6.2 A scale-invariant power asymmetry
6.2.2 The curvaton model
96
where Γσ is the curvaton decay rate [88]. Since R ∝ σ¯∗2, the power spectrum for gravitational- potential perturbations produced by the curvaton is proportional to ¯σ2∗. A variation ∆¯σ in the value of the mean curvaton field across the observable Universe therefore induces a fractional power asymmetry ∆PΨ,σ/PΨ,σ ≃ 2(∆¯σ∗/¯σ∗). For superhorizon fluctuations, δσ and ¯σ obey the same evolution equation, so the ratioδσ/¯σ is conserved [187]. We will therefore omit the “*” subscript from future expressions in this section.
First we must ensure that this inhomogeneity does not violate Eqs. (6.5) and (6.6). The potential fluctuation at the time of decoupling produced by a fluctuationδσ in the curvaton field is given by Eq. (5.40):
Ψ(τdec)≃ −R 5
Ψ(τdec)
9 10Ψp
"
2 δσ
¯ σ
+
δσ
¯ σ
2#
, (6.9)
where Ψpis the potential fluctuation in the radiation-dominated era [Ψ(τdec) = 0.937Ψpfrom Section 5.2]. In Section 5.4, we examined how this potential fluctuation induces large-scale anisotropies in the CMB, and we now briefly review how δσ is constrained by the CMB quadrupole. Consider a superhorizon sinusoidal perturbation to the curvaton fieldδ¯σ=σksin(~k·~x+̟). If we ignored the term in Eq. (6.9) quadratic inδσ, then the upper bound toδ¯σwould be obtained by setting̟= 0.
As with the inflaton, the constraint would then then arise from the CMB octupole. However, the term in Eq. (6.9) that is quadratic inδσgives rise to a term in Ψ that is quadratic in (~k·~x)—i.e., Ψquad=−(R/5)[Ψ(τdec)/(0.9Ψp)](σk/σ)¯ 2(~k·~x)2for̟= 0. Noting that (∆¯σ/¯σ) = (σk/¯σ)(~k·x~dec), the quadrupole bound in Eq. (6.5) yields an upper limit,
R ∆¯σ
¯ σ
2
∼< 5
2(5.6Q), (6.10)
just as in Eq. (5.46). While this bound was derived for̟= 0, most other values for̟yield similar constraints, as shown in Section 5.4.
Most generally, the primordial power will be some combination of that due to the inflaton and curvaton [194],PΨ=PΨ,φ+PΨ,σ≃10−9, with a fractionξ≡PΨ,σ/PΨ due to the curvaton. Since the fluctuations in Ψ from the inflaton field are not affected by the superhorizon fluctuation in the curvaton field, the power asymmetry will be diluted by the inflaton’s contribution to the power spectrum. The total power asymmetry is therefore ∆PΨ,σ/PΨ≃2ξ(∆¯σ/¯σ)≃2A. This asymmetry can be obtained without violating Eq. (6.10) by choosingR∼<14Qξ2/A2, as shown in Fig. 6.2.
The only remaining issue is the Gaussianity of primordial perturbations. The curvaton fluctu- ation δσ is a Gaussian random variable. Since the curvaton-induced density perturbation has a contribution that is quadratic inδσ, it implies a non-Gaussian contribution to the density fluctua- tion. The departure from Gaussianity can be estimated from the parameter fNL [195, 196], which for the curvaton model is fNL ≃ 5ξ2/(4R) [186, 197, 198]. The current upper limit, fNL ∼< 100
98
[199, 59, 18, 61, 62], leads to the lower limit toRshown in Fig. 6.2.
Figure 6.2 shows that there are values of R and ξ that lead to a power asymmetryA = 0.072 and are consistent with measurements of the CMB quadrupole and fNL. For any value of A, the allowed region ofR-ξparameter space is
5
4fNL,max ∼< R
ξ2 ∼<14 Q
A2, (6.11)
where fNL,max is the largest allowed value for fNL. Thus we see that measurements of the CMB
Figure 6.2: The R-ξ parameter space for the curvaton model that produces a power asymmetry A= 0.072 (top) andAp= 0.050 (bottom). The observed asymmetry isA= 0.072±0.022 [46]. Here R is 3/4 times the fraction of the cosmological density due to curvaton decay, andξ is the fraction of the power due to the curvaton. The shaded regions in this plot are excluded. The upper limit to Rcomes from the CMB-quadrupole constraint. The lower bound comes fromfNL≤100. The lower limit to ξ comes from the requirement that the fractional change in the curvaton field across the observable Universe be less than one. IfAis lowered, the lower bound toRremains unchanged, but the upper bound increases, proportional toA−2. The lower limit toξalso decreases asAdecreases, proportional toA.
