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We investigate the joint density-velocity evolution in $f(R)$ gravity using smooth, compensated spherical top-hats as a proxy for the non-linear regime. Using the Hu-Sawicki model as a working example, we solve the coupled continuity, Euler and Einstein equations using an iterative hybrid Lagrangian-Eulerian scheme. The novel aspect of this scheme is that the metric potentials are solved for analytically in the Eulerian frame. The evolution is assumed to follow GR at very early epochs and switches to $f(R)$ at a pre-determined epoch. Choosing the `switching epoch' too early is computationally expensive because of high frequency oscillations; choosing it too late potentially destroys consistency with $\Lambda$CDM. To make an informed choice, we perform an eigenvalue analysis of the background model which gives a ballpark estimate of the magnitude of oscillations. There are two length scales in the problem: the comoving Compton wavelength of the associated scalar field and the width of the top-hat. The evolution is determined by their ratio. When the ratio is large, the evolution is scale-independent and the density-velocity divergence relation (DVDR) is unique. When the ratio is small, the evolution is very close to GR, except for the formation of a spike near the top-hat edge, a feature which has been noted in earlier literature. We are able to qualitatively explain this feature in terms of the analytic solution for the metric potentials, in the absence of the chameleon mechanism. In the intermediate regime, the evolution is profile-dependent and no unique DVDR exists., Comment: 28 pages + appendix, 12 figures, accepted in MNRAS

We compute the one-point PDF of an initially Gaussian dark matter density field using spherical collapse (SC). We compare the results to other forms available in the literature and also compare the PDFs in the $\Lambda$CDM model with an early dark energy (EDE) model. We find that the skewed log-normal distribution provides the best fit to the non-linear PDF from SC for both cosmologies, from $a=0.1$ to 1 and for scales characterized by the comoving width of the Gaussian: $\sigma_G = 0.5, 1, 2$. To elucidate the effect of cosmology, we examine the linear and non-linear growth rates through test cases. For overdensities, when the two models have the same initial density contrast, the differences due to cosmology are amplified in the non-linear regime, whereas, if the two models have the same linear density contrast today, then the differences in cosmology are damped in the non-linear regime. This behaviour is in contrast with voids, where the non-linear growth becomes `self-regulatory' and is less sensitive to cosmology and initial conditions. To compare the PDFs, we examine the difference of the PDFs and evolution of the width of the PDF. The trends with scale and redshift are as expected. A tertiary aim of this paper was to check if the fitting form for the non-linear density-velocity divergence relation, derived for constant equation of state ($w$) models by Nadkarni-Ghosh holds for the EDE model. We find that it does with an accuracy of 4\%, thus increasing its range of validity., Comment: 16 pages + appendix, 11 figures, 1 tables. Accepted for publication in MNRAS

The linear Einstein-Boltzmann equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its mathematical properties. For this purpose, we focus on a simplified set of equations aimed at describing the broad features of the matter power spectrum. We first perform an eigenvalue analysis and study the onset of oscillations in the system signaled by the transition from real to complex eigenvalues. We then provide a stability criterion of different numerical schemes for this linear system and estimate the associated step-size. We elucidate the stiffness property of the Einstein-Boltzmann system and show how it can be characterized in terms of the eigenvalues. While the parameters of the system are time dependent making it non-autonomous, we define an adiabatic regime where the parameters vary slowly enough for the system to be quasi-autonomous. We summarize the different regimes of the system for these different criteria as function of wave number $k$ and scale factor $a$. We also provide a compendium of analytic solutions for all perturbation variables in 6 limits on the $k$-$a$ plane and express them explicitly in terms of initial conditions. These results are aimed to help the further development and testing of numerical cosmological Boltzmann solvers., Comment: 41 pages (21 text + appendices), 19 figures and 2 tables; accepted in MNRAS

