6.2 A scale-invariant power asymmetry
6.2.1 Single-field models
consider how a superhorizon fluctuation in the curvaton field could produce a scale-dependent power asymmetry that is more pronounced on large scales than on small scales.
It is possible to dilute the power asymmetry on smaller scales by introducing discontinuities in the inflaton potential and its derivative that change relative contributions of the curvaton and inflaton field to the primordial perturbations [165]. We examine this proposal in Appendix C and find that the discontinuity in the inflaton potential required to satisfy the quasar constraint on the asymmetry violates constraints from ringing in the power spectrum [183, 184]. In Appendix C we also find that it is not possible to sufficiently dilute the asymmetry on small scales by smoothly changing the relative contributions of the curvaton and inflaton fluctuations to the primordial power spectrum.
We then consider the dark-matter isocurvature perturbations generated by some curvaton sce- narios [89, 185, 186, 187, 188, 189, 190]. In the presence of a superhorizon fluctuation in the curvaton field, the power in these isocurvature perturbations will be asymmetric. Since isocurvature pertur- bations decay once they enter the horizon, they will contribute more to the large-scale (ℓ ∼< 100) CMB anisotropies than to the smaller scales probed by quasars. Consequently, the desired scale- dependence of the asymmetry is a natural feature of isocurvature perturbations.
We review how isocurvature perturbations are generated in the curvaton scenario in Section 6.3.1, and we review the CMB signatures of isocurvature perturbations in Section 6.3.2. In Section 6.4, we examine how a hemispherical power asymmetry could be created by a superhorizon fluctuation in the curvaton field in two limiting cases of the curvaton scenario. We find in Section 6.4.1 that it is not possible to generate the observed hemispherical power asymmetry if the curvaton decay created the dark matter because the necessary superhorizon isocurvature fluctuation induces an unacceptably large dipolar anisotropy in the CMB. In Section 6.4.2, we show that the observed asymmetry can be generated by a superhorizon curvaton fluctuation if the curvaton’s contribution to the dark matter is negligible. Our model predicts that the asymmetry will have a specific spectrum and that the current bounds on the contribution of isocurvature perturbations to the CMB power spectrum are nearly saturated. We summarize our findings and discuss these future tests of our model in Section 6.5.
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by the potential energy, the pressure is negative, and the expansion of the Universe is inflationary.
Quantum fluctuations in the inflaton give rise to primordial density perturbations characterized by a gravitational-potential power spectrum PΨ(k) ∝ V /ǫV, where V and ǫV are evaluated at the value the inflaton took when the comoving wavenumberk exited the horizon during inflation.
Differentiation of the expression forPΨ(k) suggests that the power spectrum can be approximated as PΨ(k)∝kns−1, where the scalar spectral index ns= 1−6ǫV + 2ηV is close to unity, consistent with current measurements [18, 191].
The power spectrum PΨ(k) may vary with k because different values of ksample the quantity V /ǫat different values of the inflatonφ. This suggests that the power asymmetry might be explained by a large-amplitude mode ofφwith a comoving wavelength that is long compared with the current Hubble distance (k≪H0). Then one side of the CMB sky would reflect the imprint of a different value ofφthan the other side. FromPΨ(k)∝V /ǫV, we infer a fractional power asymmetry,
∆PΨ
PΨ =−2 rπ
ǫ(1−ns)∆φ
mPl, (6.2)
where ∆φ is the change in the inflaton field across the observable Universe and m2Pl = G−1 in natural units. The observed 7.2% variation in the amplitude of the CMB temperature fluctuations [46] corresponds to a power asymmetry ∆PΨ/PΨ≃2A= 0.144, whereAis defined by Eq. (6.1).
The gravitational-potential perturbation Ψ during matter domination is related to the inflaton perturbation δφ through Ψ = (6/5)p
π/ǫ(δφ/mPl). Thus, a long-wavelength perturbation δφ ∝ sin[~k·~x+̟], withkxdec≪1 (wherexdecis the distance to the surface of last scatter), introduces a gravitational-potential perturbation with the same spatial dependence. It follows from Eq. (6.2) that
∆Ψ = 6A/[5(ns−1)]. An immediate concern, therefore, is whether this large-amplitude perturbation is consistent with the isotropy of the CMB.
