Direction dependence of the power spectrum and its effect on the cosmic microwave background radiation

Abstract

We study several anisotropic inflationary models and their implications for the observed violation of statistical isotropy in the CMBR data. In two of these models the anisotropy decays very quickly during the inflationary phase of expansion. We explicitly show that these models lead to violation of isotropy only for low l CMBR modes. Our primary aim is to fit the observed alignment of l = 2, 3 multipoles to the theoretical models. We use two measures, based on the power tensor, which contains information about the alignment of each multipole, to quantify the anisotropy in data. One of the measures uses the dispersion in eigenvalues of the power tensor. We also define another measure which tests the overall correlation between two different multipoles. We perturbatively compute these measures of anisotropy and fix the theoretical parameters by making a best fit to l = 2, 3 multipoles. We show that some of the models studied are able to consistently explain the observed violation of statistical isotropy.

Relating the inhomogeneous power spectrum to the CMB hemispherical anisotropy

“…It has also been suggested that this anisotropy may not really be in disagreement with the inflationary Big Bang cosmology, which may have a phase of anisotropic and inhomogeneous expansion at very early time. The anisotropic modes, generated during this early phase may later re-enter the horizon [51,52] and lead to the observed signals of anisotropy.…”

Quadrupole-octopole alignment of CMB related to the primordial power spectrum with dipolar modulation in anisotropic spacetime

“…3 This form of the correlation has slight difference from the original one in Ref. [29]. Here, we have only extracted the anisotropic effects.…”

“…It refers to the correlator of two power tensors with different multipoles [29]. Thus, the correlation of two power tensors is defined as…”

“…(10). As an example, we discuss the simplest case that the initial power spectrum has a dipolar modulation, namely, [29,30] …”

Noncommutative geometry and the primordial dipolar imaginary power spectrum

“…The precise operator that should arise while computing the power spectrum must be consistently prescribed by the underlying field theory. One may, for example, use perturbation theory [21,53] and compute the corrections to the power spectrum arising due to noncommutative space-time order by order in the noncommutative parameter, θ 0 . This is rather complicated and we postpone this detailed calculation to future research.…”