Professor T. Padmanabhan
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Item Zeldovich approximation and the probability distribution for the smoothed density field in the nonlinear regime(American Astronomical Society, 1993-06-20) Padmanabhan, T.; Subramanian, KandaswamyThe study of large-scale structure in the Universe is often based on the observed density distribution of matter smoothed by a suitable filter function. The probability distribution for this smoothed density field in the nonlinear regime is studied using the Zel'dovich approximation. When the shear term of the velocity field is not too large, one can obtain a reasonably good analytic approximation tho this probability distribution. The properties of this distribution are discussed and compared with other attempts along similar lines.Item Transfer of power in nonlinear gravitational clustering(Wiley-Blackwell, 1996-12-15) Bagla, J. S.; Padmanabhan, T.We investigate the transfer of power between different scales and the coupling of modes during the non-linear evolution of gravitational clustering in an expanding universe. We start with a power spectrum of density fluctuations that is exponentially damped outside a narrow range of scales, and use numerical simulations to study the evolution of this power spectrum. Non-linear effects generate power at other scales, with most power flowing from larger to smaller scales. The ‘cascade’ of power leads to equipartition of energy at smaller scales, implying a power spectrum with index n ~ - 1. We find that such a spectrum is produced in the range 1 < ð < 200 for density contrast ð. This result continues to hold even when small-scale power is added to the initial power spectrum. Semi-analytic models for gravitational clustering suggest a tendency for the effective index to move towards a critical index Nc ~-1. We find that such a spectrum is produced in the range 1< ð<200 for density contrast ð. This result continues to hold even when small-scale power is added to the initial power spectrum. Semi-analytic models for gravitational clustering suggest a tendency for the effective index to move towards a critical index Nc ~-1 in this range. For n< Nc , power in this range grows faster than linear rate, while if n> Nc , it grows at a slower rate- thereby changing the index closer to Nc. At scales larger than the narrow range of scales with initial power, a k⁴ tail is produced. We demonstrate that non-linear small scales do not affect the growth of perturbations at larger scales.Item Scaling Relations for Gravitational Clustering in two Dimensions(American Astronomical Society, 1997-10-10) Bagla, J. S.; Engineer, Sunu; Padmanabhan, T.It is known that radial collapse around density peaks can explain the key features of the evolution of a correlation function in gravitational clustering in three dimensions. The same model also makes spe-ciÐc predictions for two dimensions. In this paper we test these predictions in two dimensions with the help of N-body simulations. We Ðnd that there is no stable clustering in the extremely nonlinear regime, but a nonlinear scaling relation does exist and can be used to relate the linear and the nonlinear corre-lation function. In the intermediate regime, the simulations agree with the model.Item Scaling properties of nonlinear gravitational clustering(Royal Astronomical Society, 1994-09-16) Nityananda, R.; Padmanabhan, T.Hamilton et al. recently proposed the idea that the growth of density perturbations in an expanding universe is govemed by a general scaling law, and showed agreement with existing numerical simulations. We examine the possible origin of this scaling behaviour in more detail. The underlying equations of motion are cast in a suggestive form, and motivate a conjecture that the scaled pair velocity, h(x, α)≡ -[v/(αx)], depends on the expansion factor α and comoving coordinate x only through the density contrast ξ-(x, α) (the two-point correlation averaged over a sphere of radius x). This leads naturally to the proposed scaling law - the true non-linear density contrast is a universal function of the density contrast ξ-L(l,a), computed in the linear theory and evaluated at a scale lwhich is derived to be l =x(1 +ξ-)¹/³. Apart from basing the proposed scaling form on an explicit dynamical hypothesis, this gives a convenient solution for the scaling function in terms of the input pair velocity. Possibilities for further elaboration of this approach in interpreting simulations of non-linear gravitational clustering are briefly discussed.Item Patterns in non-linear gravitational clustering: a numerical investigation(American Astronomical Society, 1996-02-15) Padmanabhan, T.; Cen, Renyue; Ostriker, Jeremiah P.; Summers, F. J.The nonlinear clustering of dark matter particles in an expanding universe is usually studied by N-body simulations. One can gain some insight into this complex problem if simple relations between physical quantities in the linear and nonlinear regimes can be extracted from the results of N-body simulations. Hamilton and coworkers and Nityananda & Padmanabhan have made an attempt in this direction by relating the mean relative pair velocities to the mean correlation function in a useful manner. We investigate this relation and other closely related issues in detail for six different power spectra: power laws with spectral indexes n = -2 and -1; cold dark matter (CDM) and hot dark matter models with density parameter Ω = t1 a CDM model including a cosmological constant (Α) with ΩCDM = 0.4 and ΩΑ = 0.6; and an n = -1 model with Ω = 0.1. We find the following: (t) Power-law spectra lead to self-similar evolution in an Ω = 1 universe. (2) Stable clustering does not hold in an Ω = 1 universe to the extent that our simulations can ascertain. (3) Stable clustering is a better approximation in the case of an Ω < 1 universe in which structure formation freezes out at some low redshift. (4) The relation between the dimensionless pair velocity and the mean correlation function, ξ, is only approximately independent of the shape of the power spectrum. At the nonlinear end, the