Lemaıtre Workshop, Black Holes, Gravitational Waves andSpacetime Singularities, Castel Gandolfo, Vatican Observatory

Singularity crossing, transformation of

matter properties and the problem of

parametrization in field theories.

A.Yu. Kamenshchik

University of Bologna and INFN, Bologna

May 9 - May 12, 2017

Based on

Z. Keresztes, L.A. Gergely, A.Y. Kamenshchik, V. Gorini andD. Polarski,Soft singularity crossing and transformation of matterproperties,Phys. Rev. D 88 (2013) 023535

A.Y. Kamenshchik,Quantum cosmology and late-time singularities,Class. Quantum Grav. 30 (2013) 173001

A.Y. Kamenshchik, E.O. Pozdeeva, A. Tronconi, G. Venturiand S.Y. Vernov,Transformations between Jordan and Einstein frames:Bounces, antigravity, and crossing singularities,Physical Review D 94 (2016) 063510

Bianchi-I cosmological model and crossing singularities,Physical Review D 95 (2017) 083503

Content

1. Introduction

2. Description of the tachyon cosmological model

3. The Big Brake cosmological singularity and more generalsoft singularities

4. Crossing of the soft singularity in the model with theanti-Chaplygin gas and dust

5. Crossing the Big Brake singularity in the tachyon modelin the presence of dust

Big Bang – Big Crunch crossing ?

1. Crossing of “hard” singularities as a choice ofparametrization

2. Transformations between the Einstein frame and theJordan frame in Friedmann-Lemaıtre models

3. Anisotropic Bianchi - I universe

4. Conclusions and discussion

Introduction

I General relativity connects the geometrical properties ofthe spacetime to its matter content.Matter tells spacetime how to curve itself, the spacetimegeometry tells matter how to move.

I Cosmological singularities constitute one of the mainproblems of modern cosmology.

I The discovery of cosmic acceleration stimulated thedevelopment of “exotic” cosmological models of darkenergy; some of these models possess the so called soft orsudden singularities characterized by a finite value of theradius of the universe and its Hubble parameter.

I “Traditional” or “hard” singularities are associated with azero volume of the universe (or of its scale factor), andwith infinite values of the Hubble parameter, of the energydensity and of the pressure –Big Bang and Big Crunch

I In some models interplay between the geometry andmatter forces matter to change some of its basicproperties, such as the equation of state for fluids andeven the form of the Lagrangian.

I Tachyons (Born-Infeld fields) are natural candidates for adark energy

I The toy tachyon model, proposed in 2004 has twoparticular features:The tachyon field transforms itself into a pseudo-tachyonfield,The evolution of the universe can encounter a new typeof singularity - the Big Brake singularity.

I The Big Brake singularity is a particular type ofthe so called “soft” cosmological singularities -the radius of the universe is finite, the velocityof expansion is equal to zero, the deceleration isinfinite.

I The Big Brake singularity is a particular one - itis possible to cross it

I It is possible to cross other soft singularities,sometimes matter changes its properites

I Crossing of the Big Bang - Big Crunchsingularity: is it a question of a fieldparametrization ?

I What happens in anisotropic spacetimes ?

Description of the tachyon model

The flat Friedmann-Lemaıtre universe

ds2 = dt2 − a2(t)dl2

The tachyon Lagrange density

L = −V (T )√

1− T 2

The energy density

ρ =V (T )√1− T 2

The pressure

p = −V (T )√

1− T 2

The Friedmann equation

H2 ≡ a2

a2= ρ

The equation of motion for the tachyon field

T

1− T 2+ 3HT +

V,TV

= 0

In our model

V (T ) =Λ

sin2[32

√Λ (1 + k) T

]×

√1− (1 + k) cos2

[3

2

√Λ (1 + k)T

],

where k and Λ > 0 are the parameters of the model. Thecase k > 0 is more interesting.