quadrupole andfNL place an upper bound,
A∼<
s (14Q)
4fNL,max
5
, (6.12)
on the power asymmetry that may be generated by a superhorizon curvaton fluctuation. Given that fNL∼<100 andQ= 1.8×10−5, we haveA∼<0.14, which is about twice the observed value.
The allowed region ofR-ξparameter space disappears iffNL,max is too small, so Eq. (6.11) also implies a lower bound onfNL:
fNL∼> 5 4
A2 14Q
. (6.13)
ForQ= 1.8×10−5, we predict (forA= 0.072)fNL ∼>26, much larger than thefNL predicted by standard slow-roll inflation (fNL ≪1) [53]. Values as small asfNL≃5 should be accessible to the forthcoming Planck satellite [196, 200, 201], and so there should be a clear non-Gaussian signature in Planck if the power asymmetry was generated by a curvaton perturbation andA= 0.072.
If (δσ/¯σ) ≪ 1, the power due to the curvaton is PΨ,σ ≃ (2R/5)2
(δσ/¯σ)2
, and the power required from the curvaton fixesR(δσ/¯σ)rms ≃8×10−5ξ1/2. It follows that (δσ/¯σ)rms ∼<0.33 for the allowed parameter space in Fig. 6.2 for A= 0.072, thus verifying that this parameter is small.
We find from (∆¯σ/¯σ) =A/ξ ∼<1 that the required cross-horizon variation ∆¯σ/¯σ in the curvaton is large compared with the characteristic quantum-mechanical curvaton fluctuation (δσ/¯σ)rms; the required ∆¯σ is at least a∼3σfluctuation. While such a large quantum fluctuation is unlikely, we note that the observed asymmetry is a 3.3σdeviation from an isotropic primordial power spectrum;
it is therefore equally unlikely that the asymmetry is a statistical fluke. The superhorizon fluctuation in the curvaton field could also be a superhorizon inhomogeneity not completely erased by inflation.
Another possibility is that positive- and negative-value cells of ¯σcreated during inflation may be large enough to encompass the observable Universe; if so, we would observe an order-unity fluctuation in
¯
σnear the ¯σ= 0 wall that divides two cells [179].
We have considered the specific asymmetryA≃0.072 reported by Ref. [46], but our results can be scaled for different values ofA, should the measured value for the asymmetry change in the future.
In particular, thefNLconstraint (the lower bound toR) in Fig. 6.2 remains the same, but the upper bound (from the quadrupole) increases asA is decreased. The lower limit toξ also decreases asA is decreased. Here we have also considered a general model in which primordial perturbations come from some combination of the inflaton and curvaton. Although it may seem unnatural to expect the two field decays to produce comparable fluctuation amplitudes, our mechanism works even ifξ= 1 (the fluctuations are due entirely to the curvaton). Thus, the coincidence is not a requirement of the model.
If the power asymmetry can indeed be attributed to a superhorizon curvaton mode, then the
100
workings of inflation are more subtle than the simplest models would suggest. Fortunately, the theory makes a number of predictions that can be pursued with future experiments. To begin, the modulated power should produce signatures in the CMB polarization and temperature-polarization correlations [202, 203]. The curvaton model predicts non-Gaussianity, of amplitude fNL ∼>26 for A≃0.072, which will soon be experimentally accessible. However, the theory also predicts that the small-scale non-Gaussianity will be modulated across the sky by the variation in ¯σ (and thus inξ and R). The presence of curvaton fluctuations also changes other features of the CMB [198]. The ratio of tensor and scalar perturbations (r) is reduced by a factor of (1−ξ) and the scalar spectral index isns= 1−2ǫV−(1−ξ)(4ǫV−2ηV). The tensor spectral index (nT), however, is unaltered by the presence of the curvaton, and so this model alters the inflationary consistency relation between nT andr and possibly the prospects for testing it [204].