theoretical astrophysics and cosmology master degree in … · 2019. 3. 7. · • the matter and...

Homogeneous Friedman Universe 0.1

Theoretical Astrophysics and Cosmology

Master Degree in Astronomy

and Erasmus-Mundus

Alberto Franceschini

Cosmology Course


PROGRAMME FOR THE COSMOLOGY COURSE

0. Introduction to the course

• General motivation. Aims. • Origin and evolution of the cosmological large-scale structure. • Building a physical description about the origin of our world.

00. The Homogeneous and Isotropic (Friedmann) Universe

• Hubble law. The Cosmological Principle. Isotropic curved spaces. The Robertson-Walker metric. Geometrical properties of the space-time.

• Cosmic dynamics, the Newtonian and general-relativistic approach. Cosmological models and parameters.

• Fundamental observables. The redshift. Luminosity and angular diameter distances. Time-redshift relations. Hubble diagrams.

• Generalized dynamical equations. The cosmological constant. Observational evidences [4] Good knowledge of everything

1. The Large Scale Structure of the Universe. Local properties

• General and structural properties of the universe. • Large scale distribution of galaxies. Angular and spatial correlation functions. Higher order correlations.

Limber relation. • Power-spectrum of the cosmic structures. Relationship of the power-spectrum and ξ(r). Observational

data on the large scale structure. The initial power-spectrum of the perturbations. 3D mapping of galaxies, clusters, AGNs.

• Counts-in-cells. Outline of fractal and topological analyses of the large-scale structure of the universe. [2] Chap. 2 all, Cap. 14.1, 14.2, 14.4, 14.6 (see below bibliographic references)

[1] Chap. 13.1, 13.2, 13.8, 16.1, 16.2, 16.3, 16.4, 16.6, 16.7, 16.8, 16.11, 16.12

2. Deviations from homogeneity. Gravitational lensing

• Point-like lenses and isothermal spherical distributions. Lens potentials. Einstein radius. Lensing cross-sections. Lensing effects on time lags. Caustics.


• Observations of the gravitational lensing and cosmological applications. Estimate of the total galaxy cluster mass. Estimates of H0 . Effects of a cosmological constant Λ in the lensing statistics. Micro-lensing and weak-lensing. Mapping of the mass distribution.

[1] Cap. 19.1, 19.2, 19.3, 19.4, 19.5

[2] Cap. 4.6

Further reading: [3] Cap. 4.2, 4.3, 4.4, 4.5

3. Cosmological evolution of perturbations in the cosmic fluid.

• Cosmological evolution of perturbations in the thermo-dynamical parameters of the various components of the cosmic fluid. General equations in a static universe and in an expanding one.

• Evolution in a matter dominated universe. Hubble drag. Relationship of perturbations and velocity fields.

4. Perturbations in an expanding universe. Peculiar motions of galaxies and structures.

• Deviations from the Hubble flow, peculiar velocity fields in the cosmo. Observations of peculiar velocity fields. The cosmic viral theorem.

• Origin of the large scale motions. • Constraints on the cosmological parameters from the large scale motions.

[2] Chap. 11.1, 11.2, 11.3, 11.4.1

[3] Chap. 16.10

5. Brief thermal history of the Universe

• The matter and radiation content of the Universe. Energy densities and their evolution. Radiation-dominated universes. The epoch of recombination and equivalence. Time-scales of cosmic evolution.

• Cosmic entropy per baryon. • Primordial nucleo-synthesis and its consequencies.

[1] Chap. 2.1, 2.2, 2.6, 2.7, 2.8

6. The Cosmic Microwave Background

• Discovery of the CMB. Observations from ground and from space. COBE, WMAP & Planck. Origin of the CMB.

• Spatial properties, isotropy of the CMB. Statistical description of the angular structure. • Origin of the CMB angular fluctuations. Physical processes in operation on the large scales. Fluctuations

on intermediate angular scales. Contributions of sources to the anisotropies on small scales.