Some trajectories (cosmological evolutions) finish in an infinitede Sitter expansion. In other trajectories the tachyon fieldtransforms into a pseudotachyon field with the Lagrangedensity, energy density and positive pressure:

L = W (T )√

T 2 − 1,

ρ =W (T )√T 2 − 1

,

p = W (T )√

T 2 − 1,

W (T ) =Λ

sin2[32

√Λ (1 + k) T

]×

√(1 + k) cos2

[3

2

√Λ (1 + k)T − 1

]

What happens to the Universe after the

transformation of the tachyon into the

pseudotachyon ?

It encounters the Big Brake

cosmological singularity.

The Big Brake cosmological singularity

and other soft singularities

t → tBB <∞

a(t → tBB)→ aBB <∞

a(t → tBB)→ 0

a(t → tBB)→ −∞

R(t → tBB)→ +∞

ρ(t → tBB)→ 0

p(t → tBB)→ +∞

If a(tBB) 6= 0 it is a more general soft singularity.

Crossing the Big Brake singularity and the future of

the universe

At the Big Brake singularity the equations forgeodesics are regular, because the Christoffelsymbols are regular (moreover, they are equal tozero).

Is it possible to cross the Big Brake ?

Let us study the regime of approach to the BigBrake.

On analyzing the equations of motion we find that onapproaching the Big Brake singularity the tachyon fieldbehaves as

T = TBB +

(4

3W (TBB)

)1/3

(tBB − t)1/3.

Its time derivative s ≡ T behaves as

s = −(

4

81W (TBB)

)1/3

(tBB − t)−2/3,

the cosmological radius is

a = aBB −3

4aBB

(9W 2(TBB)

2

)1/3

(tBB − t)4/3,

its time derivative is

a = aBB

(9W 2(TBB)

2

)1/3

(tBB − t)1/3

and the Hubble variable is

H =

(9W 2(TBB)

2

)1/3

(tBB − t)1/3.

All these expressions can be continued in the regionwhere t > tBB ,which amounts to crossing the BigBrake singularity. Only the expression for s issingular at t = tBB but this singularity is integrableand not dangerous.

Once reaching the Big Brake, it is impossible for

the system to stay there because of the infinite

deceleration, which eventually leads to a decreaseof the scale factor. This is because after the Big

Brake crossing the time derivative of the

cosmological radius and Hubble variable change

their signs. The expansion is then followed by a

contraction, culminating in the Big Crunchsingularity.

Crossing of the soft singularity in the model with the

anti-Chaplygin gas and dustOne of the simplest cosmological models revealing a Big Brakesingularity is the model based on the anti-Chaplygin gas withan equation of state

p =A

ρ, A > 0

Such an equation of state arises in the theory of wigglystrings ( B. Carter, 1989, A. Vilenkin, 1990).

ρ(a) =

√B

a6− A

At a = a∗ =(BA

)1/6the universe encounters the Big Brake

singularity.

The anti-Chaplygin gas plus dust

The energy density and the pressure are

ρ(a) =

√B

a6− A +

M

a3, p(a) =

A√Ba6− A

.

Due to the dust component, the Hubble parameter has anon-zero value at the encounter with the singularity, thereforethe dust implies further expansion. With continued expansionhowever, the energy density and the pressure of theanti-Chaplygin gas would become ill-defined.

Change of the equation of state at soft singularity

crossingsThe abrupt transition from the expansion to the contraction ofthe universe does not look natural. There is analternative/complementary way of resolving the paradox.

One can try to change the equation of state of theanti-Chaplygin gas on passing the soft singularity.

There is some analogy between the transition from anexpansion to a contraction of a universe and the perfectlyelastic bounce of a ball from a wall in classical mechanics.There is also an abrupt change of the direction of the velocity(momentum).

However, we know that in reality the velocity is changedcontinuously due to the deformation of the ball and of the wall.

The pressure of the anti-Chaplygin gas

p =A√

Ba6− A

tends to +∞ when the universe approaches the softsingularity.Requiring the expansion to continue into the region a > aS ,while changing minimally the equation of state, we assume

p =A√| Ba6− A|

,

p =A√

A− Ba6

, for a > aS .

This implies the energy density

ρ = −√

A− B

a6.

The anti-Chaplygin gas transforms itself intoChaplygin gas with negative energy density.The pressure remains positive, expansion continues.The spacetime geometry remains continuous.The expansion stops at a = a0, where

M

a30−

√A− B

a60= 0.

