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Karim A. Malik

Publications and source records attributed to Karim A. Malik.

At least 55 records · Page 3Linked to original sources
arXiv2010

Can cosmological perturbations produce early universe vorticity?

In this special issue article, based on the talk with the same title in session B5 (Theoretical and Mathematical Cosmology) at GR19, we review the case of vorticity generation in cosmology using cosmological perturbation theory. We show that, while at linear order the vorticity evolution equation has no source term in the absence of anisotropic stress, at second order vorticity is sourced by gradients in entropy and energy density perturbations. We then present some estimates for the magnitude and scale dependence of the vorticity power spectrum using simple input power spectra for the energy density and entropy perturbations. Finally, we close with possible directions for future work followed by some hints toward the observational importance of the vorticity so generated, and the possibility of primordial magnetic field generation.

gr-qc↗
arXiv2010

Estimating the amount of vorticity generated by cosmological perturbations in the early universe

We estimate the amount of vorticity generated at second order in cosmological perturbation theory from the coupling between first order energy density and non-adiabatic pressure, or entropy, perturbations. Assuming power law input spectra for the source terms, and working in a radiation background, we calculate the wave number dependence of the vorticity power spectrum and its amplitude. We show that the vorticity generated by this mechanism is non-negligible on small scales, and hence should be taken into consideration in current and future CMB experiments.

astro-ph.CO↗
arXiv2009

Gauges and Cosmological Backreaction

We present a formalism for spatial averaging in cosmology applicable to general spacetimes and coordinates, and allowing the easy incorporation of a wide variety of matter sources. We apply this formalism to a Friedmann-LeMaitre-Robertson-Walker universe perturbed to second-order and present the corrections to the background in an unfixed gauge. We then present the corrections that arise in uniform curvature and conformal Newtonian gauges.

gr-qc↗
arXiv2009

Practical tools for third order cosmological perturbations

We discuss cosmological perturbation theory at third order, deriving the gauge transformation rules for metric and matter perturbations, and constructing third order gauge invariant quantities. We present the Einstein tensor components, the evolution equations for a perfect fluid, and the Klein-Gordon equation at third order, including scalar, vector and tensor perturbations. In doing so, we also give all second order tensor components and evolution equations in full, exhilarating generality.

astro-ph.CO↗
arXiv2009

Numerical calculation of second order perturbations

We numerically solve the Klein-Gordon equation at second order in cosmological perturbation theory in closed form for a single scalar field, describing the method employed in detail. We use the slow-roll version of the second order source term and argue that our method is extendable to the full equation. We consider two standard single field models and find that the results agree with previous calculations using analytic methods, where comparison is possible. Our procedure allows the evolution of second order perturbations in general and the calculation of the non-linearity parameter f_NL to be examined in cases where there is no analytical solution available.

astro-ph.CO↗
arXiv2009

Cosmological perturbations

We review the study of inhomogeneous perturbations about a homogeneous and isotropic background cosmology. We adopt a coordinate based approach, but give geometrical interpretations of metric perturbations in terms of the expansion, shear and curvature of constant-time hypersurfaces and the orthogonal timelike vector field. We give the gauge transformation rules for metric and matter variables at first and second order. We show how gauge invariant variables are constructed by identifying geometric or matter variables in physically-defined coordinate systems, and give the relations between many commonly used gauge-invariant variables. In particular we show how the Einstein equations or energy-momentum conservation can be used to obtain simple evolution equations at linear order, and discuss extensions to non-linear order. We present evolution equations for systems with multiple interacting fluids and scalar fields, identifying adiabatic and entropy perturbations. As an application we consider the origin of primordial curvature and isocurvature perturbations from field perturbations during inflation in the very early universe.

astro-ph↗
arXiv2009

Generalised constraints on the curvature perturbation from primordial black holes

Primordial black holes (PBHs) can form in the early Universe via the collapse of large density perturbations. There are tight constraints on the abundance of PBHs formed due to their gravitational effects and the consequences of their evaporation. These abundance constraints can be used to constrain the primordial power spectrum, and hence models of inflation, on scales far smaller than those probed by cosmological observations. We compile, and where relevant update, the constraints on the abundance of PBHs before calculating the constraints on the curvature perturbation, taking into account the growth of density perturbations prior to horizon entry. We consider two simple parameterizations of the curvature perturbation spectrum on the scale of interest: constant and power-law. The constraints from PBHs on the amplitude of the power spectrum are typically in the range 10^{-2}-10^{-1} with some scale dependence.

astro-ph.CO↗
arXiv2009

The non-adiabatic pressure in general scalar field systems

We discuss the non-adiabatic or entropy perturbation, which controls the evolution of the curvature perturbation in the uniform density gauge, for a scalar field system minimally coupled to gravity with non-canonical action. We highlight the differences between the sound and the phase speed in these systems, and show that the non-adiabatic pressure perturbation vanishes in the single field case, resulting in the conservation of the curvature perturbation on large scales.

