arXiv subjects

Karim A. Malik

Publications and source records attributed to Karim A. Malik.

At least 37 records · Page 2Linked to original sources

arXiv2016

Cosmology on all scales: a two-parameter perturbation expansion

We propose and construct a two-parameter perturbative expansion around a Friedmann-Lemaître-Robertson-Walker geometry that can be used to model high-order gravitational effects in the presence of non-linear structure. This framework reduces to the weak-field and slow-motion post-Newtonian treatment of gravity in the appropriate limits, but also includes the low-amplitude large-scale fluctuations that are important for cosmological modelling. We derive a set of field equations that can be applied to the late Universe, where non-linear structure exists on supercluster scales, and perform a detailed investigation of the associated gauge problem. This allows us to identify a consistent set of perturbed quantities in both the gravitational and matter sectors, and to construct a set of gauge-invariant quantities that correspond to each of them. The field equations, written in terms of these quantities, take on a relatively simple form, and allow the effects of small-scale structure on the large-scale properties of the Universe to be clearly identified. We find that inhomogeneous structures source the global expansion, that there exist new field equations at new orders, and that there is vector gravitational potential that is a hundred times larger than one might naively expect from cosmological perturbation theory. Finally, we expect our formalism to be of use for calculating relativistic effects in upcoming ultra-large-scale surveys, as the form of the gravitational coupling between small and large scales depends on the non-linearity of Einstein's equations, and occurs at what is normally thought of as first order in cosmological perturbations.

astro-ph.CO↗

arXiv2016

Vector and tensor contributions to the curvature perturbation at second order

We derive the evolution equation for the second order curvature perturbation using standard techniques of cosmological perturbation theory. We do this for different definitions of the gauge invariant curvature perturbation, arising from different splits of the spatial metric, and compare the expressions. The results are valid at all scales and include all contributions from scalar, vector and tensor perturbations, as well as anisotropic stress, with all our results written purely in terms of gauge invariant quantities. Taking the large-scale approximation, we find that a conserved quantity exists only if, in addition to the non-adiabatic pressure, the transverse traceless part of the anisotropic stress tensor is also negligible. We also find that the version of the gauge invariant curvature perturbation which is exactly conserved is the one defined with the determinant of the spatial part of the inverse metric.

astro-ph.CO↗

arXiv2015

Second-order cosmological perturbation theory and initial conditions for $N$-body simulations

We use gauge-invariant cosmological perturbation theory to calculate the displacement field that sets the initial conditions for $N$-body simulations. Using first and second-order fully relativistic perturbation theory in the synchronous-comoving gauge, allows us to go beyond the Newtonian predictions and to calculate relativistic corrections to it. We use an Einstein--de Sitter model, including both growing and decaying modes in our solutions. The impact of our results should be assessed through the implementation of the featured displacement in cosmological $N$-body simulations.

gr-qc↗

arXiv2015

A short note on the curvature perturbation at second order

Working with perturbations about an FLRW spacetime, we compute the gauge-invariant curvature perturbation to second order solely in terms of scalar field fluctuations. Using the curvature perturbation on uniform density hypersurfaces as our starting point, we give our results in terms of field fluctuations in the flat gauge, incorporating both large and small scale behaviour. For ease of future numerical implementation we give our result in terms of the scalar field fluctuations and their time derivatives.

astro-ph.CO↗

arXiv2014

Conserved Quantities in Lemaitre-Tolman-Bondi Cosmology

We study linear perturbations to a Lema{\^ı}tre-Tolman-Bondi (LTB) background spacetime. Studying the transformation behaviour of the perturbations under gauge transformations, we construct gauge invariant quantities. We show, using the perturbed energy conservation equation, that there are conserved quantities in LTB, in particular a spatial metric trace perturbation, ζ_{SMTP}, which is conserved on all scales. We then briefly extend our discussion to the Lema{\^ı}tre spacetime, and construct gauge-invariant perturbations in this extension of LTB spacetime.