We investigate the dynamical behavior of rings around bodies whose shapes depart considerably from that of a sphere. To this end, we have developed a new self-gravitating discrete element N-body code, and employed a local simulation method to simulate a patch of the ring. The central body is modeled as a symmetric (oblate or prolate) ellipsoid, or defined through the characteristic frequencies (circular, vertical, epicyclic) that represent its gravitational field. Through our simulations we explore how a ring's behavior -- characterized by dynamical properties like impact frequency, granular temperature, number density, vertical thickness and radial width -- varies with the changing gravitational potential of the central body. We also contrast properties of rings about large central bodies (e.g. Saturn) with those of smaller ones (e.g. Chariklo). Finally, we investigate how the characteristic frequencies of a central body, restricted to being a solid of revolution with an equatorial plane of symmetry, affect the ring dynamics. The latter process may be employed to qualitatively understand the dynamics of rings about any symmetric solids of revolution., Comment: 24 pages, 23 figures, submitted to Icarus

There currently exist many observations which are not consistent with the cosmological principle. We review these observations with a particular emphasis on those relevant for Square Kilometre Array (SKA). In particular, several different data sets indicate a preferred direction pointing approximately towards the Virgo cluster. We also observe a hemispherical anisotropy in the Cosmic Microwave Background Radiation (CMBR) temperature fluctuations. Although these inconsistencies may be attributed to systematic effects, there remains the possibility that they indicate new physics and various theories have been proposed to explain them. One possibility, which we discuss in this review, is the generation of perturbation modes during the early pre-inflationary epoch, when the Universe may not obey the cosmological principle. Better measurements will provide better constraints on these theories. In particular, we propose measurement of the dipole in number counts, sky brightness, polarized flux and polarization orientations of radio sources. We also suggest test of alignment of linear polarizations of sources as a function of their relative separation. Finally we propose measurement of hemispherical anisotropy or equivalently dipole modulation in radio sources., Comment: 19 pages, To appear in Journal of Astrophysics and Astronomy (JAA) special issue on "Science with the SKA: an Indian perspective"

We investigate the evolution of axis ratios of triaxial haloes using the phase space description of triaxial collapse. In this formulation, the evolution of the triaxial ellipsoid is described in terms of the dynamics of eigenvalues of three important tensors: the Hessian of the gravitational potential, the tensor of velocity derivatives and the deformation tensor. The eigenvalues of the deformation tensor are directly related to the parameters that describe triaxiality, namely, the minor to major and intermediate to major axes ratios ($s$ and $q$) and the triaxiality parameter $T$. Using the phase space equations, we evolve the eigenvalues and examine the evolution of the PDF (probability distribution function) of the axes ratios as a function of mass scale and redshift for Gaussian initial conditions. We find that the ellipticity and prolateness increase with decreasing mass scale and decreasing redshift. These trends agree with previous analytic studies but differ from numerical simulations. However, the PDF of the scaled parameter ${\tilde q} = (q-s)/(1-s)$ follows a universal distribution over two decades in mass range and redshifts which is in qualitative agreement with the universality for conditional PDF reported in simulations. We further show using the phase space dynamics that, in fact, ${\tilde q}$ is a phase space invariant and is conserved individually for each halo. These results, demonstrate that the phase space analysis is a useful tool that provides a different perspective on the evolution of perturbations and can be applied to more sophisticated models in the future., Comment: 10 pages, 5 figures, accepted in MNRAS

