Advanced Course in Modern Cosmology

Alexei A. Starobinsky

Landau Institute for Theoretical Physics RAS,Moscow- Chernogolovka, Russia

Part IUtrecht University, Utrecht, Netherlands

10.11.2014

Basics of cosmology

History of the Universe

Present matter content of the Universe

Dark matter and dark energy

Background evolution - ideal fluids

Background evolution - scalar field

Basics of cosmology

Subject of cosmology:

general properties of the present Universe at large scales, itspast and future

3 ”whales” on which ”old” classic cosmology is based:

1. The Einstein gravity (General Theory of Relativity)

Gravitational field is described by a space-time metricsatisfying the Einstein equations (possibly with a cosmologicalconstant)

L =R − 2Λ

16πG+ Lm

Verified with ∼ 10−4 accuracy in Solar system experiments.Gravitational radiation from double radio-pulsars.

2. Approximate homogeneity and isotropy of the Universe

a) space-time metric:

v = H0r, H0 = (70± 3)km

s ·Mpc

the Hubble law with H0 - isotropic

b) matter:isotropic spatial distribution of galaxies and clusters, isotropyof galaxy counts;isotropy of the X-ray background

c) radiation (CMB):black-body with the almost isotropicTγ = (2.72548± 0.00057)K

3. Hot past (Big Bang)

Change from the ”old” to ”new” standard

cosmology

1. Understanding that all 3 basic foundations are approximate.a) There exist natural generalizations of the Einsteinequations, and we need them.b) The Universe may be and generically is indeed veryanisotropic and inhomogeneous at very large scales notobservable now.c) The very early Universe may still be ”cold”.

2. Discovery of two new kinds of dark ”entities”: dark matterand dark energy.

3. Discovery of two new periods in the evolution of theUniverse in the very remote past and at the present time whenits expansion is accelerated.

Main epochs of the Universe evolution – before

1979H ≡ a

awhere a(t) is a scale factor of an isotropic

homogeneous spatially flat universe (aFriedmann-Lemaitre-Robertson-Walker background):

ds2 = dt2 − a2(t)(dx2 + dy 2 + dz2) + small perturbations

The history of the Universe in one line: two main epochs

? −→ FLRWRD=⇒FLRWMD −→ ?

Geometry

H =1

2t=⇒ H =

2

3t

Physicsp = ρ/3 =⇒ p ρ

Main epochs of the Universe evolution – now4 main epochs

? −→ DS=⇒FLRWRD=⇒FLRWMD=⇒DS −→ ?

Geometry

|H | << H2=⇒ H =1

2t=⇒ H =

2

3t=⇒ |H | << H2

Physics

p ≈ −ρ =⇒ p = ρ/3 =⇒ p ρ =⇒ p ≈ −ρ

Duration in terms of the number of e-folds ln(afin/ain)

> 60 ∼ 55 8 0.3

Present matter content of the Universe

In terms of the critical density

ρcrit =3H2

0

8πG≈ 0.9× 10−29 g/cm3, Ωi = ρi

ρcrit,

∑i Ωi = 1

(neglecting spatial curvature - less than 0.7%):

I Baryons (p,n) and leptons (e−) 5%No primordial antimatter.

I Photons (γ) 4× 10−5

Tγ = (2.72548± 0.00057)K

I 3 types of neutrinos (νe , νµ, ντ ) < 0.5%∑i

mνi < 0.23 eV,∑

i

mνi = 94Ωνh2 eV.

I Non-relativistic non-baryonic dark matter ≈ 25%

I Dark energy ≈ 70%

Dark matterDark matter and dark energy are seen through gravitationalinteraction only – we know the structure of their effectiveenergy-momentum tensor.

DM - non-relativistic, gravitationally clustered.DE - relativistic, unclustered.Definition of their effective EMT – through equations(conventional).DM - through the generalized Poisson equation:

4Φ

a2= 4πG (ρ− ρ0(t)).

Φ(r, t) is measured using the motion of ’test particles’ in it.a) Stars in galaxies → rotation curves.b) Galaxies → peculiar velocities.c) Hot gas in rich galaxy clusters → X-ray profiles.d) Photons → gravitational lensing (strong and weak).

Observations: DM is non-relativistic, has a dust-like EMT –p ε = ρc2, p > 0, collisionless in the first approximation –σ/m < 1 cm2/g, and has the same spatial distribution asvisible matter for scales exceeding a few Mpc.

Ground experiments: very weakly interacting with baryonicmatter, σ < 10−43 cm2 for m ∼ (50− 100) GeV.

Dark energyTwo cases where DE shows itself:1) inflation in the early Universe – primordial DE,2) present accelerated expansion of the Universe – present DE.

Quantitative and internally self-consistent definition of itseffective EMT - through gravitational field equationsconventionally written in the Einstein form:

1

8πG

(Rν

µ −1

2δνµR

)= T ν

µ (vis) + T νµ (DM) + T ν

µ (DE) ,

G = G0 = const - the Newton gravitational constantmeasured in laboratory.In the absence of direct interaction between DM and DE:

T νµ (DE);ν = 0 .

