22 ,

ijγ being the canonical metric on 3H , 3S or 3E with curvature K , and a minimally coupled scalar field φ . The matter action mS is conventional, except that the scale factor appearing is not the

Einstein-frame scale factor but instead ( )6/cosh φab =+ and ( )6/sinh φab −=− on the positive and negative tension branes respectively. Now we wish to make use of two very powerful principles. The first is the assertion that even in the absence of symmetry, the low energy modes of the five-dimensional theory should be describable with a four-dimensional effective action. The second is that since the original theory was coordinate invariant, the four dimensional effective action must be coordinate invariant too. Since the five-dimensional theory is local and causal, it is reasonable to expect these properties in the four-dimensional theory. If furthermore the relation between the four-dimensional induced metrics on the branes and the four-dimensional fields is local, then covariance plus agreement with the above results forces the relation to be

( )( ) dgg 426/cosh µνµν φ=+ ( )( ) dgg 42

6/sinh µνµν φ−=− . (2.5) When we couple matter to the brane metrics, these expressions should enter the action for matter confined to the positive and negative tension branes respectively. Likewise we can from (2.4) and covariance immediately infer the effective action for the four-dimensional theory:

( ) [ ] [ ]∫ ++−− ++⎟⎟⎠

⎞⎜⎜⎝

⎛∂−−= ggRMgxd mm SSS 2

244

21

2φµ , (2.6)

where we have defined the effective four-dimensional Planck mass ( ) LMGM 3

51

424 8 == −π .

The two brane geometries are determined according to the formulae (2.5), and the background solution relevant post-collision is assumed to consist of two flat, parallel branes with radiation

15

densities ±ρ . The corresponding four-dimensional effective theory has radiation density rρ , and a massless scalar field with kinetic energy density φρ . The four-dimensional Friedmann equation in conformal time then reads

( ) ⎟⎠⎞

⎜⎝⎛ +≡+= 2

444

442 431'

aArAaaa r φρρ , (2.7)

where we have defined the constants 4A and 4r , and used the fact that the massless scalar kinetic energy 6−∝ aφρ . The solution to (2.7) and the massless scalar field equation ( ) 0''2 =φa is:

( )ττ 442 14 rAa += , ( )⎟⎟⎠

⎞⎜⎜⎝

⎛+

=ττφ4

4

1ln

23

rA . (2.8)

From these solutions, we reconstruct the scale factors on the branes according to (2.5), obtaining: ττ 441 rAb +±=± , (2.9) so we see that with the choice of normalization for the scale factor a made in (2.7), the brane scale factors are unity at collision. We may now directly compare the predictions (2.9) with the exact five-dimensional solution, equating the terms linear in τ to obtain

( ) ( ) ( )2/tanh12

1/1 0

2

4 yrrLLA ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+= −+ , ( )

( )2/tanh12 04 y

rrLr −+ += , (2.10)

where 0y is the rapidity associated with the relative velocity of the branes at collision ( )0tanh yV = and ±r is the value of the radiation density ±ρ on each brane at collision. Thence, the eq. (2.9) can be rewritten also:

( ) ( ) ( )τ2/tanh12

1/11 0

2

yrrLLb ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+±= −+

±( )

( )τ2/tanh12 0yrrL −+ ++ . (2.10b)

Furthermore, we define the fractional density mismatch on the two branes as

−+

−+

+−

=rrrrf , (2.11)

so that we have

( )2/tanh120

4 yLfrrr =− −+ . (2.12)

Now, we describe the perturbations of the brane-world system in terms of the four-dimensional effective theory. We shall now describe the scalar perturbations, in longitudinal gauge with a spatially flat background where the scale factor and the scalar field are given by (2.8). The perturbed line element is ( ) ( ) ( )[ ]2222 2121 xddads r

Ψ−+Φ+−= ττ . (2.13)

16

Since there are no anisotropic stresses in the linearized theory, we have Ψ=Φ . A complete set of perturbation equations consists of the radiation fluid equations, the scalar field equation of motion and the Einstein momentum constraint:

( )'334' 2 Φ−−= rr vkδ Φ+= rrv δ

41' ( ) ( ) ( ) Φ+−=+ '4'2'' 2 φδφδφδφ kH

( )δφφρ '21

32' 2 +=Φ+Φ rrvaH , (2.14)

where primes denote τ derivatives, rδ is the fractional perturbation in the radiation density, rv is the scalar potential for its velocity i.e. rr vv ∇=

rr , δφ is the perturbation in the scalar field, and from (2.8) we have the background quantities

( )( )[ ]ττ

τ

4

4

1221'

rr

aa

++

=≡H , and ( )[ ]ττφ

411'

32

r+= .

We are interested in solving these equations in the long wavelength limit, 1<<τk . Solving all the above equations for alnδ , one finds

( ) ( )wwi

i

+≈

+ 11δδ , ,,...1 Ni = (2.15)

for adiabatic perturbations. The components of the background energy density in the four-

dimensional effective theory are scalar kinetic energy, with 1=φw , and radiation, with 31

=rw . It

follows that for adiabatic perturbations, at long wavelengths we must have

rδδφ 23

≈ . (2.16)

In longitudinal gauge, the fractional energy density perturbation and the velocity potential perturbation in the scalar field (considered as a fluid with 1=w ) are given by

( )⎟⎟⎠

⎞⎜⎜⎝

⎛Φ−=

''2

φδφδφ ,

'φδφ

φ =v . (2.17)

From the equations (2.14) above (and using 2' −∝ aφ ) it follows that

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛ −

'2

23 2

'

φδφδδφ rr vk . (2.18)

Maintaining the adiabaticity condition (2.16) up to order ( )2τk then requires that the fractional velocity perturbations for the scalar field and the radiation should be equal: '/φδφ≈rv . Expressing the radiation velocity in terms of δφ , the momentum constraint then yields

17

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ Φ+Φ⎟⎟⎠

⎞⎜⎜⎝

⎛+≈

−

''2

321

1

φρρδφφ

Hr , (2.19)

where 22'21 −= aφρφ .

The above equations may be used to determine the leading terms in an expansion in τk of all the quantities of interest about the singularity. We shall choose to parameterize the expressions in terms

of the parameters describing the comoving energy density perturbation, Φ−= − 22

32 km Hε , which

has the following series expansion about 0=τ : ( ) ( )τετεε EDm 20 += , (2.20) where 0ε and 2ε are arbitrary constants, and

( ) ...ln2121 22

4 +−−= ττττ kkrD , ( ) ...2 +=ττE . (2.21)

For adiabatic perturbations, we obtain

( )τττεττ

εδφ kOkk

rkk

ln,43

43

41ln

83

49

222

24

220 ++⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−= , (2.22a)

( ) ( )τττττ

εφ kOrk

v ln,14

3420 +⎟

⎠⎞

⎜⎝⎛ −= , ( )τττδδ φ kOr ln,

32 22+= , ( )τττφ kOvvr ln,+= ,

( )τττεττ

ε kOkk

rkk

ln,8

38

53ln82

38

3222

24

220 +−⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅+

⋅+−=Φ , (2.22b)

( ) ( ) ( )τττετττ

εδφ kOkk

rkrk

ln,81

813

81ln

82121

83

6 222

24

4220 +−⎟⎟⎠

⎞⎜⎜⎝

⎛++

⋅+−= , (2.22c)

( ) ( )ττττεεζ kOkrkkkM ln,ln

4116

81

21 2

42

2022,4 +⎟⎠⎞

⎜⎝⎛ +++−= , (2.22d)

where M,4ζ is the curvature perturbation on comoving slices introduced by Mukhanov. With regard the (2.22c) we note that is possible the following mathematical connection with the Aurea ratio:

( ) 409,04091477,02

1540824829,06

17/13

7/13 ≅=⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ==

−

− .

Furthermore, in the eqs. (2.22b-2.22c) there is the number 8, that is related to the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function

18

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

,

and that 2, 3, 5, 8 and 13 are Fibonacci’s numbers. Now, we consider the propagation of metric perturbations through a collision of tensionless branes where the background space-time is precisely 3

2/ RZC ×M . The form that we take for the five-dimensional cosmological background metric is ( )( ) ( ) ji

ij dxdxytbdytdtytnds δ,, 222222 ++−= , (2.23) and we write the most general scalar metric perturbation about this as ( ) ( ) ( )( )+∇+∇−Γ−+−Φ+−= i

ii

i dydxtdtdxdytWdtdydtytnds βα 222222 2221221, ( ) ( )( ) ji

jiij dxdxytb χδ ∇∇−Ψ−+ 221,2 . (2.24) For perturbations on 3RC ×M it is straightforward to find a gauge in which the metric takes the form

( ) jijiij dxdxkkkdytdtkds ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ −++−⎟

⎠⎞

⎜⎝⎛ += χδχχ 2

321

341 222222 , (2.25)

and χ satisfies a massless scalar equation of motion on 3RC ×M . To be precise, the gauge is

0== βα , χ2k−Ψ−Φ=Γ , χ2

32 k=Φ , χ2

31 k=Ψ , 0=W . (2.26)

Notice that the non-zero variables can all be related to χ according to

( ) χ2

31,

32,

32,, k⎟

⎠⎞

⎜⎝⎛ ++−=ΨΦΓ . (2.27)

We shall, henceforth, refer to these as the “Milne ratio conditions”. Furthermore, imposing the 2Z symmetry, we obtain Neumann boundary conditions on χ , ( ) 0' =±yχ , (2.28) where 2/0yy ±=± are the location of the two 2Z fixed points. In the model space-time, the lowest energy mode for χ is y -independent and has the asymptotic form ( ) ktPQyt ln, +=χ , (2.29)

19

with Q and P being arbitrary constants. We have the following relations: ( ) ininout PQQ 2ln2 −+−= γ , inout PP = . (2.30) These relations are sufficient to determine the metric fluctuations after the bounce. We are only interested in the long-wavelength part of the spectrum, and, for the cases of interest, P is suppressed by 2k compared to Q . As a result, we obtain the approximate matching rule inout QQ −= , inout PP = . (2.31) The key conditions (2.26) through (2.28) are satisfied precisely for all time in a compactified Milne

2mod Z background. Now, we wish to use the four-dimensional effective (moduli) theory to infer the boundary data for the five dimensional bulk perturbations. In any four-dimensional gauge, the four-dimensional metric perturbation µνh and scalar field perturbation δφ determined the induced metric perturbations on the branes via the formulae (2.5): ( ) µνφµνµν δφghh ,ln2 ±

± Ω+= , (2.32) where ( )6/cosh φ=Ω+ and ( )6/sinh φ−=Ω− and the metric perturbations are fractional i.e.

µνµνδ hag 2= , ±±

± = µνµνδ hbg 2 . This formula is particularly easy to use in five-dimensional longitudinal gauge. This gauge may always be chosen, and it is completely gauge fixed. In this gauge the five-dimensional metric takes the form ( ) ( ) ( )( ) ( ) ( )( ) ji

ijLLLL dxdxytbdytdtdyWdtytnds δΨ−+Γ−+−Φ+−= 21,21221, 222222 . (2.33) In the absence of anisotropic stresses the brane trajectories are unperturbed in this gauge. An immediate consequence is that the four-dimensional longitudinal gauge scalar perturbation variables

±Φ and ±Ψ describing perturbations of the induced geometry on each brane ( ) ( ) ( )( )2222 2121 xddbds r

±±±±±± Ψ−+Φ+−= ττ , (2.34) are precisely the boundary values of the five-dimensional longitudinal gauge perturbations

( )±± Φ≡Φ yL and ( )±± Ψ≡Ψ yL . Using (2.32) and (2.34), we find for the induced perturbations

δφφ⎟⎠⎞

⎜⎝⎛+Φ=Φ+ 6

tanh6

14 , δφφ

⎟⎠⎞

⎜⎝⎛−Φ=Ψ+ 6

tanh6

14 ,

δφφ⎟⎠⎞

⎜⎝⎛+Φ=Φ− 6

coth6

14 , δφφ

⎟⎠⎞

⎜⎝⎛−Φ=Ψ− 6

coth6

14 . (2.35)

The brane conformal times may be expressed in terms of t by integrating,

( )∫±

± =t

ytqdt

0 ,τ , (2.36)

20

where nbq /≡ . So for example the boundary value of the bulk metric perturbation LΦ on the positive tension brane is given explicitly by

( ) ( )( ) ( )( )( ) ( )( )dtytqdtytqdtytqytL ∫∫∫ −+

−+

−++ +Φ=Φ 111

4 ,6/,tanh6

1,, δφφ , (2.37)

where +y is the location of the positive tension brane. Also in these equations (2.35-2.37), we have the following connection with the Aurea ratio:

( ) 409,04091477,02

1540824829,06

17/13

7/13 ≅=⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ==

−

− .

Now, using (2.22) and the following equation

( ) ( ) ( )( ) ( )( ) ( ) ( )( )[ ] ( )ktttLOyyrrLyyyLk

ytt M ln,,sinh3sinh182/cosh32

2/tanh 2200

200

022200

,4 ±−+ ++−−−−+=Ψ ρεζ

(2.38) to find Q and P before and after the bounce for all components of the metric perturbations and matching according to the rule given in equation (2.31) results in M,4ζ inheriting two separate scale-invariant long wavelength contributions in the post-singularity state. The first occurs as a direct consequence of the sign change in (2.31), and is independent of the amount of radiation generated at the singularity. The second is proportional to the difference in the densities of the radiation on the two branes. At leading order in velocities we have

( ) ( ) ( )222532

2044

220

,4 ,,3264

3 LLVVrOk

VrrVVLk

outoutinM ±

−±

−+ +−

−+=∆ ρεεζ , (2.39)

where inV and outV are the relative velocities of the branes before and after collision. Note that since

0ε∝P , matching P is in fact equivalent to matching 0ε across the collision. In terms of four-dimensional parameters including 4r given in (2.12) defining the abundance of the radiation and the fractional density mismatch f defined in (2.11), we find again at leading order in velocities

( )LVfr

kVV

Lkout

outinM

34

2044

220

,4 823

643

⋅−+=∆

εεζ . (2.40)

This is the final result, relevant to tracking perturbations across the singularity in the ekpyrotic and cyclic models. The result for the long wavelength curvature perturbation amplitude in the four-dimensional effective theory, propagated into the hot Big Bang after the brane collision is:

( )( ) ( ) 22

40

2220

643sinh

2/cosh2/tanh

169

LkV

Lkcoll

Mεθθ

θθεζ ≈−= (2.41)

21

where θ is the rapidity corresponding to the relative speed collV of the branes at collision, and the

second formula assumes collV is small. L is the bulk curvature scale, and 20

16kε has a scale invariant

power spectrum. With regard the eqs. (2.40) and (2.41), we have the following mathematical connections with the Aurea ratio and the Fibonacci’s numbers: 64 = 48 + 16 = 24 + 24 + 8 + 8; 182134864 2 +++== that are Fibonacci’s numbers;

8324 ×= ; ( ) ( ) 88838364 ++×+×= , with 3 and 8 that are Fibonacci’s numbers; ( ) 342224,3068 7/53 ⋅Φ≅ − .

Furthermore, in the eq. (2.40) 2, 3 and 8 are Fibonacci’s numbers and 8 is connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

The d + 1-dimensional space-time we consider is a direct product of d – 1-dimensional Euclidean space, 1−dR , and a two-dimensional time-dependent space-time known as compactified Milne space-time, or CM . The line element for 1−× d

C RM is thus 22222 xddtdtds r

++−= θ , 00 θθ ≤≤ , ∞<<∞− t , (2.42) where xr are Euclidean coordinates on 1−dR , θ parameterizes the compact dimension and t is the time. The compact dimension may either be a circle, in which case we identify θ with 0θθ + , or a

2Z orbifold in which case we identify θ with 02θθ + and further identify θ with θθ −02 . The fixed points 0=θ and 0θθ = are then interpreted as tensionless 2Z -branes approaching at rapidity

0θ , colliding at 0=t to re-emerge with the same relative rapidity. The orbifold reduction is the case of prime interest in the ekpyrotic/cyclic models, originally motivated by the construction of heterotic M theory from eleven dimensional supergravity. In these models, the boundary branes possess nonzero tension. However, the tension is a subdominant effect near 0=t and the brane collision is locally well-modelled by 1−× d

C RM . Now consider a string loop of radius R in M theory frame. Its mass M is Rπ2 times the effective string tension L2µ , where L is the size of the extra dimension. The effective Einstein-frame gravitational coupling is given by Ldd /2

12

+= κκ . The gravitational potential produced by such a loop in d spacetime dimensions is:

22

( ) 32

2

2 −−−

−=Φ dd

d RAdMκ (2.43)

where DA is the area of the unit D -sphere, ( )( )2/1/2

12 +Γ+

= DDAD π . Specializing to the case

of interest, namely 2-branes in eleven-dimensional M theory, the tension 2µ is related to the eleven dimensional gravitational coupling by a quantization condition relating to the four-form flux, reading ( )2

1123

2 /2 κπµ n= (2.44) with n an integer. Equations (2.43) and (2.44) then imply that the typical gravitational potential around a string loop is

( ) nnRnR

/64

105 20

162262

2

θµπµ

−≈−≈−=Φ− (2.45)

up to numerical factors. With regard the (2.45) we note that are possible the following mathematical connections with the Aurea ratio and the Fibonacci’s numbers: 105 = 89 + 16 = 55 + 34 + 8 + 8; 521105 ×= ; ( ) ( ) 8883838824241648182134864 2 ++×+×=+++=+=+++== ; and 3, 5, 8, 21, 34, 55 and 89 are Fibonacci’s numbers. Furthermore, we have that

640,1640625,164

105≅= ; ( ) ( ) 646,164625,1

215

215

7/217/57/217/5 ≅=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− .

Thence, the gravitational potential on the scale of the loops is of order 2

0θ and therefore is consistently small for small collision rapidity. Since the mean separation of the loops when they are produced is of order their size R , this potential Φ is the typical gravitational potential throughout space. Multiplying the tt component of the background metric (2.42) by Φ+ 21 and redefining t , we conclude that the outgoing metric has an expansion rapidity of order ( )2

00 1 θθ C+≈ with C a constant of order unity. We conclude that for small 0θ the gravitational back-reaction due to string loop productions is small.

2.1 On some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology.

We shall employ a coordinate system in which the five-dimensional line element for the background takes the form ( )( ) ( ) 2222222 ,, xdytbdytdtytnds r

++−= , (2.46) where y parameterizes the fifth dimension and ix , ,3,2,1=i the three non-compact dimensions. Cosmological isotropy excludes idtdx or idydx terms, and homogeneity ensures n and b are independent of xr . The yt, part of the background metric may then be taken to be conformally flat.

23

We find it simplest to work in coordinates in which the brane locations are fixed but the bulk evolves. The bulk metric is therefore given by (2.46), with the brane locations fixed at 0yy ±= for all time t . The five-dimensional solution then has to satisfy both the Einstein equations and the Israel matching conditions on the branes. The bulk Einstein equations read ν

µνµ δΛ−=G , where the

bulk cosmological constant is 2/6 L−=Λ . Evaluating the linear combinations 55

00 GG + and

( ) iiGGG 2/35

500 −+ , we find

012 22,

2,,, =+−+− ντττ ββββ eyyy , (2.47) ( ) 02

31 22

,2,,, =−−+− ν

τττ ββνν eyyy , (2.48)

where ( ) τeLt =/ , bln3≡β and ( )Lnt /ln≡ν . The Israel matching conditions on the branes read

Lnt

nn

bb yy == ,, , (2.49)

where all quantities are to be evaluated at the brane locations 0yy ±= . Now we express the metric as a series of Dirichlet or Neumann polynomials in 0y and y , bounded at order n by a constant times ny0 , such that the series satisfies the Israel matching conditions exactly at every order in 0y . To implement this, we first change variables from b and n to those obeying Neumann boundary conditions. From (2.49), nb / is Neumann. Likewise, if we define ( )ytN , by

( ) yytNnt

−=

,1 , (2.50)

then one can easily check that ( )ytN , is also Neumann on the branes. Since N and nb / obey Neumann boundary conditions on the branes, we can expand both in a power series

( ) ( ) ( )∑∞

=

+=3

0n

nn yPtNtNN , ( ) ( ) ( )∑∞

=

+=3

0/n

nn yPtqtqnb , (2.51)

where ( )yPn are polynomials

( ) 20

2

2yy

nnyyP nn

n−

−−= , ,...4,3=n (2.52)

satisfying Neumann boundary conditions and each bounded by ( ) ( )2/2 0 −< nyyP n

n , for the relevant range of y . Note that the time-dependent coefficients in this ansatz may also be expanded as a power series in 0y . By construction, our ansatz satisfies the Israel matching conditions exactly at each order in the expansion. Substituting the series ansatze (2.51) into the background Einstein equations (2.47) and (2.48), we may determine the solution order by order in the rapidity 0y . At each order in 0y , one generically obtains a number of linearly independent algebraic equations, and at most one ordinary differential equation in t . The first few terms of the solution are

24

( ) ...98241

211 4

022

00 +−+−= ytttyt

N (2.53) ...2725

61 2

02

3 +⎟⎠⎞

⎜⎝⎛ −+−= ytN (2.54)

and

...87

231 4

0422

02

0 +⎟⎠⎞

⎜⎝⎛ −+−= yttytq (2.55) ...2 2

03

3 +−= ytq (2.56).