astro-ph↗
arXiv2008

A Concise Introduction to Perturbation Theory in Cosmology

We give a concise, self-contained introduction to perturbation theory in cosmology at linear and second order, striking a balance between mathematical rigour and usability. In particular we discuss gauge issues and the active and passive approach to calculating gauge transformations. We also construct gauge-invariant variables, including the second order tensor perturbation on uniform curvature hypersurfaces.

astro-ph↗
arXiv2008

A not so short note on the Klein-Gordon equation at second order

We give the governing equations for multiple scalar fields in a flat Friedmann-Robertson-Walker (FRW) background spacetime on all scales, allowing for metric and field perturbations up to second order. We then derive the Klein-Gordon equation at second order in closed form in terms of gauge-invariant perturbations of the fields in the uniform curvature gauge. We also give a simplified form of the Klein-Gordon equation using the slow-roll approximation.

astro-ph↗
arXiv2008

Non-gaussianity of inflationary field perturbations from the field equation

We calculate the tree-level bispectrum of the inflaton field perturbation directly from the field equations, and construct the corresponding f_NL parameter. Our results agree with previous ones derived from the Lagrangian. We argue that quantum theory should only be used to calculate the correlators when they first become classical a few Hubble times after horizon exit, the classical evolution taking over thereafter.

astro-ph↗
arXiv2007

Different approaches to the second order Klein-Gordon equation

We derive the Klein--Gordon equation for a single scalar field coupled to gravity at second order in perturbation theory and leading order in slow-roll. This is done in two ways: we derive the Klein--Gordon equation first using the Einstein field equations, and then directly from the action after integrating out the constraint equations. We also point out an unexpected result regarding the treatment of the field equations.

astro-ph↗
arXiv2007

Scaling Cosmologies from Duality Twisted Compactifications

Oscillating moduli fields can support a cosmological scaling solution in the presence of a perfect fluid when the scalar field potential satisfies appropriate conditions. We examine when such conditions arise in higher-dimensional, non-linear sigma-models that are reduced to four dimensions under a generalized Scherk-Schwarz compactification. We show explicitly that scaling behaviour is possible when the higher-dimensional action exhibits a global SL(n,R) or O(2,2) symmetry. These underlying symmetries can be exploited to generate non-trivial scaling solutions when the moduli fields have non-canonical kinetic energy. We also consider the compactification of eleven-dimensional vacuum Einstein gravity on an elliptic twisted torus.

hep-th↗
arXiv2007

Constraints on the primordial curvature perturbation from primordial black holes

We calculate the constraints on the primordial curvature perturbation at the end of inflation from the present day abundance of Primordial Black Holes (PBHs), as a function of the reheat temperature T_{\rm RH}. We first extend recent work on the formation of PBHs on scales which remain within the horizon during inflation and calculate the resulting constraints on the curvature perturbation. We then evaluate the constraint from PBHs that form, more conventionally, from super-horizon perturbations. The constraints apply for T_{\rm RH} < 10^{8} GeV and the inclusion of sub-horizon PBHs leads to a limit which is roughly three times tighter than the bound from super-horizon PBHs.

astro-ph↗
arXiv2006

A numerical study of non-gaussianity in the curvaton scenario

We study the curvaton scenario using gauge-invariant second order perturbation theory and solving the governing equations numerically. Focusing on large scales we calculate the non-linearity parameter f_nl in the two-fluid curvaton model and compare our results with previous analytical studies employing the sudden decay approximation. We find good agreement of the two approaches for large curvaton energy densities at curvaton decay, Omega_dec, but significant differences of up to 10 percent for small Omega_dec.

astro-ph↗
arXiv2005

Gauge-invariant perturbations at second order: multiple scalar fields on large scales

We derive the governing equations for multiple scalar fields minimally coupled to gravity in a flat Friedmann-Robertson-Walker (FRW) background spacetime on large scales. We include scalar perturbations up to second order and write the equations in terms of physically transparent gauge-invariant variables at first and second order. This allows us to write the perturbed Klein-Gordon equation at second order solely in terms of the field fluctuations on flat slices at first and second order.

astro-ph↗
arXiv2005

Forming sub-horizon black holes at the end of inflation

Previous authors have calculated the mass function of primordial black holes only on scales which are well outside the horizon at the end of inflation. Here we extend the calculation to sub-horizon scales, on which the density perturbation never becomes classical. Regarding the formation of black holes as a `measurement' of the (high peaks) of the density perturbation, we estimate a mass function by assuming that black holes form as soon as inflation ends, in those rare regions where the Bardeen potential exceeds a threshold value of $Psi_c\simeq 0.5$.

astro-ph↗