• Cosmological re-ionization and its impact on CMB. • Constraints of CMB observations on the cosmological parameters. • The CMB photon spectrum. Spectral distorsions. The Sunyaev-Zeldovich effect. Observational limits on

the spectral distorsions and their implications. • Polarization properties of CMB photons, observations and implications. [2] CHAP. 2.1. Cap. 15.1, 15.2, 15.3, 15.4, 15.5 solo cenni, 15.6, 15.7, 15.8 solo cenni, 15.9, 15.10. Cap. 18.4.1

[1] Chap. 17.1, 17.2, 17.3,17.5

7. The Primordial Universe, Big Bang, phase transitions, cosmological inflation

• Big Bang singularity. Planck time. • The problem of the cosmological horizons: propagation of the information and visibility of the universe. • Brief overview of the standard model of elementary particles. Fundamental interactions. Cosmological

phase transitions and their epochs. • Open questions about the standard Big Bang model. The horizon problem. The flatness problem.

Cosmological inflation and solutions to the problems. • The Anthropic Principle. • Current interpretation of the origin of the perturbation field and of the cosmic large-scale structure.

[1] Chap. 6.1, 6.2, 6.3, 7.1, 7.2, 7.3, 7.4, 7.5, 7.8, 7.9, 7.10, 7.11, 7.12, 7.13

8. Origin and Evolution of the Cosmological Structure

• Generation of the perturbation field (the Primordial Power Spectrum). • Fluctuations from inflation. • General composition of the cosmic fluid: the Dark Matter component. • Properties of the Cold and Hot Dark Matter fluids. • Scales and masses involved within the Cosmological Horizon. • Scale-invariant primordial spectrum and the horizon entrance of perturbations of different scales • Free-streaming and damping in the Dark Matter fluid. • HDM versus CDM. • Stagnation of the Dark Matter perturbation before the equivalence epoch. • Evolution of the DM cosmological perturbation field. The transfer function in the linear regime. CDM

cosmology. • Effects of non-linear evolution in the DM. • Evolution of the cosmic fluid after recombination: the fate of the baryon gas. • Collapse of DM halos. The Press-Schechter theory. • Galaxy formation.


9. The Post-Recombination Universe

• Intergalactic diffuse gas. Absorption-lines in quasar spectra, Lyman-alpha clouds. The missing baryon problem.

• Evolutionary history of star formation and the stellar mass function. Production of heavy elements, constraints from the background radiations.

• Cosmological evolution of Black-Hole accretion in Active Galactic Nuclei and quasars and relationship with galaxy evolution.

[2] Chap. 17, 18, 19

Main bibliographic references

[1] Coles & Lucchin: Cosmology, Second Edition, J Wiley, 2002

[2] Longair: Galaxy Formation, Second Edition. Springer, 2008

[3] Peacock, Cosmological Physics, Cambridge University Press, 2005

[4] Rowan-Robinson “Cosmology”, Clarendon Press, Oxford

and: Franceschini, Lecture Notes in Friedmann Cosmology, 2015, see

http://www.astro.unipd.it/franceschini/CosmologiaIII.html

Note: We will refer in this course to Sections, as the 8 listed above. Chapters will instead mean chapters inside these Sections.

These notes can be found in:

http://www.astro.unipd.it/franceschini/CorsoMagistrale/AstroMundus/

http://www.astro.unipd.it/franceschini/CosmologiaIII.htmlhttp://www.astro.unipd.it/franceschini/CorsoMagistrale/AstroMundus/


Section 0

THE HOMOGENEOUS FRIEDMANN UNIVERSE

0.1 Basic concepts about the fundamental cosmological metrics

It is useful here to summarize some basic concepts about the homogeneous and isotropic Friedmann

Universe. These concepts are the basis for our understanding and description of the Universe in its

present structure and evolution. As it is discussed in some good detail in Sect. 1, while strongly

structured on scales smaller than a few to several tens of Mpc, all observations show excellent

evidence for isotropy and homogeneity on larger scales, and particularly almost perfect homogeneity

on scales sufficiently larger than few hundreds Mpc.

The above is the first fundamental evidence about the global properties of the Universe, and makes

the content of the Cosmological Principle: the Universe appears as homogeneous and isotropic and in

fact looks to share the same properties to any fundamental observer at a given cosmic time. This is key

to any our attempt to describe the general properties of the Universe based on what we observe in our

past light-cone.