With regard the eqs. (2.54-2.55) we note that are possible the following mathematical connections with the Aurea ratio and the Fibonacci’s numbers:

( ) 06849207,02

1506944,0725

7/397/39 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) 87154,02

15875,087

7/27/2 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− .

Furthermore, )324(72 ×= and 8 are connected with the “modes” that correspond to the physical vibrations of the bosonic strings and to the physical vibrations of a superstring by the following Ramanujan functions:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

,

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

To calculate the affine distance between the branes along a spacelike geodesic we must solve the geodesic equations in the bulk. Let us first consider the situation in Birkhoff-frame coordinates for which the bulk metric is static and the branes are moving. The Birkhoff-frame metric takes the form ( ) ( ) 222222 xdYAdTYNdYds r

+−= , (2.57) where for Schwarzschild-AdS with a horizon at 0=Y ,

25

( ) ( )( )LY

LYYA/2cosh/2cosh

0

2 = , ( ) ( )( )

( )( )

2

0

02

/2sinh/2sinh

/2cosh/2cosh

⎥⎦

⎤⎢⎣

⎡=

LYLY

LYLYYN . (2.58)

At 0=T , the Y -coordinate of the branes is represented by the parameter 0Y ; their subsequent trajectories ( )TY± can then be determined by integrating the Israel matching conditions, which read

( ) 21/2tanh ±± −±= VLY , where ( ) ( )±±± = YNdTdYV // are the proper speeds of the positive- and negative-tension branes respectively. From this, it further follows that 0Y is related to the rapidity

0y of the collision by ( )LYhy /2sectanh 00 = . For the purpose of measuring the distance between the branes, a natural choice is to use spacelike geodesics that are orthogonal to the four translational Killing vectors of the static bulk, corresponding to shifts in xr and T . Taking the xr and T coordinates to be fixed along the geodesic then, we find that λ,Y is constant for an affine parameter λ along the geodesic. To make the connection to our original brane-static coordinate system, recall that the metric function ( ) ( )YAytb 22 , = , and thus

( ) ( )2

,22

,2

24

2,,,,2

, λλλλ

λ θyttn

bybbtbb

Y yt +−=−+

= , (2.59)

where we have introduced the constant cVy /tanh 0 ==θ . Adopting y now as the affine parameter, we have ( )( ) ( )( )242222

,,2

,,2,

24222, 20 θθ −−++−+= btnbbtbbbtbnbb yyytyt , (2.60)

where t is to be regarded now as a function of y . We can solve this equation order by order in 0y using the series ansatz

( ) ∑∞

=

=0n

nn ycyt , (2.61)

where the constants nc are themselves series in 0y . Using the series solution for the background geometry given in the eqs. (2.53)-(2.56), and imposing the boundary condition that ( ) 00 tyt = , we obtain

( ) ( ) ( ) ( )80

70

40

20

206

050

3005

020

20

40

3003

020

200

00 60795250134

417

72051

24362

2yOytttytttyttyttytyttc +

++−⎟⎟

⎠

⎞⎜⎜⎝

⎛++++−

++−+=

(2.62)

( )70

60

60

40

205

030

40

40

202

0201 4

536

231809185

352 yOytttytyttytc +⎟⎟

⎠

⎞⎜⎜⎝

⎛+++−⎟⎟

⎠

⎞⎜⎜⎝

⎛++= (2.63)

( ) ( ) ( )60

50

40

204

050

30

030

20

20

2000

2 22342

48461

2yOyttytttytytttc +

++⎟

⎠⎞

⎜⎝⎛ +−−+

+−−= (2.64)

( ) ( )50

40

20

20

20

20

3 1820125

35 yOyttytc +

+−−= (2.65)

( )40

30

202

0

3000

4 125

47

485

245 yOytytttc +−⎟⎟

⎠

⎞⎜⎜⎝

⎛++= (2.66)

26

( )30

20

20

5 6061 yOytc += (2.67)

( )20

06 720

61 yOtc +−= (2.68)

( )07 0 yOc += . (2.69) Substituting 000 / yxt = and 0yy ω= we find ( ) ( )000 / yOyxx +=ω , i.e. to lowest order in 0y , the geodesics are trajectories of constant time lying solely along the ω direction. Hence in this limit, the affine and metric separation of the branes, defined with the following equation

( ) ( )20

1

14

420

4

4

11ln

1yO

xxLyOd

xxLdm +⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

=+−

= ∫− ωω

, (2.70)

must necessarily agree. To check this, the affine distance between the branes is given by

( ) ( ) ( )+

+−

+−+−

++=−= ∫− 3

30260

159104352'

60

40

20

50

50

3004

020

30

300

00220

0

yttytttytyttytdyttnLd y

ya

( ) ( )80

70

70

50

300

252012795552331115 yOytttt

++−+

+ , (2.71)

which to lowest order in 0y reduces to

( ) ( )20

80

70

50

30

0 168853

2053

352 yOxOxxxx

Lda +++++= , (2.72)

in agreement with the series expansion of (2.70). We obtain also the following equation:

( ) ( )( )∫−

−=

−==

1

1 224

440421

24404

122

4

xxxde

bxnx

Ld x

m ξωξδ

( ) ( ) ( )( ) ( ) ( )( )⎥⎦⎤

⎢⎣⎡ +−+

−= 41041040040042

4

~~4~~~38

11 xkYBxkJAxkYBxkJAxk

x. (2.72b)

To evaluate the perturbation adδ in the affine distance between the branes, consider

( )∫ ∫∫ +⎥⎥⎦

⎤

⎢⎢⎣

⎡=++= λ

δδδδδλλδσρ

ρσ

νµµν

σρρσ

νννµ

µννµκ

κµννµ

µνσρρσ

νµµν d

xxg

xxg

xxgxxxxgxxxgxxg

xxgddxxg

&&

&&

&&

&&&&&&&

&&&&

212

21

,

(2.73) where dots indicate differentiation with respect to the affine parameter λ , and in going to the second line we have integrated by parts and made use of the background geodesic equation

νµσµνσ xxgx &&&& ,2

1= and the constraint 1=νµ

µν xxg && . If the endpoints of the geodesics on the branes

are unperturbed, this expression is further simplified by the vanishing of the surface term. Converting to coordinates where 000 / yxt = and 0yy ω= , to lowest order in 0y the unperturbed

27

geodesics lie purely in the ω direction, and so the perturbed affine distance is identical to the following perturbed metric distance

∫ ∫− −

−=Γ=

0

0

241

121

244

y

y

x

Lm de

bnxdynt

Ld ωξδ . (2.74)

Explicitly, we find

( ) ( ) ( ) ( ) ++−+⎥⎦

⎤⎢⎣

⎡ ++

+−

+−= 4

020

30

300

0

20

0

020 44

39

12342 yAtByttA

ttB

tyAtB

Ldaδ

( ) ( ) ( ) ( )+⎥

⎦

⎤⎢⎣

⎡ +−++⎥

⎦

⎤⎢⎣

⎡ +++++− 6

0

20

20

205

00

40

20

20

40

20

65744214

12037511101213521692 ytBtAty

tttAtttB

( ) ( ) ( )80

70

0

60

40

20

20

60

40

20

10080360152935171524811428875952866888854 yOy

ttttAttttB

+⎥⎦

⎤⎢⎣

⎡ ++−++++− ,

(2.75) which, substituting 000 / yxt = and dropping terms of ( )2

0yO , reduces to

( )80

70

50

50

30

3000

0 24343~

96275

425~

893

4

~2

~2 xOAxxBAxxBAxxBAxxB

Lda +−−−−−−−−=δ , (2.76)

where 2

0~ ByB = . Also this expression is in accordance with the series expansion of (2.74). However,

the perturbed affine and metric distance do not agree at ( )20yO .

With regard the eqs. (2.72) and (2.76) we have the following mathematical connections with Aurea ratio and with Fibonacci’s numbers:

( ) ( ) 663054757,12

152

15666,135

7/207/57/207/5 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅=

−

− ;

( ) ( ) 663697,22

152

1565,22053

7/147/127/147/12 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅=

−

−

( ) ( ) ( ) 0718,52

152

152

15077381,5168853

7/107/167/217/107/167/21 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ≅=

−−

−−

( ) ( ) 12089,12

152

15125,189

7/107/77/107/7 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅=

−−

−− ;

( ) ( ) 236067977,62

152

1525,6425

7/77/287/77/28 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ−Φ≅=

−

− ;

( ) ( ) 870901,22

152

15864583,296275

7/207/147/207/14 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅=

−

− ;

( ) ( ) 3078,142

152

152916,1424343

7/177/357/177/35 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅= .

28

Furthermore, we have that: 53 = 21 + 34 – 2; 853 = 2 + 8 + 233 + 610; where 2, 8, 21, 34, 233 and 610 are Fibonacci’s numbers;

239 = ; 233425 −= ; 275 = 233 + 34 + 8; 343 = 233 + 34 + 8 + 13 + 55; where 3, 8, 13, 34, 55 and 233 are Fibonacci’s numbers.

3. On some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang. [6] [7] [8] [16]

With regard the ekpyrotic perturbations including gravity, we consider the action for N decoupled fields interacting only through gravity:

( ) ( )∫ ∑ ∑ ⎥⎦

⎤⎢⎣

⎡−∂−−

= =

N

i

N

iiii VRgxd

1 1

24

21

21 φφ , (3.1)

where we have chosen units in which 18 2 =≡ −

PlMGπ . In a flat Friedmann-Robertson-Walker background with line element ( )dtadtds 222 +−= x2, the scalar field and Friedmann equations are given by 03 , =++

iiii VH φφφ &&& (3.2) and

( )⎥⎦

⎤⎢⎣

⎡+= ∑ ∑

i iiii VH φφ 22

21

31 & , (3.3)

where aaH /&= and ( )iii VV

iφφ ∂∂= /, with no summation implied. Another useful relation is

∑−=i

iH 2

21 φ&& . (3.4)

If all the fields have negative exponential potentials ( ) iic

iii eVV φφ −−= then as is well-known, the Einstein-scalar equations admit the scaling solution

( )pta −= , ( )tAc i

ii −= ln2φ , 2

22i

ii c

AV = , ∑=i ic

p 22 . (3.5)

Thus, if 1>>ic for all i , we have a very slowly contracting universe with 1<<p . We focus on the entropy perturbation since this is a local, gauge-invariant quantity, and on the case of only two scalar fields. The entropy perturbation equation

29

032

2

22

21

,1,222

21

,2

1,21,222 21222111 =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

++−

++ sVVVVV

ks δφφφφ

φφφφφφ

δ φφφφφφφφ

&&

&&

&&

&&&&&& (3.6)

in flat spacetime is replaced by

Φ+

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

++−

+++22

21

2

22

22

21

,1,222

21

,2

1,21,22

2

2 432

3 21222111

φφθδ

φφφφ

φφφφφφ

δδ φφφφφφφφ

&&

&

&&

&&

&&

&&&&&&&

aks

VVVVVaksHs . (3.7)

For simplicity we will focus attention on straight line trajectories in scalar field space. Since 0=θ& , the entropy perturbation is not sourced by the Newtonian potential Φ and we can solve the equations rather simply. We shall assume that the background solution obeys scaling symmetry so that 12 φγφ && = . Denoting τ derivatives with primes, and introducing the re-scaled entropy field ( ) saS δτδ = , (3.8) eq. (3.7) becomes

0'''' ,22 =⎟

⎠⎞

⎜⎝⎛ +−+ SVa

aakS δδ φφ . (3.9)

The crucial term governing the spectrum of the perturbations is then

⎟⎠⎞

⎜⎝⎛ − 2

,2 '' aV

aa

φφτ . (3.10)

When this quantity is approximately 2, we will again get nearly scale-invariant perturbations. It is customary to define the quantity

( ) ( )2

22

2

22

21

21

21

23

HHw φγφφε

&&& +=

+≡+≡ . (3.11)

In the background scaling solution,

( )2

2

12 γε

+=

c . (3.12)

We proceed by evaluating the quantity in (3.10) in an expansion in inverse powers of ε and its derivatives with respect to N , where ( )endaaN /ln= , where enda is the value of a at the end of the ekpyrotic phase. Note that N decreases as the fields roll downhill and the contracting ekpyrotic phase proceeds. We obtain the first term in (3.10) by differentiating (3.4), obtaining

⎟⎠⎞

⎜⎝⎛ −= ε

2112'' 22aH

aa . (3.13)

The second term in (3.10) is found by differentiating (3.11) twice with respect to time and using the background equations and the definition of N . We obtain

30

( )0,

222,

2

2562 εεεεφφ OHaVa N +⎟

⎠⎞

⎜⎝⎛ −−−= . (3.14)

Finally, need to express ( ) aHaa =≡ /'H in terms of the conformal time τ . From (3.13) we obtain ( )ε−= 1' 2HH , (3.15) which integrates to

( )∫ −=− τετ

0

1 1dH . (3.16)

Now, inserting ( ) ττ dd /1= under the integral and using integration by parts we can re-write this as

( ) ⎟⎠⎞

⎜⎝⎛ −−= ∫−− τ

ττεετε

ετ0

11 '11 dH . (3.17)

Using the same procedure once more, the integral in this expression can be written as

( ) ( ) ( )∫ ∫−− −=τ τ

τττετ

ετετεττεετ

0 0

11 ''' dddd . (3.18)

Now using the fact that N,' εε H= , and that to leading order in ε/1 , H can be replaced by its value

in the scaling solution (with constant ε ), 1−= ετH , we can re-write the second term on the right-hand side as

( ) ( ) ( )∫ ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=− −− τ τ

εε

ετεττττε

τετ

0 0,

,11 1'N

Ndddd , (3.19)

which shows that this term is of order 2/1 ε and can thus be neglected. Altogether we obtain

( )∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−−≈−=− τ

εε

εετετ

0 2,1 111 NdH . (3.20)

Using (3.13) and (3.14) with (3.20) we can calculate the crucial term entering the entropy perturbation equation,

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=⎟

⎠⎞

⎜⎝⎛ − 2

,2,

2

43

2312''

εε

ετ φφ

NaVaa . (3.21)

The deviation from scale-invariance in the spectral index of the entropy perturbation is then given by

2,21εε

εN

sn −=− . (3.22)

The first term on the right-hand side is the gravitational contribution, which, being positive, tends to make the spectrum blue. The second term is the non-gravitational contribution, which tends to make the spectrum red.

31

Now defining R to be the curvature perturbation on comoving spatial slices, for N scalar fields with general Kahler metric ( )φijg on scalar field space, the linearized Einstein-scalar field equations lead to

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ−−= 2

2

2

2

aks

DtDg

HH j

i

ijφ

&&R , (3.23)

where the 1−N entropy perturbations

( )( ) ml

lm

kjjkiii

gg

sφφφδφφφ

φδφ &&

&&−= (3.24)

are just the components of iδφ orthogonal to the background trajectory, and the operator 22 / DtD is just the geodesic operator on scalar field space. Things simplify because the scalar field space is flat, so the metric is ijijg δ= , and DtD / reduces to an ordinary time derivative. Considering only two scalar fields, we have 2

22

121 / φφδφ &&& +−= ss , 2

22

112 / φφδφ &&& ++= ss . (3.25)

For a straight line trajectory in field space, the right-hand side of (3.24) vanishes even if the entropy perturbation is nonzero. We assume that the scalar field bounce occurs after the ekpyrotic potentials are turned off, so that the universe is kinetic-dominated from the 4d point-of-view. The scalar field trajectory is

12~φγφ && −= , for btt < , and 12

~φγφ && = , for btt > , with 1φ& constant and negative in the vicinity of the bounce. The bounce leads to a delta function on the right-hand side of (3.23),

( ) ( )+−= bb tttDtD

222

2

2φδφ & , (3.26)

where bt is the time of the bounce of the negative-tension brane. As can be readily seen from (3.23), if the entropy perturbations already have acquired a scale-invariant spectrum by the time bt , then the bounce leads to their instantaneous conversion into curvature perturbations with precisely the same long wavelength spectrum. We can estimate the amplitude of the resulting curvature perturbation by integrating equation (3.24) using (3.26). Since we have assumed the universe is kinetic-dominated at this time, ( )tH 3/1= . Since the entropy perturbation ( ) 2

22

11221 / φφδφφδφφδ &&&& +−≡s (3.27) is canonically normalized, its spectrum is given by

∫= 232

22 1

4 tkdkkπ

δφ h (3.28)

up to non-scale invariant corrections. This expression only holds as long as the ekpyrotic behaviour is still underway: the ekpyrotic phase ends at a time endt approximately given by ( )22

min /2 endtcV = .

After endt , the entropy perturbation obeys 01 =+ − sts δδ &&& , which has the solution ( )tBAs −+= lnδ .

32

Matching this solution to the growing mode solution 1−t in the ekpyrotic phase, one finds that by bt the entropy grows by an additional factor of ( )bend tt /ln1+ . Employing the Friedmann equation to relate 12

~φγφ && = to H , putting everything together and restoring the Planck mass, we find for the variance of the spatial curvature perturbation in the scale-invariant case,

( ) ( )( ) ( )∫ ∫ ∆≡++

= kk

dkk

dkttM

Vcbend

Pl

2222

2

22min

22 /ln1~1

~

3 RRγγ

πh (3.29)

for the perfectly scale-invariant case. Notice that the results depends only logarithmically on bt : the main dependence is on the minimum value of the effective potential and the parameter c . Observations on the current Hubble horizon indicate ( ) 92 102.2 −×≈∆ kR . Ignoring the logarithm in

(3.29), this requires PlMVc 321

min 10−≈ , or approximately the GUT scale. This is of course entirely consistent with the heterotic M-theory. With regard the eqs. (3.29) we have the following mathematical connections with the Aurea ratio:

( ) 03329,02

1552

152

103377,03

17/16

7/162 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +⋅

=Φ⋅

≅=−

−

π. If we take 4340,0=PlM , then we

have that

( ) 1793145665,02

15179307948,03

17/25

7/2522 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

−

PlMπ. Furthermore, we obtain

( ) ( ) 13455906702,793289,101801793145665,0arccos1801793145665,0arcsin ++=≅+=⋅+⋅ππ

,

with 34 and 55 Fibonacci’s numbers. If the entropic perturbations are suddenly converted to curvature perturbations, the curvature perturbations inherit the spectral tilt given in (3.22). We now begin by re-expressing eq. (3.22) in terms of N , the number of e-folds before the end of the ekpyrotic phase (where ( )Nd 1−= εN and 1>>ε ):

Nd

dnsε

εln21 −=− . (3.30)

This expression is identical to the case of the Newtonian potential perturbations, except that the first term has the opposite sign. In this expression, )(Nε measures the equation of state during the ekpyrotic phase, which must decrease from a value much greater than unity to a value of order unity in the last N e-folds. If we estimate αε N≈ , then the spectral tilt is

NNα

α −≈−21sn . (3.31)

Here we see that the sign of the tilt is sensitive to α . For nearly exponential potentials ( )1≈α , the spectral tilt is 02.1/11 ≈+≈ Nns , slightly blue, because the first term dominates. However, there are well-motivated examples in which the equation of state does not decrease linearly with N . We have introduced α to parameterize these cases. If 14.0>α , the spectral tilt is red. For example,

33

97.0=sn for 2≈α . These examples represent the range that can be achieved for the entropically-induced curvature perturbations in the simplest models, roughly 02.197.0 << sn . For comparison, if we use the same estimating procedure for the Newtonian potential fluctuations in the cyclic model (assuming they converted to curvature fluctuations before the bounce through 5d effects), we obtain 97.095.0 << sn . This range agrees with the estimate obtained by an independent analysis based on studying inflaton potentials directly. With regard the values of sn , i.e. 0.95, 0.97 and 1.02, we have the following mathematical connections with the Aurea ratio:

( )7/67,0

7/67,0

2159552046220,095,0

−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ=≅ ;

( )7/33,0

7/33,0

2159773457024,097,0

−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ=≅ ;

( )7/33,0

7/33,0

2150231794109,102,1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ=≅ ;

A second way of analyzing the spectral tilt is to assume a form for the scalar field potential. Consider the case where the two fields have steep potentials that can be modelled as

( ) ∫−=− φ

φcd

eVV 01 and 12 φγφ && = . Then eq. (3.22) becomes

( )2,

22

2 4141cc

Mcn

Pls

φγ−

+=− , (3.32)

where we have used the fact that ( )φc has the dimensions of inverse mass and restored the factors of Planck mass. The presence of PlM clearly indicates that the first term on the right is a gravitational term. It is also the piece that makes a blue contribution to the spectral tilt. The second term is the non-gravitational term and agrees precisely with the following flat space-time result

2,41

cc

nsφ−=− , (3.33)

although the agreement is not at all obvious at intermediate steps of the calculation. For a pure exponential potential, which has 0, =φc , the non-gravitational contribution is zero, and the spectrum is slightly blue. For plausible values of 20=c and 2/1=γ , say, the gravitational piece is about one percent and the spectral tilt is 01.1≈sn , also consistent with our earlier estimate. We note that 20=c , is related with the Aurea ratio by the following mathematical formula:

( ) ( )[ ] 562,2023708,13618,2090,11

23

215

215

23

7/147/357/147/35 =⋅=+≅⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=⋅Φ+Φ .

