Well-proportioned universes suppress CMB quadrupole

A widespread myth asserts that all small universe models suppress the CMB quadrupole. In actual fact, some models suppress the quadrupole while others elevate it, according to whether their low-order modes are weak or strong relative to their high-order modes. Elementary geometrical reasoning shows that a model's largest dimension determines the rough value ell_min at which the CMB power spectrum ell(ell + 1)C_ell/(2pi) effectively begins; for cosmologically relevant models, ell_min < 4. More surprisingly, elementary geometrical reasoning shows that further reduction of a model's smaller dimensions -- with its largest dimension held fixed -- serves to elevate modes in the neighborhood of ell_min relative to the high-ell portion of the spectrum, rather than suppressing them as one might naively expect. Thus among the models whose largest dimension is comparable to or less than the horizon diameter, the low-order C_ell tend to be relatively weak in well-proportioned spaces (spaces whose dimensions are approximately equal in all directions) but relatively strong in oddly-proportioned spaces (spaces that are significantly longer in some directions and shorter in others). We illustrate this principle in detail for the special cases of rectangular 3-tori and spherical spaces. We conclude that well-proportioned spaces make the best candidates for a topological explanation of the low CMB quadrupole observed by COBE and WMAP.Comment: v1: 10 pages, 1 figure. v2: improved exposition of competing mode-suppression and mode-enhancement effects, coincides with published version, 12 pages, 1 figur

Constraints on the Detectability of Cosmic Topology from Observational Uncertainties

Recent observational results suggest that our universe is nearly flat and well modelled within a $\Lambda$CDM framework. The observed values of $\Omega_{m}$ and $\Omega_{\Lambda}$ inevitably involve uncertainties. Motivated by this, we make a systematic study of the necessary and sufficient conditions for undetectability as well as detectability (in principle) of cosmic topology (using pattern repetition) in presence of such uncertainties. We do this by developing two complementary methods to determine detectability for nearly flat universes. Using the first method we derive analytical conditions for undetectability for infinite redshift, the accuracy of which is then confirmed by the second method. Estimates based on WMAP data together with other measurements of the density parameters are used to illustrate both methods, which are shown to provide very similar results for high redshifts.Comment: 16 pages, 1 figure, LaTeX2

Cosmic microwave background anisotropies in multi-connected flat spaces

This article investigates the signature of the seventeen multi-connected flat spaces in cosmic microwave background (CMB) maps. For each such space it recalls a fundamental domain and a set of generating matrices, and then goes on to find an orthonormal basis for the set of eigenmodes of the Laplace operator on that space. The basis eigenmodes are expressed as linear combinations of eigenmodes of the simply connected Euclidean space. A preceding work, which provides a general method for implementing multi-connected topologies in standard CMB codes, is then applied to simulate CMB maps and angular power spectra for each space. Unlike in the 3-torus, the results in most multi-connected flat spaces depend on the location of the observer. This effect is discussed in detail. In particular, it is shown that the correlated circles on a CMB map are generically not back-to-back, so that negative search of back-to-back circles in the WMAP data does not exclude a vast majority of flat or nearly flat topologies.Comment: 33 pages, 19 figures, 1 table. Submitted to PR

Simulating Cosmic Microwave Background maps in multi-connected spaces

This article describes the computation of cosmic microwave background anisotropies in a universe with multi-connected spatial sections and focuses on the implementation of the topology in standard CMB computer codes. The key ingredient is the computation of the eigenmodes of the Laplacian with boundary conditions compatible with multi-connected space topology. The correlators of the coefficients of the decomposition of the temperature fluctuation in spherical harmonics are computed and examples are given for spatially flat spaces and one family of spherical spaces, namely the lens spaces. Under the hypothesis of Gaussian initial conditions, these correlators encode all the topological information of the CMB and suffice to simulate CMB maps.Comment: 33 pages, 55 figures, submitted to PRD. Higher resolution figures available on deman

The spectral action and cosmic topology

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincare' dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and see we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincare' homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces.Comment: 55 pages, LaTe

Radiative Corrections to the Inflaton Potential as an Explanation of Suppressed Large Scale Power in Density Perturbations and the Cosmic Microwave Background

The Wilkinson Microwave Anisotropy Probe microwave background data suggest that the primordial spectrum of scalar curvature fluctuations is suppressed at small wavenumbers. We propose a UV/IR mixing effect in small-field inflationary models that can explain the observable deviation in WMAP data from the concordance model. Specifically, in inflationary models where the inflaton couples to an asymptotically free gauge theory, the radiative corrections to the effective inflaton potential can be anomalously large. This occurs for small values of the inflaton field which are of the order of the gauge theory strong coupling scale. Radiative corrections cause the inflaton potential to blow up at small values of the inflaton field. As a result, these corrections can violate the slow-roll condition at the initial stage of the inflation and suppress the production of scalar density perturbations.Comment: 20 pages, 2 figures, v2: refs added, v3: JCAP versio