Testing physical models for dipolar asymmetry: From temperature to $k$ space to lensing

One of the most intriguing hints of a departure from the standard cosmological model is a large-scale dipolar power asymmetry in the cosmic microwave background (CMB). If not a statistical fluke, its origins must lie in the modulation of the position-space fluctuations via a physical mechanism, which requires the observation of new modes to confirm or refute. We introduce an approach to describe such a modulation in $k$ space and calculate its effects on the CMB temperature and lensing. We fit the $k$-space modulation parameters to Planck 2015 temperature data and show that CMB lensing will not provide us with enough independent information to confirm or refute such a mechanism. However, our approach elucidates some poorly understood aspects of the asymmetry, in particular that it is weakly constrained. Also, it will be particularly useful in predicting the effectiveness of polarization in testing a physical modulation.

Figure 1

Limber approximation kernels (in arbitrary units) for various cosmological observations (contours) out to last scattering ($r=r_{\mathrm{LS}}$). The grey box indicates very roughly the reach of the planned Euclid survey [27]. Dotted magenta curves correspond to fixed multipole scales, with the red hatched region geometrically inaccessible. The vertical cyan line corresponds approximately to the scale $\ell =65$ in the primary CMB (narrow green box at the top); scales roughly to the left of it exhibit dipolar asymmetry in the CMB. To test a modulation model, many more modes are available in principle in our observable volume than in the primary CMB source region. Adapted from [25].

Figure 2

Marginalized posteriors for the parameter set $\{p_i,\mathrm{\Delta }X,\mathrm{\Delta }Y,\mathrm{\Delta }Z\}$; dark and light blue (solid) contours enclose 68% and 95% of the likelihood, respectively. The black and grey (dashed) contours and curves represent the theoretical distributions of the parameters coming solely from cosmic variance in statistically isotropic skies. The values $(k_c,\mathrm{\Delta }\ln k)=(5\times 10^{-3}{\mathrm{Mpc}}^{-1},0)$ would correspond roughly to the often-considered $\ell$-space modulation to $\ell \simeq 65$.

Figure 3

Marginalized posteriors for the parameter set $\{p_i,A_{\mathcal{R}}\}$; dark and light blue (solid) contours enclose 68% and 95% of the likelihood, respectively. The black and grey (dashed) contours and curve represent the theoretical distributions of the parameters coming solely from cosmic variance in statistically isotropic skies.

Figure 4

Temperature anisotropy isotropic power spectrum, $C_{\ell }^{\mathrm{\Lambda }\mathrm{CDM}}$ (solid black curve), anisotropic power spectrum, $C_{\ell }^{\text{lo}}$ (dashed red curve), and power spectrum increment from equator to pole, $\mathrm{\Delta }{C}_{\ell }$ (dot-dashed green curve), for the case of the maximum-likelihood modulation from Table 1, which fits the observed temperature asymmetry. The modulation in amplitude is at a level of roughly 7% to $\ell \simeq 50$.

Figure 5

Lensing potential isotropic power spectrum, $C_{\ell }^{\text{lens}}$ (black curve), predicted anisotropic power spectrum, $\delta C_{\ell \ell }^{\text{lens}}$ (red curve), and predicted power spectrum increment from equator to pole, $\mathrm{\Delta }{C}_{\ell }^{\text{lens}}$ (green curve), for the case of the maximum-likelihood modulation from Table 1, which fits the observed temperature asymmetry. The modulation in amplitude is at a level of roughly 1.5% or less to $\ell \simeq 50$.