Statistical anisotropies in temperature and polarization fluctuations from a scale-dependent trispectrum

We study statistical anisotropies generated in the observed two-point function of the cosmic microwave background (CMB) fluctuations if the primordial statistics are non-Gaussian. Focusing on the dipole modulations of the anisotropies, we find that the hemispherical power asymmetry observed in the CMB temperature fluctuations can be modeled by a local-type trispectrum with amplitude ${\tau }_{\mathrm{NL}}(k_p=0.05{\mathrm{Mpc}}^{-1})\approx 2\times {10}^4$ and a large red tilt $n\approx -0.68$. We numerically evaluate the non-Gaussian covariance of the modulation estimators for both temperature and E-mode polarization fluctuations and discuss the prospects of constraining the model using Planck satellite data. We then discuss other effects of the scale-dependent trispectrum that could be used to distinguish this scenario from other explanations of the power asymmetry: higher-order modulations of the two-point function and the non-Gaussian angular power spectrum covariance. As an important consequence of the non-Gaussian power spectrum covariance, we discuss how the CMB-inferred spectral index of primordial scalar fluctuations can be significantly biased in the presence of a scale-dependent local-type trispectrum.

Figure 1

The expected $\ell$ dependent amplitude of dipole modulation from the scale-dependent trispectrum model given in Eq. (13) for our fiducial parameters. For reference, we have plotted the best-fit $\ell$—dependent dipole modulation amplitude obtained by Aiola et al. [50] using Planck temperature data. The $⟨A(\ell ){⟩}_{\mathrm{nG}}$ TT (solid blue) values only include the non-Gaussian contribution (and no Gaussian noise) which could explain its smaller magnitudes than that of the best-fit Aiola et al. model (black dotted).

Figure 2

Distributions (smoothed histograms) of scale-dependent power asymmetry parameters $(A_{{\ell }_0=300},n)$ for two different models: (i) a Gaussian and isotropic model (brown, contour with smaller amplitudes), and (ii) our fiducial non-Gaussian isotropic model (blue, contour with larger amplitudes). For all the distributions, we have used temperature multipoles $\ell =30–600$, and assumed $f_{\mathrm{sky}}=0.65$. These smoothed distributions are generated from 25,000 modulation data realizations and the two contours indicate $1\sigma$ and $2\sigma$ intervals. The dashed lines show the marginalized median values obtained from fit to Planck data by [50].

Figure 3

The log-likelihood improvement as a function of maximum multipole used in the analysis using Planck-like temperature $\mathrm{\Delta }{\widehat{X}}_M^T$ realizations only, and in combination with Planck-like E-mode $\mathrm{\Delta }{\widehat{X}}_M^E$ realizations. We find that, for our fiducial trispectrum model, adding E-mode polarization data from Planck increases the $\mathrm{\Delta }\ln \mathcal{L}\sim 3$ over the temperature-only data. The data points in the figure are median $\mathrm{\Delta }\ln \mathcal{L}$ values from fit to realizations while the band shows the 68% spread around it.

Figure 4

Left: The diagonal component of the non-Gaussian term in the power spectrum covariance as a fraction of the Gaussian term for $({\tau }_{\mathrm{NL}},n)=(2\times {10}^4,-0.68)$ model with $\mathcal{N}=40$. Right: Example of how the inferred spectral index $n_s$ can be significantly biased high if the observed large-scale power deficit is due to a scale-dependent trispectrum. In this example, a set of $C_{\ell }$ realizations was generated using the non-Gaussian model with spectral index (at the pivot wave number $k_0=0.05{\mathrm{Mpc}}^{-1}$) $n_s=0.93$, and the particular realization in the figure is a typical ($\sim 1\sigma$) realization of the non-Gaussian model with $\sim 12\%$ power deficit for $\ell =10–40$. The Gaussian fit value of spectral index was obtained by only varying $A_s$, $n_s$ and keeping other nonprimordial parameters fixed.