34

In the cyclic model, the steepness of the potential must decrease as the field rolls downhill in order that the ekpyrotic phase comes to an end, which corresponds to 0, >φc . If ( )φc changes from some initial value 1>>c to some value of order unity at the end of the ekpyrotic phase after φ changes by an amount φ∆ , then φφ ∆≈ /, cc . When c is large, the non-gravitational term in eq. (3.32) typically dominates and the spectral tilt is a few per cent towards the red. For example, suppose βφ∝c and ( )∫ ≈125φφ dc ; then, the spectral tilt is

β

β+

−=−1

03.01sn , (3.34)

which corresponds to 197.0 << sn for positive ∞<< β0 , in agreement with our earlier estimate. With regard the value 125, we have the following mathematical connection with the Fibonacci’s numbers: 125 = 2 + 5 + 8 + 21 + 34 +55. We note that negative potentials of this type with very large values of c have been argued to arise naturally in string theory. Our expression for the spectral tilt of the entropically induced curvature spectrum can also be expressed in terms of the customary “fast-roll” parameters

2

2

,

1cV

V=⎟

⎟⎠

⎞⎜⎜⎝

⎛≡

φ

ε φφ

η,,⎟⎟⎠

⎞⎜⎜⎝

⎛≡

VV . (3.35)

Note that ( )( )εγε 212/1 += . Then, the spectral tilt is

( ) ηεγ 4141 2

2

−+

=−Pl

s Mn . (3.36)

This result can be compared with the spectral index of the time-delay (Newtonian potential) perturbation, where the corresponding formula is

ηε 441 2 −−=−Pl

s Mn . (3.37)

Here, the first term is again gravitational, but it has the opposite sign of the gravitational contribution to the entropically induced fluctuation spectrum. So, the tilt is typically a few per cent redder. Finally, for inflation, the spectral tilt is ηε 261 +−=−sn (3.38) where the result is expressed in terms of the slow-roll parameters ( )( )2, /2/1 VVM Pl φε ≡ and

VVM Pl /,2

φφη ≡ . Here we have revealed the factors of PlM to illustrate that both inflationary contributions are gravitational in origin. This gives the same range for sn as the Newtonian potential perturbations in the cyclic model.

35

3.1 On some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N = 1 five-dimensional supergravity with four-dimensional boundaries.

We will now briefly review the effective description of strongly coupled heterotic string theory as 11-dimensional supergravity with boundaries given by Horava and Witten. The bosonic part of the action is of the form YMSG SSS += (3.39) where SGS is the familiar 11-dimensional supergravity

⎥⎦

⎤⎢⎣

⎡++−−= ∫ 11874321

11111 ......

...2 1728

2241

21

IIIIIIIIIIJKL

IJKLMSG GGCGGRgS εκ

(3.40)

and YMS are the two 8E Yang-Mills theories on the orbifold planes explicitly given by

( )( ) ( )( )( )( )∫ ∫ ⎭⎬⎫

⎩⎨⎧ −−⎟

⎠⎞

⎜⎝⎛−

⎭⎬⎫

⎩⎨⎧ −−⎟

⎠⎞

⎜⎝⎛−=

110

210

2223/2

2221

3/2

2 21

481

21

481

M MYM trRFtrgtrRFtrgSπκ

πκπκ

πκ.

(3.41) Here ( )i

JIF are the two 8E gauge field strengths and IJKC is the 3-form with field strength

[ ]JKLIIJKL CG ∂= 24 . In order for the above theory to be supersymmetric as well as anomaly free, the Bianchi identity for G should receive a correction such that

( ) ( ) ( ) ( ) ( ) LKJILKJI xJxJdG πρδδπκ

π−+⎟

⎠⎞

⎜⎝⎛−= 112111

3/2

11 4221 (3.42)

where the sources are given by

( ) ( ) ( ) ⎟⎠⎞

⎜⎝⎛ ∧−∧= RtrRFtrFJ iii

21 . (3.43)

With regard the eqs. (3.40) and (3.42), we have the following mathematical connections with the aurea ratio and the Ramanujan modular equations:

( ) 4101875,12

15414213562,127/5

7/5 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ≅= ; 3243242444321728 2 ⋅=⋅⋅=⋅= ;

4101875817,1342,306432 ⋅= Hz ; 832432 3 ⋅⋅= ;

( ) 1133912969,02

15112539539,022

17/67,31

7/67,31 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

−

π We note that 2, 3 and 8 are

Fibonacci’s numbers, while 24 is related to the physical vibrations of the bosonic strings by the following Ramanujan function:

36

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

While the standard embedding of the spin connection into the gauge connection ( ) ( ) RtrRFtrF ∧=∧ 11 (3.43b) leads to vanishing source terms in the weakly coupled heterotic string Bianchi identity, in the present case, one is left with non-zero sources RtrR ∧± on the two hyperplanes. As a result, the antisymmetric tensor field G and, hence, the second term in the gravitino supersymmetry variation

( ) ...8288

2+Γ−Γ+=Ψ ηηδ JKLM

KLMIJIJKLMII GgD (3.44)

do not vanish. With regard the mathematical connections with the Aurea ratio and Fibonacci’s numbers, we note that:

( ) 4101875817,1342,3062

15342,306342,3064327/5

7/5 ⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛ +⋅=Φ⋅= ; 432 – 288 = 144 and 144

is a Fibonacci’s number. Furthermore, 1224288 ⋅= and 24 is related to the physical vibrations of the bosonic strings by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

Now, let us start with the zeroth order metric ( ) BA

AB dxdxVdxRdxdxds Ω++= 3/10

21120

211

νµµνη , (3.45)

where ABΩ is a Calabi-Yau metric with Kahler form baba iΩ=ω . (Here a and b are holomorphic and anti-holomorphic indices). To keep track of the scaling properties of the solution, we have introduced moduli 0V and 0R for the Calabi-Yau volume and the orbifold radius, respectively. To order 3/2κ , the metric can be written in the form ( ) ( )( ) ( ) BA

ABAB dxdxhVdxRdxdxbds +Ω++++= 3/10

21120

211 ˆ1ˆ1 γη νµ

µν (3.46)

37

where the functions b , γ and ABh depend on 11x and the Calabi-Yau coordinates. Furthermore,

ABCDG and 11ABCG receive a contribution of order 3/2κ from the Bianchi identity source terms. The general explicit form of the corrections are

( )2/32ˆ 113/2

00 πρα −−= − xVRb (3.47a) ( )2/3

22ˆ 113/200 πραγ −= − xVR (3.47b)

( ) ABAB xVRh Ω−= − 2/32 113/2

00 πρα (3.47c) ( )11

61 xG EF

EFABCDABCD εωαε= (3.47d)

011 =ABCG (3.47e) with

( ) ( )∫ ΩΩ ∧∧⎟⎠⎞

⎜⎝⎛−=

XRtrR

vω

πκ

πα

3/2

4281 , ∫ Ω=

Xv . (3.48)

Here ( )11xε is the step function which is ( )11 −+ for 11x positive (negative). With regard the eqs. (3.47) and (3.48), we have the following mathematical connections with the Aurea ratio:

( ) 469451,02

1547140452,032

7/117/11 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) ( ) =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅=

−−

−− 090170,0618034,02

152

15942809041,03

227/357/7

7/357/7

708204,0= ; 944272,034708204,0 =⋅ ;

( ) 1674018269,02

151666,061

7/267/26 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) 02835642,02

1552

152

10281348,0281

7/33,187/33,18 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +⋅

=Φ⋅

≅=−

−

π.

In the five-dimensional space 5M of the reduced theory, the orbifold fixed planes constitute four-dimensional hypersurfaces which we denote by ( )iM 4 , 2,1=i . There will be an 6E gauge field ( )1

µA

accompanied by gauginos and gauge matter fields on the orbifold plane ( )14M . We will set these

gauge matter fields to zero in the following. The field content of the orbifold plane ( )24M consists of

an 8E gauge field ( )2µA and the corresponding gauginos. In addition, there is another important

boundary effect which results from the non-zero internal gauge field and gravity curvatures. More precisely, note that

( ) ( )∫ ∫ ⎟⎠⎞

⎜⎝⎛−=Ω=Ω

X X

ABAB

ABAB vRtrRFtrF α

κππ

3/211 4216 , ( ) 02 =ABF . (3.49)

38

In view of the boundary action (3.41), it follows that we will retain cosmological type terms with opposite signs on the two boundaries. Note that the size of those terms is set by the same constant α , given by eq. (3.48), which determines the magnitude of the non-zero mode. The boundary cosmological terms are another important ingredient in reproducing the 11-dimensional background as a solution of the five-dimensional theory. We can perform the Kaluza-Klein reduction on the metric BA

AB dxdxVdxdxgVds Ω+= − 3/13/2211

βααβ . (3.50)

The complete configuration for the antisymmetric tensor field that we use in the reduction is given by

αβγC , [ ]βγδααβγδ CG ∂= 24 , ABABC ωαα Α=61 , ABABG ωαβαβ F= , αββααβ Α∂−Α∂=F ,

ABCABCC ξω61

= , ABCABCG ξωαα ∂= (3.51)

and the non-zero mode is

( )11

6xG EF

EFABCDABCD εωεα

= , (3.52)

where α was defined in eq. (3.48). We can now compute the five-dimensional effective action of Horava-Witten theory. Using the field configuration (3.49) – (3.52) we find from the action (3.39) – (3.41) that boundhypergrav SSSS ++=5 (3.53) where

⎥⎦⎤

⎢⎣⎡ Α++−−= ∫ δεβγα

αβγδεαβαβ ε

κFFFF

21

23

21

525

RgSMgrav (3.54a)

⎢⎣⎡ ++∂∂+∂∂−−= −−∫ αβγδ

αβγδα

αα

α ξξκ

GGVVVVVgSMhyper

21225 24

1221

21

5

( )( ) ⎥⎦

⎤+Α+∂−∂+ − 22

312

242 ααξξξξε εεεαβγδ

αβγδε ViG (3.54b)

( )( ) ( )( )∑∫∫ ∫

=

−− −−−+−−−=2

1

1125 4

2

14

24 16

122222

1i

M

i

GUTM Mbound i

VtrFgVgVgS µνπααα

κ. (3.54c)

In this expression, we have now dropped higher-derivative terms. The 4-form field strength αβγδG is subject to the Bianchi identity

( ) ( ) ( ) ( ) ( ) µνρσµνρσ πρδδπακ

−+−= 11211125

11 24xJxJdG

GUT

(3.55)

which follows directly from the 11-dimensional Bianchi identity (3.42). The currents ( )iJ have been defined in eq. (3.43). The five-dimensional Newton constant 5κ and the Yang-Mills coupling GUTα are expressed in terms of 11-dimensional quantities as

39

v

225

κκ = , 3/22 4

2⎟⎠⎞

⎜⎝⎛=κπκα

vGUT . (3.56)

Since we have compactified on a Calabi-Yau space, we expect the bulk part of the above action to have eight preserved supercharges and, therefore, to correspond to minimal 1=N supergravity in five dimensions. Accordingly, let us compare the result (3.54) to the known 1=N supergravity-matter theories in five dimensions. In these theories, the scalar fields in the universal hypermultiplet parameterize a quaternionic manifold with coset structure ( ) ( ) ( )12/1,2 USUSUQ ×=M . Hence, to compare our action to these we should dualize the three-form αβγC to a scalar field σ by setting (in the bulk)

( )( )εεεεαβγδεαβγδ αξξξξσε Α−∂−∂−∂= − 2

21 2 iVG . (3.57)

Then the hypermultiplet part of the action (3.54b) can be written as

⎥⎦⎤

⎢⎣⎡ +∇∇−−= −∫ 22

2 31

2 5

ακ

αα VqqhgvS vu

uvMhyper (3.58)

where ( )ξξσ ,,,Vqu = . The covariant derivative α∇ is defined as uuu kqq ααα αΑ+∂=∇ with

( )0,0,2,0 −=uk . The sigma model metric Qvuuv Kh ∂∂= can be computed from the Kahler potential ( )CCSSKQ 2ln −+−= , σξξ iVS ++= , ξ=C . (3.59) Consequently, the hypermultiplet scalars uq parameterize a Kahler manifold with metric uvh . It can be demonstrated that uk is a Killing vector on this manifold. To analyze the supersymmetry properties of the solutions shortly to be discussed, we need the supersymmetry variations of the fermions associated with the theory (3.53). They can be obtained either by a reduction of the 11-dimensional gravitino variation (3.44) or by generalizing the known five-dimensional transformations by matching onto gauged four-dimensional 2=N theories. It is sufficient to keep the bosonic terms only. Both approaches lead to

( ) ( ) ( )( ) ++∂−−∂−−+= − jij

ij

iii iiViD εττξττξεγδγεδψ ααβγγβ

αβγααα 2121

2/1

214

82

F

( ) ( ) ( ) jj

ijj

i xVGVi ετγεαετε αβγδεβγδεα 3

1113 12

2962 −−−

( ) ( )( ) ++∂+−∂−= − jij

ij

ii iiViGV εττξττξγεγεδζ ααα

εαβγδαβγδε

21212/1

2482

( )( ) jj

ii xViVVi ετεαεγ ββ 3

1111

22−− −∂+ (3.60)

where iτ are the Pauli spin matrices. Thence, we see that the relevant five-dimensional effective theory for the reduction of Horava-Witten theory is a gauged 1=N supergravity theory with bulk and boundary potentials.

40

The theory (3.53) has all of the prerequisites necessary for such a three-brane solution to exist. Generally, in order to have a ( )2−D -brane in a D -dimensional theory, one needs to have a ( )1−D -form field, or, equivalently, a cosmological constant. This cosmological term is provided by the bulk potential term in the action (3.53). From the viewpoint of the bulk theory, we could have multi three-brane solutions with an arbitrary number of parallel branes located at various places in the 11x direction. As is well known, however, elementary brane solutions have singularities at the location of the branes, needing to be supported by source terms. The natural candidates for those source terms, are the boundary actions. Given the anomaly-cancellation requirements, this restricts the possible solutions to those representing a pair of parallel three-branes corresponding to the orbifold planes. It is clear that in order to find a three-brane solution, we should start with the Ansatz ( ) ( ) 2222

5 dyybdxdxyads += µννµ η (3.61)

( )yVV = where a and b are functions of 11xy = and all other field vanish. The general solution for this Ansatz, satisfying the equations of motion derived from the action (3.53), is given by

2/10Haa = , 2

0Hbb = , 30HbV = and 03

2 cyH += α (3.62)

where 0a , 0b and 0c are constants. We note that the boundary source terms have fixed the form of the harmonic function H in the above solution. Without specific information about the sources, the function H would generically be glued together from an arbitrary number of linear pieces with

slopes α32

± . The edges of each piece would then indicate the location of the source terms. The

necessity of matching the boundary sources at 0=y and πρ , however, has forced us to consider only two such linear pieces, namely [ ]πρ,0∈y and [ ]0,πρ−∈y . These pieces are glued together at

0=y and πρ . Therefore, we have

( ) ( )( )πρδδα −−=∂ yyHy 3222 (3.63)

which shows that the solution represents two parallel three-branes located at the orbifold planes. We stress that this solution solves the five-dimensional theory (3.53) exactly, whereas the original deformed Calabi-Yau solution was only an approximation to order 3/2κ . It is straightforward to show that the linearized version of (3.62), that is, the expansion to first order in ( )3/2κα O= , coincides with Witten’s solution (3.46) – (3.47) upon appropriate matching of the integration constants. Hence, we have found an exact generalization of the linearized 11-dimensional solution. We still have to check that our solution preserves half of the supersymmetries. When αβg and V are the only non-zero fields, the supersymmetry transformations (3.60) simplify to

( ) ( ) jj

iii VyD ετγαεεδψ ααα 31

122 −−= ( ) ( ) j

jiii VyiVVi εταεεγδζ β

β 311

22−− −∂= . (3.64)

With regard the eqs. (3.54), (3.55), (3.57), (3.60) and (3.62-3.64), we have the following mathematical connections with Aurea ratio and with the Ramanujan’s modular equations:

41

( ) 709125,02

15707106781,02

17/5

7/5 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) 0590213661,02

1552

152

1058925565,024

27/67,7

7/67,7 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +⋅

=Φ⋅

≅=−

− ;

( ) 8043399,22

15828427125,2227/15

7/15 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ≅= ;

( ) 17525,02

1517677,024

17/33,25

7/33,25 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) 05640,02

1552

152

1056269769,024

17/33,8

7/33,8 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +⋅

=Φ⋅

≅=−

−

π;

( ) 179314,02

15176776695,082

7/257/25 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) 118708,02

1511785113,012

27/31

7/31 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− ;

( ) 0290137,02

1552

152

1029462782,048

27/18

7/18 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +⋅

=Φ⋅

≅=−

− ;

( ) 0145898,02

1552

152

1014731391,096

27/28

7/28 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +⋅

=Φ⋅

≅=−

− ;

( ) 469451,02

1547140452,032

5/115/11 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅=

−

− .

