Matter bounces are bouncing scenarios wherein the universe contracts as in a matter dominated phase at early times. Such scenarios are known to lead to a scale invariant spectrum of tensor perturbations, just as de Sitter inflation does. In this work, we examine if the tensor bi-spectrum can discriminate between the inflationary and the bouncing scenarios. Using the Maldacena formalism, we analytically evaluate the tensor bi-spectrum in a matter bounce for an arbitrary triangular configuration of the wavevectors. We show that, over scales of cosmological interest, the non-Gaussianity parameter hNL that characterizes the amplitude of the tensor bi-spectrum is quite small when compared to the corresponding values in de Sitter inflation. During inflation, the amplitude of the tensor perturbations freeze on super-Hubble scales, a behavior that results in the so-called consistency condition relating the tensor bi-spectrum and the power spectrum in the squeezed limit. In contrast, in the bouncing scenarios, the amplitude of the tensor perturbations grow strongly as one approaches the bounce, which suggests that the consistency condition will not be valid in such situations. We explicitly show that the consistency relation is indeed violated in the matter bounce. We discuss the implications of the results. © 2015 IOP Publishing Ltd and Sissa Medialab srl .

L. Sriramkumar

Department Of Physics

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IIT Madras

Journal of Cosmology and Astroparticle Physics

Certain features in the primordial scalar power spectrum are known to improve the fit to the cosmological data. We examine whether bouncing scenarios can remain viable if future data confirm the presence of such features. In inflation, the fact that the trajectory is an attractor permits the generation of features. However, bouncing scenarios often require fine-tuned initial conditions, and it is only the ekpyrotic models that allow attractors. We demonstrate, for the first time, that ekpyrotic scenarios can generate specific features that have been considered in the context of inflation. © 2019 American Physical Society.

Physical Review D

Primordial features from ekpyrotic bounces

Primordial Gravitational waves leave polarization imprints on the Cosmic Microwave Background (CMB) . In this article, we investigate polarization bispectrum, which is also referred to as the B-mode auto bispectrum, due to a matter bounce Universe. For simplicity, we consider a minimally coupled Einstein frame and obtain an analytical integral expression for the bispectrum and numerically perform the integration. We find that the signal-to-noise ratio is small, when compared with the same in the inflationary paradigm and hence quite difficult to detect in the future experiments. Thus a detection of tensor mode bispectrum in future will be helpful in ruling out matter bounce model. Also, to ease the numerical evaluation of the bispectrum, we develop and use various techniques. We believe that these techniques can be used in various other contexts. © 2019 IOP Publishing Ltd and Sissa Medialab.

B-mode auto-bispectrum due to matter bounce

In a recent work, we constructed a model consisting of two fields - a canonical scalar field and a noncanonical ghost field - that sourced a symmetric matter bounce scenario. The model involved only one parameter, viz., the scale associated with the bounce. For a suitable value of the parameter, the model led to strictly scale-invariant power spectra with a COBE normalized scalar amplitude and a rather small tensor-to-scalar ratio. In this work, we extend the model to achieve near-matter bounces, which contain a second parameter apart from the bounce scale. As the new model does not seem to permit analytical evaluation of the scalar modes near the bounce, with the aid of techniques that we used in our earlier work, we compute the scalar and the tensor power spectra numerically. For appropriate values of the additional parameter, we find that the model produces red spectra with a scalar spectral tilt and a small tensor-to-scalar ratio, which are consistent with the recent observations of the anisotropies in the cosmic microwave background by Planck. © 2019 American Physical Society.

