The cosmological dynamics

Using the line element written for the maximally symmetric Universe (1.9) we obtain the cosmological dynamics by solving the Einstein equations,

G^µ_v = 8πGT_v^µ, (1.21)

where T_v^µ is the energy momentum of the components and G^µ_v is the Einstein tensor and is defined by the Ricci scalar and the Ricci tensor:

G_µν ≡R_µν− 1

2g_µνR . (1.22)

The Christoffel symbols, the Ricci scalar and the the Ricci tensor are defined similar to the their 3D versions in (1.5) and (1.4) while now the indices span the space-time coordinates,

Γ^ρ_µν ≡ g^ρλ

2 (∂_µg_νλ+∂_νg_µλ−∂_λg_µν) , (1.23) R_µν ≡∂_αΓ^α_µν −∂_νΓ^α_µα+ Γ^α_βαΓ^β_µν−Γ^α_βνΓ^β_µα, (1.24)

R ≡R^µ_µ=g^µνRµν. (1.25)

The left hand side of the Einstein equation (1.22) specifies the geometric prop-erties of the space-time and the right hand side characterizes the energies and momenta of the particles. For the FLRW metric with zero curvature k = 0 the

1.2. The cosmological dynamics

Christoffel symbols Γ^ρ_µν using the Eq. (1.23) are:

Γ⁰₀₀ = 0, Γ⁰_0i = Γ⁰_i0 = 0, Γ⁰_ij =aa δ˙ _ij, Γⁱ_0j = Γⁱ_j0 = ^a_a^˙δ_jⁱ,

Γ¹₁₁ = 0, Γ¹₂₂ =−r, Γ¹₃₃ =−rsin²θ,

Γ²₃₃ =−sinθcosθ, Γ²₁₂ = Γ²₂₁ = Γ³₃₁ = Γ³₁₃ = ¹_r, Γ³₂₃ = Γ³₃₂ = cotθ,

(1.26)

where ˙a ≡ ^da_at and t is the cosmic time. Note that g_αν satisfies the relation g^µρg_ρν =δ^µ_ν,whereδ_µ^νis Kronecker’s deltaδ_µ^ν = 1 for µ=ν and δ_µ^ν = 0 for µ6=ν. Unlike the Kronecker’s deltaδ_µν is not a tensor⁵, however it shares similar prop-erties i. e. (δ_µν = 1 for µ=ν and δ_µν = 0 for µ6=ν) which is useful to write the results in a compact way. The Ricci tensor and the Ricci scalar are computed by equations (1.24) and (1.25)

R₀₀=−3H²+ ˙H, R_0i =R_i0 = 0, R_ij =a²3H²+ ˙Hδ_ij, (1.27)

R = 62H²+ ˙H , (1.28)

where H ≡ a/a˙ is the Hubble parameter and represents the expansion rate of the Universe. The Einstein tensor components, using the equation (1.22) reads

G⁰₀ =−3H², G⁰_i =Gⁱ₀ = 0, Gⁱ_j =−3H²+ 2 ˙Hδ_jⁱ, (1.29) where we have used G^µ_ν =g^µαG_αν.

1.2.1 The stress energy tensor

The stress energy tensor contains all the information about the energy content of the Universe. This tensor is symmetric as a result of the Einstein equation and the symmetry of the Einstein tensor G_µν. The assumption of isotropy and homogeneity implies thatT₀^µ must be zero andT_i^j must be diagonal with equal values T₁¹ = T₂² = T₃³. Thus, in the FLRW space-time the energy-momentum tensor can only take the perfect fluid form:

T_v^µ= (ρ+P)u^µuν +P δ_v^µ, (1.30) where u^µ = (−1,0,0,0) is the four velocity in the comoving coordinates, ρ and P correspond respectively to the density and the pressure of the perfect fluid.

5Under coordinate transformation this object does not transform correctly.

1.2.2 Friedmann Equations

From the (0,0) and (i, i) components of the Einstein equations Eq. (1.21) we obtain,

where the sums are over all cosmological species. Combining the two equations we can write an equation for ¨a,

¨ a

a =−4πG

3 (ρ+ 3P), (1.33)

which is equivalent to the continuity equation if we multiply by a² and take a time derivative,

˙

ρ+ 3H(ρ+P) = 0. (1.34)

It is worth mentioning that the equivalence of the two equations comes from the fact that the Einstein tensor satisfies the Bianchi identities, i.e., the covariant derivative of the Einstein tensor vanishes, ∇_µG^µ_ν = 0, and from the Einstein equations the same symmetry should be hold in the stress-energy tensor i.e.,

∇_µT_ν^µ = 0. The conservation of the stress-energy tensor gives the same equation as Eq. (1.34) in the FLRW background. That is why the equation (1.34) is called the conservation or continuity equation.