Gravitational-potential perturbations give rise to temperature fluctuations in the CMB through the Sachs-Wolfe effect [173] (δT /T ≃ Ψ/3). A large-scale potential perturbation might thus be expected to produce a CMB temperature dipole of similar magnitude. However, for the Einstein-de Sitter universe, the potential perturbation induces a peculiar velocity whose Doppler shift cancels the intrinsic temperature dipole [85, 159]. The same is true for a flat universe with a cosmological constant, as we saw in Chapter 5.
Although the dipole vanishes, measurements of the CMB temperature quadrupole and octupole constrain the cosmological potential gradient [85, 159, 162]. Here we will briefly review how these constraints are derived before applying them to the superhorizon mode necessary to create the observed power asymmetry; the full calculation is presented in Chapter 5. Sincekxdec≪1, we first expand the sinusoidal dependence Ψ(~x) = Ψ~ksin(~k·~x+̟) in powers of~k·~x as in Eq. 5.15. The
terms that contribute to the CMB quadrupole and octupole are
Ψ(~x) =−Ψ~k
((~k·~x)2
2 sin̟+(~k·~x)3 6 cos̟
)
. (6.3)
The CMB temperature anisotropy produced by the potential in Eq. (6.3) is
∆T
T (ˆn) =−Ψ~k
µ2
2 (kxdec)2δ2sin̟+µ3
6 (kxdec)3δ3cos̟
, (6.4)
where µ≡kˆ·nˆ and Ψ~k is evaluated at the time of decoupling (τdec). As described in Section 5.3, theδ2 andδ3coefficients account for the Sachs-Wolfe (including integrated) effect and the Doppler effect induced by Ψ~k; for a ΛCDM Universe with ΩM = 0.28, matter-radiation equality redshift zeq = 3280, and decoupling redshift zdec = 1090, we find thatδ2 = 0.33 and δ3 = 0.35. Choosing ˆk= ˆz, Eq. (6.4) gives nonzero values for the spherical-harmonic coefficientsa20anda30. The relevant observational constraints are therefore given by Eqs. (5.34) and (5.35):
(kxdec)2
Ψ~k(τdec) sin̟
∼< 5.8Q (6.5)
(kxdec)3
Ψ~k(τdec) cos̟
∼< 32O (6.6)
where Q and O are upper bounds on |a20| and |a30|, respectively, in a coordinate system aligned with the power asymmetry.
As in Section 5.3, we takeQ= 3√
C2 ∼<1.8×10−5 and O = 3√
C3 ∼<2.7×10−5, three times the measured rms values of the quadrupole and octupole [178], as 3σ upper limits; this accounts for cosmic variance in the quadrupole and octupole due to smaller-scale modes. The temperature quadrupole and octupole induced by the superhorizon mode can be made arbitrarily small for fixed
∆Ψ≃Ψ~k(kxdec) cos̟by choosingkto be sufficiently small. However, we also demand that Ψ~k∼<1 everywhere, and this sets a lower bound on (kxdec).
We now return to the power asymmetry generated by an inflaton perturbation. The largest value of ∆Ψ is obtained if̟= 0, in which case the perturbation produces no quadrupole. The octupole constraint [Eq. (6.6)] combined with (kxdec) ∼> |∆Ψ| [i.e., the requirement Ψ~k ∼< 1] implies that
|∆Ψ| ∼<(32O)1/3. Given that (1−ns)∼<0.06, we see that the maximum possible power asymmetry obtainable with a single superhorizon mode is Amax ≃ 0.05(32O)1/3 ≃0.0048. This is too small, by more than an order of magnitude, to account for the observed asymmetry (A= 0.072±0.022).
The limit can be circumvented if a number of Fourier modes conspire to make the density gradient across the observable Universe smoother. This would require, however, that we live in a very special place in a very unusual density distribution.
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