Velocity Divergence Power Spectrum Research Articles

Evolution mapping II: describing statistics of the non-linear cosmic velocity field.

Abstract We extend the evolution mapping approach, introduced in the first paper of this series to describe non-linear matter density fluctuations, to statistics of the cosmic velocity field. This framework classifies cosmological parameters into shape parameters, which determine the shape of the linear matter power spectrum, PL(k, z), and evolution parameters, which control its amplitude at any redshift. Evolution mapping leverages the fact that density fluctuations in cosmologies with identical shape parameters but different evolution parameters exhibit similar non-linear evolutions when expressed as a function of clustering amplitude. We analyse a suite of N-body simulations sharing identical shape parameters but spanning a wide range of evolution parameters. Using a method for estimating the volume-weighted velocity field based on the Voronoi tesselation of simulation particles, we study the non-linear evolution of the velocity divergence power spectrum, Pθθ(k), and its cross-power spectrum with the density field, Pδθ(k). We demonstrate that the evolution mapping relation applies accurately to Pθθ(k) and Pδθ(k). While this breaks down in the strongly non-linear regime, deviations can be modelled in terms of differences in the suppression factor, g(a) = D(a)/a, similar to those for the density field. Such modelling describes the differences in Pθθ(k) between models with the same linear clustering amplitude to better than 1percnt accuracy at all scales and redshifts considered. Evolution mapping simplifies the description of the cosmological dependence of non-linear density and velocity statistics, streamlining the sampling of large cosmological parameter spaces for cosmological analysis.

The effect of baryons on redshift space distortions and cosmic density and velocity fields in the EAGLE simulation

Abstract We use the Evolution and Assembly of GaLaxies and their Environments (EAGLE) galaxy formation simulation to study the effects of baryons on the power spectrum of the total matter and dark matter distributions and on the velocity fields of dark matter and galaxies. On scales k ≳ 4 h Mpc−1 the effect of baryons on the amplitude of the total matter power spectrum is greater than 1 per cent. The back-reaction of baryons affects the density field of the dark matter at the level of ∼3 per cent on scales of 1 ≤ k/( h Mpc−1) ≤ 5. The dark matter velocity divergence power spectrum at k ≲ 0.5 h Mpc−1 is changed by less than 1 per cent. The 2D redshift space power spectrum is affected at the level of ∼6 per cent at $|\boldsymbol {k}|\gtrsim 1\,h\,{\rm Mpc}^{-1}$ (for μ > 0.5), but for $|\boldsymbol {k}|\le 0.4\,h\,{\rm Mpc}^{-1}$ it differs by less than 1 per cent. We report vanishingly small baryonic velocity bias for haloes: the peculiar velocities of haloes with M200 > 3 × 1011 M⊙ (hosting galaxies with M* > 109 M⊙) are affected at the level of at most 1 km s−1, which is negligible for 1 per cent-precision cosmology. We caution that since EAGLE overestimates cluster gas fractions it may also underestimate the impact of baryons, particularly for the total matter power spectrum. Nevertheless, our findings suggest that for theoretical modelling of redshift space distortions and galaxy velocity-based statistics, baryons and their back-reaction can be safely ignored at the current level of observational accuracy. However, we confirm that the modelling of the total matter power spectrum in weak lensing studies needs to include realistic galaxy formation physics in order to achieve the accuracy required in the precision cosmology era.

Modified gravityN-body code comparison project

Self-consistent ${\it N}$-body simulations of modified gravity models are a key ingredient to obtain rigorous constraints on deviations from General Relativity using large-scale structure observations. This paper provides the first detailed comparison of the results of different ${\it N}$-body codes for the $f(R)$, DGP, and Symmetron models, starting from the same initial conditions. We find that the fractional deviation of the matter power spectrum from $\Lambda$CDM agrees to better than $1\%$ up to $k \sim 5-10~h/{\rm Mpc}$ between the different codes. These codes are thus able to meet the stringent accuracy requirements of upcoming observational surveys. All codes are also in good agreement in their results for the velocity divergence power spectrum, halo abundances and halo profiles. We also test the quasi-static limit, which is employed in most modified gravity ${\it N}$-body codes, for the Symmetron model for which the most significant non-static effects among the models considered are expected. We conclude that this limit is a very good approximation for all of the observables considered here.