Furthermore, the number 8, 12, 24, 48 and 96 are connected with the “modes” that correspond to the physical vibrations of a superstring and to physical vibrations of the bosonic strings by the following Ramanujan functions:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

,

42

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

We note also that 8 is a Fibonacci’s number. The Killing spinor equations 0=i

αδψ , 0=iδζ are satisfied for the solution (3.62) if we require that the spinor iε is given by ii H 0

4/1 εε = , ( ) jij

i03011 ετεγ = (3.65)

where i

0ε is a constant symplectic Majorana spinor. This shows that we have indeed found a BPS solution preserving four of the eight bulk supercharges. 3.2 On some equations concerning the colliding Branes and the Origin of the Hot Big Bang We have derived the five-dimensional effective action of heterotic M-theory in the precedent subsection (3.1). Now, we shall use a simplified action describing gravity γδg , the universal “breathing” modulus of the Calabi-Yau three-fold φ , a four-form gauge field γδεζΑ with field strength Α= dF and a single bulk M5-brane. It is given by

( ) +⎟⎟⎠

⎞⎜⎜⎝

⎛−∂−−= ∫ !52

321

2

2225

35

5

FR

M

φ

φ egxdMS

( ) ( ) ( ) ( ) ( ) ( )( )∑ ∫=

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂∂∂Α−−−

3

1

435

4 !43

iiiiiiii i

XXXXehdMM

ζλ

εκ

δν

γµγδεζ

µνκλφ εξα , (3.66)

where ,4,...,0,,, =ζεδγ .3,...,0...., =νµ The space-time is a five-dimensional manifold 5M with coordinates γx . The four-dimensional manifolds ( )i

4M , 3,2,1=i are the visible, hidden, and bulk branes respectively, and have internal coordinates ( )

µξ i and tension 35Miα . Note that iα has

dimension of mass. If we denote αα −≡1 , βαα −≡2 , and βα ≡3 , then the visible brane has tension 3

5Mα− , the hidden brane ( ) 35Mβα − , and the bulk brane 3

5Mβ . It is straightforward to show that the tension of the bulk brane, 3

5Mβ , must always be positive. Furthermore, one can easily deduce that the tension on the visible brane, 3

5Mα− , can be either positive or negative. We will take 0>α , so that the tension on the visible brane is negative. Furthermore, we will choose β such that

0>− βα , that is, the tension of the hidden brane is positive. The tensor ( )ihµν is the induced metric

on ( )i4M . The functions ( ) ( )( )µγ ξ iiX are the coordinates in 5M of a point on ( )i

4M with coordinates ( )µξ i .

In other words, ( ) ( )( )µγ ξ iiX describe the embedding of the branes into 5M . The BPS solution of Lukas, Ovrut and Waldram is then given by

43

( )( ) ( ) 24222222 dyyDBxdAdNyDds ++−=

rτ ; ( )yBDe 3=φ ; ( )yDNBAY

2130123

−−−= αF for Yy < ( ) ( )yDNBAY

2130123

−−−−= βαF for Yy > , (3.67) where ( ) CyyD +=α for Yy < ( ) ( ) YCyyD ββα ++−= for Yy > , (3.68) and NCBA ,,, and Y are constants. Note that NCBA ,,, are dimensionless and Y has the dimension of length. The visible and hidden boundary branes are located at 0=y and Ry = , respectively, and the bulk brane is located at Yy = , RY ≤≤0 . We assume that 0>C so that the curvature singularity at 0=D does not fall between the boundary branes. Note that 0=y lies in the region of smaller volume while Ry = lies in the region of larger volume. Note that inserting the solution of the four-form equation of motion into eq. (3.66) yields precisely the bulk action with charge α− in the interval Yy ≤≤0 and charge βα +− in the interval RyY ≤≤ . The formulation of the action eq. (3.66) using the four-form Α is particularly useful when the theory contains bulk branes, as is the case in ekpyrotic theory. The following equation

( )kHaHHa

cc

ccck τδ ∆⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 2

214&

, (3.69)

expresses the density perturbation in terms of the time delay at the time of collision, ( )kτ∆ . If we consider the exponential potential YmveV α−−= , then the eq. (3.69) yields

( )kmC

vmk ταδ ∆

+≈

224 2

. (3.70)

Now we compute the spectrum of quantum fluctuations of the brane kYδ and use the result to compute the time delay, ( )kτ∆ . For the calculation of quantum fluctuations, it is sufficient to work at the lowest order in αβ / . Without loss of generality, we can therefore set 1== NA . In that case, the bulk brane Lagrangian is given by

( ) ( )⎥⎦⎤

⎢⎣⎡ −∂∂= YVYYYDBM νµ

µνβ ηβ 23

5 213L . (3.71)

Note that this agrees with βL given in the following equation

( ) ( )⎥⎦⎤

⎢⎣⎡ −= YVNYYD

NBAM 222

335

213 &β

βL , (3.72)

when we set 1== NA and spatial gradients of Y to zero. Let us first consider the spatially homogeneous motion of the brane which will be described by ( )τ0Y . It is governed by the following equation of motion

44

( ) ( ) EYVYYD =+ 02

02

021 & , (3.73)

where E is a constant. Eq. (3.73) is simply the statement that the energy E of the bulk brane is conserved to this order in αβ / . Since we have chosen the visible brane to lie at 0=y and the hidden universe to lie at Ry = , we focus on the branch 0<Y& in which case the bulk brane moves towards the visible brane. The solution to eq. (3.73) is then given by

( ) ( )( )( )∫ −

=− 0

0 '2''Y

YVEdYYDτ (3.74)

with 0≤τ , and with the collision occurring at 0=τ . Let us now consider fluctuations around the background solution ( )τ0Y . Namely, if ( ) ( )xYYY r,0 τδτ += , with ( ) ( )ττδ 0, YxY <<

r , we can expand the action to quadratic order in Yδ

( )( )[ ] ( ) ( )220

02

0

0100

20

22220 2

121 Y

dYVd

dYdVDEVDYYDfluc δααδδ ⎥

⎦

⎤⎢⎣

⎡+−−+∂+−≈ −−

r&L , (3.75)

where we have used eq. (3.73), and where we have introduced ( )00 YDD ≡ and ( )00 YVV ≡ for simplicity. The key relation is the fluctuation equation as derived from the action (3.75)

0222

22 =

⎥⎥⎦

⎤

⎢⎢⎣

⎡−− k

pert

pertk fxaa

dxfd

x rr &&

τ ; ( )τ−≡ kxr

, (3.76)

where kk YDf rr δ⋅≡ 0 and where perta is defined by

⎟⎟⎠

⎞⎜⎜⎝

⎛−≡ −

20

02

00

030 dY

VdDdYdVD

aa

pert

pert α&&

. (3.77)

The fluctuation eq. (3.76), can be compared with the corresponding equation for the perturbations of a scalar field with no potential and minimally coupled to an FRW background with scale factor ( )τa

02 2 =++ kkk kaa

rrr &&&& δφφδφδ . (3.78)

Defining kk af rr δφ⋅= , eq. (3.78) becomes

0222

22 =⎥⎦

⎤⎢⎣⎡ −− k

k fxaa

dxfd

x rr &&

τ . (3.79)

Let us now discuss the Hubble horizon for the perturbations. Recall that in usual 4d cosmology (see eq. (3.79)), we have

( ) ( ) ( ) 1−≈−⋅=−⋅⋅⎟⎠⎞

⎜⎝⎛=−= Hkaka

akkx physphys τττ , (3.80)

45

where aaH &/21 ≡− is the Hubble radius as derived from the scale-factor a . By definition, a mode is said to be outside the Hubble horizon when its wavelength is larger than the Hubble radius. From eq. (3.80), we see that this occurs when 1<x . Therefore, a mode with amplitude kf r crosses outside the horizon when ( )1O≈x . Similarly, in our scenario we can write ( ) ( ) 12/1

0−≡−=−= pertphysphys HkDkkx ττ , (3.81)

where 2/1

0/ Dkk phys = . The role of the Hubble radius is replaced by

( ) ( )( )∫ −

=−≡− 0

0

2/10

2/10

1

)(2Y

pert YVEdYYDDDH τ , (3.82)

which is to be thought of as an effective Hubble radius for the perturbations. So, the length scale at which amplitudes freeze depends on Ba (rather than perta ), but the amplitude itself, as derived from eq. (3.76), depends on perta . The feature of two different scale factors is novel aspect of ekpyrotic scenario. With regard the comparison to inflationary cosmology, we have that in inflation, the wavelengths are stretched superluminally while the horizon is nearly constant. In the ekpyrotic scenario, the wavelengths are nearly constant while the horizon shrinks. We can obtain a spectrum which is scale-invariant. Writing the equation for the perturbations in the form of eq. (3.76) is useful since one can read off from it the spectral slope of the power spectrum. It is determined by the value of ( ) 2/ τpertpert aa&& . In particular, one obtains a scale-invariant spectrum if ( ) 2/ 2 =τpertpert aa&& when the modes observed on the CMB cross outside the horizon. Combining eqs. (3.74) and (3.77), we find

( )( )( )

2

020

02

00

030

2

'2''

⎥⎦

⎤⎢⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∫− Y

pert

pert

YVEdYYD

dYVdD

dYdVD

aa

ατ&&

. (3.83)

The spectrum will be scale-invariant if the right hand side of eq. (3.83) equals 2 when the modes of interest cross outside the horizon. Thence, we have:

( )( )( )

2'2

''2

020

02

00

030

2 =⎥⎦

⎤⎢⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∫− Y

pert

pert

YVEdYYD

dYVdD

dYdVD

aa

ατ&&

. (3.83b)

With regard the eq. (3.83b), we have the following mathematical connections with the Aurea ratio:

( ) ( ) 3381966,0618034,22

152

157/147/14

7/147/14 =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− ; 2332

=⋅ ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

−

7/287/77/217/287/77/21

215

215

215

6145898,0618034,1236068,4 =++= ; 2631

=⋅ . We note also that 2 is prime number and

Fibonacci’s number.

46

Thence we can rewrite the eq. (3.83b) also as follow:

( )( )( )

=⎥⎦

⎤⎢⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∫−

2

020

02

00

030

2

'2''Y

pert

pert

YVEdYYD

dYVdD

dYdVD

aa

ατ&&

( ) ( ) 2)381966,0618034,2(32

215

215

32][

32

7/147/147/147/14 =+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ=

−

− ;

We can add a potential ( )YV of the form that might result from the exchange of wrapped M2-branes. We would like to think of V as the potential derived from the superpotential W for the modulus Y in the 4d low energy theory. Typically, superpotentials for such moduli are of exponential form, for example, cYeW −≈ , (3.84) where c is a positive parameter with dimension of mass. The corresponding potential is constructed from W and the Kahler potential K according to the usual prescription

⎥⎥⎦

⎤

⎢⎢⎣

⎡−= WW

MWWDDKeV

plji

ijMK pl

2/ 32

. (3.85)

where 2// pli

ii MKD +∂∂= φ is the Kahler covariant derivative, i

i KK φ∂∂= / , jiij KK φφ ∂∂∂= /2

and a sum over each superfield iφ is implicit. Eqs. (3.84) and (3.85) imply that V decays exponentially with Y . Here it will suffice to perform the calculation using a simple exponential potential, namely ( ) YmveYV α−−= , (3.86) where v and m are positive, dimensionless constants. Note that, in the case where the potential is generated by the exchange of wrapped M2-branes, the parameter m is of the form αν /3cTm = , where c is a constant, 3T is the tension of the M2-brane, and ν is the volume of the curve on which it is wrapped. The perturbation modes of interest are those which are within the current Hubble horizon. As the wavelengths corresponding to those modes passed outside the effective Hubble horizon on the moving bulk-brane, the amplitudes became fixed. Scale invariance will require

1>>mD during this period. We know that, if the potential V is negligible compared to E , the spectrum of fluctuations is not scale-invariant. Hence, we consider the limit where 0VE << . This

condition, as seen from the equation of motion for 0Y , eq. (3.73), is satisfied if 00 =Y& initially, or, equivalently, if the bulk brane begins nearly at rest. For the brane to be nearly at rest, one must have

0VE ≈ initially. As the brane traverses the fifth dimension, V increases exponentially, whereas

E is constant. Hence, the condition 0VE << is automatically satisfied. The bulk brane beginning nearly at rest is precisely what we expect for a nearly BPS initial state. Applying the condition

0VE << , eq. (3.74) reduces to

47

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡≈ −∫

022

20

2

0

2/'2 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

ατ , (3.87)

where we have neglected the endpoint contribution at 0=Y . On the other hand, eq. (3.77) gives

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−

020

22 110

mDDvem

aa Ym

pert

pertαα&&

. (3.88)

Combining the above two expressions, we obtain:

pert

pertYm a

amDvem

D &&/11

022

20

0 ⎟⎟⎠

⎞⎜⎜⎝

⎛+=− αα

; ;21/11200

2⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

mDaa

mD pert

pert&&τ

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

00

2 21112mDmDa

a

pert

pert τ&&

. (3.89)

The right hand side of eq. (3.89) is approximately equal to 2 in the limit of large 0mD . Thence, we have

22111200

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

mDmDaa

pert

pert τ&&

. (3.89b)

Also here, with regard the numerical result of eq. (3.89b), we have the following mathematical connections with Aurea ratio:

( ) ( ) 3381966,0618034,22

152

157/147/14

7/147/14 =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− ; 2332

=⋅ ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

−

7/287/77/217/287/77/21

215

215

215

6145898,0618034,1236068,4 =++= ; 2631

=⋅ .

Hence, the exponential potential of eq. (3.86) results in a nearly scale-invariant spectrum of perturbations provided that 0VE << and 10 >>mD are satisfied when modes pass outside the effective Hubble horizon. We next compute the perturbation amplitude, by using eq. (3.76) to calculate kY∆ . The conditions

0VE << and 10 >>mD must be satisfied when wavelengths pass outside the horizon. These conditions can be relaxed once the mode is well outside the horizon. In the limit that 10 >>mD when the relevant modes cross outside the horizon, eq. (3.76) reduces to

[ ] 02 22

22 =−− k

k fxdx

fdx r

r

, (3.90)

48

with solution ( ) ( ) ( ) ( )( )xJkCxJkCxfk 2/322/31

2/1−+= , (3.91)

where 2/3±J are Bessel functions. The coefficients ( )kC1 and ( )kC2 are fixed by requiring that modes well-within the horizon (i.e., 1>>x ) be Minkowskian vacuum fluctuations, that is

τ

βik

k eBMk

f −=356

1r for 1>>x . (3.92)

Using this initial condition, we find the following amplitude for modes outside the horizon (with

1<<x )

( ) ( )( ) BM

ifkf kk 3

52/32/3

2/3

622 βπτπ −

−=≡∆ . (3.93)

Substituting eq. (3.87) and using kk YDf δ0= , we find

( ) 2

035

2/3

0

322 Dve

BMmY

Ym

k

α

βπα −

=∆ . (3.94)

Finally, we define the time-delay ( )kτ∆ by

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

∆=∆

035

2/3

2

0

2316 mDBM

mYYk k

βπατ & , (3.95)

where we have used the equation of motion for 0Y , eq. (3.73). Note that the time-dependence of

( )kτ∆ is mild, a necessary condition for the validity of the time-delay formalism. The factor of ( )( )τ00 YmDmD ≡ is to be evaluated at time τ when a given mode crosses outside the horizon

during the motion of the bulk brane. Let kD denote the value of 0D at horizon crossing for mode k . Since horizon crossing occurs when 1=x , or, equivalently, when ( ) 1−=− kτ , eq. (3.87) gives

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

22log2 2 mC

kve

km

mD α . (3.96)

Substituting eqs. (3.95) and (3.96) into eq. (3.70), we find

( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+=

kk mDmCBM

vm 2234

235

2/3

42

βπαδ . (3.97)

This expression for kδ increases gradually with increasing k , corresponding to a spectrum tilted slightly towards the blue. The blue tilt is due to the fact that, in this example, D is decreasing as the brane moves. That is, the spectral index,

49

k

ks mDkd

dn 41

loglog

12

+≈+≡δ

, (3.98)

exceeds unity. The current CMB data constrains the spectral index to lie in the range about

2.18.0 << sn . Therefore, for our results to be consistent with experiments, we must have 20>kmD , (3.99) a constraint that is easily satisfied. With regard the value s of sn , i.e. 0.8 and 1.2, we have the following mathematical connections with Aurea ratio:

( ) 618034,1618033987,12

157/7

7/7 ≅=⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ ; 2135255,1

43618034,1 =⋅ ;

( ) ( ) ( ) =++=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−

−− 021286,0055728,012

152

152

157/567/420

7/567/420

077014,1= ; 8077605,043077014,1 =⋅ .

For the value of the eq. (3.99), we have the following mathematical connections:

( ) ( ) ( ) =++=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

− 381966,0236068,4090170,112

152

152

157/147/217/35

7/147/217/35

944272,2034708204,15 =⋅= .

Thence, we obtain: 20>kmD =

( ) ( ) ( ) =++=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ=

−

− 381966,0236068,4090170,112

152

152

157/147/217/35

7/147/217/35

944272,2034708204,15 =⋅= . We note that 20,944272 21≅ ; 21 = 13 + 8, where 8

and 13 are Fibonacci’s numbers. Furthermore, the number 8 is connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

50

Furthermore, we consider the power-law potential ( ) ( ) ( )qq CYvYvDYV +−=−= α , (3.100) where 0>v and 0<q are constants. In this case, eq. (3.83) gives

241

212 2

2 ≈

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

≈

q

qaa

pert

pert τ&&

(3.101)

for 1>>q . Hence, a power-law potential can also lead to a nearly scale-invariant spectrum provided that its exponent is sufficiently large. We can straightforwardly extend our analysis to an arbitrary potential ( )YV . Let us suppose that ( )YV satisfies

( ) ( )YVdYdVYD α>> , ( )

dYdV

dYVdYD α>>2

2

. (3.102)

Then, eq. (3.83) reduces to

⎟⎠⎞

⎜⎝⎛≈ 2

2

'''2

VVV

aa

pert

pert τ&&

. (3.103)

Hence, the conditions for scale invariance are eqs. (3.102) as well as

1'

''2 ≈

VVV . (3.104)

Also for the numerical value of eqs. (3.101) and (3.104), we have the following mathematical connections with the Aurea ratio:

( ) ( ) 3381966,0618034,22

152

157/147/14

7/147/14 =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− ; 2332

=⋅ ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

−

7/287/77/217/287/77/21

215

215

215

6145898,0618034,1236068,4 =++= ; 2631

=⋅ ;

( ) ( ) 3381966,0618034,22

152

157/147/14

7/147/14 =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− ; 1331

=⋅ .

51

If µνg is the unperturbed, homogeneous metric (see eq. (3.67) with A and N functions of time), the perturbed 5d metric can be written as ( ) ( ) ( )txhtyDtAgg ,,2 r

µνµνµν += , (3.105) where 3,...,0, =νµ . We can treat the tensor perturbations µνh as functions of xr and t only. We are

interested in the tensor perturbations which satisfy the conditions: 00 =µh , 0=ijh , and 0=∂ ij

ih . The perturbed 5d Einstein action to quadratic order is

( )∫ ∫ ∂∂−=−≡ νµ

µν

νµ

µν hhhhxadMRgxdMS i

iTfluct

&&24255

35

82 (3.106)

where the second expression is obtained by integrating over y . The tensor action is analogous to the scalar action given in eq. (3.75). From the action, we can derive the tensor analogue of the scalar fluctuation equation of motion, eq. (3.76)

0222

22 =⎥⎦

⎤⎢⎣⎡ −− T

k

Tk fx

aa

dxfd

x rr &&

τ , (3.107)

where

( )

( )∫≡ τεπ

µν

νµ khkdh 3

3

2 (3.108)

and k

Tk ahf rr ≡ . (3.109)

The critical difference between this tensor equation and the scalar fluctuation equation, eq. (3.76), is that the effective scale factor perta in eq. (3.76) has been replaced by a . We introduced a potential to insure that perta led to a nearly scale-invariant spectrum,

( ) 2/ 2 ≈τpertpert aa&& . However, ( )τa in the tensor equation is approximately constant (recall that ( )( ) ( )αβ /2/1

50

3 Ο+= MBIa ). Consequently, the root mean square tensor fluctuation amplitude

( ) ( ) 2/32/3

2/3

22 ππkhk

h kk ≈≡∆

rr . (3.110)

is not scale-invariant. With regard the eq. (3.110), we have obtained the following mathematical connections with the Aurea ratio: ( ) 74960995,152 2/3 =π ;

( ) ( ) 75,1575004243,152

152

157/407/32

7/407/32 ≅=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− ;

52

( ) 2708618458,02

157/19

7/19 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ

−

− ; ( ) 715558,151802708618458,0arcsin =⋅π

;

( ) ( ) ( ) 472136,102

152

152

1507/147/28

07/147/28 =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ ;

708204,1523472136,10 =⋅ .