We investigate the dynamics of triaxial collapse in terms of eigenvalues of the deformation tensor, the velocity derivative tensor and the gravity Hessian. Using the Bond-Myers model of ellipsoidal collapse, we derive a new set of equations for the nine eigenvalues and examine their dynamics in phase space. The main advantage of this form is that it eliminates the complicated elliptic integrals that appear in the axes evolution equations and is more natural way to understand the interplay between the perturbations. This paper focuses on the density-velocity dynamics. The Zeldovich approximation implies that the three tensors are proportional; the proportionality constant is set by demanding `no decaying modes'. We extend this condition into the non-linear regime and find that the eigenvalues of the gravity Hessian and the velocity derivative tensor are related as ${\tilde q}_d + {\tilde q}_v=1$, where the triaxiality parameter ${\tilde q} = (\lambda_{\mathrm{max}} - \lambda_{\mathrm{inter}})/(\lambda_{\mathrm{max}} - \lambda_{\mathrm{min}})$. This is a {\it new universal relation} holding true over all redshifts and a range of mass scales to within a few percent accuracy. The mean density-velocity divergence relation at late times is close to linear, indicating that the dynamics is dictated by collapse along the largest eigendirection. This relation has a scatter, which we show, is intimately connected to the velocity shear. Finally, as an application, we compute the PDFs of the two variables and compare with other forms in the literature., Comment: 23 pages (16 text+appendix); 11 figures, revised version accepted for publication in MNRAS

We study the dipole anisotropy in number counts and flux density weighted number counts {or sky brightness} in the NRAO VLA Sky Survey (NVSS) data. The dipole anisotropy is expected due to our local motion with respect to the CMBR rest frame. We analyse data with an improved fit to the number density, n(S), as a function of the flux density S, which allows deviation from a pure power law behaviour. We also impose more stringent cuts to remove the contribution due to clustering dipole. In agreement with earlier results, we find that the amplitude of anisotropy is significantly larger in comparison to the prediction based on CMBR measurements. The extracted speed is found to be roughly 3 times the speed corresponding to CMBR. The significance of deviation is smaller, roughly 2 sigma, in comparison to earlier estimates. For the cut, S>30 mJy, the speed is found to be $1110\pm370$ Km/s using the source count analysis. The direction of the dipole anisotropy is found to be approximately in agreement with CMBR. We find that the results are relatively insensitive to the lower as well as upper limit imposed on the flux density. Our results suggest that the Universe is intrinsically anisotropic with the axis of anisotropy axis pointing roughly towards the CMBR dipole direction. Finally we present a method which may allow an independent extraction of the local speed and an intrinsic dipole anisotropy, provided a larger data set becomes available in future., Comment: 27 pages, 9 figures, substantially revised version, no change in conclusions, to be published in Astroparticle physics

We present a new application of Lagrangian Perturbation Theory (LPT): the stability analysis of fluid flows. As a test case that demonstrates the framework we focus on the plane Couette flow. The incompressible Navier-Stokes equation is recast such that the particle position is the fundamental variable, expressed as a function of Lagrangian coordinates. The displacement due to the steady state flow is taken to be the zeroth order solution and the position is formally expanded in terms of a small parameter (generally, the strength of the initial perturbation). The resulting hierarchy of equations is solved analytically at first order. We find that we recover the standard result in the Eulerian frame: the plane Couette flow is asymptotically stable for all Reynolds numbers. However, it is also well established that experiments contradict this prediction. In the Eulerian picture, one of the proposed explanations is the phenomenon of `transient growth' which is related to the non-normal nature of the linear stability operator. The first order solution in the Lagrangian frame also shows this feature, albeit qualitatively. As a first step, and for the purposes of analytic manipulation, we consider only linear stability of 2D perturbations but the framework presented is general and can be extended to higher orders, other flows and/or 3D perturbations., Comment: 22 pages, accepted in Frontiers in Physics (Mathematical Physics)