Possible forms of DE

I Physical DE.New non-gravitational field of matter. DE proper place –in the rhs of gravity equations.

I Geometrical DE.Modified gravity. DE proper place – in the lhs of gravityequations.

I Λ - intermediate case.

Generically, DE can be both physical and geometrical, e.g. inthe case of a non-minimally coupled scalar field or, moregenerically, in scalar-tensor gravity. So, there is no alternative”(either) dark energy or modified gravity”.

Determination of DE EMT from observations

Neglecting the spatial curvature (less than 0.7% of the criticaldensity):

ds2 = dt2 − a2(t)(dx2 + dy 2 + dz2)

The reconstruction programme: determination of the Universeevolution in the past from observational data.The basic quantity to be found: the Hubble parameter H ≡ a

a

as a function of redshift z ≡ a(t0)a− 1.

All components of the Riemann tensor can be expressedthrough H(z) and dH(z)

dz.

EMT of present DE from the definition above:

ρDE =3H2

0

8πG(h2(z)− Ωm0(1 + z)3)

pDE =3H2

0

8πG

(−h2(z) +

1

3

dh2(z)

dz

)where h(z) = H(z)

H0, H0 = H(t0) is the Hubble constant and

Ωm0 is the present density of non-relativistic matter in termsof the critical one.

The DE effective equation of state wDE ≡ pDE

ρDE.

wDE > −1 – normal case,wDE < −1 – phantom case,wDE ≡ −1 – the exact cosmological constant.

Background evolution - ideal fluids

Friedmann-Lemaitre-Robertson-Walker background

ds2 = dt2 − a2(t)dl2

H2 +Ka2

=8πGρ + Λ

3

Ideal fluid: p = p(ρ)

Energy-momentum tensor conservation: ρ + 3H(ρ + p) = 0

3 ln a = −∫

dρ

ρ + p

Some simple solutionsThe spatially flat case: K = 0

I. One ideal fluid with the linear equation of state: p = αρ

a(t) ∝ tq , q =2

3(1 + α)

α = 0 a(t) ∝ t2/3 – matter-dominated stage;α = 1/3 a(t) ∝ t1/2 – radiation-dominated stage.

II. Dust (CDM and baryons) + radiation:ρ = ρm + ρr , pm = 0, pr = ρr/3

a(η) = a1η(η + η1), η =

∫dt

a(t)

t(η) = a1η2(η

3+

η1

2

), ηeq = η1

√2− 1

2≈ 0.207η1

III. Cosmological constant + spatial curvature = de Sitterspace-time

Rαβγδ = H20 (gαγgβδ − gαδgβγ )

4 most popular forms of its space-time metric (only the firstmetric covers the whole space-time):

ds2 = dt2c − H−2

0 cosh2(H0tc) (dχ2c + sin2 χcdΩ2)

ds2 = dt2 − a21e

2H0t (dr 2 + r 2dΩ2), a1 = const

ds2 = dt2o − H−2

0 sinh2(H0to) (dχ2o + sinh2 χodΩ2)

ds2 = (1− H20R2)dτ 2 − (1− H2

0R2)−1dR2 − R2dΩ2

dΩ2 = dθ2 + sin2 θdφ2

IV. Dust + positive cosmological constant – the standardΛCDM model (neglecting radiation).

a(t) = a1

(sinh

3H1t

2

)2/3

, Λ = 3H21

Consequence for the first statefinder parameter, or jerk:

s =

...a a2

a3≡ 1

Home problem 1

Find a(t) for:1) dust + radiation + spatial curvature,2) radiation + cosmological constant.

FLRW dynamics with a scalar field

In the absence of spatial curvature and other matter:

H2 =κ2

3

(φ2

2+ V (φ)

)

H = −κ2

2φ2

φ + 3Hφ + V ′(φ) = 0

where κ2 = 8πG (~ = c = 1).

Reduction to the first order equationIt can be reduced to the first order Hamilton-Jacobi-likeequation for H(φ). From the equation for H , dH

dφ= −κ2

2φ.

Insering this into the equation for H2, we get

2

3κ2

(dH

dφ

)2

= H2 − κ2

3V (φ)

Time dependence is determined using the relation

t = −κ2

2

∫ (dH

dφ

)−1

dφ

However, during oscillations of φ, H(φ) acquires non-analyticbehaviour of the type const +O(|φ− φ1|3/2) at the pointswhere φ = 0, and then the correct matching with anothersolution is needed.Home problem 2Solve this equation for V = V0 exp(ακφ).

Inflationary slow-roll dynamics

Slow-roll occurs if: |φ| H |φ|, φ2 V , and then |H | H2.

Necessary conditions: |V ′| κV , |V ′′| κ2V . Then

H2 ≈ κ2V

3, φ ≈ − V ′

3H, N ≡ ln

af

a≈ κ2

∫ φ

φf

V

V ′ dφ

First obtained in A.A. Starobinsky, Sov. Astron. Lett. 4, 82(1978) in the V = m2φ2

2case and for a bouncing model.