The cyclic story can be described in terms of an ordinary four-dimensional field theory, which can be obtained by taking the long wavelength limit of the brane picture. The distance between branes becomes a moduli (scalar) field φ . The interbrane interaction is replaced by a scalar field potential, ( )φV . The different stages in the cyclic model in the brane picture are in one-to-one correspondence

to the motion of the scalar field along the potential. Then, the action S describing gravity, the scalar field φ , and the matter-radiation fluid is:

( ) ( ) ( )∫ ⎥⎦⎤

⎢⎣⎡ +−∂−−= RV

GgxdS ρφβφφ

π424

21

161R , (3.111)

where g is the determinant of the Friedmann-Robertson-Walker metric µνg , G is Newton’s constant and R is the Ricci scalar. The β factor has the property that ∞→β as 0→a such that →βa constant. The revised solution to the equation of motion is ( )4/1 βρ aR ∝ which approaches a constant as 0→a . The energy, once thinned out during the dark energy dominated phase, remains thinned out at the bounce. The β -factor simply reflects the fact that the extra-dimension collapses but our three-dimensions do not. As a result, entropy produced during one cycle is not concentrated at the crunch and does not contribute significantly to the entropy density at the beginning of the next cycle. Hence, cycles can continue for an arbitrarily long time and there is no practical way of distinguishing one cycle from the next. If the cyclic model can be described in terms of ordinary field theory, then it may seem surprising that it is possible to generate a nearly scale invariant spectrum density perturbations. There are actually three distinct ways of producing a nearly scale-invariant spectrum, and that inflation represents only one of them. The three ways can be characterized by

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −

≡V

Vw

2

2

2121

φ

φ

&

&

,

the effective equation of state of the scalar field. Case I is where 1≈w and the universe is expanding, the example of inflation. Case II is a contracting universe with 0≈w . Case III is a contracting universe with 1>>w , that is the situation that applies in the cyclic model. What is required to obtain 1>>w ? From the expression for w , it is apparent that this is only possible if the potential is negative. In particular, for a negative exponentially steep potential

( )φcV exp−≈ , the solutions to the equation of motion have a scaling solution in which V2/2φ& is constant and approximately – 1. Consequently, w is much greater than unity and nearly constant. The generation of fluctuations for 1>>w can be understood heuristically by examining the perturbed Klein-Gordon equation:

53

kk Vaak δφδφ φφ ⎟

⎠⎞

⎜⎝⎛ ++−= ,

2 '''' (3.112)

where φ (x, t ) has been expanded in Fourier components ( )tkδφ with wavenumber k and prime is derivative with respect to conformal time η . The aa /'' term is due to gravitational expansion, and the last term is due to the self-interaction of the scalar field. This equation applies equally to inflation and to cyclic models. The cyclic model corresponds to the limit where the gravity term is negligible and, instead, the perturbation equation is driven by the potential term. For the negative exponential potential, for example, the scaling solution corresponds to 2

, /2 ηφφ ≈V . We have defined that 1>>w . We take the following values: 97.4=w and 23.4=w . We obtain the following mathematical connections with the Aurea ratio: ( ) ( ) ( ) ( ) 485292,2013156,0236068,0618034,0618034,17/637/217/77/7 =+++=Φ+Φ+Φ+Φ −−− ;

970584,42485292,2 =⋅ (for ⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ

215 );

( ) ( ) 236067977,42

157/21

37/21 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ=Φ ; ( ) ( ) 236067977,0

215

337/21 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ=Φ

−

−− ;

( ) 654585,13180236067977,0arcsin =⋅π

; ( ) 1329,1318023,0arcsin ≅=⋅π

.

We note that 3, 21 and 13 are Fibonacci’s numbers. For inflation, the most stringent constraints are on the flat part of the potential, the range of the inflaton field where the density perturbations are generated. The constraints are commonly expressed as bounds on two “slow-roll” parameters:

1' 2

<<⎟⎠⎞

⎜⎝⎛=

VVε (3.113)

and

1''<<=

VVη . (3.114)

For the cyclic model, the analogous constraints are on the steep portion of the potential where perturbations are generated. The constraints can be expressed in terms of two “fast-roll” parameters:

1'

2

<<⎟⎠⎞

⎜⎝⎛=VVε (3.115)

and

( )

1'''1 2 <<−=

VVVη . (3.116)

The first constraint forces the slope to be steep and the second fixes the curvature, where each applies to the range of φ where the fluctuations are generated that are within the horizon today. The result is that the constraints in the two models are remarkably similar.

54

We note that if we take for 090170,0=η , we have the following mathematical connections with the Aurea ratio:

( ) ( ) 270510,0034442,0236068,02

152

157/497/21

7/497/21 =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−−

−− ;

090170,031270510,0 =⋅ ;

( ) 090169944,02

157/35

7/35 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ

−

− ; ( ) 5173384,5180090169944,0arcsin ≅=⋅π

that is a

Fibonacci’s number.

4. On some equations concerning the “null energy condition” (NEC) violation regarding the inflationary models. [9] [10] [16]

The metric of the higher dimensional theory is R -flat (RF) or R -flat up to a conformal factor (CRF): ( )dtadteds 2222 ( +−= Ω x2) nm

mn dydyg+ , (4.1) where the x are the non-compact spatial dimensions; myy ≡ are the extra dimensions; ( )ta is the usual FRW scale factor; and ( ) mnmn geytg Ω−= 2, (4.2) where mng has Ricci (scalar) curvature 0=R , as evaluated in the compact dimensions. We call the metric R -flat (RF) if =Ω const. and conformally R -flat (CRF) if ( ) ( )ytyt ,, Ω=Ω . Now we develop some basic relations that make it possible to detect easily when a higher dimensional theory is forced to violate the NEC. To describe a spatially-flat FRW spacetime after dimensional reduction, the metric ( )ytgmn , and warp function ( )yt,Ω must be functions of time t and extra-dimensional coordinates my only. We parameterize the rate of change of mng using quantities ξ and mnσ defined by

mnmnmn g

kdtdg σξ +=

121 (4.3)

where 0=mn

mng σ and where ξ and σ are functions of time and the extra dimensions. The space-space components of the energy-momentum tensor are block diagonal with a 33× block describing the energy-momentum in the three non-compact dimensions and kk × block for the k compact directions. The 0-0 component is the higher dimensional energy density ρ . Associated with the two blocks of space-space components of IJT are two trace averages:

µνµνγ Tp 33 3

1≡ and mn

mnkk T

kp γ1

≡ , (4.4)

55

where k,3γ are respectively the 33× and kk × blocks of the higher dimensional space-time metric.

Violating the NEC means that 0<NMMN nnT for at least one null vector Mn and at least one space-

time point. We find simple methods for identifying a subset of cases where the NEC must be violated. For this purpose, the following two lemmas are very useful: Lemma 1: If 3p+ρ or kp+ρ is less than zero for any space-time point, then the NEC is violated. The second lemma utilizes the concept of A-averaged quantities: ( ) ( )∫∫ ΩΩ= ydgeydgQeQ kAkA

A/ ; (4.5)

that is, quantities averaged over the extra dimensions with weight factor ΩAe where, for simplicity, we restrict ourselves to constant A . Using the fact that the weight function in the A -average is positive definite, a straightforward consequence is: Lemma 2: If 03 <+

Apρ or 0<+

Akpρ for any A and any ,t x , then the NEC must be violated. To illustrate the utility of A -averaging, we introduce the CRF metric into the higher-dimensional Einstein equations, and then try to express terms dependent on a in terms of the 4d effective scale factor using the relation ( ) ( )taeta 2/φ≡ , where: ∫ Ω−≡ ydgee kk 2lφ (4.6) and l is the 4+k-dimensional Planck length. The 4d effective scale factor, ( )ta , obeys the usual 4d Friedmann equations:

daa

4

2

31 ρ=⎟

⎠⎞

⎜⎝⎛ & (4.7)

dpaa

aa

4

2

2 −=+⎟⎠⎞

⎜⎝⎛ &&&

. (4.8)

Note that the 4d effective energy density d4ρ and pressure dp4 are generally different from ρ and 3p in the higher dimensional theory if the warp factor is non-trivial. Then, using the Einstein equations, we obtain

( ) ( ) ( )AAAAddA k

kk

kppee 222443

2

22

22 σξξξρρφ −−

+−

+−+=+Ω− (4.9)

( ) ( ) ( ) +−+

−−+

⎟⎠⎞

⎜⎝⎛ −++=+Ω−

AAAAddAk kk

kkAppee 222

442

22

221

423

21 σξξξρρφ

( ) ( )AA

adtd

akke

kAk

kξ3

322 1

2263105 +

+Ω∂⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−+++−+ Ω . (4.10)

There is a range where

56

∗≡−+−

≥≥ Ak

kkA63

51042

, (4.11)

which is the case for 313 ≥≥ k (for CRF). Some theorems below rely on choosing 2=A ; for this value to be within the range given in eq. (4.11), it is necessary that 38 ≥≥ k . Since this includes the relevant string and M-theory models, we will implicitly assume this range of k for CRF models. We note that 3, 8 and 13 are Fibonacci’s numbers. The two relations in eq. (4.9) can be rewritten

( ) ( ) ++

−+=+Ω− 243

2

221

AdA kkwpee ξρρφ non-positive terms for all A (4.12)

( ) ( ) ( )++++=+Ω−

AdAk adtd

akkwpee ξρρφ 3

342 1

2231

21 non-positive terms for some A , (4.13)

where the values of A that make the last term non-positive are those that are in the range in eq. (4.11). Recall that w represents the ratio of the total 4d effective pressure dp4 to the total 4d effective energy density d4ρ . On the left hand side of eqs. (4.12) and (4.13), both φ and

A... depend on the warp factor, Ω , but

the combination is invariant under shifts C+Ω→Ω , where C is a constant. Furthermore, the combination tends to have a weak dependence on Ω . For example, if kp+ρ is homogeneous in my , the left hand side reduces to ( )kpK +ρ , where the dimensionless coefficient K is not very sensitive to Ω or A ; in particular, ( ) ( ) ( )2/2 IAIAIK k += l , where ( ) ∫ Ω≡ ydgeAI kA . (4.14) In this notation, the k -dimensional volume of the compact space is ( )0IVk = ; then, K is equal to

kk V/l , a coefficient which is strictly less than unity. Similarly, if kp+ρ is smooth and Ω has a

sharp maximum on some subspace of dimension m and volume mv , then the left hand side of eq. (4.13) is ( )( )( )max/1 km

m pv +Ο ρl , where ( )maxkp+ρ is the value of kp+ρ evaluated on the subspace where Ω is maximal. If the NEC is violated, it must be violated in the compact dimensions; it must be violated strongly ( kw significantly below the minimally requisite value for NEC violation); and the violation in the compact dimensions must vary with time in a manner that precisely tracks the equation-of-state in the 4d effective theory. The magnitude of the NEC violation is proportional to d4ρ according to eq. (4.13), which is roughly 10010 times greater during the inflationary epoch than during the present dark energy dominated epoch. Hence, the source of NEC violation for inflation must be different and 10010 stronger. The fact that NEC violation is required to have inflation in theories with extra dimensions is unexpected since this was not a requirement in the original inflationary models based on four dimensions only. Curiously, a criticism raised at times about models with bounces from a contracting phase to an expanding phase, such as the ekpyrotic and cyclic alternatives to inflationary cosmology, is that the bounce requires a violation of the NEC (or quantum gravity corrections to GR as the FRW scale factor ( ) 0→ta that serve the same function).

57

If is true that the violation of the condition NEC (condition of null energy) is required for the inflationary universe model and for the cyclic universe model, then it is possible that for the cyclic model the acceleration and initial exponential expansion of the inflationary phase, is equivalent to the collision between the two Brane-worlds and to the consequent acceleration of the expansion of space immediately after the Big Bang. This could be the explanation of the various cosmological and mathematical connections between the two models. Then, the inflation and the Big Bang would be only phases of the cyclic universe. Every cycle has its phase of Big Bang and its phase of inflation. 4.1 On some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with 1>w . With regard the evolution to a smooth universe in an ekpyrotic contracting phase with 1>w , we find that the ekpyrotic smoothing mechanism is robust in the sense that the ratio ot the proper volume of the smooth region to the mixmaster-like region grows exponentially fast along time slices of constant mean curvature. In this system the spacetime is described in terms of a coordinate system ( )ixt, and a tetrad (e0 , eα ) where both the spatial coordinate index i and the spatial tetrad index α go from 1 to 3. Choose e0 to be hypersurface orthogonal with the relation between tetrad and coordinates of the form e0 tN ∂= −1 and e i

ie ∂= αα where N is the lapse and the shift is chosen to be zero. Choose the spatial frame αe to be Fermi propagated along the integral curves of e0. The commutators of the tetrad components are decomposed as follows: [ ] ( ) β

βα

βααα σδ eHeuee +−= 00 , & (4.15) [ ] [ ]( ) γ

δγαβδ

γβαβα εδ enaee += 2, (4.16)

where αβn is symmetric, and αβσ is symmetric and trace free. The scale invariant tetrad variables are defined by He /00 ≡∂ and He /αα ≡∂ while scale invariant versions of the other gravitational variables are given by HnaeNAE ii /,,,,,, αβ

ααβααβ

ααβα σ≡Σ . (4.17)

Note that the relation between the scale invariant tetrad variables and the coordinate derivatives is t∂=∂ −1

0 N (4.18) iiE ∂=∂ αα , (4.19)

where NH=N is the scale invariant lapse. The matter model is a scalar field φ with potential V of the form ( ) φφ ceVV −−= 0 , (4.20) where 0V and c are positive constants. The scale invariant matter variables are given by φ0∂=W (4.21) φαα ∂=S (4.22) 2/ HVV = . (4.23) The time coordinate t is chosen so that He t 3=− . (4.24)

58

Note that this means that the surfaces of constant time are constant mean curvature surfaces. Note also that the singularity is approached as −∞→t . Due to equation (4.24) the scale invariant lapse satisfies an elliptic equation ( ) 332 2 =−+ΣΣ++∂+∂∂− VWA αβ

αβαα

αα NNN . (4.25)

We note that 3 is a Fibonacci’s number. Furthermore, we have the following mathematical connection with the Aurea ratio:

( ) ( ) ( ) 4381966,01618034,22

152

152

157/1407/14

7/1407/14 =++=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

− ;

3434 =⋅ .

The gravitational quantities iEα , αA , αβN and αβΣ satisfy the following hyperbolic evolution equations ( )iiii

t EEEE ββαααα Σ+−=∂ N (4.26)

⎟⎠⎞

⎜⎝⎛ Σ−−Σ∂+∂−∂Σ+=∂ β

βαα

βαβαβ

βααα AAAAt 2

121

NNN (4.27)

( ) ( ) ( )( )βδγ

αγδγβγ

ααβγ

βδ

αγδαβαβ εε Σ∂−Σ+−+∂Σ−=∂ NNNNt 2NN (4.28)

( ) [ +−∂−Σ−+∂+∂+∂∂+Σ=Σ∂ ><><><>< γβγ

αβααβγδ

βαγδβαβααβαβ ε NNANAt 23NNNN

( ) )( ) ]><>< +−∂++ βαδ

βγδ

βγ

αγδαβγγ ε SSNANNN 2 . (4.29)

Here parentheses around a pair of indices denote the symmetric part, while angle brackets denote the symmetric trace-free part. The equations of motion for the matter variables are as follows: Wt N=∂ φ (4.30) ( )[ ]β

βαααααα SSWWSSt Σ+−∂+∂+=∂ NN (4.31)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−−−∂+∂+=∂φα

αα

αα

α VSAWSSWWt 23NN . (4.32)

In addition, the variables are subject to the vanishing of the following constraint quantities [ ] iiii

com ENEAE γλγ

βαβααβλλ ε −−∂=)(C (4.33)

αγαβα

αβγαγα

γ ε NAANJ 2)( −∂+∂=C (4.34)

αγδ

βδαβγβ

βα

βαβα ε WSNAC −Σ−Σ−Σ∂= 3)(C (4.35)

( ) VSSWNNNAAAG 31

61

61

61

121

61

321 22

−−−ΣΣ−+−−∂+= αα

αβαβ

γγ

αβαβ

ααα

αC (4.36)

φααα ∂−= SS )(C . (4.37)

59

With regard the value 12 of eq. (4.36) we have the mathematical connection with the following Ramanujan’s modular equation: The number 12 = 24 / 2, is related to the physical vibrations of the bosonic strings by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

For simplicity, we choose the initial conformal metric to be flat and ( )zyx ,, to be the usual cartesian coordinates for that metric, and we choose the spatial triad to lie along those spatial directions. Thus, the scale free spatial triad becomes ii HE αα δψ 21 −−= . (4.38) It then follows from equation (4.16) that ψψ αα i

iEA ∂−= −12 (4.39) 0=αβN . (4.40) The shear is essentially the trace-free part of the extrinsic curvature, and the constraint equations simplify for a particular rescaling of the trace-free part of the extrinsic curvature with the conformal factor. We therefore introduce the quantity αβZ defined by αβαβ ψ Z6−=Σ . (4.41) Similar considerations apply to the matter variables, leading us to define the quantity Q given by QW 6−=ψ . (4.42) Here we will specify Q , φ and a part of ikZ and solve the constraint equations for the conformal factor ψ and the rest of ikZ . From equation (4.35) and our ansatz for the scale invariant variables we obtain φkik

i QZ ∂=∂ . (4.43) In the vacuum case this equation simply becomes the conditions that ikZ is divergence-free, which is in turn simply an algebraic condition on the Fourier coefficients of ikZ . Note that since αβΣ must be trace-free, so must ikZ . A simple, but still fairly general divergence-free and trace-free ikZ is the following:

60

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−+=

xabbxaxabxa

bZik

coscos0coscos0

1212

211

2

κκ

, (4.44)

where κ , 1a , 2a , 1b and 2b are constants. We still keep this divergence-free part of ikZ but now add to it a piece that has a non-zero divergence. We simply specify the Fourier coefficients of φ and Q via

( ) ( )111 cos0, dxm

HftxQ +== (4.45) ( ) ( )222 cos0, dxmftx +==φ , (4.46)

where 1f , 1m , 1d , 2f , 2m and 2d are constants. This turns equation (4.43) into an algebraic equation for the Fourier coefficients of this non-zero divergence piece of ikZ which we then solve. Now imposing equation (4.36) our ansatz yields

( ) ( ) 72252

81

81

41

43 −+−∂∂−⎟

⎠⎞

⎜⎝⎛ −=∂∂ ψψφφψψ HZZQVH ik

iki

ii

i , (4.47)

which is solved for the conformal factor ψ using the numerical methods. With regard the eq. (4.47), the number 8 is connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

The constraint equations (4.33) and (4.34) are automatically satisfied by this ansatz. We then satisfy equation (4.37) by using the given value of φ to compute the initial value of αS . Now we show results from a single example that demonstrates the generic behaviour: evolution from a highly inhomogeneous, anisotropic universe with significant curvature at the initial time to a universe containing distinct volumes of either smooth, homogeneous 1>>w matter dominated regions, or 1=w mixmaster-like regions. Whenever a 1>>w region forms it grows exponentially fast in proper volume relative to 1=w regions. The particular initial conditions for this example are (4.44 – 4.46)

70.01 =a , 10.02 =a , 01.0=κ , 80.11 =b , 15.02 −=b , 00.21 =f , 11 =m , 7.11 −=d , 15.02 =f , 22 =m , 0.12 −=d , and 1.00 =V , 10=c (4.48)

for the scalar field potential parameters (4.20). With regard the values of (4.48), we take the following: 0,10 0,15 0,70 1,70 1,80 and 10. We have the following mathematical connections with the Aurea ratio:

61

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−−

−−−

7/637/497/357/637/497/35

215

215

215

137767,0013156,0034442,0090170,0 =++= ; 103326,043137767,0 =⋅ ;

( ) ( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ+Φ

−−−−

−−−−

7/847/567/427/287/847/567/427/28

215

215

215

215

150679,032226018,0003106,0021286,0055728,0145898,0 =⋅=+++= ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−−

−−−

7/357/217/77/357/217/7

215

215

215

708204,043944272,0090170,0236068,0618034,0 =⋅=++= ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−

−−

7/427/77/77/427/77/7

215

215

215

718847,143291796,2055728,0618034,0618034,1 =⋅=++= ;

( ) ( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ+Φ

−−−−

−−−−

7/847/497/217/77/847/497/217/7

215

215

215

215

80,1290,0783298,12891649,0003106,0034442,0236068,0618034,0 =⋅≅=⋅=+++= ;

( ) ( ) 281153,1043708204,13618034,2090170,11

215

215

7/147/357/147/35 =⋅=+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ .

Now, let ( )n1χ be a complex character to the modulus 5 such that ( ) i=21χ , and let

284078227,015

25210=

−−−

=κ . (4.48b)

The function

( ) ( ) ( )11 ,2

1,2

1 χκχκ sLisLisf ++

−= , (4.48c) where ( ) ( )∑

∞

=

=1

,n

snnsL χχ ,

is called the Davenport-Heilbronn function and satisfies the Riemann-type equation

( ) ( ) ( )sgsfss

−=⎟⎠⎞

⎜⎝⎛ +

Γ⎟⎠⎞

⎜⎝⎛

−

12

15

2/π . (4.48d)

We note that 10281153,10 ≅−κ . Furthermore:

62

( ) 381966,02

157/14

7/14 =⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Φ

−

− ; 286475,043381966,0 =⋅ . Thence, we can write also:

284078227,015

25210=

−−−

=κ 2865,043

2152841,0

7/14

≅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ +≅≅

−

.

It is enlightening to visualize the evolution via the behaviour of the matter ( )mΩ , shear ( )sΩ and curvature ( )kΩ contributions to the normalized energy density, defined as

VSSWm 31

61

61 2 ++≡Ω α

α (4.49) αβαβΣΣ≡Ω

61

s (4.50)

( )2121

61

32

γγ

αβαβ

ααα

α NNNAAAk −++∂−≡Ω , (4.51)

where 1=Ω+Ω+Ω ksm by (4.36). We note that the eq. (4.51), i.e. 12 = 24/2, is related to the physical vibrations of the bosonic strings by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

The effective equation of state parameter w takes the following form in Hubble normalized variables:

VSSW

VSSWw

++

−+=

αα

αα

21

21

21

21

2

2

. (4.52)

It is evident that at late times the region that has smoothed out and become matter dominated coincides with 1>>w , whereas the mixmaster-like regime evolves to 1=w . We can calculate the behaviour of the solution in the asymptotic matter dominated region as follows. At late times, all spatial derivatives have become negligible. The constraint (4.36) then reduces to

016

22

≈−+ VW , (4.53)

and slicing condition for N (4.25) becomes

63

V−

≈3

33N , (4.54)

Furthermore, V is finite and non-zero. This implies from (4.20, 4.23, 4.24) that φ takes the asymptotic form

( ) ( )ctxtx 2, 0 +≈ φφ (4.55)

and thus W (4.21) tends to

Nc

W 2≈ . (4.56)

Combining these relations gives

cW ≈ , (4.57) 2

32cV −≈ , (4.58) 2

2c

≈N , (4.59)

and from (4.52) 13/2 −≈ cw . (4.60) We have that 414213562,12 ==c ; cW ≅ and 2=V . Thence, we have the following mathematical connections with the Aurea ratio:

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−−

−−−

7/357/217/77/357/217/7

215

215

215

416408,123944272,0090170,0236068,0618034,0 =⋅=++= ;

( ) ( ) 3381966,0618034,22

152

157/147/14

7/147/14 =+=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ

−

− ; 2332

=⋅ ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

−

7/287/77/217/287/77/21

215

215

215

6145898,0618034,1236068,4 =++= ; 2631

=⋅ ;

Let S denote the proper spatial volume element associated with the spatial metric ijh of =t const.

slices i.e., hS det= . The fractional change of S with respect to time is

ijtijt hhS ∂−=∂

21ln , (4.61)

which can be written as N3ln =∂ St . (4.62)

64

In the asymptotic regime where spatial gradients are negligible, N approaches a constant (4.54), and thus (4.62) can be integrated to give

2/6 ctm eS ∝ , 1>>w (4.63) t

v eS ∝ , 1=w (4.64) where we have used (4.54) where 1>>w , and note that 0≈V when 1=w . Thus, at late times the ratio R of the proper volume of matter to mixmaster-like regions of the universe grows as

( )2/61 ct

v

m edxS

dxS−−∝=

∫∫R . (4.65)

Thus, as long as 6>c (which is equivalent to 1>w ), ∞→R as −∞→t . We note that 449489743,26 = , is related to the following mathematical connection with the Aurea ratio:

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−−

−−−

7/427/287/77/427/287/7

215

215

215

458980,23819660,0055728,0145898,0618034,0 =⋅=++= . 5. On some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. [11] [16] The action of p-adic string theory is given by

∫∫ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++−≡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++−= +

−+

−14

2

4124

2

4

11

21

11

21 22 pm

p

spm

p

s

pexd

gm

ppxd

gmS ps φφφφφφ (5.1)

where 22 ∇+−∂= t in the flat space and we have defined

1

11 2

22 −≡

pp

gg sp

and p

mm sp ln

2 22 ≡ . (5.2)

The dimensionless scalar field ( )xφ describes the open string tachyon, sm is the string mass scale and sg is the open string coupling constant. Though the action (5.1) was originally derived for p a prime number, it appears that it can be continued to any positive integer and even makes sense in the limit 1→p . Setting 0= in the action, the resulting potential takes the form

65

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−⎟⎟⎠

⎞⎜⎜⎝

⎛= +12

2

4

11

21 p

p

s

pgmV φφ . (5.2b)

The action (5.1) is a simplified model of the bosonic string which only qualitatively reproduces some aspects of a more realistic theory. That being said, there are several nontrivial similarities between p-adic string theory and the full string theory. The field equation that results from (5.1) is pm pe φφ =− 2/ (5.3) We are interested in perturbing around the solution 1=φ , which is a critical point of the potential, representing the unstable tachyonic maximum. One may wonder whether the field theory (5.1) naively allows for slow roll inflation in the conventional sense. Naively one might expect that for a slowly rolling field the higher powers of in the kinetic term are irrelevant and one may approximate (5.1) by a local field theory. The action (5.1) can be rewritten as

( )∫ ⎥⎦⎤

⎢⎣⎡ +−= ...214 χχχ VxdS (5.4)

where we have defined the field χ as

φχχ 0= (5.5) ( ) ss

mp

pgp

12ln

0 −=χ (5.6)

and the potential is

( )1

02

2

2

42

2

1ln

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=p

s

ss

pp

gm

pmV

χχχχ . (5.7)

In (5.4) the . . . denotes terms with higher powers of . Thence, the eq. (5.4) can be rewritten also

( ) ( )∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−−−

=+

...1ln12

ln12

ln21

1

02

2

2

42

24

p

s

sss

ss

s pp

gm

pmm

pp

gpm

pp

gpxdS

χχχφφ . (5.7b)

Working in the context of the action (5.4) let us consider the slow roll parameters describing the flatness of the potential (5.7) about the unstable maximum 0χχ = . It is straightforward to show that

( )

( ) 012 0

2

20

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=χχχ

χχ

VV

M p (5.8)

( )( ) 2

2

22

2

2

0

2 1ln41

0 ⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

∂∂

=s

psp m

Mp

pp

gVV

M χχχχ

χ. (5.9)

With regard the approximate solution for the classical background, we must solve the Friedmann equation

66

φρ22

31

pMH = (5.10)

to second order in u . To find the energy density φρ , we turn to the stress energy tensor for the p-adic scalar field. A convenient expression for µνT is:

( ) ( )

⎢⎢

⎣

⎡+

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−= ∫∫

−−−

−−−

+− 1

0

1

2

1

0

1

21

2

4 22222 111

22

φφτφφτφφφτ

α

τ

α

ττ

µνµνppppp mm

p

mm

p

pm

p

s eedm

eedmp

egg

mT

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂−

−−−

∫ φφττ

ν

τ

µ

221

1

022

4pp mm

pp

s eedgm

m . (5.11)

One may verify that the µνT is symmetric by changing the dummy integration variable ττ −→1 in the last term. For homogeneous ( )tφ the above expression simplifies, and for 00T we find

( )

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−=−=

−−−

+−

∫ φφτφφφρττ

φ

2221

1

021

2

4

001

12

2ppp mm

p

pm

p

s eedmp

eg

mT

( )

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂+

−−−

∫ φφτττ22

11

021 pp m

tm

tp

eedm

. (5.12)

One can evaluate the above expression term by term, keeping up to ( ) 22 ue t ≈Ο λ . The final result reads

( ) ( )[ ] ( ) ( ) ( )( ) ( )2

2

42

2

4

00 1211

111211

211 u

pgpmueu

pupeu

gmT

p

s

p

s Ο++−

=Ο+⎥⎦

⎤⎢⎣

⎡−+

++−

−+−= µµ . (5.13)

The ( )uΟ terms cancel out and matching the coefficients in the Friedmann equation gives us the simple results

( )11

6 22

420 +

−=

pgp

MmH

pp

s (5.14)

and 01 =H (5.15) for zeroth and first order respectively. The ( )2uΟ contribution to 00T is quite complicated but once we use (5.15) it simplifies greatly. Matching coefficient at order ( )2uΟ in the Friedmann equation gives

λλ µ

23

2222

4

2 1ln

81

41

⎟⎟⎠

⎞⎜⎜⎝

⎛

−==

p

s

sppp

s

Mm

ppp

ge

MmgmH . (5.16)

67

We note that the number 8 in the eq (5.16) is a Fibonacci’s number and can be connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

Because of our sign convention for rH , the fact that 02 >H means that the expansion is slowing as φ rolls from the unstable maximum, as one would expect in a conventional inflationary model. We are approximating the background dynamics as de Sitter which amounts to working in the limit

0→u so that

( )11

6 22

420

2

+−

=≡pg

pMmHH

pp

s (5.17) 10 ≡≡ φφ . (5.18)

We expand the p-adic tachyon field in perturbation theory as ( ) ( )( ) ( ) ( )xtxttxt rrr ,1,, 0 δφδφφφ +=+= . (5.19) The perturbed Klein-Gordon equation (5.3) takes the form δφδφ pe pm =− 2/ . (5.20) One can construct solutions by taking δφ to be an eigenfunction of the operator. If we choose δφ to satisfy δφδφ B+=− (5.21) then this is also a solution to (5.20) if 22 2ln sp mpmB == (5.22) where in the second equality we have used (5.2). For fields which are on-shell (that is, when (5.21) is solved) the field obeys

( ) ( ) ( )( )( ) ( )( ) δφδφδφδφδφ 2////

21111111

2222

s

mBmBmBm

mp

BeB

Beee pppp −

=−

−=−−

−=−=− − . (5.23)

Thus, for on-shell fields the kinetic term in the Lagrangian can be written as

( ) ...21...

21

21...1

21

22

4/

2

4 2

+=+−

=+−= −− ϕϕφφφφ

sp

sm

p

sshellon m

pgme

gm pL (5.24)

68

In (5.24) we have defined the “canonical” field φϕ A≡ (5.25) where

s

s

gpmA

2≡ . (5.26)

The field ϕ has a canonical kinetic term in the action, at least while (5.21) is satisfied. Now, let us return to the task of solving (5.21), bearing in mind that δφδϕ A= is the appropriate canonically normalized field. We write the quantum mechanical solution in term of annihilation/creation operators as

( )( )

( )[ ]∫ += ..2

1, 32/3 chetakdxt ikx

kkϕπδϕ r (5.27)

and the mode functions ( )tkϕ are given by

( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛=

+

0

12/12

032

1aH

kHeHa

ti

k ν

νππϕ (5.27b)

where the order of the Hankel functions is

20

2

20

249

49

Hm

HB s+=+=ν (5.28)

and of course tHea 0= . In the second equality in (5.28) we have used (5.22) and (5.2). In writing (5.27) we have used the usual Bunch-Davies vacuum normalization so that on small scales,

0aHk >> , one has

k

ak 2

1−

≅ϕ

which reproduces the standard Minkowski space fluctuations. This is the usual procedure in cosmological perturbation theory. On large scales, 0aHk << , the solutions (5.27) behave as

ν

ϕ−

⎟⎟⎠

⎞⎜⎜⎝

⎛≅

2/3

032 aH

kk

Hk

which gives a large-scale power spectrum for the fluctuations

1

0

20

2

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

sn

aHkHP

πδϕ

with spectral index ν231 −=−sn .

69

From (5.28) it is clear that to get an almost scale-invariant spectrum we require 0Hms << . In this limit we have

2

0341 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−≅−

Hmn s

s (5.29)

which gives a red tilt to the spectrum, in agreement with the latest WMAP data. For 95.0≅sn one has 02.0 Hms ≅ . Comparing (5.27) to the corresponding solution in a local field theory we see that the p-adic tachyon field fluctuations evolve as though the mass-squared of the field was 22 sm− which may be quite different from the mass scale which one would infer by truncating the infinite series of derivatives: 22 / χ∂∂ V ( 0χχ = ).

We note that for 95.0=sn , from the eq. (5.29), we obtain that 075,02 20

2

=Hms . Thence, we have the

following mathematical connections with the Aurea ratio:

( ) ( ) 944272,034708204,090170,0618034,0

215

21595,0

7/357/77/357/7 =⋅=+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅

−−

−−

From the eq. (5.28), we have that:

524795068,1325,2075,025,2075,0492

49

20

2

==+=+=+=Hmsν ;

( ) ( ) 527864,134145898,1145898,01

215

21552480,1

7/2807/280 =⋅=+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ≅

−

− .

We note that for the eqs. (4.48b-4.48c-4.48d), we have that

70028407822,015

25210100

=−

−−=

κ ; 1,527864 – 0,002840 = 1,525024.

We now want to fix the parameters of the model by comparing to the observed features of the CMB perturbation spectrum. There are three dimensionless parameters, sg , p and the ratio ps Mm / . The important question is whether there is a sensible parameter range which can account for CMB observations, i.e., the spectral tilt and the amplitude of fluctuations. Using (5.14) in (5.29), we can relate the tilt to the model parameters via

( ) ( )1

181812

2

2

22

2 −+

=⎟⎟⎠

⎞⎜⎜⎝

⎛⇔⎟⎟

⎠

⎞⎜⎜⎝

⎛+=−

s

s

p

ss

s

ps n

gpp

Mmg

mM

ppn . (5.30)

Also for this equation, we note that the number 8 is a Fibonacci’s number and is connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

70

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

318

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

Thus one can have a small tilt while ensuring that the string scale is smaller than the Planck scale, provided that 1/2 <<pgs . Henceforth we will use (5.30) to determine ps Mm / in terms of p , sg ,

and 05.01 ≅−sn . All the dimensionless parameters in our solution, 0/ Hms , sm/λ , 2φ and

smH /2 , are likewise functions of 1−sn , p and sg . From (5.14) and (5.30) we see that for 1>>p ,

132166

0

−≅≅≅ ss

ps

s

pp

s nmM

pg

mM

gHm . (5.31)

It may seem strange to have H exceeding sm since that means the energy density exceeds the fundamental scale, but this is an inevitable property of the p-adic tachyon at its maximum, as shown in eq. (5.13). This is similar to other attempts to get tachyonic or brane-antibrane inflation from string theory, since the false vacuum energy is just the brane tension which goes like ss gm /4 . Next we determine sm/λ , where λ is the mass scale appearing in the power series in teλ which provides the ansatz for the background solutions. We consider the following equation for λ in the

smH >>0 limit

( ) pee pmH == + 20

21 /3λµ .

The positive root for λ gives

3

1−≅ s

s

nmλ . (5.31b)

In order to fix the amplitude of the density perturbations we consider the curvature perturbation ζ . We assume that

δϕϕ

ζ&

H−≈

as in conventional inflation models. To evaluate the prefactor ϕ&/H we must work beyond zeroth order in the small u expansion. We take u−=1φ to evaluate the prefactor, even though the perturbation δχ is computed in the limit that 1=φ . This should reproduce the full answer up to ( )uΟ corrections. The prefactor is

12/3

0 11

12 −

−≅≅− s

s

s munp

guA

HHλϕ&

.

71

With regard the eqs. (5.31) and (5.31b), we have the following mathematical connections with the Aurea ratio:

193649167,0132166

0

=−≅≅≅ ss

ps

s

pp

s nmM

pg

mM

gHm ;

( ) 194531,034145898,0

21519365,0

7/287/28 =⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ≅

−

− .

129099444,03

1=

−≅ s

s

nmλ ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ≅

−−−

−−−

7/637/497/287/637/497/28

215

215

21512901,0

1290,0128997,032193496,0013156,0034442,0145898,0 ≅=⋅=++= .

We should evaluate u at the time of horizon crossing, ∗t , defined to be approximately 60 e-foldings before the end of inflation endt , assuming that the energy scale of inflation is high. The inflation ends when 2/1/1 pu ≈ . From eqs. (5.31-5.31b) we see that 1/2/0 −= snH λ ; therefore we can write

the scale factor ( ) tHeta 0≅ in the form ( ) ( ) 1/2 −≅ sntuta (5.32) so that endaea 60−

∗ = corresponds to

2/1130130 1

peueu ss n

endn −−−−

∗ ≡= . (5.33)

We note that 1/2/0 −= snH λ for 95.0=sn is equal to 40. This value can be related with the following mathematical connections with the Aura ratio:

( ) ( ) ( ) =++=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−

− 090170,01618034,22

152

152

157/3507/14

7/3507/14

944272,434708204,3 =⋅= ;

( ) ( ) 944272,1634708204,12618034,1090170,11

215

215

7/77/357/77/35 =⋅=+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ ;

( ) ( ) 277605,1834708204,13618034,2090170,11

215

215

7/147/357/147/35 =⋅=+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ ;

1/2/0 −= snH λ = 40; 4,9 + 16,9 + 18,2 = 40.

72

The power spectrum of the curvature perturbation is given by

1

0

22 −

⎟⎟⎠

⎞⎜⎜⎝

⎛≡=

sn

aHkAPHP ζδϕζ ϕ&

(5.34)

where the amplitude of fluctuations ζA can now be read off as

3

1602

22

138

−=

−

s

ns

ne

pgA

s

πζ . (5.35)

Thence, we can rewrite the eq. (5.34) as follows

≡= δϕζ ϕPHP

2

& 3

1602

2 138

−

−

s

ns

ne

pg s

π. (5.35b)

As an example, taking 95.0≅sn one can fix the amplitude of the density perturbations 102 10−≅ζA by choosing

71048.0 −×≅p

gs . (5.36)

Setting 102 10−=ζA and using (5.35) we obtain an expression for sg in terms of p and 1−sn

52/31302

1018

3 −−− ×−= sn

s nepg sπ . (5.37)

Combining (5.37) with (5.30), we also obtain

51302 10113 −−− ×−+

= sn

p

s nep

pMm sπ . (5.38)

The string scale is bounded from above as 61094.0/ −×≤ps Mm and that for typical values of p , sn

it is close to 61061.0/ −×≅ps Mm . Furthermore, from (5.37) that sg is unconstrained and that sg , p are not independent parameters.

Now we define the Hubble slow roll parameters Hε , Hη by

2

2

221

HM pH

ϕε&

≡ , (5.39) ϕϕηε&

&&

HHH ≡− . (5.40)

73

These are the appropriate parameters to describe the rate of time variation of the inflaton as compared to the Hubble scale. Using the solution u−≅ 1φ (recall that φϕ A= , ( )ss gpmA 2/= ) we find that

1121 160 −

+≅ −−

sn

H nep

pε , (5.41) 2

1−−≅ s

Hn

η . (5.42)

We see that the Hubble slow-roll parameters are small. This means that p-adic tachyon field rolls slowly in the conventional sense. One reaches the same conclusion if one defines the potential slow roll parameters using the correct canonical field, which is ϕ (5.25):

012

22

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=Ap V

VM

ϕϕ, (5.43) 1

211

2

22 −−=

∂∂

= sAp nVV

M ϕϕ. (5.44)

With regard the eqs. (5.42) and (5.44), we have that:

025,0121

=−− sn ;

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=Φ+Φ+Φ

−−−

−−−

7/707/357/217/707/357/21

215

215

215

250776,043334369,0008131,0090170,0236068,0 =⋅=++= ; 0250776,0250776,0

521

=×⋅

.

On the other hand, consider the potential slow roll parameter which one would naively define using the derivative truncated action (5.4):

012 0

22

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=χχχV

VM p , (5.45) 1

21

ln11

02

22 −⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−=

∂∂

= sp np

pVV

M χχχ (5.46)

where in (5.46) we have used equations (5.9) and (5.30). We see that (5.46) can be enormous, though the tachyon field rolls slowly. Taking the largest allowed value of p , 1410≈p , and

95.0≅sn we have 112212 10/ ≈∂∂− χVVM p . Since large values of p are required if one wants to

obtain 1≈sg , it follows that it is somewhat natural for p-adic inflation to operate in the regime where the higher derivative corrections play an important role in the dynamics.