Viable scalar spectral tilt and tensor-to-scalar ratio in near-matter bounces

Bouncing scenarios offer an alternative to the inflationary paradigm for the generation of perturbations in the early universe. Recently, there has been a surge in interest in examining the issue of primordial magnetogenesis in the context of bouncing universes. As in the case of inflation, the conformal invariance of the electromagnetic action needs to be broken in bouncing scenarios in order to generate primordial magnetic fields which correspond to observed strengths today. The non-minimal coupling, which typically depends on a scalar field that possibly drives the homogeneous background, leads to a cross-correlation at the level of the three-point function between the perturbation in the scalar field and the magnetic fields. This has been studied in some detail in the context of inflation and, specifically, it has been established that the three-point function satisfies the so-called consistency relation in the squeezed limit. In this work, we study the cross-correlation between the magnetic fields and the perturbation in an auxiliary scalar field in a certain class of bouncing scenarios. We consider couplings that lead to scale invariant spectra of the magnetic field and evaluate the corresponding cross-correlation between the magnetic field and the perturbation in the scalar field. We find that, when compared to de Sitter inflation, the dimensionless non-Gaussianity parameter that characterizes the amplitude of the cross-correlations proves to be considerably larger in bouncing scenarios. We also show that the aforementioned consistency condition governing the cross-correlation is violated in the bouncing models. We discuss the implications of our results. © 2019 IOP Publishing Ltd and Sissa Medialab.

Cross-correlations between scalar perturbations and magnetic fields in bouncing universes

Recently, we numerically showed that, for a nonminimal coupling that is a simple power of the scale factor, scale invariant magnetic fields arise in a class of bouncing universes. In this work, we analytically evaluate the spectrum of magnetic and electric fields generated in a subclass of such models. We illustrate that, for cosmological scales which have wave numbers much smaller than the wave number associated with the bounce, the shape of the spectrum is preserved across the bounce. Using the analytic solutions obtained, we also illustrate that the problem of backreaction is severe at the bounce. Finally, we show that the power spectrum of the magnetic field remains invariant under a two-parameter family of transformations of the nonminimal coupling function. © 2016 American Physical Society.

Duality and scale invariant magnetic fields from bouncing universes

We consider the generation of primordial magnetic fields in a class of bouncing models when the electromagnetic action is coupled non-minimally to a scalar field that, say, drives the background evolution. For scale factors that have the power law form at very early times and non-minimal couplings which are simple powers of the scale factor, one can easily show that scale invariant spectra for the magnetic field can arise before the bounce for certain values of the indices involved. It will be interesting to examine if these power spectra retain their shape after the bounce. However, analytical solutions for the Fourier modes of the electromagnetic vector potential across the bounce are difficult to obtain. In this work, with the help of a new time variable that we introduce, which we refer to as the e - fold, we investigate these scenarios numerically. Imposing the initial conditions on the modes in the contracting phase, we numerically evolve the modes across the bounce and evaluate the spectra of the electric and magnetic fields at a suitable time after the bounce. As one could have intuitively expected, though the complete spectra depend on the details of the bounce, we find that, under the original conditions, scale invariant spectra of the magnetic fields do arise for wavenumbers much smaller than the scale associated with the bounce. We also show that magnetic fields which correspond to observed strengths today can be generated for specific values of the parameters. But, we find that, at the bounce, the backreaction due to the electromagnetic modes that have been generated can be significantly large calling into question the viability of the model. We briefly discuss the implications of our results. © 2015 IOP Publishing Ltd and Sissa Medialab srl .

Generation of scale invariant magnetic fields in bouncing universes

As is well known, the non-Gaussianity parameter fNL, which is often used to characterize the amplitude of the scalar bi-spectrum, can be expressed completely in terms of the scalar spectral index ns in the squeezed limit, a relation that is referred to as the consistency condition. This relation, while it is largely discussed in the context of slow roll inflation, is actually expected to hold in any single field model of inflation, irrespective of the dynamics of the underlying model, provided inflation occurs on the attractor at late times. In this work, we explicitly examine the validity of the consistency relation, analytically as well as numerically, away from slow roll. Analytically, we first arrive at the relation in the simple case of power law inflation. We also consider the non-trivial example of the Starobinsky model involving a linear potential with a sudden change in its slope (which leads to a brief period of fast roll), and establish the condition completely analytically. We then numerically examine the validity of the consistency relation in three inflationary models that lead to the following features in the scalar power spectrum due to departures from slow roll: (i) a sharp cut off at large scales, (ii) a burst of oscillations over an intermediate range of scales, and (iii) small, but repeated, modulations extending over a wide range of scales. It is important to note that it is exactly such spectra that have been found to lead to an improved fit to the CMB data, when compared to the more standard power law primordial spectra, by the Planck team. We evaluate the scalar bi-spectrum for an arbitrary triangular configuration of the wavenumbers in these inflationary models and explicitly illustrate that, in the squeezed limit, the consistency condition is indeed satisfied even in situations consisting of strong deviations from slow roll. We conclude with a brief discussion of the results. © 2015 IOP Publishing Ltd and Sissa Medialab srl .