Then the contraction of the universe begins.

At the moment when the energy density of theChaplygin gas becomes equal to zero

(again a soft singularity), the Chaplygin gastransforms itself into the anti-Chaplygin gas

and the contraction continues culminating in anencounter with the Big Crunch singularitya = 0.

Crossing the Big Brake singularity and the future of

the universe in the tachyon model in the presence of

dust.

What happens with the Born-Infeld typepseudo-tachyon field in the presence of a dustcomponent? Does the universe still run into a softsingularity?Yes !

T = TS ±√

2

3HS

√tS − t, HS =

√ρm,0a3S

.

How can the universe cross this singularity ?

A pseudo-tachyon field with a constant potentialis equivalent to the anti-Chaplygin gas.To the change of the equation of state of theanti-Chaplygin gas corresponds the followingtransformation of the Lagrangian of thepseudo-tachyon field:

L = W0

√T 2 + 1,

p = W0

√T 2 + 1

ρ = − W0√T 2 + 1

.

It is a new type of Born-Infeld field, which we maycall “quasi-tachyon”.

For an arbitrary potential the Lagrangian

reads

L = W (T )√T 2 + 1, a > aS

T

T 2 + 1+ 3HT − W,T

W= 0,

ρ = − W (T )√T 2 + 1

,

p = W (T )√T 2 + 1.

In the vicinity of the soft singularity thefriction term 3HT in the equation of motiondominates over the potential term W,T/W .Hence, the dependence of W (T ) on itsargument is not essential and apseudo-tachyon field approaching thissingularity behaves like one with a constantpotential. Thus, it is reasonable to assumethat upon crossing the soft singularity thepseudo-tachyon transforms itself into aquasi-tachyon for any potential W (T ).

Big Bang – Big Crunch crossing ?

I The idea that the Big Bang - Big Crunchsingularity can be crossed appears verycounterintuitive.

I Some approaches to the description of thiscrossing were elaborated during the last decade(I. Bars, S.H. Chen, P.J. Steinhardt andN. Turok, C. Wetterich, P. Dominis Prester).

I There is an analogy with the horizon whicharises due to a certain choice of the spacetimecoordinates: the singularity arises because ofsome choice of the field parametrization.

I On choosing some convenient fieldparametrization one can provide a matchingbetween the characteristics of the universebefore and after the singularity crossing.

I Analogy to the Kruskal coordinates for theSchwarzschild metric.

I On choosing appropriate combinations of thefield variables we can describe the passagethrough the Big Bang - Big Crunch singularity,but this does not mean that the presence ofsuch a singularity is not essential. Indeed,extended objects cannot survive this passage.

Friedmann-Lemaıtre cosmology in the presence of a

scalar field: Einstein frame versus Jordan frame

S =

∫d4x√−g[U(σ)R − 1

2gµνσ,µσ,ν + V (σ)

]

Conformal coupling

U(σ) = U0 −1

12σ2

A conformal transformation of the metric

gµν =U1

Ugµν,

A new scalar field φ:

dφ

dσ=

√U1(U + 3U ′2)

U⇒ φ =

∫ √U1(U + 3U ′2)

Udσ.

φ =√

3U1 ln

[√12U0 + σ√12U0 − σ

]σ =

√12U0 tanh

[φ√

12U1

].

The action then becomes the action for a minimallycoupled scalar field:

S =

∫d4x√−g[U1R(g)− 1

2gµνφ,µφ,ν + W (φ)

],

W (φ) =U21V (σ(φ))

U2(σ(φ)).

This is called the transformation from the Jordan

frame to the Einstein frame.

In a flat Friedmann-Lemaıtre universe

ds2 = N2dτ 2 − a2dl2,

ds2 = N2dτ 2 − a2dl2.

N =

√U

U1N , a =

√U

U1a, t =

∫ √U1

Udt,

where t and t are the cosmic time parameters inthe Jordan and the Einstein frames.

a = a

√U1

U0cosh

(φ√

12U1

).

In the vicinity of the singularity in the Einsteinframe:

a ∼ t13 → 0, when t → 0.