Velocity and mass bias in the distribution of dark matter haloes

The non-linear, scale-dependent bias in the mass distribution of galaxies and the underlying dark matter is a key systematic affecting the extraction of cosmological parameters from galaxy clustering. Using 95 million halos from the Millennium-XXL N-body simulation, we find that the mass bias is scale independent only for $k<0.1 h{\rm Mpc}^{-1}$ today ($z=0$) and for $k<0.2 h{\rm Mpc}^{-1}$ at $z=0.7$. We test analytic halo bias models against our simulation measurements and find that the model of Tinker et al. 2005 is accurate to better then 5% at $z=0$. However, the simulation results are better fit by an ellipsoidal collapse model at $z=0.7$. We highlight, for the first time, another potentially serious systematic due to a sampling bias in the halo velocity divergence power spectra which will affect the comparison between observations and any redshift space distortion model which assumes dark matter velocity statistics with no velocity bias. By measuring the velocity divergence power spectra for different sized halo samples, we find that there is a significant bias which increases with decreasing number density. This bias is approximately 20% at $k=0.1h$Mpc$^{-1}$ for a halo sample of number density $\bar{n} = 10^{-3} (h/$Mpc$)^3$ at both $z=0$ and $z=0.7$ for the velocity divergence auto power spectrum. Given the importance of redshift space distortions as a probe of dark energy and the on-going major effort to advance models for the clustering signal in redshift space, our results show this velocity bias introduces another systematic, alongside scale-dependent halo mass bias, which cannot be neglected.

Nonlinear structure formation in nonlocal gravity

We study the nonlinear growth of structure in nonlocal gravity models with the aid of N-body simulation and the spherical collapse and halo models. We focus on a model in which the inverse-squared of the d'Alembertian operator acts on the Ricci scalar in the action. For fixed cosmological parameters, this model differs from ΛCDM by having a lower late-time expansion rate and an enhanced and time-dependent gravitational strength ∼ 6% larger today). Compared to ΛCDM today, in the nonlocal model, massive haloes are slightly more abundant (by ∼ 10% at M ∼ 1014 M⊙/h) and concentrated ≈ 8% enhancement over a range of mass scales), but their linear bias remains almost unchanged. We find that the Sheth-Tormen formalism describes the mass function and halo bias very well, with little need for recalibration of free parameters. The fitting of the halo concentrations is however essential to ensure the good performance of the halo model on small scales. For k ≳ 1 h/Mpc, the amplitude of the nonlinear matter and velocity divergence power spectra exhibits a modest enhancement of ∼ 12% to 15%, compared to ΛCDM today. This suggests that this model might only be distinguishable from ΛCDM by future observational missions. We point out that the absence of a screening mechanism may lead to tensions with Solar System tests due to local time variations of the gravitational strength, although this is subject to assumptions about the local time evolution of background averaged quantities.

An improved model for the non-linear velocity power spectrum

Abstract The velocity divergence power spectrum is a key ingredient in modelling redshift-space distortion effects on quasi-linear and non-linear scales. We present an improved model for the z=0 velocity divergence auto and cross power spectrum which was originally suggested by Jennings et al. Using numerical simulations we measure the velocity fields using a Delaunay tessellation and obtain an accurate prediction of the velocity divergence power spectrum on scales k &amp;lt; 1 h Mpc−1. We use this to update the model which is now accurate to 2 per cent for both Pθθ and Pθδ at z= 0 on scales k &amp;lt; 0.65 h Mpc−1 and k &amp;lt; 0.35 h Mpc−1, respectively. We find that the formula for the redshift dependence of the velocity divergence power spectra proposed by Jennings et al. recovers the measured z &amp;gt; 0 P(k) to markedly greater accuracy with the new model. The non-linear Pθθ and Pθδ at z=1 are recovered accurately to better than 2 per cent on scales k &amp;lt; 0.2 h Mpc−1. Recently, it was shown that the velocity field shows larger differences between modified gravity cosmologies and Λ cold dark matter (ΛCDM) compared to the matter field. An accurate model for the velocity divergence power spectrum, such as the one presented here, is a valuable tool for analysing redshift-space distortion effects in future galaxy surveys and for constraining deviations from general relativity.

The non-linear matter and velocity power spectra in f(R) gravity

We study the matter and velocity divergence power spectra in a f(R) gravity theory and their time evolution measured from several large-volume N-body simulations with varying box sizes and resolution. We find that accurate prediction of the matter power spectrum in f(R) gravity places stronger requirements on the simulation than is the case with LCDM, because of the nonlinear nature of the fifth force. Linear perturbation theory is shown to be a poor approximation for the f(R) models, except when the chameleon effect is very weak. We show that the relative differences from the fiducial LCDM model are much more pronounced in the nonlinear tail of the velocity divergence power spectrum than in the matter power spectrum, which suggests that future surveys which target the collection of peculiar velocity data will open new opportunities to constrain modified gravity theories. A close investigation of the time evolution of the power spectra shows that there is a pattern in the evolution history, which can be explained by the properties of the chameleon-type fifth force in f(R) gravity. Varying the model parameter |f_R0|, which quantifies the strength of the departure from standard gravity, mainly varies the epoch marking the onset of the fifth force, as a result of which the different f(R) models are in different stages of the same evolutionary path at any given time

