Growth of density perturbations

MP .1.8×1016GeV, H'3.1×10−5 r 0.07 1/2 MP .7.6×1013GeV. (1.59)

1.4.2

Growth of density perturbations

The scalar perturbations produced during inflation are pushed outside the Hubble patch due to the quasi-exponential expansion. After inflation, the comoving Hubble patch starts to expand and consequently the perturbations modes which are frozen on super-Hubble scales successively re-enter the Hubble patch. Once a perturbation mode is inside the comoving Hubble patch, it starts evolving.

Let us now consider the evolution of the perturbation modes governed by the Einstein field equations which relate the scalar metric perturbations to density and pressure fluctuations in the cosmic fluid. This is most conveniently done in the conformal Newtonian gauge. In this gauge, there are only two scalar modes (Φ and Ψ) which are actually equal in the absence of anisotropic stresses (perfect fluid approximation) 1_{. On sub-Hubble scales, Φ acts as a Newtonian potential. For} the purpose of analytical estimation, a single component domination is assumed2.

1_{This is true provided that Φ and Ψ are Gaussian, random fields which is actually the case}

according to the measurement of CMB [49]. Note that, some components of the cosmic fluid such as neutrinos which decouple from the plasma when the temperature drops to a few MeV, do not behave as ideal fluids and can give rise to small anisotropic stresses. However, this effect becomes negligible once matter comes to dominate the Universe. We do not discuss this issue here.

The cosmic fluid is made of multiple components such as non-relativistic mat- ter dominated by DM and plasma which consists of photons, baryons and lep- tons1_{. On super-Hubble scales, the scalar perturbations are constant and related} to the gauge-invariant curvature perturbations through the relation:

Φk' −

3 + 3ω

5 + 3ωRk, (1.60)

which provides the initial condition for the subsequent sub-Hubble evolution. Here, the curvature perturbation, Rk, is given by Eq. (1.50). The density per-

turbation of the dominant fluid component is

δk≡ δρk ¯ ρ =− 2 3 k2 H2Φk−2Φk− 2 HΦ 0 k, (1.61)

where δ≡δρ/ρ¯denotes the density contrast. On super-Hubble scales (kH),

δk ' −2Φk'constant. For modes that re-enter the Hubble patch during RD,

Φk' −2₃Rk on super-Hubble scales, whereas deep inside the Hubble patch

Φk(τ) = 6Rk " (kτ) cos(kτ /√3)−√3 sin(kτ /√3) (kτ)3 # , (1.62)

i.e. the gravitational potential exhibits an oscillatory behaviour with a decaying amplitude and a well-defined phase. Using Eq. (1.61), one then obtains the sub- Hubble radiation density contrast

δrad,k(τ)'6 Φ (i) k cos(kτ / √ 3)' −4Rkcos(kτ / √ 3). (1.63)

Clearly, the radiation density fluctuations do not grow during RD; instead, they oscillate aroundδrad,k= 0 with a constant amplitude (acoustic oscillations). They

continue to oscillate even after matter comes to dominate the energy density of the Universe, albeit with a shifted equilibrium pointδrad,k' −4ΦDM,k because of

the potential due to the fluctuation in DM (and baryons) density. The acoustic

oscillations in the radiation density show up in the CMB temperature angular power spectrum as peaks and troughs.

On the other hand, the perturbations in the sub-dominant DM fluid, which are constant on super-Hubble scales, δDM,k =Rk 1, start growing once they re-

enter the Hubble patch. DM density perturbation modes that re-enter the Hubble patch during RD grow only logarithmically till the end of RD,

δDM,k(τ)' −9 Φ (i) k ln(kτ /√3) +γE− 1 2 '6Rk ln(kτ /√3) +γE− 1 2 ,(1.64) where γE= 0.577 is the Euler constant. This is due to the fact that during RD the gravitational potential is mostly dominated by the rapidly-decaying density fluctuation in the relativistic fluid [see Eq. (1.62)] 2_{. However, as it is clear} from Eq. (1.64), besides their logarithmic growth during RD, the DM density perturbations are enhanced by a significant numerical factor over the initial value,

δ_DM(i) _,k =Rk.