6. On some equations concerning p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. [12] [13] [14] [15] [16]

Consider the standard Minkowski signature minisuperspace model of a homogeneous isotropic universe with a cosmological constant λ . The usual parametrization of the metric 2

32222 Ω+−= dadtNds (6.1)

leads to classical solutions which are trigonometrical functions of time. In the p-adic case we prefer to work with rational functions. We shall use the following ansatz

74

( )( ) ( ) 2

32

22 Ω+−= dtqdt

tqtNds . (6.2)

Here N and a are functions of time and 2

3Ωd is the metric on the unit 3-sphere. The action for this metric is the same as the corresponding usual case

( )[ ] ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−−= 2

1

142

1) 22

2t

tq

NqdtNtqS λ&

. (6.3)

We assume that the cosmological constant λ is a rational number. The classical equations of motions have the form λ2=q&& . (6.4) The solution of this equation for the boundary conditions ( ) 10 qq = , ( ) 2qTq = , (6.4b) is the following

( ) 1122 qtT

Tqqttq +⎥⎦

⎤⎢⎣⎡ −

−+= λλ . (6.5)

Here ( )tq , ( ) pQtp ∈ . The Green function corresponding to the transition from the point 1q to the point 2q has the form ( ) ( )∫=

pQ Tp TqqdTKqq ,0,, 2121G (6.6)

where ( )TqqKT ,0, 21 is the propagator ( )TqqKT ,0, 21 = ( ) ( )∫ ∏

tp tdqSχ . (6.7)

In the path integral one integrates over trajectories with the boundary conditions (6.4b). One can perform the Gaussian path integral (6.7) in the usual way using shifting to the classical solution. One gets ( ) ( ) ( )clpT STcTqqK χ=,0, 21 (6.8) where clS is the action calculated on the trajectories (6.5).

( ) ( )[ ] ( )TqqTqqTTqqSS clcl 84

224

,,2

2121

32

12−

+−++−== λλ . (6.9)

The factor ( )Tc is the same as for a free particle

75

( ) ( )2/1

p

p

TT

tcλ

= . (6.9b)

Therefore one has the Green function

( ) ( ) ( )[ ] ( )∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+−++−=

pQ p

p

pp T

qqTqqTT

TdTqq

842

24,

221

21

32

2/121 λλχλ

G . (6.10)

The corresponding wave function has the form

( ) ( ) ( )∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−−+=Ψ

pQPl

p

p

p TTqT

qlT

TdTq

2442

81 322

22/1λλχ

λ (6.11)

where we restore the explicit dependence on the Planck length. We note that the number 24 in the eq. (6.11) can be related to the physical vibrations of the bosonic strings by the following Ramanujan function:

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−

∞ −∫

42710

421110log

'142

'

cosh'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

φ

ππ

π

π

.

Now let us estimate the integral on T applying the stationary phase approximation. The saddle-point equation has the form

021

488' 2

222

=−+−−=q

TqTS λλ (6.12)

which yields

( )

2

222,1

11λ

λqT −±−= . (6.12b)

As is known, for ( )4mod1≡p there is the square root of – 1 in pQ , so we get nontrivial saddle points. For ( )4mod3≡p we have no saddle point at all. To make sense of the saddle points in the case ( )4mod1≡p we should be sure that the square root qλ−1 also has a sense. For this purpose we have to assume that 1<

pqλ . The corresponding actions have the form

( )[ ]2/31131 qS λλ

−±−= , ( )[ ]2/31131 qS λλ

−±= , (6.13)

76

In order that these expressions be rational we have to assume that λ is rational as well as that 2ξλ =q is such a rational that the solution of the equation

122 =+ηξ (6.14) in respect to η is also rational. Let us consider the Euclidean metric

( )( ) ( ) 2

32

22 Ω+= dtqdt

tqtNds . (6.15)

The Euclidean action for this metric is the same as the corresponding action in the usual case

( )[ ] ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−+−= 2

1

142

1) 22

2t

tq

NqdtNtqS λ&

. (6.16)

We shall prove that it is possible to restore

( ) ∫ ∏∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛=Ψ dqS

lidTq E

PlE 2

2exp π (6.17)

when 0→Pll in the corresponding p-adic wave function. Indeed, in the p-adic case for Euclidean metric, we get a basic Green function

( ) ( ) ( )[ ] ( )∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−−−+=

TqqTqqT

TT

dTqq p

p

pp 84

224

,2

2121

32

2/121 λλχλ

G . (6.18)

Now let us estimate the integral on T in

( ) ( )∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−+

pQ pp

TqTqT

TT

dT84

224

232

2/1 λλχλ

(6.19)

applying the stationary phase approximation. Formally there are the following saddle points

( )[ ]2/1111 qT λλ

−±= , ( )[ ]2/1111 qT λλ

−±−= , (6.19b)

for 1<

pqλ

with corresponding actions

( )[ ]2/31131 qS λλ

−±−= , ( )[ ]2/31131 qS λλ

−±= . (6.20)

77

Note that now these stationary points have sense for all p and for q satisfying (6.15) according our general formula for the wave function of the universe ( ) ( ) ( )∑∫∏=Ψ

am pppp gS µνχ DG3 , (6.21)

we write

( ) ( )[ ] ( )( )[ ]

⎭⎬⎫

⎩⎨⎧ ±−±

±−=±−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛±−

=Ψ ∏=

1132exp

11)11

311

2/31

2/1

,...5,3,2

2/31 qiq

qq

qp

p

ppE λ

λπ

λ

λλ

λλ

λλ.

(6.22) 6.1 Zeta strings and zeta nonlocal scalar fields. The exact tree-level Lagrangian for effective scalar field ϕ which describes open p-adic string tachyon is

⎥⎦

⎤⎢⎣

⎡+

+−−

= +− 122

2 11

21

11 p

p pp

pp

gϕϕϕL , (6.23)

where p is any prime number, 22 ∇+−∂= t is the D-dimensional d’Alambertian and we adopt metric with signature ( )++− ... . Now, we want to show a model which incorporates the p-adic string Lagrangians in a restricted adelic way. Let us take the following Lagrangian

∑ ∑ ∑ ∑≥ ≥ ≥ ≥

+−

⎥⎦

⎤⎢⎣

⎡+

+−=−

==1 1 1 1

1222 1

12111

n n n n

nnnn n

ngn

nCL φφφLL . (6.24)

Recall that the Riemann zeta function is defined as

( ) ∑ ∏≥

−−==

1 111

n pss pn

sζ , τσ is += , 1>σ . (6.25)

Employing usual expansion for the logarithmic function and definition (6.25) we can rewrite (6.24) in the form

( )⎥⎦

⎤⎢⎣

⎡−++⎟

⎠⎞

⎜⎝⎛−= φφφφζ 1ln

2211

2gL , (6.26)

where 1<φ . ⎟⎠⎞

⎜⎝⎛

2ζ acts as pseudodifferential operator in the following way:

( )( )

( )dkkkex ixkD φζ

πφζ ~

221

2

2

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛ , ε+>−=− 222

02 kkk

r, (6.27)

78

where ( ) ( ) ( )dxxek ikx φφ ∫ −=~ is the Fourier transform of ( )xφ .

Dynamics of this field φ is encoded in the (pseudo)differential form of the Riemann zeta function. When the d’Alambertian is an argument of the Riemann zeta function we shall call such string a “zeta string”. Consequently, the above φ is an open scalar zeta string. The equation of motion for the zeta string φ is

( )

( )∫ +>− −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛

ε φφφζ

πφζ

2

2

220 1

~22

12 kk

ixkD dkkker (6.28)

which has an evident solution 0=φ . For the case of time dependent spatially homogeneous solutions, we have the following equation of motion

( ) ( ) ( ) ( )( )t

tdkkketk

tikt

φφφζ

πφζ

ε −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∂−∫ +>

−

1~

221

2 00

20

2

2

0

0 . (6.29)

With regard the open and closed scalar zeta strings, the equations of motion are

( )

( )( )

∫ ∑≥

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛ n

n

nnixk

D dkkke φθφζπ

φζ1

212 ~

221

2, (6.30)

( )( ) ( )

( )( )

( )∫ ∑≥

+−−

⎥⎦

⎤⎢⎣

⎡−

+−

+=⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛

1

112

12

1121~

421

42

n

nnn

nixkD n

nndkkke φθθθζπ

θζ , (6.31)

and one can easily see trivial solution 0==θφ . The exact tree-level Lagrangian of effective scalar field ϕ , which describes open p-adic string tachyon, is:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−

−= +

−12

2

2 11

21

12 pm

p

Dp

p pp

pp

gm

p ϕϕϕL , (6.32)

where p is any prime number, 22 ∇+−∂= t is the D-dimensional d’Alambertian and we adopt metric with signature ( )++− ... , as above. Now, we want to introduce a model which incorporates all the above string Lagrangians (6.32) with p replaced by Nn∈ . Thence, we take the sum of all Lagrangians nL in the form

∑ ∑∞+

=

+−∞+

= ⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−

−==

1

12

1

2

2 11

21

12

n

nm

n n

Dn

nnn nn

nn

gmCCL n φφφL , (6.33)

whose explicit realization depends on particular choice of coefficients nC , masses nm and coupling constants ng . Now, we consider the following case

79

hn nnC +

−= 2

1 , (6.34)

where h is a real number. The corresponding Lagrangian reads

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−= ∑∑

+∞

=

+−+∞

=

−−

1

1

1

22 12

1 2

n

nh

n

hm

D

h nnn

gmL φφφ (6.35)

and it depends on parameter h . According to the Euler product formula one can write

∏∑−−

+∞

=

−−

−=

p hmn

hm

pn

2

2

21

2

1

1 . (6.36)

Recall that standard definition of the Riemann zeta function is

( ) ∑ ∏+∞

=−−

==1 1

11n p

ss pnsζ , τσ is += , 1>σ , (6.37)

which has analytic continuation to the entire complex s plane, excluding the point 1=s , where it has a simple pole with residue 1. Employing definition (6.37) we can rewrite (6.35) in the form

⎥⎦

⎤⎢⎣

⎡+

+⎟⎠⎞

⎜⎝⎛ +−= ∑

+∞

=

+−

1

122 122

1n

nhD

h nnh

mgmL φφφζ . (6.38)

Here ⎟⎠⎞

⎜⎝⎛ + h

m22ζ acts as a pseudodifferential operator

( )( )

( )dkkhmkexh

mixk

D φζπ

φζ ~22

12 2

2

2 ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=⎟

⎠⎞

⎜⎝⎛ + , (6.39)

where ( ) ( ) ( )dxxek ikx φφ ∫ −=

~ is the Fourier transform of ( )xφ . We consider Lagrangian (6.38) with analytic continuations of the zeta function and the power series

∑ +−

+1

1n

h

nn φ , i.e.

⎥⎦

⎤⎢⎣

⎡+

+⎟⎠⎞

⎜⎝⎛ +−= ∑

+∞

=

+−

1

122 122

1n

nhD

h nnACh

mgmL φφφζ , (6.40)

where AC denotes analytic continuation. Potential of the above zeta scalar field (6.40) is equal to hL− at 0= , i.e.

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−= ∑+∞

=

+−

1

12

2 12 n

nhD

h nnACh

gmV φζφφ , (6.41)

80

where 1≠h since ( ) ∞=1ζ . The term with ζ -function vanishes at ,...6,4,2 −−−=h . The equation of motion in differential and integral form is

∑+∞

=

−=⎟⎠⎞

⎜⎝⎛ +

122 n

nhnAChm

ϕφζ , (6.42)

( )

( ) ∑∫+∞

=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+−

12

2 ~22

1n

nh

R

ixkD nACdkkh

mke

Dφφζ

π, (6.43)

respectively. Now, we consider five values of h , which seem to be the most interesting, regarding the Lagrangian (6.40): ,0=h ,1±=h and 2±=h . For 2−=h , the corresponding equation of motion now read:

( )

( ) ( )( )∫ −

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−=⎟

⎠⎞

⎜⎝⎛ −

DR

ixkD dkk

mke

m 32

2

2 11~2

2212

2 φφφφζ

πφζ . (6.44)

This equation has two trivial solutions: ( ) 0=xφ and ( ) 1−=xφ . Solution ( ) 1−=xφ can be also shown taking ( ) ( )( )Dkk πδφ 2~

−= and ( ) 02 =−ζ in (6.44). For 1−=h , the corresponding equation of motion is:

( )

( )( )∫ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−=⎟

⎠⎞

⎜⎝⎛ −

DR

ixkD dkk

mke

m 22

2

2 1~1

2211

2 φφφζ

πφζ . (6.45)

where ( )1211 −=−ζ .

The equation of motion (6.45) has a constant trivial solution only for ( ) 0=xφ . For 0=h , the equation of motion is

( )

( )∫ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛

DR

ixkD dkk

mke

m φφφζ

πφζ

1~

221

2 2

2

2 . (6.46)

It has two solutions: 0=φ and 3=φ . The solution 3=φ follows from the Taylor expansion of the Riemann zeta function operator

( )( )( )∑

≥⎟⎠⎞

⎜⎝⎛+=⎟

⎠⎞

⎜⎝⎛

122 2!

002 n

nn

mnmζζζ , (6.47)

as well as from ( ) ( ) ( )kk D δπφ 32~

= . For 1=h , the equation of motion is:

( )

( ) ( )∫ −−=⎟⎟⎠

⎞⎜⎜⎝

⎛+−

DR

ixkD dkk

mke 2

2

2

1ln21~1

221 φφζπ

, (6.48)

where ( ) ∞=1ζ gives ( ) ∞=φ1V . In conclusion, for 2=h , we have the following equation of motion:

81

( )

( ) ( )∫ ∫

−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

DR

ixkD dw

wwdkk

mke

φφζ

π 0

2

2

2

21ln~2

221 . (6.49)

Since holds equality

( ) ( )∫ ∑∞

===

−−

1

0 1 2 211lnn n

dww

w ζ

one has trivial solution 1=φ in (6.49).

Now, we want to analyze the following case: 2

2 1n

nCn−

= . In this case, from the Lagrangian (6.33),

we obtain:

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ −−=

φφφζζφ

121

221 2

222 mmgmL

D

. (6.50)

The corresponding potential is:

( ) ( )2

124731 φφφφ

−−

−=g

mVD

. (6.51)

The equation of motion is:

( )[ ]( )2

2

22 111

21

2 −+−

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ −

φφφφζζ

mm. (6.52)

Its weak field approximation is:

022

12 22 =⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ − φζζ

mm, (6.53)

which implies condition on the mass spectrum

22

12 2

2

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

mM

mM ζζ . (6.54)

From (6.54) it follows one solution for 02 >M at 22 79.2 mM ≈ and many tachyon solutions when

22 38mM −< . With regard the extension by ordinary Lagrangian, we have the Lagrangian, potential, equation of

motion and mass spectrum condition that, when 2

2 1n

nCn−

= , are:

⎥⎦

⎤⎢⎣

⎡−

++⎭⎬⎫

⎩⎨⎧

−⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛ −−=

φφφφφζζφ

1ln

21

21

22

22

2

2222 mmmgmL

D

, (6.55)

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡−

−−++−=φ

φζζφφ1

1ln1012

22

2gmV

D

, (6.56)

82

( )2

22

222 12ln1

21

2 φφφφφφφζζ

−−

++=⎥⎦

⎤⎢⎣

⎡+−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ −

mmm, (6.57)

2

2

2

2

2

2

21

2 mM

mM

mM

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛− ζζ . (6.58)

In addition to many tachyon solutions, equation (6.58) has two solutions with positive mass:

22 67.2 mM ≈ and 22 66.4 mM ≈ .

Now, we describe the case of ( ) 21

nnnCn−

= µ . Here ( )nµ is the Mobius function, which is defined

for all positive integers and has values 1, 0, – 1 depending on factorization of n into prime numbers p . It is defined as follows:

( ) ( )⎪⎩

⎪⎨

⎧

−=,1

,1

,0knµ

( )⎪⎩

⎪⎨

⎧

==

≠==

.0,1,...21

2

knpppppn

mpn

jik (6.59)

The corresponding Lagrangian is

( ) ( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

++−+= ∑ ∑

∞+

=

∞+

=

+

1 1

1

2200 12

12n n

n

m

D

nn

n

ngmCL φµφµφµ L (6.60)

Recall that the inverse Riemann zeta function can be defined by

( )( )∑

+∞

=

=1

1n

snn

sµ

ζ, its +=σ , 1>σ . (6.61)

Now (6.60) can be rewritten as

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎟⎠⎞

⎜⎝⎛

−+= ∫∞

0

2

200

2

121 φφφ

ζφµ d

mgmCL

D

ML , (6.62)

where ( ) ( )∑+∞

=−−+−+−−−==

1111076532 ...

nnn φφφφφφφφφµφM The corresponding potential,

equation of motion and mass spectrum formula, respectively, are:

( ) ( ) ( ) ( ) ⎥⎦⎤

⎢⎣⎡ −−−==−= ∫

φ

µµ φφφφφφ0

22202 ln1

20 dC

gmLV

D

M , (6.63)

( ) 0ln2

2

1020

2

=−−−⎟⎠⎞

⎜⎝⎛

φφφφφζ

Cm

C

m

M , (6.64)

83

012

2

102

2

0

2

2=−+−

⎟⎟⎠

⎞⎜⎜⎝

⎛C

mMC

mMζ

, 1<<φ , (6.65)

where usual relativistic kinematic relation 222

02 Mkkk −=+−=

r is used.