The second ingredient for

our formal description of

it is the Hubble law, the

recession velocities of

galaxies are proportional

to their distances,

0v H d=d

d

. These two

general properties, and the symmetry properties that they manifest, can be viewed heuristically with

the analogy of the inflating balloon, see figure. This can be very effectively formalized into a metrics,

Global size of the Universe proportional to the scale factor R(t) function only of time

R(t)

R(t)


called the Robertson – Walker (RW). This is the general cosmological metrics, a foundation of every

cosmological “measurement”

( ) ( )

++

−−= 22222

2

2

222 sin

1ϕϑϑ ddr

krdr

ctadtds (0.1)

where ( ) ( ) 0a t R t R≡

and where ( )R t is the scale factor detailing the expansion law of the Universe, 0R the scale factor at

the present time. This metrics involves only the symmetry properties of space and nothing else, like

the precise law of gravitation (like General Relativity). k is the curvature constant, it does not vary

with time, and for the time being can be set at either 0, -1 or +1 (corresponding to flat Euclidean, open

hyperbolic of closed spherical models). However its precise physical meaning and value will become

clear after setting up the dynamical conditions for the universe in Sect. 0.5 below. The radial r and

angular coordinates are expressed in the so-called comoving units, that is they make a time invariant

coordinate grid to identify cosmic objects at a given time. For complete self-consistency, the radial

comoving coordinate should coincide with the proper coordinate distance at the present time. Proper

(or physical) coordinates are simply scaled from the comoving ones as

( )d a t r= ⋅ .

In these terms, the Hubble constant is simply: ( )( )

0

0

t t

a tH

a t=

=

Concerning the geometry of the Universe, the

three cases of

positive,

negative and

zero constant k

correspond to


positive, negative and flat geometries. These Universes are called closed, open and flat, respectively.

Surfaces and volumes are finite in the first cases and infinite in the others.

In the case of a closed universe a light signal, or a space ship, traveling in an arbitrary direction, will

travel straight away, but its path will be deflected by gravity onto a circular orbit all around the

Universe.

0.2 The cosmological redshift

The first application of the Robertson-Walker metrics is a fundamental generalization of the Hubble

law 0v cz H r= = and the concept of the relation between recession velocity and distance. Let us

consider a photon travelling from a distant source and a fundamental observer (an observer

“comoving” with the Hubble flow). Then 2 0ds = in eq. (0.1) for these photons. Now, by separating

the radial (spatial) and temporal components in (0.1), we have

00 2 ( )1

e

e

r t

t

dr dta tc kr

=−

∫ ∫ (0.01)

and also, independently of time:

002

0

0

0 0

0

, where ( ) 1; so that( ) ( )1

( )( )

that is, in terms of frequency:( )

e

e

e e

e

e e

dt dtdr a ta t a tc kr

a t dta t dt

a t

nn

n n

= = =−

= =

= ⋅

and because of the definition of redshift from special relativity:

10 0 0

10

( ) / / 1 ( ) 1

we finally get (1 ) ( ) /e e e

e e

z a t

z a t

n n n n n

n n

−

−

− = − ≡ = −

+ = =


a relation generalizing the concept of redshift (in the classical special-relativistic limit and/or for low

redshift, this is just v c z= ⋅ ) to sources at any distances in space-time. The conclusion is that the cosmological redshift z is a measure of how much the scale factor has changed from the epoch when

the signal has been sent to that it has been received. This introduces a substantial modification of the

standard Doppler redshift. For large distances, the redshift is a very fundamental measure of distance,

and expresses the ratio of the scale factor at the time the photon is emitted to that when it is received.

0.3 Simple geometrical cosmological models

1- The Milne vacuum model, ( )R t t∝

k = ─1, 0ρ =

2- The Einstein de-Sitter model

k=0 , q0=0.5 Rc → ∞

For the Einstein-de Sitter flat

model the time dependence is:

2/3

0

( ) ta tt

= ±


3- The open model

k= ─ 1,

q00.5

0.4 Approaching the cosmological dynamics from a Newtonian analogue

Now, if we want to define the

appropriate values of k and the

function R(t) we need to insert a

specific model to treat the

gravitational effects in the Universe

and how these and the universal

geometry are affected by the various

mass-energy components (non-

relativistic matter, including Dark

O: observer (coordinates t=t0 , r=0)

Scheme of Newtonian treatment of dynamics. Shell of particles at proper distance x to observer O.