astro-ph.CO↗

arXiv2014

Effects of non-linearities on magnetic field generation

Magnetic fields are present on all scales in the Universe. While we understand the processes which amplify the fields fairly well, we do not have a "natural" mechanism to generate the small initial seed fields. By using fully relativistic cosmological perturbation theory and going beyond the usual confines of linear theory we show analytically how magnetic fields are generated. This is the first analytical calculation of the magnetic field at second order, using gauge-invariant cosmological perturbation theory, and including all the source terms. To this end, we have rederived the full set of governing equations independently. Our results suggest that magnetic fields of the order of $10^{-30}$ G can be generated (although this depends on the small scale cut-off of the integral), which is largely in agreement with previous results that relied upon numerical calculations. These fields are likely too small to act as the primordial seed fields for dynamo mechanisms.

astro-ph.CO↗

arXiv2014

Formation of subhorizon black holes from preheating

We study the production of primordial black holes (PBHs) during the preheating stage that follows a chaotic inflationary phase. The scalar fields present in the process are evolved numerically using a modified version of the HLATTICE code. From the output of the numerical simulation, we compute the probability distribution of curvature fluctuations, paying particular attention to sub-horizon scales. We find that in some specific models these modes grow to large amplitudes developing highly non-Gaussian probability distributions. We then calculate PBH abundances using the standard Press-Schechter criterion and find that overproduction of PBHs is likely in some regions of the chaotic preheating parameter space.

astro-ph.CO↗

arXiv2013

Comments on gauge-invariance in cosmology

We revisit the gauge issue in cosmological perturbation theory, and highlight its relation to the notion of covariance in general relativity. We also discuss the similarities and differences of the covariant approach in perturbation theory to the Bardeen or metric approach in a non-technical fashion.

astro-ph.CO↗

arXiv2013

The Poisson equation at second order in relativistic cosmology

We calculate the relativistic constraint equation which relates the curvature perturbation to the matter density contrast at second order in cosmological perturbation theory. This relativistic "second order Poisson equation" is presented in a gauge where the hydrodynamical inhomogeneities coincide with their Newtonian counterparts exactly for a perfect fluid with constant equation of state. We use this constraint to introduce primordial non-Gaussianity in the density contrast in the framework of General Relativity. We then derive expressions that can be used as initial conditions of N-body codes for structure formation which probe the observable signature of primordial non-Gaussianity in the statistics of the evolved matter density field.

astro-ph.CO↗

arXiv2013

Quantifying the behaviour of curvature perturbations during inflation

How much does the curvature perturbation change after it leaves the horizon, and when should one evaluate the power spectrum? To answer these questions we study single field inflation models numerically, and compare the evolution of different curvature perturbations from horizon crossing to the end of inflation. In particular we calculate the number of efolds it takes for the curvature perturbation at a given wavenumber to settle down to within a given fraction of their value at the end of inflation. We find that e.g. in chaotic inflation, the amplitude of the comoving and the curvature perturbation on uniform density hypersurfaces differ by up to 180 % at horizon crossing assuming the same amplitude at the end of inflation, and that it takes approximately 3 efolds for the curvature perturbation to be within 1 % of its value at the end of inflation.

astro-ph.CO↗

arXiv2013

Quantifying the behaviour of curvature perturbations near Horizon Crossing

How much does the curvature perturbation change after it leaves the horizon, and when should one evaluate the power spectrum? To answer these questions we study single field inflation models numerically, and compare the evolution of different curvature perturbations from horizon crossing to the end of inflation. We find that e.g. in chaotic inflation, the amplitude of the comoving and the curvature perturbation on uniform density hypersurfaces differ by up to 180 \% at horizon crossing assuming the same amplitude at the end of inflation, and that it takes approximately 3 efolds for the curvature perturbation to be within 1 \% of its value at the end of inflation.

astro-ph.CO↗

arXiv2012

Modelling non-dust fluids in cosmology

Currently, most of the numerical simulations of structure formation use Newtonian gravity. When modelling pressureless dark matter, or `dust', this approach gives the correct results for scales much smaller than the cosmological horizon, but for scenarios in which the fluid has pressure this is no longer the case. In this article, we present the correspondence of perturbations in Newtonian and cosmological perturbation theory, showing exact mathematical equivalence for pressureless matter, and giving the relativistic corrections for matter with pressure. As an example, we study the case of scalar field dark matter which features non-zero pressure perturbations. We discuss some problems which may arise when evolving the perturbations in this model with Newtonian numerical simulations and with CMB Boltzmann codes.

