Isocurvature modes in the cosmic microwave background

6.3 Review of isocurvature perturbations

6.3.2 Isocurvature modes in the cosmic microwave background

104

When we recall that ˜λ <1 is required to make the dark matter freeze-out prior to curvaton decay, we see that the first term in Eq. (6.27) is proportional to ˜λ/B, which is much smaller than˜ R if BR˜ ≫1. The first term is therefore negligible, and we are left with

˜lim

BR≫1T^SS= 1−R. (6.28)

In the opposite limit, in which the curvaton’s contribution to the dark matter density is negligible, we have

˜lim

BR≪1T^SS =

"

(α−3)˜λ

2(α−2) + ˜B−1

#

R≡κR. (6.29)

The first two terms in Eq. (6.29) are positive by definition, soκ∼>−1. The first term is always less than 0.5 since ˜λ < 1, but ˜B = (4/3)Bm/Ω^(bd)_cdm could be much larger than unity since Ω^(bd)_cdm ≪1.

The only upper limit onκis given by ˜BR≪1 which implies thatκ≪1/R.

and the curvaton field are perturbed, Sσγ ≃2δσ∗/σ¯∗, where δσ∗ and ¯σ∗ are evaluated at horizon exit [194]. However, as mentioned in Section 6.2.2, the ratio δσ/¯σ is conserved for superhorizon perturbations [187]. Given thatP^δσ= [Hinf/(2π)]², we have

B²= H_inf²

π²σ¯²_∗. (6.36)

Sinceζ⁽ⁱ⁾is determined by the inflaton fluctuation andSσγ is determined by the curvaton fluctuation, Sσγ andζ⁽ⁱ⁾are uncorrelated. From Eq. (6.17), we see that

P^ζ(k) = A²+TζS²B², (6.37)

P^S(k) = TSS² B², (6.38)

C^ζS(k) = T^ζST^SSB². (6.39)

The CMB power spectrum may be divided into contributions from adiabatic and isocurvature perturbations [206]:

Cℓ= A²+TζS²B²Cˆ_ℓ^ad+TSS² B²Cˆ_ℓ^iso+T^ζST^SSB²Cˆ_ℓ^cor. (6.40)

In this decomposition, ˆC_ℓ^ad is the CMB power spectrum derived from a flat spectrum of adiabatic fluctuations with P^ζ(k) = 1, and ˆC_ℓ^iso is the CMB power spectrum derived from a flat spectrum of dark-matter isocurvature perturbations with P^S(k) = 1. If both isocurvature and adiabatic perturbations are present, with P^ζ(k) = P^S(k) = C^ζS(k) = 1, then the CMB power spectrum is Cˆ_ℓ^ad+ ˆC_ℓ^iso+ ˆC_ℓ^cor. Figure 6.3 shows these three component spectra, as calculated by CMBFast [207]

with WMAP5 best-fit cosmological parameters [18]: Ωb = 0.0462,Ωcdm= 0.233,ΩΛ = 0.721 and H0= 70.1 km/s/Mpc.

Figure 6.3 clearly shows that isocurvature perturbations leave a distinctive imprint on the CMB power spectrum. It is therefore possible to constrain the properties of P^ζ(k) and P^S(k) using CMB data. These constraints are often reported as bounds on the isocurvature fractionαand the correlation parameterγ:

α ≡ TSS² B²

A²+TζS²B²+TSS² B², (6.41) γ ≡ sign(T^ζST^SS) TζS²B²

A²+TζS²B². (6.42)

We will find it useful to continue to defineξas the fraction of adiabatic power from the curvaton:

ξ≡ TζS²B²

A²+TζS²B², (6.43)

106

Figure 6.3: CMB power spectra for unit-amplitude initial perturbations. The solid curve is ˆC_ℓ^ad: the power spectrum derived fromP^ζ(k) = 1. The long-dashed curve is ˆC_ℓ^iso: the power spectrum derived fromP^S(k) = 1. The short-dashed curve is ˆC_ℓ^cor: the difference between the power spectrum derived fromP^ζ(k) =P^S(k) =C^ζS(k) = 1 and ˆC_ℓ^ad+ ˆC_ℓ^iso.

withT^ζS =R/3. We then see that

α = 9(ξ/R²)TSS²

1 + 9(ξ/R²)TSS²

(6.44)

γ = sign(T^SS)ξ. (6.45)

Ideally, we would like to use constraints for α and γ that were derived assuming only that nad ≃ niso ≃ 1, where nad and niso are the spectral indices for P^ζ(k) and P^S(k) respectively.