Modelling redshift space distortions in hierarchical cosmologies

The anisotropy of clustering in redshift space provides a direct measure of the growth rate of large scale structure in the Universe. Future galaxy redshift surveys will make high precision measurements of these distortions, and will potentially allow us to distinguish between different scenarios for the accelerating expansion of the Universe. Accurate predictions are needed in order to distinguish between competing cosmological models. We study the distortions in the redshift space power spectrum in $\Lambda$CDM and quintessence dark energy models, using large volume N-body simulations, and test predictions for the form of the redshift space distortions. We find that the linear perturbation theory prediction by Kaiser (1987) is a poor fit to the measured distortions, even on surprisingly large scales $k \ge 0.05 h$Mpc$^{-1}$. An improved model for the redshift space power spectrum, including the non-linear velocity divergence power spectrum, is presented and agrees with the power spectra measured from the simulations up to $k \sim 0.2 h$Mpc$^{-1}$. We have found a density-velocity relation which is cosmology independent and which relates the non-linear velocity divergence spectrum to the non-linear matter power spectrum. We provide a formula which generates the non-linear velocity divergence $P(k)$ at any redshift, using only the non-linear matter power spectrum and the linear growth factor at the desired redshift. This formula is accurate to better than 5% on scales $k<0.2 h $Mpc$^{-1}$ for all the cosmological models discussed in this paper. Our results will extend the statistical power of future galaxy surveys.

How to evade the sample variance limit on measurements of redshift-space distortions

We show how to use multiple tracers of large-scale density with different biases to measure the redshift-space distortion parameter β ≡ b−1f ≡ b−1d ln D/d ln a (where D is the growth factor and a the expansion factor), to, as the signal-to-noise (S/N) of a survey increases, much better precision than one could achieve with a single tracer (to arbitrary precision in the low noise limit). In combination with the power spectrum of the tracers this would allow a more precise measurement of the bias-free velocity divergence power spectrum, f2Pm, with the ultimate, zero noise limit, being that f2Pm can be measured as well as would be possible if velocity divergence was observed directly, with maximum rms improvement factor ∼ [5.2(β2+2β+2)/β2]1/2 (e.g., ≃ 10 times better than a single tracer with β = 0.4). This would allow a determination of fD as a function of redshift with an error as low as ∼ 0.1% (again, in the idealized case of the zero noise limit). Theratio b2/b1 can be determined with an even greater precision than β, potentially producing, when measured as a function of scale, an exquisitely sensitive probe of the onset of non-linear bias. We also extend in more detail previous work on the use of the same technique to measure non-Gaussianity. Currently planned redshift surveys are typically designed with S/N ∼ 1 on scales of interest, which severely limits the usefulness of our method. Our results suggest that there are potentially large gains to be achieved from technological or theoretical developments that allow higher S/N, or, in the long term, surveys that simply observe a higher number density of galaxies.

Generation of vorticity and velocity dispersion by orbit crossing

We study the generation of vorticity and velocity dispersion by orbit crossing using cosmological numerical simulations, and calculate the backreaction of these effects on the evolution of large-scale density and velocity divergence power spectra. We use Delaunay tessellations to define the velocity field, showing that the power spectra of velocity divergence and vorticity measured in this way are unbiased and have better noise properties than for standard interpolation methods that deal with mass-weighted velocities. We show that high resolution simulations are required to recover the correct large-scale vorticity power spectrum, while poor resolution can spuriously amplify its amplitude by more than 1 order of magnitude. We measure the scalar and vector modes of the stress tensor induced by orbit crossing using an adaptive technique, showing that its vector modes lead, when input into the vorticity evolution equation, to the same vorticity power spectrum obtained from the Delaunay method. We incorporate orbit-crossing corrections to the evolution of large-scale density and velocity fields in perturbation theory by using the measured stress tensor modes. We find that at large scales ($k\ensuremath{\simeq}0.1h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$) vector modes have very little effect in the density power spectrum, while scalar modes (velocity dispersion) can induce percent-level corrections at $z=0$, particularly in the velocity divergence power spectrum. In addition, we show that the velocity power spectrum is smaller than predicted by linear theory until well into the nonlinear regime, with little contribution from virial velocities.