O x


Matter, and relativistic particles like photons and neutrinos in particular). We can get a first insight

into this potentially complicated problem by considering a heuristic approach of using a simple

Newtonian treatment, which is illustrated in the figure. Let us imagine an observer at arbitrary

position O and an expanding shell of particles at distance x. Assuming the origin at the position O

(this is very arbitrary), these particles will feel the gravitational pull of the matter content inside x, but,

because of the Gauss theorem, will not be affected by all the rest of the surrounding universe. Then we

have, in terms of the universal scale factor:

3

30

20

4 4 ( ) ( ) and in comoving coord.: ( )3 3

Now, considering the mass conservation ( ) ( ) ,4 ( ) ( ) 4( ) ( ) .

3 3

Gm V x Gm x G t a tmx a tx

t a tG t a ta t G a t

ρ π ρ π ρ

ρ ρπ ρ π ρ

−

−

− − −= = = =

=

−⇒ = = −

rF r

An integral of this one is the famous Friedmann law:

2 208( )

3 ( )a t G k c

a tπ ρ= − .

0.5 Friedmann Cosmological dynamics and the expansion history of the Universe

The approach above, however, is not complete and not completely self-consistent, for many different

reasons. The most important is that it cannot include into the analysis the relativistic particle

components of the Universe (e.g. photons) that can be relevant during some epochs. Another aspect

not resolved by the classical treatment is the fact that in the classical approach the gravitational

information travels at infinite speed, so that the local dynamics may be influenced by events at infinite

distances, while a relativistic approach and the finite speed of light would naturally limit this sphere

of influence to a relatively small environment of any random point. For these and other reasons, we

need to refer to the more general framework provided us by the General Relativity and its field

equations:


4

1 8 ,2ik ik ik

GR g R Tcπ

− =

with lik ilkR R= the Ricci tensor given by the contraction of the Riemann tensor lijkR , and with

ikikR g R= a scalar quantity, the curvature scalar, and where the quantity gik is the metric tensor that

is synthetically given by eq. (0.1) in its diagonal form. In turn, the Riemann tensor contains linearly

first and second order space-time derivatives of the metrics gik . The quantity

2( )ik i k ikT p c U U pgρ= + −

is the energy-momentum tensor for a perfect fluid, where p and ρ are the pressure and energy

density of matter and Ui the four-velocity of the cosmic fluid. An even more general form of the field

equations has also been considered by Einstein that is still generally co-variant (it is the most general

co-variant form of the equations):

4

1 8 ,2ik ik ik ik

GR g R g Tcπ

− =Λ−

where Λ is an absolute constant, a scalar that is fully independent on position and time, called by Einstein the cosmological constant, in relation with his cosmological static model of the Universe.

The 16 field equations become immediately 4 equations if we consider the diagonal form of the

metrics gik in (0.1). Once applied to our cosmological situation and to the RW metrics brings to the

following two dynamical laws (the second is the famous Friedmann equation, from which the

universal models based on these eqs. are called Friedmann universes):

( ) ( )

( ) ( ) ( )

2

2 2 2 2

4 3 ( ) 1( )3 3

8 13 3

G p ta a t t a tc

G ta a t kc a t

π ρ

π ρ

= − + + Λ

= − + Λ

(0.2)


with Λ the cosmological constant and k the curvature constant, that is the same k constant appearing

in the metrics (0.1) and defining the universal geometry. Note that these equations, without the

relativistic terms p and Λ, are perfectly consistent with what we previously found by simple

application of the Newtonian gravity theory. Often in the first equation of the (0.2) we assume 0=p

to treat the so-called dust universes, or matter-dominated universes, or universes without relativistic

components.