asymptotic value of the dimensionless pair velocity decreases with increasing small-scale power because the stable clustering assumption is not universally true. (5) The relation between the evolved ξ and the linear regime ξ is also not universal but shows a weak spectrum dependence. We present simple theoretical arguments for these conclusions.Item Non-linear density evolution from an improved spherical collapse model(Wiley-Blackwell, 1999-11-30) Engineer, Sunu; Kanekar, Nissim; Padmanabhan, T.We investigate the evolution of non-linear density perturbations by taking into account the effects of deviations from spherical symmetry of a system. Starting from the standard spherical top hat model in which these effects are ignored, we introduce a physically motivated closure condition which specifies the dependence of the additional terms on the density contrast, δ. The modified equation can be used to model the behaviour of an overdense region over a sufficiently large range of δ. The key new idea is a Taylor series expansion in (1/δ) to model the non-linear epoch. We show that the modified equations quite generically lead to the formation of stable structures in which the gravitational collapse is halted at around the virial radius. The analysis also allows us to connect up the behaviour of individual overdense regions with the non-linear scaling relations satisfied by the two point correlation function. Comment: 11 pages, 6 figures. Final version, contains added discussion and modified figures to match the accepted version.Item New statistical indicator to study nonlinear gravitational clustering and structure formation(American Astronomical Society, 1996-04-22) Bagla, J. S.; Padmanabhan, T.In an = 1 universe dominated by nonrelativistic matter, velocity field and gravitational force field are proportional to each other in the linear regime. Neither of these quantities evolve in time and these can be scaled suitably so that the constant of proportionality is unity and velocity and force field are equal. The Zeldovich approximation extends this feature beyond the linear regime, until formation of pancakes. Nonlinear clustering which takes place after the breakdown of Zeldovich approximation, breaks this relation and the mismatch between these two vectors increases as the evolution proceeds. We suggest that the difference of these two vectors could form the basis for a powerful, new, statistical indicator of nonlinear clustering. We define an indicator called velocity contrast, study its behaviour using N-Body simulations and show that it can be used effectively to delineate the regions where nonlinear clustering has taken place.We discuss several features of this statistical indicator and provide simple analytic models to understand its behaviour. Particles with velocity contrast higher than a threshold have a correlation function which is biased with respect to the original sample. This bias factor is scale dependent and tends to unity at large scales.Item Neutral hydrogen at high redshifts as a probe of structure formation- II. Line profile of a protocluster(Wiley-Blackwell, 1994-09-05) Kumar, A.; Padmanabhan, T.; Subramanian, KandaswamyThe formation of structures at z ≤ 10.0 can be probed using the 21-cm line emisssion from the neutral hydrogen. Two of us (KS and TP, Paper I) previously computed the expected abundance of protoclusters as a function of the flux density at various redshifts, in the cold dark matter (CDM) and the hot dark matter (HDM) models. As a complement to Paper I, here we work out in detail how the H1 line profile from a spherically symmetric protocluster evolves as it decouples from Hubble expansion structures form hierarchically. Neutral hydrogen, in the small-scale clumps that from the protocluster, is the source of H1line profile in this model are typically of order 0.5-0.7 mJy, while the widths (FWHM) are of order 0.3-1.8 MHz. The major uncertainty in our calculations is the fraction of mass of the protocluster in the form of neutral hydrogen. If the neutral hydrogen fraction f is of the order of the value we have adopted (f=0.025) in our calculations or greater, then a typical protocluster could indeed be detectable by future facilities, like the Giant Metrewave Radio Telescope (GMRT) which is currently being built in India. If the neutral hydrogen fraction is much less than the value we have adopted, then a more sensitive instrument is needed to detect the H1 line emission from a typical protocluster.Item Neutral hydrogen at high redshifts as a probe of structure formation: 1. Post-COBE analysis of CDM and HDM models(Royal Astronomical Society, 1993-05-11) Subramanian, Kandaswamy; Padmanabhan, T.The structures that form in the Universe at redshifts z ≲ 10 can be detected and studied using the redshifted 21-cm line emission from neutral hydrogen. We compute the expected comoving number density, N, of protocondensates that will emit a flux higher than S, at various redshifts, in the CDM and 11DM models. The models are normalized using COBE results. Our results are compared with the present and expected future sensitivities of various telescopes for the detection of protocondensates-. In the CDM models the predicted maximum fluxes at a redshift z ≃ 3.3 are about (1.5-3) mJy and N≃(10-8-10-7)Mpc-3 . These protocondensates cannot be detected with present sensitivities, but will become detectable in the near future with improved sensitivities. At lower redshifts, the detectability of these structures critically depends on their neutral hydrogen content. In the redshift range around z≃5, individual protocondensates will not be detectable. The excess variance due to fluctuations with small density contrasts will, however, be detectable with somewhat large (say, about 60-h) integration time. At still higher redshifts, it will be virtually impossible to see any signal, even with such a large integration time. Biased CDM models predict larger fluxes, but somewhat lower abundances. Finally, the 11DM models - when normalized using COBE results - do not lead to a detectable number of sources (`pancakes') at redshifts z≳2.