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Register a new userCircles-in-the-sky searches and observable cosmic topology in the inflationary limit
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Marcelo J. Reboucas
Publication date
2008
fields
Physics
and research's language is
English
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While the topology of the Universe is at present not specified by any known fundamental theory, it may in principle be determined through observations. In particular, a non-trivial topology will generate pairs of matching circles of temperature fluctuations in maps of the cosmic microwave background, the so-called circles-in-the-sky. A general search for such pairs of circles would be extremely costly and would therefore need to be confined to restricted parameter ranges. To draw quantitative conclusions from the negative results of such partial searches for the existence of circles we need a concrete theoretical framework. Here we provide such a framework by obtaining constraints on the angular parameters of these circles as a function of cosmological density parameters and the observers position. As an example of the application of our results, we consider the recent search restricted to pairs of nearly back-to-back circles with negative results. We show that assuming the Universe to be very nearly flat, with its total matter-energy density satisfying the bounds $ 0 <|Omega_0 - 1| lesssim 10^{-5}$, compatible with the predictions of typical inflationary models, this search, if confirmed, could in principle be sufficient to exclude a detectable non-trivial cosmic topology for most observers. We further relate explicitly the fraction of observers for which this result holds to the cosmological density parameters.
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[Abridged] In a Universe with a detectable nontrivial spatial topology the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations - the so-called circles-in-the-sky. Searches for nearly antipodal circles in maps of cosmic microwave background have so far been unsuccessful. This negative outcome along with recent theoretical results concerning the detectability of nearly flat compact topologies is sufficient to exclude a detectable nontrivial topology for most observers in very nearly flat positively and negatively curved Universes ($0<|Omega_{tot}-1| lesssim 10^{-5}$). Here we investigate the consequences of these searches for observable nontrivial topologies if the Universe turns out to be exactly flat ($Omega_{tot}=1$). We demonstrate that in this case the conclusions deduced from such searches can be radically different. We show that for all multiply-connected orientable flat manifolds it is possible to directly study the action of the holonomies in order to obtain a general upper bound on the angle that characterizes the deviation from antipodicity of pairs of matching circles associated with the shortest closed geodesic. This bound is valid for all observers and all possible values of the compactification length parameters. We also show that in a flat Universe there are observers for whom the circles-in-the-sky searches already undertaken are insufficient to exclude the possibility of a detectable nontrivial spatial topology. It is remarkable how such small variations in the spatial curvature of the Universe, which are effectively indistinguishable geometrically, can have such a drastic effect on the detectability of cosmic topology.
[Abridged] An observable signature of a detectable nontrivial spatial topology of the Universe is the circles-in-the-sky in the CMB sky. In the most general search, pairs of circles with deviation from antipodality $0^circ leq theta leq 169^circ$ and radii $10^circ leq lambda leq 90^circ$ were investigated, but no matching circles were found. Assuming this negative result, we examine the question as to whether there are nearly flat universes with compact topology that would give rise to circles whose observable parameters $lambda$ and $theta$ fall o outside the ranges covered by this search. We derive the expressions for the deviation from antipodality and for the radius of the circles associated to a pair elements ($gamma,$,$gamma^{-1}$) of the holonomy group $Gamma$ which define the spatial section of any positively curved universe with a nontrivial topology. We show that there is a critical position that maximizes the deviation from antipodality, and prove that no matter how nearly flat the Universe is, it can always have a nontrivial spatial topology that gives rise to circles whose deviation from antipodality $theta$ is larger than $169^circ$, and whose radii of the circles $lambda$ are smaller than $10^circ$ for some observers. This makes apparent that slightly positively curved universes with cosmological parameters within Planck bounds can be endowed with a nontrivial spatial topology with values of the parameters $lambda$ and $theta$ outside the ranges covered by the searches for circles carried out so far. Thus, these circles searches so far undertaken are not sufficient to exclude the possibility of a universe with a detectable nontrivial cosmic topology. We present concrete examples of such nearly flat universes, and discuss the implications of our results in view of unavoidable practical limits of the circles-in-the-sky method.