A third and fourth equations that can be inferred from the field equations are the mass-energy

conservation rules (the first has already been mentioned previously):

3 40 0( ) ( ) ; ( ) ( )matter radiationt a t t a tρ ρ ρ ρ− −= =

It is customary to use the following parameters to treat the cosmological dynamics:

( )( )

( )( )

( ) ( )0

0 0 0 02 20

8 8( ) ; ( ) ; ( ) ; ( )3 ( ) 3

mm m m m

t t

a t a t G t GH t H H t t t ta t a t H t H

π ρ π ρ=

= = = Ω = Ω ≡ Ω =

that are the Hubble and density parameters. These are called the cosmological parameters. All this can

be also expressed as a function of the so-called critical density

( )200

0

3( ) such that ( )8 ( )

mC m

C

tHt tG t

ρρ

π ρ= Ω = .

In all these relations we indicate with the symbol (t) the time-dependence quantities. Quantities

appearing without the symbol are defined to be the corresponding values at present time.

As an important note, from eq. (0.6)

( ) ( )

2,0 02 2 2 2

,0 0m Ha kc H a ta t Λ

Ω= − + Ω

we have that, back in the past, when a(t) is sufficiently small, clearly while the second addendum

keeps constant and the third decreases fast because of 2a∝ , the first addendum gets larger and


larger, and dominates the other two. The consequence is that during most of the Hubble time the

Einstein de Sitter model ( 01, 0.5, 0m q kΩ = = = ) applies with good precision.

The Λ term. As for the Λ terms in the above equations, the cosmological constant, this is an

hypothetical term constant in time introduced by Einstein for consistency with his assumed static

Universe solution of the field equations that he tried to model. The Λ parameter needs to be

independent on cosmic time to maintain the general covariance (general independence from the

reference system) of the field equations (0.2). See below in this Chapter for more details and

explanations.

To achieve an accelerating expansion, as needed by the observations, the Λ constant must be positive.

From a more physical point of view, this constant can be interpreted as a structural property of the

cosmic geometry (like interpreted by Einstein). Alternatively, it is possible to interpret it as an energy

density of the vacuum Vρ by assuming that:

VGρπ8=Λ (0.3)

with associated density parameter, similarly to what we defined for the matter density:

( )0

,02 20 0

83 3

VG tH H

π ρΛ Λ

ΛΩ ≡ = = Ω

(0.4a)

hence contributing to the dynamics of the expansion at the current time 0t , together with the density

parameter of gravitating matter

( ) 0,0203

8mmm tH

GΩ==Ω ρ

π. (0.4b)

In this way the first dynamical equation in (0.2) calculated at the present time becomes

( ) ( )3

43

8 00

tGGta mV

ρπρπ−= .

(0.5)


After the above definitions and using the cosmological parameters, the dynamical equations can be

expressed as

( ) ( ) ( )2

,0 0 2022

m Ha t H a ta t Λ

Ω= − + Ω

( )( )

( )taHkctaH

ta m 2202

200,2

ΛΩ+−Ω

= .

(0.6)

Relation between dynamics and geometry. Putting these two equations together and resolving for

the term 2kc , and calculating them at the present time (k does not vary with time, as we have seen) for 1=a and 0Ha = , we get the following fundamental relation between the curvature constant and

the density parameters

[ ]2 20 1mc k H Λ= Ω + Ω − (0.7)

A further cosmological parameter measuring the deceleration or the acceleration of the cosmological

expansion is

0

0 2 2m

t

aaqa Λ

Ω−≡ = − Ω

(0.8)

that is naturally positive (to indicate deceleration under self-gravity), but may become negative

(acceleration) for a sufficiently large cosmological constant.

We see from eqs. (0.7) and (0.8) the profound relationship between the geometrical and dynamical

properties of the Universe, and their relation with its kinematical properties. A flat Universe can be

obtained when the sum of the density parameters 1m ΛΩ + Ω = , corresponding to a sort of balance

between the kinetic energy of the expansion, the self-gravitational energy, and the Λ repulsion term.


Similar relations are reported in Galaxy Formation, (MS Longair), where the constant

( ) 1/20cR t k −ℜ = = is the curvature radius of the RW metrics at the present time. The same radius at generic time t will be given by

( ) ( )tatRc ℜ= hence:

( ) ( ) ktatRc 1222 ==ℜ

(0.9)

222

kcc =ℜ

. (0.10)

In any case, eq. (0.7) sets the correct normalization and value for the curvature parameter k, which is

then proportional to the inverse Hubble radius:

[ ] [ ]

2 2 20

1 1m mH

kc H R

Λ ΛΩ + Ω − Ω + Ω −= = (0.7bis)

and has physical dimension of length-2 . Then the different geometrical solutions discussed in Sect. 0.3

above depend on the combinations of values of the various density parameters.

