A scale-dependent power asymmetry from isocurvature perturbations

If the hemispherical power asymmetry observed in the cosmic microwave background (CMB) on large angular scales is attributable to a superhorizon curvaton fluctuation, then the simplest model predicts that the primordial density fluctuations should be similarly asymmetric on all smaller scales. The distribution of high-redshift quasars was recently used to constrain the power asymmetry on scales $k\simeq 1.5h{\mathrm{Mpc}}^{-1}$, and the upper bound on the amplitude of the asymmetry was found to be a factor of 6 smaller than the amplitude of the asymmetry in the CMB. We show that it is not possible to generate an asymmetry with this scale dependence by changing the relative contributions of the inflaton and curvaton to the adiabatic power spectrum. Instead, we consider curvaton scenarios in which the curvaton decays after dark matter freezes out, thus generating isocurvature perturbations. If there is a superhorizon fluctuation in the curvaton field, then the rms amplitude of these perturbations will be asymmetric, and the asymmetry will be most apparent on large angular scales in the CMB. We find that it is only possible to generate the observed asymmetry in the CMB while satisfying the quasar constraint if the curvaton’s contribution to the total dark matter density is small, but nonzero. The model also requires that the majority of the primordial power comes from fluctuations in the inflaton field. Future observations and analyses of the CMB will test this model because the power asymmetry generated by this model has a specific spectrum, and the model requires that the current upper bounds on isocurvature power are nearly saturated.

Figure 1

CMB power spectra for unit-amplitude initial perturbations. The solid curve is ${\widehat{C}}_{\ell }^{\mathrm{ad}}$: the power spectrum derived from ${\mathcal{P}}_{\zeta }(k)=1$. The long-dashed curve is ${\widehat{C}}_{\ell }^{\mathrm{iso}}$: the power spectrum derived from ${\mathcal{P}}_S(k)=1$. The short-dashed curve is ${\widehat{C}}_{\ell }^{\mathrm{cor}}$: the difference between the power spectrum derived from ${\mathcal{P}}_{\zeta }(k)={\mathcal{P}}_S(k)={\mathcal{C}}_{\zeta S}(k)=1$ and ${\widehat{C}}_{\ell }^{\mathrm{ad}}+{\widehat{C}}_{\ell }^{\mathrm{iso}}$.

Figure 2

Measurements of temperature fluctuations in the CMB show that the rms temperature-fluctuation amplitude is larger in one side of the sky than in the other. We propose that this asymmetry is generated by a large-amplitude fluctuation in the initial value of the curvaton field ${\sigma }_{*}$. The fluctuation in ${\sigma }_{*}$ across the observable Universe is $\Delta {\overline{\sigma }}_{*}$.

Figure 3

$K_{\ell }$ for scenarios in which most of the dark matter comes from curvaton decay. The power asymmetry is given by $\Delta C_{\ell }/C_{\ell }=-2(\Delta {\overline{\sigma }}_{*}/\overline{\sigma })K_{\ell }$. The solid black curve corresponds to $\xi /R^2=0.0086$, which saturates the current bound on power from isocurvature perturbations. The lower curves have $\xi /R^2=0.007$ (long-dashed), 0.006 (short-dashed) and 0.005 (dotted). For descending values of $\xi /R^2$, these curves correspond to asymmetry amplitudes $\tilde{A}=0.055$, 0.045, 0.039, and 0.033.

Figure 4

$K_{\ell }$ for scenarios in which the curvaton’s contribution to the dark matter density is negligible and ${\mathcal{T}}_{SS}=\kappa R$. The power asymmetry is given by $\Delta C_{\ell }/C_{\ell }=2(\Delta {\overline{\sigma }}_{*}/\overline{\sigma })K_{\ell }$. All of the curves have ${\kappa }^2\xi =0.0086$, which saturates the upper limit on isocurvature power. The top three curves have $\kappa =0.6$ (top, solid), $\kappa =1$ (long-dashed), and $\kappa =3$ (short-dashed). The maximal scale-averaged asymmetries possible for these curves are $\tilde{A}=0.11$, $\tilde{A}=0.081$, and $\tilde{A}=0.062$. The dotted curve is the limit as $\kappa \to \infty$, and it has $\tilde{A}=0.055$. The bottom solid curve has $\kappa =-1$ and $\tilde{A}=0.043$.

Figure 5

The ratio $\tilde{A}/\xi$, with ${\ell }_{\max }=64$, as a function of $\xi$ for four values of ${\kappa }^2\xi$: ${\kappa }^2\xi =0.0086$ (solid), 0.007 (long-dashed), 0.006 (short-dashed), and 0.005 (dotted). Since $K_{\ell }\to \xi$ as $\ell \to \infty$, this ratio illustrates the fractional enhancement in the asymmetry on large scales ($\ell \le 64$) compared to small scales. Since we require the asymmetry to be about 6 times larger on large scales than on small scales, we see that we require $\xi \lesssim 0.016$.

Figure 6

The $\xi -\kappa$ parameter space for models in which the curvaton does not contribute significantly to the dark matter density. The shaded region in the upper-right corner is excluded by the upper bound on isocurvature power ($\alpha <0.072$), and the left shaded region is excluded by the scale invariance of the resulting asymmetry ($\tilde{A}/\xi <6$). The dotted curves show where the maximal possible asymmetry $\tilde{A}$ equals the observed asymmetry $\pm 1\sigma$; the bottom shaded region cannot produce an asymmetry within $1\sigma$ of the observed value.