CI & BI modes

5.2 Evolution of photon and baryon perturbations

5.2.1 Evolution of photons and baryons prior to decoupling

5.2.1.4 CI & BI modes

whereψhas been replaced byφsince there is no shear in the tight-coupling regime (see equation (5.15)). The solution of equation (5.44) up to leading order is given by

φ =− 4Rν

5 + 4Rν

1

kτ. (5.45)

The potentialφ decreases with time outside the horizon in the radiation-dominated era. This is shown in Figure 5.8(a) for differentk−modes.

Through horizon crossing, as we emphasized for the neutrino isocurvature density model, it is difficult to obtain the potentialφexactly. A similar treatment to that for the neutrino isocurvature density model can be applied here. The equations (5.37) and (5.38) that we found for the neu- trino isocurvature density model are still valid and applicable for we did not make any restrictive assumption related to the nature of the primordial perturbations for their derivation. Therefore prior to matter-radiation equality, the potential oscillates and is sourced by the photons. After matter-radiation equality, it becomes constant. These features are shown in Figure 5.8(b) where the time evolution of the potentialφ is shown for four different wavenumbers corresponding to different epochs of horizon crossing.

(a) (b)

(c) (d)

Figure 5.9: Evolution of the photon density contrast prior to decoupling for the CI mode:

Comparison of numerical and semi-analytic solutions for some wavenumbers. We consider k = 0.019h Mpc⁻¹for the top-left panel,0.047h Mpc⁻¹for the top-right panel,0.088h Mpc⁻¹ for the bottom-left panel and0.25h Mpc⁻¹ for the bottom-right panel respectively. The vertical dashed lines mark the matter-radiation equality and decoupling.

today. The driving term is small in the radiation domination era as the photon and the neutrino densities are initially unperturbed but becomes important in the matter domination era as the matter perturbation sources the gravitational potential [73].

(a) (b)

(c) (d)

Figure 5.10: Evolution of the photon velocity divergence prior to decoupling for the CI mode:

Comparison of numerical and semi-analytic solutions for some wavenumbers. We considerk = 0.019h Mpc⁻¹ for the top-left panel,0.047h Mpc⁻¹ for the top-right panel, 0.088h Mpc⁻¹ for the bottom-left panel and 0.25h Mpc⁻¹ for the bottom-right panel respectively. The vertical dashed lines mark the matter-radiation equality and decoupling.

The time evolution of the photon and baryon density contrasts for the CI and BI modes are given

by [73]

δ^CI_γ =−8 3Ω_c,0

√3

k sinkr_s(τ)×e^−k2/kD2 +

√3 k

Z τ 0

(1 +R(τ^′))^1/2sin [krs(τ)−krs(τ^′)]×F^CI(τ^′)dτ^′, (5.48) δ^CI_b =−2Ωc,0

√3

k sinkrs(τ)×e^−k2/k2D + 3

4

√3 k

Z τ 0

(1 +R(τ^′))^1/2sin [kr_s(τ)−kr_s(τ^′)]×F^CI(τ^′)dτ^′, (5.49) for the CI mode, and by

δ_γ^BI =−8 3Ωb,0

√3

k sinkrs(τ)×e^−k2/k2D +

√3 k

Z τ 0

(1 +R(τ^′))^1/2sin [krs(τ)−krs(τ^′)]×F^BI(τ^′)dτ^′, (5.50) δ_b^BI = 1−2Ω_b,0

√3

k sinkr_s(τ)×e^−k2/k2D + 3

4

√3 k

Z τ 0

(1 +R(τ^′))^1/2sin [krs(τ)−krs(τ^′)]×F^BI(τ^′)dτ^′, (5.51) for the BI mode. Equations (5.48-5.51) are exact but require a perfect knowledge of the gravi- tational driving term. One thing to notice is thek⁻¹ dependence of the photon density contrast for the CI and BI modes that washes out perturbations on small scales while amplifying them on large scales. As we show in the next section, this redistribution of power boosts the ISW effect of the CMB temperature power spectrum (large scales) and suppresses anisotropies on small scales, for these modes. On small scales, the effect of thek⁻¹ factor can be compared to Silk damping as they both they both suppress perturbations on these scales. However there are two main differ- ences. Firstly, Silk damping does not act on large scales while thek⁻¹factor amplifies large scale perturbations. Secondly, Silk damping only becomes significant around recombination while the k⁻¹factor redistributes the power at all times.