Relationship of scale factor with redshift, time, radial comoving and proper coordinates. Using

(0.7) in the second of (0.6), the Friedmann equation can be written quite effectively as

( )

Ω++Ω−Ω−Ω

= ΛΛ22

02 1 a

aHta mm

or, collecting 2a

( )2

2 20 3 2

1m ma H t Ha a a

ΛΛ

Ω Ω + Ω − = = − + Ω

(0.11)

and passing to the redshift z:


( )( )

( ) ( ) ( )[ ]2111

22

202 +Ω−+Ω+

+= Λ zzzzz

Hta m . (0.12)

( ) ( )( ) ( )

( ) ( ) ( )[ ] 212021

21111

11 zzzzz

Hdtdz

zdtzd

dttda

m +Ω−+Ω++=

+−=

+= Λ

−

(0.13)

and eventually

( ) ( ) ( ) ( )1 222

0( )(1 ) 1 1 1 2mdz a t z H z z z z zdt Λ

= − + = − + + Ω + − Ω + (0.14)

Then the time – redshift relation is

( ) ( )

( ) ( ) ( )1 22

,020

1 1 211 ( )

mdz dzdt z z z z

H zz a t

−

Λ = − = − + Ω + − Ω + ++

with ( )00, tmm Ω=Ω and ( ),0 0tΛ ΛΩ = Ω .

(0.15)

We can further express the relations between time, redshift and the comoving radial coordinate dr in

the Robertson – Walker metrics using

( )1drc dt a t dr

z= = −

+

(0.16)

dr being the radial distance increment. Setting to )1( zdrdx += the proper radial increment, we

get from (0.15)

( )

( ) ( ) ( )[ ] dzzzzzzH

cdzdzdtcdx m

212

0

2111

−Λ +Ω−+Ω++

=−= (0.17)

( ) ( ) ( ) ( )( ) ( )( )zzzzzzzzz mmm +Ω−−+Ω++=+Ω−+Ω+ Λ 21112211 22 .

At the same time (1 )dr z dx= + ⋅ that can be resolved for r from eq. (0.17):

1/220

0

( ) (1 ) ( 1) (2 )z

Mcr z z z z z dz

H−

Λ′ ′ ′ ′ ′ = + Ω + − Ω + ∫ (0.18)


For a flat universal geometry 1=Ω+ΩΛ m we get 1 for the comoving radial coordinate

( )[ ] 2130 1 ΛΩ++Ω

=zdz

Hcdr

m

(0.19)

Note that, particularly for treating precisely the epochs before the Recombination (see Sect. 5 later),

eqs. (0.11), (0.17) and (0.19) have to be completed with the contributions of relativistic particles and

radiation to the cosmic dynamics. We can easily see (Sect. 5) that (0.11) is substituted in such a more

general case by:

( )2 2 2 2 20 2( ) 1m ma H t a t H aa aγ

γΛ Λ

Ω Ω≡ = + + Ω − Ω − Ω − Ω +

(0.20)

where mΩ , ΛΩ , γΩ are the density parameters of the gravitating matter, the Λ term, and radiation

all calculated at the present cosmic time (for simplicity we have omitted the undersigned 0), all

defined at the present time. Note that the different factors containing mΩ and γΩ come from the

different dependences of the matter and radiation densities with redshift: 3( )m a tρ−∝ , and

4( )a tγρ−∝ .

From observations of the background radiations, at the present time we have:

1 For a flat Universe geometry 1=Ω+ΩΛ m we have

( ) ( ) ( ) ( )( ) ( )( )zzzzzzzzz mmm +Ω−−+Ω++=+Ω−+Ω+ Λ 21112211 22 .

