Bayesian analysis of sparse anisotropic universe models and application to the 5-yr WMAP data

TL;DR

The paper extends the CMB Gibbs sampling framework to exact Bayesian analysis of anisotropic, sparse covariance models and applies it to the 5-year WMAP temperature data, enabling robust inference on non-diagonal CMB covariances. It introduces a sparse, non-diagonal covariance implementation for the ACW anisotropy model, uses LDL-based Cholesky methods, and employs Metropolis within Gibbs to sample anisotropy parameters $g_*$ and the preferred direction $\hat{\mathbf{n}}$. Validation on simulations confirms exact recovery of input anisotropy and shows how priors mitigate local maxima; application to WMAP5 finds a tentative nonzero $g_*$ with a preferred axis near $(l,b) \approx (110^{\circ},10^{\circ})$ in the W-band, though systematics such as correlated noise and beams require deeper investigation before cosmological interpretation. The work provides a general, scalable Bayesian framework for testing anisotropic early-universe models with high-resolution CMB data and highlights the importance of comprehensive systematic studies. The results demonstrate both the potential for detecting rotational invariance violations and the need for realistic simulations to separate cosmology from instrumental effects.

Abstract

We extend the previously described CMB Gibbs sampling framework to allow for exact Bayesian analysis of anisotropic universe models, and apply this method to the 5-year WMAP temperature observations. This involves adding support for non-diagonal signal covariance matrices, and implementing a general spectral parameter MCMC sampler. As a worked example we apply these techniques to the model recently introduced by Ackerman et al., describing for instance violations of rotational invariance during the inflationary epoch. After verifying the code with simulated data, we analyze the foreground-reduced 5-year WMAP temperature sky maps. For l <= 400 and the W-band data, we find tentative evidence for a preferred direction pointing towards (l,b) = (110 deg, 10 deg) with an anisotropy amplitude of g_* = 0.15 +/- 0.039. Similar results are obtained from the V-band data [g_* = 0.10 +/- 0.04; (l,b) = (130 deg, 20 deg)]. Further, the preferred direction is stable with respect to multipole range, seen independently in both l=[2,100] and [100,400], although at lower statistical significance. We have not yet been able to establish a fully satisfactory explanation for the observations in terms of known systematics, such as non-cosmological foregrounds, correlated noise or asymmetric beams, but stress that further study of all these issues is warranted before a cosmological interpretation can be supported.

Figures (12)

• Figure 1: Covariance elements, $C_{\ell,\ell}$ and $C_{\ell,\ell+2}$, used in the construction of the ACW covariance matrix. These are computed by modifying CAMB, a publicly available Boltzmann code.
• Figure 2: Temperature maps showing isotropic fluctuations (top row), while the two lower rows depict anisotropic contributions with $g_*=0.9999$ (middle row) and $g_*=-0.9999$ (bottom row). The maps in the left column are presented in Mollweide projection, while the right row is Cartesian. The anisotropy direction was chosen to be $(l,b) = (0^{\circ}, 90^{\circ})$. Note the subtle tendency for stripes along the equator for the positive $g_*$, and perpendicular to the equator for negative $g_*$.
• Figure 3: Power spectra of simulated anisotropic sky maps with $g_* = 3$, with (green) and without (black) rescaling. Red curve shows the power spectrum for an isotropic simulation with $g_* = 0$.
• Figure 4: Marginal likelihood functions for $\mathcal{L}(g_*)$ (top) and $\mathcal{L}(\hat{\mathbf{n}})$ (bottom) for a simulated data set with uniform noise and full-sky coverage, shown in logarithmic units. The input values of $g_* = 0.8$ and $(l,b) = (57^{\circ}, 33^{\circ})$ are accurately reproduced. Notice the shallow local maximum at $g_* \sim -0.5$ and the secondary peaks in the marginal direction map.
• Figure 5: Evolution of Gibbs chains mapping the posterior of a simulated data set. Note how chains trapped in the local maximum at negative anisotropy amplitude eventually converges to the positive maximum.