Modeling non-linear effects in the redshift space two-point correlation function and its implications for the pairwise velocity dispersion

The anisotropies in the galaxy two-point correlation function measured from redshift surveys exhibits deviations from the predictions of the linear theory of redshift space distortion on scales as large 20 Mpc/h where we expect linear theory to hold in real space. Any attempt at analyzing the anisotropies in the redshift correlation function and determining the linear distortion parameter \beta requires these deviations to be correctly modeled and taken into account. These deviations are usually attributed to galaxy random motions and these are incorporated in the analysis through a phenomenological model where the linear redshift correlation is convolved with the random pairwise velocity distribution function along the line of sight. We show that a substantial part of the deviations arise from non-linear effects in the mapping from real to redshift space caused by the coherent flows. Models which incorporate this effect provide a better fit to N-body results as compared to the phenomenological model which has only the effect of random motions. We find that the pairwise velocity dispersion predicted by all the models that we have considered are in excess of the values determined directly from the N-body simulations. This indicates a shortcoming in our understanding of the statistical properties of peculiar velocities and their relation to redshift distortion.Comment: Minor Revisions, Accepted to MNRA

Chasing Unbiased Spectra of the Universe

The cosmological power spectrum of the coherent matter flow is measured exploiting an improved prescription for the apparent anisotropic clustering pattern in redshift space. New statistical analysis is presented to provide an optimal observational platform to link the improved redshift distortion theoretical model to future real datasets. The statistical power as well as robustness of our method are tested against 60 realizations of 8 Gpc/h^3 dark matter simulation maps mocking the precision level of upcoming wide--deep surveys. We showed that we can accurately extract the velocity power spectrum up to quasi linear scales of k~0.1 h/Mpc at z = 0.35 and up to k~0.15 h/Mpc at higher redshifts within a couple of percentage precision level. Our understanding of redshift space distortion is proved to be appropriate for precision cosmology, and our statistical method will guide us to righteous path to meet the real world.Comment: 9 pages, 7 figure

Baryon Acoustic Oscillations in 2D: Modeling Redshift-space Power Spectrum from Perturbation Theory

We present an improved prescription for matter power spectrum in redshift space taking a proper account of both the non-linear gravitational clustering and redshift distortion, which are of particular importance for accurately modeling baryon acoustic oscillations (BAOs). Contrary to the models of redshift distortion phenomenologically introduced but frequently used in the literature, the new model includes the corrections arising from the non-linear coupling between the density and velocity fields associated with two competitive effects of redshift distortion, i.e., Kaiser and Finger-of-God effects. Based on the improved treatment of perturbation theory for gravitational clustering, we compare our model predictions with monopole and quadrupole power spectra of N-body simulations, and an excellent agreement is achieved over the scales of BAOs. Potential impacts on constraining dark energy and modified gravity from the redshift-space power spectrum are also investigated based on the Fisher-matrix formalism. We find that the existing phenomenological models of redshift distortion produce a systematic error on measurements of the angular diameter distance and Hubble parameter by 1~2%, and the growth rate parameter by ~5%, which would become non-negligible for future galaxy surveys. Correctly modeling redshift distortion is thus essential, and the new prescription of redshift-space power spectrum including the non-linear corrections can be used as an accurate theoretical template for anisotropic BAOs.Comment: 18 pages, 10 figure

Wide Angle Effects in Future Galaxy Surveys

Current and future galaxy surveys cover a large fraction of the entire sky with a significant redshift range, and the recent theoretical development shows that general relativistic effects are present in galaxy clustering on very large scales. This trend has renewed interest in the wide angle effect in galaxy clustering measurements, in which the distant-observer approximation is often adopted. Using the full wide-angle formula for computing the redshift-space correlation function, we show that compared to the sample variance, the deviation in the redshift-space correlation function from the simple Kaiser formula with the distant-observer approximation is negligible in galaxy surveys such as the SDSS, Euclid and the BigBOSS, if the theoretical prediction from the Kaiser formula is properly averaged over the survey volume. We also find corrections to the wide-angle formula and clarify the confusion in literature between the wide angle effect and the velocity contribution in galaxy clustering. However, when the FKP method is applied, substantial deviations can be present in the power spectrum analysis in future surveys, due to the non-uniform distribution of galaxy pairs.Comment: 17 pages, 11 figures, accepted for publication in MNRA

Power Spectrum of Velocity Fluctuations in the Universe

We investigate the power spectrum of velocity fluctuations in the universe, $V^2(k)$, starting from four different measures of velocity: (1) the power spectrum of velocity fluctuations from peculiar velocities of galaxies; (2) the rms peculiar velocity of galaxy clusters; (3) the power spectrum of velocity fluctuations from the power spectrum of density fluctuations in the galaxy distribution; (4) and the bulk velocity from peculiar velocities of galaxies. We show that measures (1) and (2) are not consistent with each other and either the power spectrum from peculiar velocities of galaxies is overestimated or the rms cluster peculiar velocity is underestimated. The amplitude of velocity fluctuations derived from the galaxy distribution (measure 3) depends on the parameter $\beta$. We estimate the parameter $\beta$ on the basis of measures (2) and (4). The power spectrum of velocity fluctuations from the galaxy distribution in the Stromlo-APM redshift survey is consistent with the observed rms cluster velocity and with the observed large-scale bulk flow when the parameter $\beta$ is in the range 0.4-0.5. In this case the value of the function $V(k)$ at wavelength $\lambda=120h^{-1}$Mpc is $\sim 350$ km s$^{-1}$ and the rms amplitude of the bulk flow at the radius $r=60h^{-1}$ Mpc is $\sim 340$ km s$^{-1}$. The velocity dispersion of galaxy systems originates mostly from the large-scale velocity fluctuations with wavelengths $\lambda >100h^{-1}$ Mpc.Comment: Astrophysical Journal, Vol. 493, in press: 23 pages, uses AAS Latex, and 14 separate postscript figure

Integrality gaps of semidefinite programs for Vertex Cover and relations to ℓ1\ell_1 embeddability of Negative Type metrics

We study various SDP formulations for {\sc Vertex Cover} by adding different constraints to the standard formulation. We show that {\sc Vertex Cover} cannot be approximated better than $2-o(1)$ even when we add the so called pentagonal inequality constraints to the standard SDP formulation, en route answering an open question of Karakostas~\cite{Karakostas}. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution is an $\ell_1$ metric, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar \cite{Char02} to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than 8/7-\eps. To this end we prove a new isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof of (current) Theorem