By developing all expressions and products, and all simplifications, this becomes

( ) ( )33 23 3 1 1m mz z z z ΛΩ + + + = Ω + + Ω , where we have taken 1=Ω+Ω Λm

allowing us to write for the radial coordinate ( )[ ] 2130 1 ΛΩ++Ω

=zdz

Hcdr

m


310 mγ−Ω Ω (0.21a)

We often use the following notation in terms of a curvature parameter kΩ :

1k m γΛΩ ≡ − Ω − Ω − Ω . (0.21b)

Then we are in a Universe with flat geometry if 0kΩ = . Then the fundamental relation allowing us to

calculate all relevant quantities, like times, distances, redshift, horizons, ecc., is

( )2 2 20 2m ka t H aa aγ

Λ

Ω Ω= + + Ω + Ω

(0.22)

For example, if we want to calculate the cosmic time corresponding to a given redshift (time from Big

Bang to that corresponding to a given redshift z), using (0.15) we have

( )

( ) ( ) ( ) ( )

2

1 23 4 2

0

( )1 ( )

1 1 11

z

m kz

dzt zz a t

dz z z zH z γ

∞

−∞

Λ

= =+

= + Ω + + Ω + + Ω + Ω +

∫

∫

(0.22.b)

a relation easy to solve numerically. The cosmic look-back time is instead the complementary

quantity:

( )20

( )1 ( )

z

look backdzt zz a t

− =+∫

.

and is illustrated in the following graph.


Time evolution of the density parameters.

The figure below shows the evolution with time and redshift of the various density parameters for a

flat Universe. This shows that their sum remains constant to unity in spite that all quantities vary with

time. This is also required by the constancy of the k parameter.

Cosmic look-back time versus redshift, in a limited redshift interval, and for standard values of the cosmological parameters (see Sect. 0.6 below).

Extrapolating to the Big Bang ( ) we obtain the present time (time lapsed from Big Bang to now):

.


0.4 Distances in cosmology and cosmological observables

Distances in a curved space-time are not so simply defined as in a flat Euclidean space. The mere

integral of the comoving distance r or of the proper distance is not related to any observation we can

imagine. So, depending on the way you make the observation, you have a different distance measure.

There are at least two kinds of distances obviously related to observations: that relating the observed

angular scale and the intrinsic size (the angular size distance), and that relating observed flux and

intrinsic luminosity (the luminosity distance).

In what follows we illustrate the approach of Carroll et al. (1992). We define as the angulare diameter

distance the quantity

ADddϑ

= (0.23)

where D is the intrinsic size of the source and dϑ its apparent angular size. Instead we define as luminosity distance :

21

4

=

SLdL π

(0.24)

where L is the source luminosity and S the flux.

Let us first define as a theoretical measure of distance the radial comoving coordinate

Md r= (Distance Measure) (0.25)

where 1r is the comoving radial distance to the observer, that is the integral of the element

in (0.16) and (0.19). The distance in (0.25) has purely a theoretical significance, because it

cannot be measured with observations. This latter can be re-written for example for a flat

Universe 1=Ω+ΩΛ m , from integration of (0.19),


( )1 2( ) 3

0 00

1r z z

M mcd r dr z dz

H

−

Λ ′ ′ ′ = = = Ω + + Ω ∫ ∫

that for z>>1 and finite Ωm , is easily integrated and gives

1/21/20

2 1Mm

cd zH

− = − Ω (0.26)

In general, again from the RW metrics

( )2

(1 )1

dr c dt c z dta tkr

= = +−

From (0.14) and (0.15) we easily obtain

( ) ( ) ( )1 22

0 00

1 1 2r z

M m

cdr r d z z z z dzH

−

Λ ′ ′ ′ ′ ′= = = + Ω + − Ω + ∫ ∫

(0.28)

Then using the RW metrics, it is easy to find the following expressions for the distances: for

the angular diameter distance, the proper source size in a reference system coherent with

Observer in r=0, t=t0

(r0, θ+d θ, φ, te.)

(r0, θ, φ, te)

dθ D


the observer is 2 2 1/20

( ) ( )edtc ds a t r d Dθ=− = ⋅ = , so that

( )1

MA e

dDd r a td zϑ

= = =+

(0.27)

This graph here illustrates the situation with respect to the Robertson-Walker metrics and

coordinate system:

Effects of the angular diameter distance on the apparent angular sizes of cosmic sources as a function of redshift.