Let us now consider the evolution of DM perturbation during MD. On super- Hubble scales (kH), δDM,k' −2Φ

(i)

k '

6

5Rk'constant [see Eq. (1.60)]. It can be shown that the Newtonian potential, Φk, is also constant on sub-Hubble scales

(see e.g. [29, 30]). Thus, using Eq. (1.61), the DM density contrast on sub-Hubble scales (kH) is given by δDM,k(τ)' − 2 3 k2 H2Φ (i) k ' − 2 3Φ (i) k a(τ) a(τ×) ' −2 5Rk a(τ) a(τ×) = 1 3δ (i) DM,k a(τ) a(τ×) , (1.65) where we substituted k = H× with the subscript ‘×’ referring to the second

Hubble crossing (i.e. the time at which a particular perturbation mode re-enters the Hubble sphere), and H=H×[a(τ×)/a(τ)]1/2 during MD. Further, for DM

density perturbation modes that re-enter the Hubble patch during RD, these keep growing during MD but now linearly instead of logarithmically as they do during RD,

1_{Here, we assume adiabatic initial conditionδ}(i) DM,k=

3 4δ

(i)

rad,k, see Section 1.4.3 below.

2_{The gravitational potential also receives a contribution from the perturbations in the DM}

fluid ΦDM,k=−(a/k)2( ¯ρDM/2MP2)δDM which is small compared to Eq. (1.62) deep inside RD

δDM,k(τ)' − 27 2 a(τ) a(τeq) Φ(_ki)ln(0.15kτeq)'9Rk a(τ) a(τeq) ln(0.15kτeq). (1.66) It is worth noting here that baryons, which are already non-relativistic during MD, contribute to both the total energy density and matter perturbations which gives rise to a contribution to the DM density perturbations growth equations, albeit relatively small (see e.g. [30]). Let us further consider the evolution of matter deep inside the era of DE domination in which casea∝ |τ|−1_{, i.e.} H_=|τ|−1_. Assuming that DE is constant in space and time,δρ_Λ=δP_Λ= 0,

Φk ∝

(

τ ∝a−1

τ3 _∝a−3_, (1.67)

and since δρ_Λ= 0, i.e. δρ=δρm, the matter density contrast is given by

δm ' − 2M2 P ¯ ρm k2 a2Φ∝ ( a3k2 a2a −1 _{= constant} a3k2 a2a −3 _∝a−2_, (1.68)

i.e. the density fluctuations stop growing once the Universe starts expanding quasi-exponentially. However, in reality the present DE domination in the Uni- verse is not sufficient to cause it to expand quasi-exponentially. Instead the scale factor still increases with a power law. Thus, DM density perturbation modes that re-enter the Hubble patch today can still grow.

To sum up this section, all perturbations modes are constant on super-Hubble scales and start evolving once they re-enter the Hubble patch. The radiation den- sity perturbations oscillate with a decaying amplitude whereas the perturbations in DM density grow logarithmically in the scale factor during RD and linearly during MD. Note that baryons, which are already non-relativistic when the per- turbation modes of cosmological interest re-enter the Hubble patch, are coupled to photons through Compton scattering and hence their perturbations do not grow until they decouple from photons, which occurs at redshift zdec'1190 [13]. By that time, δDMδb, where δb denotes the baryons’ density contrast. As a result, baryons fall into the DM potential wells.

long time ago and have grown into the structure we observe today. However, matter perturbations with longer wavelengths which re-enter the Hubble patch during MD (or shortly before radiation-matter equality) are still in the linear regime. Therefore, the Universe on these scales and larger (_&100 Mpc) is still mostly homogeneous. In other words, there are no gravitationally bound systems with masses_&1016_M

, whereM'1.988×1033g is the solar mass (see e.g. [30]).

Matter perturbations with shorter wavelengths grow according to Eq. (1.66), which upon using Eq. (1.12) can be re-written as

δDM,k(z)'9Rk

1+zeq

1+z ln (0.15kτeq) . (1.69)

Using the measured values provided by the Planck collaboration: ∆2

R'2×10−9

and zeq '3393 [13], one can see that for δDM,k to become &O(1), ln(0.15kτeq) has to be_&1 which implies that modes with k/a(t0)&(10 Mpc)−1 have already become non-linear. This corresponds to length scales _.30 Mpc and masses _. 1015M. Of particular interest is the formation of galaxies, mass∼(1011–1012)M,

which starts atz∼4 (see e.g. [30]). Figure 1.1 shows a sketch of the time evolution of density perturbations of the different components of the cosmic fluid that re- enter the Hubble patch during RD along with the Newtonian potential.