We present a new method to calculate formation of cosmological structure in the Newtonian limit. The method is based on Lagrangian perturbation theory plus two key theoretical extensions. One advance involves identifying and fixing a previously ignored gauge-like degree of freedom relating quantities calculated in LPT to those measured by a preferred Friedmann-Robertson-Walker (FRW) observer. Handling this connection between calculational and observer frames is physically essential and ensures a momentum conserving description. The second extension is to systematically re-expand the equations of motion to increase LPT's radius of convergence to the maximum future time prior to orbit crossing. The paper implements a complete algorithm and performs extensive "proof of principle" tests of the new method, including direct comparison to known solutions, evaluation of conserved quantities and formal convergence studies. All are satisfactory. We show convergence is exponential in grid size and Lagrangian order and polynomial in step size. There are three {\it powerful advantages} afforded by the new technique: (1) it employs a smooth representation of all fields and the results are not limited by particle induced shot-noise errors, (2) it permits the numerical error to be controlled by changing Lagrangian order and/or number of steps allowing, in principle, arbitrarily small errors to be achieved prior to orbit crossing and (3) it handles generic cold initial data (any periodic density and velocity fields, including those with initial rotational components). Together, these properties make the new technique well-suited to handle quasi-linear scales where analytic methods and/or numerical simulations fail to provide suitably accurate answers., Comment: 31 pages, 8 figrues, 2 tables. Accepted in MNRAS

We obtain the non-linear relation between cosmological density and velocity perturbations by examining their joint dynamics in a two dimensional density-velocity divergence phase space. We restrict to spatially flat cosmologies consisting of pressureless matter and non-clustering dark energy characterized by a constant equation of state $w$. Using the spherical top-hat model, we derive the coupled equations that govern the joint evolution of the perturbations and examine the flow generated by this system. In general, the initial density and velocity are independent, but requiring that the perturbations vanish at the big bang time sets a relation between the two. This relation, which we call the `Zel'dovich curve', acts like an attracting solution for the phase space dynamics and is the desired non-linear extension of the density-velocity divergence relation. We obtain a fitting formula for the curve, which is a generalization of the formulae by Bernardeau and Bilicki & Chodorowski, and find that as in the linear regime, the explicit dependence on the dark energy parameters stays weak even in the non-linear regime. Although the result itself is somewhat expected, the new feature of this work is the interpretation of the relation in the phase space picture and the generality of the method. Finally, as an observational implication, we examine the evolution of galaxy cluster profiles using the spherical infall model for different values of $w$. We demonstrate that using only the density or the velocity information to constrain $w$ is subject to degeneracies in other parameters such as $\sigma_8$ but plotting observations onto the joint density-velocity phase space can help lift this degeneracy., Comment: 19 pages, 8 figures, updated version (V2) incorporates comments from the referee. Accepted for publication in MNRAS, abstract slightly abridged

We investigate convergence of Lagrangian Perturbation Theory (LPT) by analyzing the model problem of a spherical homogeneous top-hat in an Einstein-deSitter background cosmology. We derive the formal structure of the LPT series expansion, working to arbitrary order in the initial perturbation amplitude. The factors that regulate LPT convergence are identified by studying the exact, analytic solution expanded according to this formal structure. The key methodology is to complexify the exact solution, demonstrate that it is analytic and apply well-known convergence criteria for power series expansions of analytic functions.This analysis fully explains the previously reported observation that LPT fails to predict the evolution of an underdense, open region beyond a certain time. It also implies the existence of other examples, including overdense, closed regions, for which LPT predictions should also fail. We show that this is indeed the case by numerically computing the LPT expansion in these problematic cases. The formal limitations to the validity of LPT expansion are considerably more complicated than simply the first occurrence of orbit crossings as is often assumed. Evolution to a future time generically requires re-expanding the solution in overlapping domains that ultimately link the initial and final times, each domain subject to its own convergence criterion. We demonstrate that it is possible to handle all the problematic cases by taking multiple steps (LPT re-expansion). We characterize how the leading order numerical error for a solution generated by LPT re- expansion varies with the choice of Lagrangian order and of time step size. Convergence occurs when the Lagrangian order increases and/or the time step size decreases in a simple, well-defined manner. We develop a recipe for time step control for LPT re-expansion based on these results., Comment: 40 pages, 24 figures. Abstract abbreviated. Accepted for publication in MNRAS. Updated version with minor comments from referee incorporated