For the luminosity distance,

the photon collecting area of

the telescope as seen by the

source is:

20 0 0sindA r d r d r dθ θ φ= ⋅ ⋅ = Ω . The observed flux, which is the energy flown per

unit area, will correspond to the fraction of the emitted radiant energy within dΩ with

1dA = , 201d rΩ = : 2 2 20

14 4 (1 )L

L LSd r zπ π

= = ⋅+

where the 2(1 )z −+ factor comes from consideration that both the rate of arrival and the

energy are decreased by 1+z. So for the luminosity distance, having set 0 1r r= in (0.25),

we have

( )20 (1 ) (1 ) 1L M Ad r z d z d z= + = + = + . (0.28)

from which

( )

( ) ( ) ( )[ ] dzzzzzzH

cz

ddz

mM

A ∫−

Λ +Ω−+Ω++=

+=

0

212

0

21111

. (0.29)

observer with coordinates:

(r=r0, θ0, φ0, t0)

Q

O

dθ

dθ

dΩ

source with coordinates (0, θ0, φ0, te)


and analogous expression from the luminosity distance. This integral can be easily solved

numerically, for example using the Cosmology Calculator by Edward Wright, University of California,

Los Angeles (http://www.astro.ucla.edu/~wright/CosmoCalc.html).

The need for the term and the present time accelerated expansion. This comes from the analysis

of the Hubble diagram, as reported below for type-1a supernovae at high redshift. These are sources

for which we know the absolute magnitude (the intrinsic luminosity).

http://www.astro.ucla.edu/~wright/CosmoCalc.html


These are the best standard candles for cosmology), whose Hubble diagram showed very faint

magnitudes for their given redshift. To explain such faint observed fluxes (large values of the

apparent magnitudes in the figure, cfr. eq. (0.28 below)) based on standard cosmology would require

negative mass density ρ (!), corresponding to an accelerating Hubble expansion, instead of a

naturally decelerating one under the effect of self-gravity. For example, the behavior of a standard

Milne universe (the “empty” 0Ω = model) is illustrated as a dotted horizontal line, inconsistent with

the observed fainter magnitudes between z=0 and z=1. The question is that they look too faint for the

redshift where they are observed, for a standard model of the Universe without the Λ term, as

illustrated in the graph above.


Note that, to first approximation, for 1mzΩ >> (e.g. during Recombination, see Sects. 5 and 6), and

neglecting the ΛΩ term in the denominator, the angular-diameter, distance-measure integral relation

(0.29) simplifies to

0.502A md c H zΩ (0.30)

and correspondingly dM ≈ 2c/H0Ωm1/2 ≈ 15.65 Gpc at the last-scattering surface at z=1000,

although a more precise determination requires the full integration of the (0.28) and (0.29), see Sect. 6.

Note that the angular diameter and luminosity distances at z=1000, for example, differ by a factror 106.

Figure 0.1 Illustration of the behavior of angular diameters and comoving distances as a function of redshift, for cosmological models with and without curvatures and the cosmological constant term. [Figure taken from Coles & Lucchin 2000].


A nice illustration of the effects of the various cosmological parameters in shaping the comoving

radial and angular size distances is offered in Figure 0.1 Concerning the radial distance, the

cosmological constant has the effect to stretch it, for a given redshift, compared to the case of a flat

matter dominated Universe. Similar stretching is produced by an open Universe. A flat Lambda-

dominated Universe shows an intermediate-level of radial stretching, but a very large one in the

direction orthogonal to the line-of-sight, such that a given angular size subtends a very large proper-

size source.

0.6 The current standard dynamical model of the Universe

We can anticipate here that a vast variety of high-precision cosmological observations, that will be

discussed in good datails during later parts of the Course, have determined the cosmological

parameters to assume the following values:

0 70 / sec/0.30.70.001

m

H Km Mpc

γ

Λ

Ω

ΩΩ

Based on this, it is immediate to plot the evolution of the universal scale factor as a function of time, as

illustrated in the

figure.

Cosmological scale

factor on a linear scale

for the standard model

of cosmology.


... on a log scale...

theoretical astrophysics and cosmology master degree in … · 2019. 3. 7. · • the matter and...

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