Now, we take the pure numbers concerning the eqs. (6.54) and (6.58). They are: 2.79, 2.67 and

4.66. We note that all the numbers are related with 2

15 +=Φ , thence with the aurea ratio, by the

following expressions: ( ) 7/1579,2 Φ≅ ; ( ) ( ) 7/217/1367,2 −Φ+Φ≅ ; ( ) ( ) 7/307/2266,4 −Φ+Φ≅ . (6.66) 6.2 p-Adic and adelic quantum cosmology Adelic quantum cosmology is an application of adelic quantum theory to the universe as a whole. In the path integral approach to standard quantum cosmology starting point is Feynman’s idea that the amplitude to go from one state with intrinsic metric ijh , and matter configuration φ on an initial hypersurface Σ , to another state with metric ijh' , and matter configuration 'φ on a final hypersurface 'Σ , is given by a functional integral of [ ]( )Φ− ∞∞ ,µνχ gS over all four-geometries µνg , and matter configurations Φ , which interpolate between the initial and final configurations, i.e. ( ) ( ) [ ]( )∫ Φ−Φ=ΣΣ ∞∞∞∞∞

,,,',',' µνµν χφφ gSghh ijij DD . (6.67)

The [ ]Φ∞ ,µνgS is the usual Einstein-Hilbert action

[ ] ( ) ( )[ ]∫∫ ∫ Φ+Φ∂Φ∂−−⎟⎟⎠

⎞⎜⎜⎝

⎛+Λ−−=Φ

∂ MM M

VggxdKhxdRgxdG

gS νµµν

µν π434

2122

161, (6.68)

for the gravitational field and matter fields Φ . In (6.68), R is scalar curvature of four-manifold M , Λ is cosmological constant, K is trace of the extrinsic curvature ijK at the boundary M∂ of the manifold M . To perform p-adic and adelic generalization we first make p-adic counterpart of the action (6.68) using form-invariance under change of real to the p-adic number fields. Then we generalize (6.67) and introduce p-adic complex-valued cosmological amplitude ( ) ( ) [ ]( )∫ Φ−Φ=ΣΣ ,,,',',' µνµν χφφ gSghh pppppijij DD . (6.69)

The space of all 3-metrics and matter field configurations ( ) ( )( )xxhij

rr φ, on a 3-surface is called superspace (this is the configuration space in quantum cosmology). Superspace is the infinite dimensional one with a finite number of coordinates ( ) ( )( )xxhij

rr φ, at each point xr of the 3-surface. One useful approximation is to truncate the infinite degrees of freedom to a finite number, thereby obtaining some particular minisuperspace model. Usually, one restricts the four-metric to be of the form ( ) ji

ij dxdxhdttNds +−= 222 , where ( )tN is the laps function. For such minisuperspaces,

84

functional integrals (6.67) and (6.69) are reduced to functional integration over three-metrics, matter configurations and to one usual integral over the laps function. If one takes boundary condition ( ) ( ) αααα

1122 , qtqqtq == then integral in (6.67) and (6.69), in the gauge 0=N , is a minisuperspace propagator. In this case it holds ( )∫= 0,, 1212

αααα qNqdNqq vvK , (6.70)

where ( ) [ ]( )∫ −= αααα χ qSqqNq vvv DK 0,, 12 (6.71) is an ordinary quantum-mechanical propagator between fixed αq in fixed time N . For quadratic classical action ( )0,, 12 qNqS cl

p , (6.71) becomes

( ) ( )( )0,,2

0,, 12

2/1

12

2

12

2

12 qNqSqq

Sqq

SqNq cl

pp

p

clp

clp

pp −∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂

−= χλK . (6.72)

If system has n decoupled degrees of freedom, its p-adic kernel is a product

( ) ( )( )0,,2

0,, 12

2/1

1 12

2

12

2

12αα

ααααα χλ qNqS

qqS

qqS

qNq clpp

p

n clp

clp

pp −∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂

−=∏=

K . (6.73)

p-Adic and adelic wave functions of the universe may be found by means of the following equation ( ) ( ) ( ) ( )xtExtU αβααβ ψχψ = , (6.73b) where ( )xαβψ are adelic wave eigenfunctions, ( ),...,...,, 2 pEEEE ∞= is the corresponding adelic energy, ( ),...,...,, 2 pαααα ∞= and ( ),...,...,, 2 pββββ ∞= are indicies for energy levels and their degeneration, respectively. The corresponding adelic eigenstates have the form ( ) ( ) ( ) ( )∏ ∏

∈ ∉∞∞ Ω=Ψ

Sp Sppppp qqqq αααα ψψ . (6.74)

A necessary condition to construct an adelic model is existence of the p-adic (vacuum) state ( )

pqαΩ , which satisfies

( ) ( )∫

≤

Ω=1

2112

1

0,,p

qpp qdqqNq

α

ααααK (6.75)

for all but a finite number of p . Now we describe the p-adic and adelic model with cosmological constant in D = 3 dimensions. This model have the metric

85

( ) ( )( )( )22222222 sin ϕθθσ ddtadttNds ++−= , (6.76) where G=σ . The corresponding v -adic action is

[ ] ( )∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

1

0222

22 1

21 λ

aaNatdtNaaSv&

, (6.77)

where 2σλ Λ= . The Euler-Lagrange equation of motion 02 =− λaNa&& has the solution

( ) ( ) ( ) ( )( )tNNtNN eaeaeeaaN

ta λλλλ

λ−− −+−= 2112sinh2

1 , (6.78)

where the boundary conditions are ( ) 10 aa = , ( ) 21 aa = . For the classical action it gives

( ) ( ) ( ) ⎥⎦⎤

⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+=

λλλλ

λ Naa

NaaNaNaS cl

v tanhsinh2

210,,

22

2121

12 . (6.79)

Quantum-mechanical propagator has the form

( ) ( ) ( ) ( )( )0,,sinhsinh2

0,, 12

2/1

12 aNaSNN

aNa clvv

v

vv −⎟⎟⎠

⎞⎜⎜⎝

⎛−= χ

λλ

λλλK . (6.80)

The equation (6.75), in a more explicit form, reads

( ) ( ) ( ) ( ) ×⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Ω 2

2

2/1

2 tanh22sinhsinh2a

NN

NNa p

p

pp λλχ

λλ

λλλ

( ) ( ) 11

1221

1sinhtanh2

daaaN

aN

pap∫

≤⎟⎟⎠

⎞⎜⎜⎝

⎛−×

λλ

λλχ . (6.81)

We note that the p-adic Gauss integral over the region of integration v

ppx −≤ is

( ) ( )∫

−≤

−−− ≤Ω=+v

p px

vpp

vvp pppdxxx 1, 22 αββαχ ; (6.81a)

( ) ( ) 1,24

2 22

2/12 >⎟⎟⎠

⎞⎜⎜⎝

⎛Ω⎟⎟⎠

⎞⎜⎜⎝

⎛−=+ −

≤

−

∫−

vp

p

v

pxpppp ppdxxx

vp

ααβ

αβχααλβαχ ; (6.81b)

where ( )uΩ is defined as follows:

86

( ) 1=Ω u , 1≤u ; ( ) 0=Ω u , 1>u . Using (6.81b), for 0=v , we obtain

( ) ( ) ( )ppp

aaNNa 2222 tanh

22Ω⎟⎟⎠

⎞⎜⎜⎝

⎛+−=Ω λλχ (6.82)

with condition ( ) 1tanh2

>p

N λλ , i.e. 1<

pN . For 2≠p , left hand side is equal to ( )

pa2Ω if

1≤p

λ holds. Applying also the (6.81a) to (6.81), we have

( ) ( ) ( ) ( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Ω⎟⎟⎠

⎞⎜⎜⎝

⎛+−×⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Ω

p

p

p

pp Na

NaN

NNa

λλ

λλχ

λλ

λλλ

sinhcoth22sinhsinh22

22

2/1

2 . (6.83)

It becomes an equality if condition 1≤

pN take place.

Thence, we can rewrite the eq. (6.81) as follow:

( ) ( ) ( ) ( ) ×⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Ω 2

2

2/1

2 tanh22sinhsinh2a

NN

NNa p

p

pp λλχ

λλ

λλλ

( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛−× ∫

≤1

112

21

1sinhtanh2

daaaN

aN

pap λ

λλ

λχ

( ) ( ) ( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Ω⎟⎟⎠

⎞⎜⎜⎝

⎛+−×⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

p

p

p

p Na

NaN

NN λλ

λλχ

λλ

λλλ

sinhcoth22sinhsinh22

22

2/1

. (6.84)

The de Sitter minisuperspace model in quantum cosmology is the simplest, nontrivial and exactly soluble model. This model is given by the Einstein-Hilbert action with cosmological term (6.68) without matter fields, and by Robertson-Walker metric ( ) ( )( )2

322222 Ω+−= dtadttNds σ , (6.85)

where π

σ322 G

= and ( )ta is the scale factor. Instead of (6.85) we shall use

( )( ) ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛Ω+−= 2

32

222 dtqdt

tqtNds σ . (6.86)

The corresponding v -adic action for this one-dimensional minisuperspace model is

[ ] ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=

2

1

142

12

2t

tv q

NqdtNqS λ&

, (6.87)

87

where 3

2σλ Λ= . The classical equation of motion ( 1=N ) λ2=q&& , with the boundary conditions

( ) 10 qq = and ( ) 2qTq = , ( )12 ttT −= gives

( ) 1122 qtT

Tqqttq +⎟

⎠⎞

⎜⎝⎛ −

−+= λλ . (6.88)

After substitution (6.88) into (6.87) and integration, one obtains that the classical action is

( ) ( )[ ] ( )TqqTqqTqTqS cl

v 842

240,,

212

21

32

12−

−−+−= λλ . (6.89)

Since (6.89) is quadratic in 2q and 1q , quantum-mechanical propagator has the form

( ) ( ) ( )( )0,,4

80,, 122/112 qTqST

TqTq clvv

v

vv −

−= χλ

K . (6.90)

The equation (6.75) reads

( ) ( )1

11

221

222

32

2/12

1448842244

8dqq

TqT

Tq

TqTqTT

TT

qpq

pp

p

pp ∫

≤⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+×⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

−=Ω

λχλλχλ

. (6.91)

7. Mathematical connections. Now, we describe some possible mathematical connections. We take the eq. (1.26) of Section 1. We note that can be related with the eqs. (5.11), (5.12) of Section 5, hence we have the following connections:

⇒≈= ∫ 2, c

eVVdN

Cc

e

φ

φ

φ

( ) ( )

⎢⎢

⎣

⎡+

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−⇒ ∫∫

−−−

−−−

+− 1

0

1

2

1

0

1

21

2

4 22222 111

22

φφτφφτφφφτ

α

τ

α

ττ

µνppppp mm

p

mm

p

pm

p

s eedm

eedmp

egg

m

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂−

−−−

∫ φφττ

ν

τ

µ

221

1

022

4pp mm

pp

s eedgm

m ;

∫ ≈= 2, c

eVVdN

Cc

e

φ

φ

φ( )

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−⇒

−−−

+−

∫ φφτφφφττ222

11

021

2

4 11

22

ppp mm

p

pm

p

s eedmp

eg

m

( )

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂+

−−−

∫ φφτττ22

11

021 pp m

tm

tp

eedm

. (7.1)

88

Thence, mathematical connections between the slow-roll formula regarding the number of e-foldings eN of inflation and the equations of the stress energy tensor for the p-adic scalar field in p-adic inflation. Now, we take the eqs. (2.4), (2.6), (2.72b) and (2.74) of Section 2. We note that can be related with the eq. (6.91) of Section 6, hence we obtain the following connections:

( ) ⇒+⎥⎦⎤

⎢⎣⎡ +−−= ∫ maKaaLMxdtd SS 22223

53

213 φ&&

( )

11

12

21

222

32

2/11

4488422448

dqqT

qTT

qT

qTqTTT

T

pqpp

p

p ∫≤

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+×⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

−⇒

λχλλχλ

; (7.2)

( ) [ ] [ ]⇒++⎟⎟⎠

⎞⎜⎜⎝

⎛∂−−= ∫ ++−− ggRMgxd mm SSS 2

244

21

2φµ

( )

11

12

21

222

32

2/11

4488422448

dqqT

qTT

qT

qTqTTT

T

pqpp

p

p ∫≤

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+×⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

−⇒

λχλλχλ

; (7.3)

( ) ( )( )∫−

−=

−==

1

1 224

440421

24404

122

4

xxxde

bxnx

Ld x

m ξωξδ

( ) ( ) ( )( ) ( ) ( )( ) ⇒⎥⎦⎤

⎢⎣⎡ +−+

−= 41041040040042

4

~~4~~~38

11 xkYBxkJAxkYBxkJAxk

x

( )

11

12

21

222

32

2/11

4488422448

dqqT

qTT

qT

qTqTTT

T

pqpp

p

p ∫≤

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+×⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

−⇒

λχλλχλ

; (7.4)

⇒=Γ= ∫ ∫− −

−0

0

241

121

244

y

y

x

Lm de

bnxdynt

Ld ωξδ

( )

11

12

21

222

32

2/11

4488422448

dqqT

qTT

qT

qTqTTT

T

pqpp

p

p ∫≤

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+×⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

−⇒

λχλλχλ

; (7.5)

Thence, mathematical connections between some equations concerning cosmological perturbations in a Big Crunch/Big Bang space-time and M-theory model of a Big Crunch/Big Bang transition (2.4-2.6), some equations concerning the solution of a braneworld Big Crunch/Big Bang cosmology (2.72b-2.74) and the equation concerning the de Sitter minisuperspace model in p-adic quantum cosmology (6.91). Now, we take the eqs. (5.11), (5.12), (5.29), (5.35) and (5.46) of Section 5. We note that can be related with the eqs. (3.30), (3.32), (3.34), (3.87), (3.96) and (3.98) of Section 3, hence we obtain the following mathematical connections:

89

( ) ⇒⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡≈ −∫

022

20

2

0

2/'2 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

ατ

( ) ( )

⎢⎢

⎣

⎡+

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−⇒ ∫∫

−−−

−−−

+− 1

0

1

2

1

0

1

21

2

4 22222 111

22

φφτφφτφφφτ

α

τ

α

ττ

µνppppp mm

p

mm

p

pm

p

s eedm

eedmp

egg

m

( )

⇒⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂−

−−−

∫ φφττ

ν

τ

µ

221

1

022

4pp mm

pp

s eedgm

m( )

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−

−−−

+−

∫ φφτφφφττ222

11

021

2

4 11

22

ppp mm

p

pm

p

s eedmp

eg

m

( )

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂+

−−−

∫ φφτττ22

11

021 pp m

tm

tp

eedm

; (7.6)

2

0341 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−≅−

Hmn s

s Ndd ε

εln2

−⇒β

β+

−⇒1

03.0 ×+≈+⇒mkd

d k 41log

log1

2δ

⇒⎟⎟⎠

⎞⎜⎜⎝

⎛× ]

22log2/[1

2 mCvek

mm

α ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡

−∫0

22

20

2

0

2/' 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

α; (7.7)

3

1602

22

138

−=

−

s

ns

ne

pgA

s

πζ Ndd ε

εln2

−⇒( )

2,

22

2 414cc

Mc Pl

φγ−

+⇒ ⇒

+−⇒

ββ

103.0

×+≈+⇒mkd

d k 41log

log1

2δ⇒⎟

⎟⎠

⎞⎜⎜⎝

⎛]

22log2/[1

2 mCvek

mm

α

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡⇒ −∫

022

20

2

0

2/' 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

α; (7.8)

121

ln11

02

22 −⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−=

∂∂

= sp np

pVV

M χχχ Ndd ε

εln2

−⇒( )

2,

22

2 414cc

Mc Pl

φγ−

+⇒ ⇒

+−⇒

ββ

103.0

×+≈+⇒mkd

d k 41log

log1

2δ⇒⎟

⎟⎠

⎞⎜⎜⎝

⎛]

22log2/[1

2 mCvek

mm

α

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡⇒ −∫

022

20

2

0

2/' 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

α. (7.9)

Thence, mathematical connections between some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang, some equations concerning the colliding branes and the origin of the hot Big Bang and some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. Now, we take the eqs. (5.2), (5.11) and (5.12) of Section 5 and the eqs. (6.28), (6.43) and (6.62), of Section 6. We obtain the following mathematical connections:

90

⇒⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++−≡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++−= ∫∫ +

−+

−14

2

4124

2

4

11

21

11

21 22 pm

p

spm

p

s

pexd

gm

ppxd

gmS ps φφφφφφ

( ) ( )

⎢⎢

⎣

⎡+

⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−⇒ ∫∫

−−−

−−−

+− 1

0

1

2

1

0

1

21

2

4 22222 111

22

φφτφφτφφφτ

α

τ

α

ττ

µνppppp mm

p

mm

p

pm

p

s eedm

eedmp

egg

m

( )

⇒⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛∂−

−−−

∫ φφττ

ν

τ

µ

221

1

022

4pp mm

pp

s eedgm

m( )

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+−

−−−

+−

∫ φφτφφφττ222

11

021

2

4 11

22

ppp mm

p

pm

p

s eedmp

eg

m

( )

⇒⎥⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂⎟⎟

⎠

⎞

⎜⎜

⎝

⎛∂+

−−−

∫ φφτττ22

11

021 pp m

tm

tp

eedm ( )

( ) ⇒−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠⎞

⎜⎝⎛ ∫ +>− ε φ

φφζπ

φζ2

2

220 1

~22

12 kk

ixkD dkkker

( )

( ) ⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛+−⇒ ∑∫

+∞

=

−

12

2 ~22

1n

nh

R

ixkD nACdkkh

mke

Dφφζ

π

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎟⎠⎞

⎜⎝⎛

−+⇒ ∫∞

0

2

200

2

121 φφφ

ζφ d

mgmC

D

ML . (7.10)

Thence, mathematical connections between some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory and some equations concerning the zeta strings and the zeta nonlocal scalar fields. In conclusion, with regard the Section 6 we have the following mathematical connections between the eqs. (6.84) and (6.91) and the eq. (3.87) of the Section 3:

( ) ( ) ( ) ( ) ×⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Ω 2

2

2/1

2 tanh22sinhsinh2a

NN

NNa p

p

pp λλχ

λλ

λλλ

( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛−× ∫

≤1

112

21

1sinhtanh2

daaaN

aN

pap λ

λλ

λχ

( ) ( ) ( ) ( ) ⇒⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Ω⎟⎟⎠

⎞⎜⎜⎝

⎛+−×⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

p

p

p

p Na

NaN

NN λλ

λλχ

λλ

λλλ

sinhcoth22sinhsinh22

22

2/1

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡⇒ −∫

022

20

2

0

2/' 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

α; (7.11)

( ) ( )⇒⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+×⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

−=Ω ∫

≤1

11

221

222

32

2/12

1448842244

8dqq

TqT

Tq

TqTqTT

TT

qpq

pp

p

pp

λχλλχλ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎥⎦

⎤⎢⎣⎡⇒ −∫

022

20

2

0

2/' 212''21

0

0

mDvemDdYeYD

v Ym

Y Ymα

α

α. (7.12)

91

Thence, mathematical connections between the some equations concerning the p-adic quantum cosmology and the fundamental equation concerning the colliding branes and the origin of the hot Big Bang. Acknowledgments I would like to thank Paul J. Steinhardt Of Department of Physics of Princeton University for his availability and the very useful references and advices. Furthermore, I would like to thank also Branko Dragovich of Institute of Physics of Belgrade (Serbia) for his availability and friendship with regard me and Gianmassimo Tasinato of Heidelberg University for his availability and the useful reference concerning the p-adic Inflation. References [1] Paul J. Steinhardt and Neil Turok – “Cosmic Evolution in a Cyclic Universe” – arXiv:hep- th/0111098v2 – Mar 2002. [2] Andrew J. Tolley, Neil Turok and Paul J. Steinhardt – “Cosmological Perturbations in a Big Crunch/Big Bang Space-time” – arXiv:hep-th/0306109v1 – Giu 2003. [3] Paul J. Steinhardt and Neil Turok – “The Cyclic Model Simplified” – arXiv:astro- ph/0404480v1 – Apr 2004. [4] Neil Turok, Malcolm Perry and Paul J. Steinhardt – “M Theory Model of a Big Crunch/Big Bang Transition” – arXiv:hep-th/0408083v1 – Aug 2004. [5] Paul L. McFadden, Neil Turok and Paul J. Steinhardt – “Solution of a Braneworld Big Crunch/ Big Bang Cosmology” – arXiv:hep-th/0512123v2 – Jan 2006. [6] Jean-Luc Lehners, Paul McFadden, Neil Turok and Paul J. Steinhardt – “Generating Ekpyrotic Curvature Perturbations Before the Big Bang” – arXiv:hep-th/0702153v1 – Feb 2007. [7] André Lukas, Burt A. Ovrut, K. S. Stelle and Daniel Waldram – “The Universe as a Domain Wall” – arXiv:hep-th/9803235v2 – May 1998. [8] Justin Khoury, Burt A. Ovrut, Paul J. Steinhardt and Neil Turok – “The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang” – arXiv:hep-th/0103239v3 – Aug 2001. See also the paper: Paul J. Steinhardt and Neil Turok – “The Origin of Density Fluctuations in a Cyclic Universe”. [9] Paul J. Steinhardt and Daniel Wesley – “Dark Energy, Inflation and Extra Dimensions” – arXiv:hep-th/0811.1614v1 – Nov 2008. [10] David Garfinkle, Woei Chet Lim, Frans Pretorius and Paul J. Steinhardt – “Evolution to a smooth universe in an ekpyrotic contracting phase with 1>w ” – arXiv:hep-th/0808.0542v1 – Aug 2008. [11] Neil Barnaby, Tirthabir Biswas and James M. Cline – “p-Adic Inflation” – arXiv:hep- th/0612230v3 – Mar 2007. [12] I. Ya . Aref’eva, B. Dragovich, P.H. Frampton and I. V. Volovich – “Wave Function of the Universe and p-adic Gravity” – IFP-382-UNC – 90-7-373 – Jul 1990. [13] Branko Dragovich – “Zeta Strings” – arXiv:hep-th/0703008.

92

[14] Branko Dragovich – “Zeta Nonlocal Scalar Fields” – arXiv:hep-th/0804.4114v1. [15] Branko Dragovich, Ljubisa Nesic – “p-Adic and Adelic Generalization of Quantum Cosmology” – arXiv:gr-qc/0005103v1 – May 2000. [16] Christian Lange, Michele Nardelli e Giuseppe Bini – “Sistema Musicale Aureo 7/nΦ e connessioni matematiche tra Numeri Primi e Paesaggio della Teoria delle Stringhe” – http://150.146.3.132/679/01/NardLanBin02.pdf - Ott 2008. Printed in February 2009 by DI. VI. Service Via Miranda, 50 – 80131 – Naples (Italy) All right reserved