For the photon-baryon velocity divergence, BS = BC = 0 for both CI and BI modes. The velocity is solely determined by the driving term of equation (5.23). The photon-baryon velocity

(a) (b)

(c) (d)

Figure 5.11: Evolution of the photon density contrast prior to decoupling for the BI mode:

Comparison of numerical and semi-analytic solutions for some wavenumbers. We consider k = 0.019h Mpc⁻¹for the top-left panel,0.047h Mpc⁻¹for the top-right panel,0.088h Mpc⁻¹ for the bottom-left panel and0.25h Mpc⁻¹ for the bottom-right panel respectively. The vertical dashed lines mark the matter-radiation equality and decoupling.

divergence for the CI and BI modes is given by θ^CI,BI_bγ =− k

6(1 +R) Z τ

0

√3(1 +R(τ^′))^3/2sink(rs(τ)−rs(τ^′)) ˙h(τ^′)dτ^′, (5.52)

as for the adiabatic mode. However, the metric field in the CI and BI modes differs from the met- ric field in the adiabatic mode. Figures 5.9-5.10 and 5.11-5.12 compare the semi-analytic and the numerical solution for the evolution of the photon density contrast and velocity divergence

for the CI and BI modes respectively.

(a) (b)

(c) (d)

Figure 5.12: Evolution of the photon velocity divergence prior to decoupling for the BI mode:

Comparison of numerical and semi-analytic solutions for some wavenumbers. We considerk = 0.019h Mpc⁻¹ for the top-left panel,0.047h Mpc⁻¹ for the top-right panel, 0.088h Mpc⁻¹ for the bottom-left panel and 0.25h Mpc⁻¹ for the bottom-right panel respectively. The vertical dashed lines mark the matter-radiation equality and decoupling.

For the super horizon evolution of the gravitational potential in CI and BI models, we proceed as previously in the NID mode. One should note that for these modes,α= 0to leading order, caus- ing the synchronous gauge densities to be equal to their conformal gauge counterparts. Equation

(5.12) simplifies for the CI and BI modes respectively as τφ˙+φ= 4

3Ωc,0τ, (5.53)

τφ˙+φ= 4

3Ω_b,0τ, (5.54)

which admit the general solutions

φ^CI =Bτ⁻¹− 4

3Ω_c,0τ, (5.55)

φ^BI =Cτ⁻¹−4

3Ω_b,0τ, (5.56)

where B and C are some constants. As for the NID mode, the first terms of these solutions represent decaying modes which vanishes rapidly with time. The second terms, proportional to τ, are the growing modes. Thus we can omit the decaying modes and write the solutions as

φ^CI =−4

3Ω_c,0τ, (5.57)

φ^BI =−4

3Ω_b,0τ. (5.58)

(a) (b)

Figure 5.13: Evolution of the gravitational potential φ in the CI mode. (a): Super horizon evolution; (b): Sub-horizon evolution in the radiation era. These curves are obtained using CAMB. The vertical dashed lines mark the matter-radiation equality and decoupling.

Thus outside the horizon, in the radiation-dominated era, the gravitational potentialφin the CI and BI modes grows linearly with time in magnitude, and does not even depend on the wavenum-

ber of the considered mode, but depends only on the dark matter and baryon densities. This be- haviour is shown in Figures 5.13(a) and 5.14(a) where we have respectively represented the super horizon evolution of the gravitational potential in CI and BI models for different wavenumbers.

For the sub-horizon evolution, the derivation done in the case of the NID mode holds.

(a) (b)

Figure 5.14: Evolution of the gravitational potential φ in the BI mode. (a): Super horizon evolution; (b): Sub-horizon evolution in the radiation era. These curves are obtained using CAMB. The vertical dashed lines mark the matter-radiation equality and decoupling.

Therefore prior to matter-radiation equality, the potential oscillates and is sourced by the pho- tons. After matter-radiation equality, it becomes constant. These features are shown in Figures 5.13(b) and 5.14(b) where the sub-horizon evolution of the potentialφis shown for four different wavenumbers.