We can rewrite the first Friedman equation Eq. (1.31) in the following form,

Ω_r+ Ω_m+ Ω_DE = 1, (1.35)

where ΩX ≡ ^8πGρ_3H2^X. The density parameters respectively correspond to the relativistic particles, non-relativistic matter and dark energy. Once we have the equation of the state P = P(ρ) we can solve the aforementioned equations to obtain a(t), ρ(t) and P(t).

For a constant linear equation of state P = wρ the continuity equation Eq. (1.34) gives , ρ ∝ a^−3(1+w). As a result for non-relativistic matter (w ' 0) ρm ∝a⁻³, for relativistic particles (w= 1/3) isρm ∝a⁻⁴and for the cosmological constant (w=−1)ρ is a constant. The negative pressure from the cosmological results in the late time cosmic acceleration which has been approved by many cosmological observations. However, the cosmic acceleration could be produced with values ofwother than−1 and yet be consistent with the cosmological data.

Using the density relation for each component and Eq. (1.35) we can rewrite the equation in the following form,

H² =H₀²Ω⁰_Λ+ Ω⁰_ka⁻²+ Ω⁰_ma⁻³+ Ω⁰_ra⁻⁴, (1.36) where H₀ is the current value of the Hubble constant.

1.2. The cosmological dynamics

1.2.3 The Hubble Law and cosmic redshift

The light emitted by a distant observer is stretched while traveling due to the expansion of the Universe. This effect is similar to the Doppler shift in the frequency of a wave in classical physics but this happens because the emitter is receding from us and according to General Relativity the cosmic expansion dilutes the photons’ energy. In the 1920s Slipher and Hubble realised that the measured wavelength λ_o of absorption lines of distant astronomical objects is larger than the wavelength λ_e measured in the laboratories. We define the red-shift as follows

z ≡ λ_o

λ_e −1 = a_o

a_e −1, (1.37)

wherea₀ anda_e are respectively the scale factors at the observation time and at the emission time.

For the small values of recessional velocityv compared to the speed of light, we have λ₀ '(1 +v/c)λ_e from the Doppler effect which results in,

z 'v/c . (1.38)

In an expansing Universe the physicalr and comovingχdistances of the objects are related by the scale factor at each time,

~r=a(t)~χ . (1.39)

Taking the time derivative of the previous relation gives,

~r˙ =H~r+aχ ,~˙ (1.40) where the first term appears due to the cosmic expansion and the second term is peculiar velocityvp and is the movement of an object with respect to the Hubble flow. It is obvious that in a maximally symmetric Universe the peculiar velocities should vanish because these velocities are different at different positions which is in contradiction with the homogeneity and isotropy of the Universe.

Slipher in 1912 for the first time measured the spectrum of a galaxy, M31, which for many years was the highest measured velocity for any object which was about −300kms⁻¹. He continued measuring velocities of “nebulas” for several years, his “catalogue” appearing in Eddington’s book “The Mathematical Theory of Relativity”, published in 1924.

However, only after Robertson’s work, Slipher’s measurement was interpreted as being due to the cosmic expansion. The main problem at the time was ig-norance of the distances to the galaxies and also Robertson’s work was largely diregarded.

Hubble, based on previous works, reported an extragalactic distance for 18 galaxies and was able to plot the relationship between distance and recession

velocity. Most of Hubble’s data for velocity came from Slipher’s measurements.

Hubble, in fact, wrongly interpreted the velocity-distance relationship to the de-Sitter effect in a static de de-Sitter Universe⁶, not to cosmic expansion - Robertson had already done that. De Sitter in 1933 wrote the linear expansion law taking into account of redshift-distance relation in the de Sitter model. But then the Friedmann-Lemaître model was acknowledged by physicists. That acceptance was mainly due to the later paper of Eddington in 1930 in which he “rediscovers”

the papers of Friedmann and Lemaître. This of course was after Robertson had published his paper in 1929 claiming that the redshift was due to cosmic expansion, and also after Hubble’s redshift-distance relationship (see Jones [1997]

for the history of cosmology).

1.2.4 Comoving distance

The lights we observe from distant objects are traveling on a light cone and satisfy the geodesic equation

ds² =−c²dt²+a²(t)dχ² = 0. (1.41) Considering the case in which light travels from t = te at distance χ = χe

(redshiftz) and reaches us att =t_o atχ= 0 (z = 0), and using the line element, gives us a distance which is defined as,

dc≡χ1 =

Z χ1

0

dχ=−

Z t1

t0

c

a(t)dt , (1.42)

where d_c is called the comoving distance.

In this section we have studied a homogenous and isotropic Universe and we derived the FLRW metric and the Friedmann equations. The homogeneity and isotropy are a good approximation of the Universe at large scales. However, we know that these symmetries are not respected at smaller scales anymore and in the next section we will perturb the FLRW metric and we study the evolution of the perturbations in such an Universe.