An important, and potentially detectable, signature of a non-trivial topology for the universe is the presence of so called circles-in-the-sky in the cosmic microwave background (CMB). Recent searches, confined to antipodal and nearly antipodal circles, have however failed to detect any. This outcome, coupled with recent theoretical results concerning the detectability of very nearly flat universes, is sufficient to exclude a detectable non-trivial cosmic topology for most observers in the inflationary limit ($0< |Omega_{tot}-1| lesssim 10^{-5}$). In a recent paper we have studied the consequences of these searches for circles if the Universe turns out to be exactly flat ($Omega_{tot} = 1 $) as is often assumed. More specifically, we have derived the maximum angles of deviation possible from antipodicity of pairs of matching circles associated with the shortest closed geodesic for all multiply-connected flat orientable $3$-manifolds. These upper bounds on the deviation from antipodicity demonstrate that in a flat universe for some classes of topology there remains a substantial fraction of observers for whom the deviation from antipodicity of the matching circles is considerably larger than zero, which implies that the searches for circles-in-the-sky undertaken so far are not enough to exclude the possibility of a detectable non-trivial flat topology. Here we briefly review these results and discuss their consequences in the search for circles-in-the-sky in a flat universes.
In a Universe with a detectable nontrivial spatial topology the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations --- the so-called circles-in-the-sky. Searches undertaken for nearly antipodal pairs of such circles in cosmic microwave background maps have so far been unsuccessful. Previously we had shown that the negative outcome of such searches, if confirmed, should in principle be sufficient to exclude a detectable non-trivial spatial topology for most observers in very nearly flat ($0<midOmega_{text{tot}}-1mid lesssim10^{-5}$) (curved) universes. More recently, however, we have shown that this picture is fundamentally changed if the universe turns out to be {it exactly} flat. In this case there are many potential pairs of circles with large deviations from antipodicity that have not yet been probed by existing searches. Here we study under what conditions the detection of a single pair of circles-in-the-sky can be used to uniquely specify the topology and the geometry of the spatial section of the Universe. We show that from the detection of a emph{single} pair of matching circles one can infer whether the spatial geometry is flat or not, and if so we show how to determine the topology (apart from one case) of the Universe using this information. An important additional outcome of our results is that the dimensionality of the circles-in-the-sky parameter space that needs to be spanned in searches for matching pair of circles is reduced from six to five degrees of freedom, with a significant reduction in the necessary computational time.
Inflation is an early period of accelerated cosmic expansion, thought to be sourced by high energy physics. A key task today is to use the influx of increasingly precise observational data to constrain the plethora of inflationary models suggested by fundamental theories of interactions. This requires a robust theoretical framework for quantifying the predictions of such models; helping to develop such a framework is the aim of this thesis. We provide the first complete quantization of subhorizon perturbations for the well-motivated class of multi-field inflationary models with a non-trivial field metric, which we show may yield interesting signatures in the bispectrum of the Cosmic Microwave Background (CMB). The subsequent evolution of perturbations in the superhorizon epoch is then considered, via a covariant extension of the transport formalism. To develop intuition about the relationship between inflationary dynamics and the evolution of cosmic observables, we investigate analytic approximations of superhorizon perturbation evolution. The validity of these analytic results is contingent on reaching a state of adiabaticity which we discuss and illustrate in depth. We then apply our analytic methods to elucidate the types of inflationary dynamics that lead to an enhanced CMB non-Gaussianity, both in its bispectrum and trispectrum. In addition to deriving a number of new simple relations between the non-Gaussianity parameters, we explain dynamically how and why different shapes of inflationary potential lead to particular observational signatures. Candidate theories of high energy physics such as low energy effective string theory also motivate single-field modifications to the Einstein-Hilbert action. We show how a range of such corrections allow for consistency of single-field chaotic inflationary models that are otherwise in tension with observational data.
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