CMB as a probe of new physics and old times

Cosmic birefringence (CB)---a rotation of photon-polarization plane in vacuum---is a generic signature of new scalar fields that could provide dark energy. Previously, WMAP observations excluded a uniform CB-rotation angle larger than a degree.

In this thesis, we develop a minimum-variance--estimator formalism for reconstructing direction-dependent rotation from full-sky CMB maps, and forecast more than an order-of-magnitude improvement in sensitivity with incoming Planck data and future satellite missions. Next, we perform the first analysis of WMAP-7 data to look for rotation-angle anisotropies and report null detection of the rotation-angle power-spectrum multipoles below L=512, constraining quadrupole amplitude of a scale-invariant power to less than one degree. We further explore the use of a cross-correlation between CMB temperature and the rotation for detecting the CB signal, for different quintessence models. We find that it may improve sensitivity in case of marginal detection, and provide an empirical handle for distinguishing details of new physics indicated by CB.

We then consider other parity-violating physics beyond standard models---in particular, a chiral inflationary-gravitational-wave background. We show that WMAP has no constraining power, while a cosmic-variance--limited experiment would be capable of detecting only a large parity violation. In case of a strong detection of EB/TB correlations, CB can be readily distinguished from chiral gravity waves.

We next adopt our CB analysis to investigate patchy screening of the CMB, driven by inhomogeneities during the Epoch of Reionization (EoR). We constrain a toy model of reionization with WMAP-7 data, and show that data from Planck should start approaching interesting portions of the EoR parameter space and can be used to exclude reionization tomographies with large ionized bubbles.

In light of the upcoming data from low-frequency radio observations of the redshifted 21-cm line from the EoR, we examine probability-distribution functions (PDFs) and difference PDFs of the simulated 21-cm brightness temperature, and discuss the information that can be recovered using these statistics. We find that PDFs are insensitive to details of small-scale physics, but highly sensitive to the properties of the ionizing sources and the size of ionized bubbles.

Finally, we discuss prospects for related future investigations.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.

A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.

For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.