On the scalar consistency relation away from slow roll

It is well known that the non-Gaussianity parameter fNL characterizing the scalar bi-spectrum can be expressed in terms of the scalar spectral index in the squeezed limit, a property that is referred to as the consistency relation. In contrast to the scalar bi-spectrum, the three-point cross-correlations involving scalars and tensors and the tensor bi-spectrum have not received adequate attention, which can be largely attributed to the fact that the tensors had remained undetected at the level of the power spectrum until very recently. The detection of the imprints of the primordial tensor perturbations by BICEP2 and its indication of a rather high tensor-to-scalar ratio, if confirmed, can open up a new window for understanding the tensor perturbations, not only at the level of the power spectrum, but also in the realm of non-Gaussianities. In this work, we consider the consistency relations associated with the three-point cross-correlations involving scalars and tensors as well as the tensor bi-spectrum in inflationary models driven by a single, canonical, scalar field. Characterizing the cross-correlations in terms of the dimensionless non-Gaussianity parameters CNL and CNLγ that we had introduced earlier, we express the consistency relations governing the cross-correlations as relations between these non-Gaussianity parameters and the scalar or tensor spectral indices, in a fashion similar to that of the purely scalar case. We also discuss the corresponding relation for the non-Gaussianity parameter hNL used to describe the tensor bi-spectrum. We analytically establish these consistency relations explicitly in the following two situations: a simple example involving a specific case of power law inflation and a non-trivial scenario in the so-called Starobinsky model that is governed by a linear potential with a sharp change in its slope. We also numerically verify the consistency relations in three types of inflationary models that permit deviations from slow roll and lead to scalar power spectra with features which typically result in an improved fit to the data than the more conventional, nearly scale invariant, spectra. We close with a summary of the results we have obtained. © 2014 IOP Publishing Ltd and Sissa Medialab srl .

Examining the consistency relations describing the three-point functions involving tensors

We present a new and accurate Fortran code, the Bl-spectra and Non-Gaussianity Operator (BINGO), for the efficient numerical computation of the scalar bi-spectrum and the non-Gaussianity parameter fNL in single field inflationary models involving the canonical scalar field. The code can calculate all the different contributions to the bi-spectrum and the parameter fNL for an arbitrary triangular configuration of the wavevectors. Focusing firstly on the equilateral limit, we illustrate the accuracy of BINGO by comparing the results from the code with the spectral dependence of the bi-spectrum expected in power law inflation. Then, considering an arbitrary triangular configuration, we contrast the numerical results with the analytical expression available in the slow roll limit, for, say, the case of the conventional quadratic potential. Considering a non-trivial scenario involving deviations from slow roll, we compare the results from the code with the analytical results that have recently been obtained in the case of the Starobinsky model in the equilateral limit. As an immediate application, we utilize BINGO to examine of the power of the non-Gaussianity parameter f NL to discriminate between various inflationary models that admit departures from slow roll and lead to similar features in the scalar power spectrum. We close with a summary and discussion on the implications of the results we obtain. © 2013 IOP Publishing Ltd and Sissa Medialab srl.

Journal	Journal of Cosmology and Astroparticle Physics
Publisher	Institute of Physics Publishing
ISSN	14757516
Open Access	No