However, in the Jordan frame:

a ∼ t13

(t

13 + t−

13

)→ const 6= 0.

Meanwhile, the scalar field σ crosses the value±√

12U0 and the coupling function U changes itssign.Thus, the evolution in the Jordan frame is regular,and we can use this fact to describe the crossing ofthe Big Bang - Big Crunch singularity in theEinstein frame.

If one considers the expansion of the universefrom the Big Bang with normal gravitydriven by the standard scalar field, thecontinuation backward in time shows that itwas preceded by the contraction towards aBig Crunch singularity in the antigravityregime, driven by a phantom scalar field witha negative kinetic term.

The possibility of a change of sign of the effectivegravitational constant in the model with aconformaly coupled scalar field was analyzed in 1981by A. Starobinsky, following the earlier suggestionmade by A. Linde in 1980.

It was shown that in a homogeneous and isotropicuniverse, one can indeed cross the point where theeffective gravitational constant changes sign.However, the presence of anisotropies changes thesituation: these anisotropies grow indefinitely whenthis constant is equal to zero.

Singularity crossing in a Bianchi - I universe

ds2 = N(τ)2dτ 2 − a2(τ)(e2β1(τ)dx21 + e2β2(τ)dx22 + e2β3(τ)dx23 ),

ds2 = N(τ)2dτ 2 − a2(τ)(e2β1(τ)dx21 + e2β2(τ)dx22 + e2β3(τ)dx23 ),

β1 + β2 + β3 = 0.

βi =βi0a3, θ0 = β2

10 + β220 + β2

30.

φ =φ0

a3, φ =

φ0(3θ02

+3φ204U1

) 12

ln t.

In the vicinity of the singularity in the Einsteinframe

a ∼ t13 .

In the Jordan frame

a ∼ t13 (tγ + t−γ)→ 0,

because

γ =φ0

3√φ20 + 2θ0U1

<1

3.

Thus, one also encounters the Big Bang singularityin the Jordan frame.

Mixing between geometrical and matter degrees of

freedom and the singularity crossingThe Friedmann-Lemaıthe model with a massless scalar fieldcan be described by the Lagrangian

L =1

2x2 − 1

2y 2,

where

x =4√U1√3

a32 cosh

√3

4√U1

φ, y =4√U1√3

a32 sinh

√3

4√U1

φ,

and the Friedmann equation is

x2 − y 2 = 0.

Inversely,

a3 =3(x2 − y 2)

16U1,

φ =4√U1√3

arctanhx

y.

Initiallyx > |y |.

The solution

x = x1t + x0, y = y1t + y0, x21 = y 2

1 .

Choosing the constants as

x0 = y0 = A > 0, x1 = −y1 = B > 0,

we have

a3 =3ABt

4U1.

We can make a continuation in the plane (x , y), tox < |y | or, in other words, to t < 0. Such acontinuation implies an antigravity regime and thetransition to the phantom scalar field, just as in themore complicated schemes, discussed before.

How can we generalize these considerationsto the case when the anistropy term ispresent ?

L =1

2r 2 − 1

2r 2(ϕ2 + ϕ2

1 + ϕ22),

ϕ1 =

√3

8α1, ϕ2

√3

8α2,

β1 =1√6α1 +

1√2α2, β2 =

1√6α1 −

1√2α2, β3 = − 2√

6α1.

We can again consider the plane (x , y) as

x = r cosh Φ,

y = r sinh Φ,

where a new hyperbolic angle Φ is defined by

Φ =

∫dt√ϕ21 + ϕ2

2 + ϕ2.

We have reduced a four-dimensional problem to theold two-dimensional one, on using the fact that thevariables α1, α2 and φ enter into the equation ofmotion for the scale factor a only through thesquares of their time derivatives.

The behaviour of the scale factor before and afterthe crossing of the singularity can be matched byusing the transition to the new coordinates x and y ,which mix geometrical and scalar field variables in aparticular way.

To describe the behaviour of the anisotropic factorsit is enough to fix the constants βi0.

Conclusions and discussion

I General relativity contains many surprises

concerning relations between matter and

geometry. It is enough to take it seriously.

I There is no need to be afraid of

singularities!