astro-ph.CO↗

arXiv2012

Vorticity from isocurvature in the early universe

Vorticity is ubiquitous in nature however, to date, studies of vorticity in cosmology and the early universe have been quite rare. In this paper, based on a talk in session CM1 of the 13th Marcel Grossmann Meeting, we consider vorticity generation from scalar cosmological perturbations of a perfect fluid system. We show that, at second order in perturbation theory, vorticity is sourced by a coupling between energy density and entropy gradients, thus extending a well-known feature of classical fluid dynamics to a relativistic cosmological framework. This induced vorticity, sourced by isocurvature perturbations, may prove useful in the future as an additional discriminator between inflationary models.

astro-ph.CO↗

arXiv2012

The magnitude of the non-adiabatic pressure in the cosmic fluid

Understanding the non-adiabatic pressure, or relative entropy, perturbation is crucial for studies of early-universe vorticity and Cosmic Microwave Background observations. We calculate the evolution of the linear non-adiabatic pressure perturbation from radiation domination to late times, numerically solving the linear governing equations for a wide range of wavenumbers. Using adiabatic initial conditions consistent with WMAP seven year data, we find nevertheless that the non-adiabatic pressure perturbation is non-zero and grows at early times, peaking around the epoch of matter/radiation equality and decaying in matter domination. At early times or large redshifts (z=10,000) its power spectrum peaks at a comoving wavenumber k~0.2h/Mpc, while at late times (z=500) it peaks at k~0.02 h/Mpc.

astro-ph.CO↗

arXiv2011

Second Order Perturbations During Inflation Beyond Slow-roll

We numerically calculate the evolution of second order cosmological perturbations for an inflationary scalar field without resorting to the slow-roll approximation or assuming large scales. In contrast to previous approaches we therefore use the full non-slow-roll source term for the second order Klein-Gordon equation which is valid on all scales. The numerical results are consistent with the ones obtained previously where slow-roll is a good approximation. We investigate the effect of localised features in the scalar field potential which break slow-roll for some portion of the evolution. The numerical package solving the second order Klein-Gordon equation has been released under an open source license and is available for download.

astro-ph.CO↗

arXiv2011

Comparing different formulations of non-linear cosmological perturbation theory

We compare and contrast two different metric based formulations of non- linear cosmological perturbation theory: the MW2009 approach in [K. A. Malik and D. Wands, Phys. Rept. 475 (2009), 1.] following Bardeen and the recent approach of the paper KN2010 [K. Nakamura, Advances in Astronomy 2010 (2010), 576273]. We present each formulation separately. In the MW2009 approach, one considers the gauge transformations of perturbative quantities, choosing a gauge by requiring that certain quantities vanish, rendering all other variables gauge invariant. In the KN2010 formalism, one decomposes the metric tensor into a gauge variant and gauge invariant part from the outset. We compare the two approaches in both the longitudinal and uniform curvature gauges. In the longitudinal gauge, we find that Nakamura's gauge invariant variables correspond exactly to those in the longitudinal gauge (i.e., for scalar perturbations, to the Bardeen potentials), and in the uniform curvature gauge we obtain the usual relationship between gauge invariant variables in the flat and longitudinal gauge. Thus, we show that these two approaches are equivalent.

astro-ph.CO↗

arXiv2010

How the curvaton scenario, modulated reheating and an inhomogeneous end of inflation are related

In this paper we analyse three models of the early universe, for which the respective mechanisms for generating the curvature perturbation are considered disparate. We find that in fact the mechanisms are very similar, and hence explain why they give rise to a large non-gaussianity. We show that the mechanism for generating the primordial curvature perturbation, and hence the observable non-gaussianity, is similar in both the Curvaton and Modulated Reheating models. In both cases the model can be written in terms of an energy transfer between the constituting fluids. We then show that this is also true for the mechanism of generating the curvature perturbation by symmetry breaking the end of inflation. We then relate this to the non-gaussian contribution to the curvature perturbation and find that it is inversely proportional to the efficiency with which the curvature perturbation is transferred between the fluids. For the first time, we generalise models of modulated reheating to allow for a non-linear energy transfer rate.

astro-ph.CO↗

arXiv2010

The Quantum Origin of Cosmic Structure

In this concise, albeit subjective review of structure formation, I shall introduce the cosmological standard model and its theoretical and observational underpinnings. I will focus on recent results and current issues in theoretical cosmology, in particular in cosmological perturbation theory and its applications.

astro-ph.CO↗