Unfortunately, such an analysis does not exist. The most general analyses [206, 208, 209] make no assumptions regardingniso and conclude that models withniso≃2 provide the best fit to the data.

Since their bounds onαandγ are marginalized over a range ofniso values that are unreachable in the curvaton scenario, these constraints are not applicable to our model.

There are analyses that specifically target the curvaton scenario, but they assume that the curvaton generates all of the primordial fluctuations (i. e., A² ≪ TζS²B²) [208, 18]. In this case,

ξ= 1, and the isocurvature and adiabatic fluctuations are completely correlated or anti-correlated, depending on the sign ofT^SS. Furthermore, Eq. (6.44) shows thatα∼>0.9 if ξ= 1 andTSS² ∼> R². Since this high value for αis thoroughly ruled out, these analyses of isocurvature perturbations in the curvaton scenario disregard the possibility that ˜BR ≪ 1 and assume that most of the dark matter is created by curvaton decay. In this case,T^SS is given by Eq. (6.28) and the derived upper bound onα (α <0.0041 from Ref. [18]) implies thatR >0.98. Since we require R ≪1, we can conclude that we will be restricted to mixed-perturbation scenarios in which both the curvaton and the inflaton contribute to the adiabatic perturbation spectrum andξ <1.

Finally, some analyses constrain completely uncorrelated (γ = 0) isocurvature and adiabatic perturbations (a.k.a. axion-type isocurvature) withniso= 1 [210, 18]. These constraints are relevant to our models, however, because we will see that ξ = |γ| must be small to create an asymmetry that vanishes on small scales. (The discussion in the previous paragraph also foreshadows the fact that ξ≪1 will be necessary to obtain R≪1.) We will therefore use the bound on αderived for uncorrelated adiabatic and isocurvature in our analysis. WMAP5 data alone constrainsα <0.16 at 95% confidence, but the upper bound onαis significantly reduced if BAO and SN data are used to break a degeneracy betweenαand nad [18]. With the combined WMAP5+BAO+SN dataset, the 95% C.L. upper bound onαis

α <0.072, (6.46)

with a best-fit value ofnad≃1. Ref. [210] found a similar bound: α <0.08 at 95% C.L.

The other observable effect of isocurvature fluctuations that we must consider is non-Gaussianity.

Following Ref. [211], we definef_NL^(iso) through

Smγ =η+f_NL^(iso) η²− hη²i

, (6.47)

whereη is drawn from a Gaussian probability spectrum. This is analogous to the definition of fNL

for adiabatic perturbations [196]. For isocurvature perturbations from the curvaton,

Smγ =T^SSSσγ =T^SS

"

2δσ∗

¯ σ∗

+ δσ∗

¯ σ∗

2#

, (6.48)

and we can setη= 2T^SSδσ∗/σ¯∗. Thus we see that f_NL^(iso)= 1

4T^SS (6.49)

for the curvaton model. Given the current upper bound on α, f_NL^(iso) ≃ 10⁴ produces a CMB bispectrum that is equal in magnitude to the CMB bispectrum if fNL ≃ 20 for purely adiabatic perturbations [211]. Since the current upper limit on fNL from the CMB is fNL ∼<100 [199, 59,

108

18, 61, 62], we see that the non-Gaussianity of the isocurvature fluctuations is undetectable for T^SS ∼> 10⁻⁵. Recall from Section 6.2.2 that the curvaton also introduces non-Gaussianity in the adiabatic perturbations; fNL for mixed perturbations from the inflaton and curvaton is given by [186, 198]

fNL=5ξ²

4R. (6.50)