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Register a new userFitting functions on the cheap: the relative nonlinear matter power spectrum
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We propose an alternative approach to the construction of fitting functions to the nonlinear matter power spectrum extracted from $N$-body simulations based on the relative matter power spectrum $delta(k,a)$, defined as the fractional deviation in the absolute matter power spectrum produced by a target cosmology away from a reference $Lambda$CDM prediction. From the computational perspective, $delta(k,a)$ is fairly insensitive to the specifics of the simulation settings, and numerical convergence at the 1%-level can be readily achieved without the need for huge computing capacity. Furthermore, $delta(k,a)$ exhibits several interesting properties that enable a piece-wise construction of the full fitting function, whereby component fitting functions are sought for single-parameter variations and then multiplied together to form the final product. Then, to obtain 1%-accurate absolute power spectrum predictions for any target cosmology only requires that the community as a whole invests in producing one single ultra-precise reference $Lambda$CDM absolute power spectrum, to be combined with the fitting function to produce the desired result. To illustrate the power of this approach, we have constructed the fitting function RelFit using only five relatively inexpensive $w$CDM simulations (box length $L=256 h^{-1}$Mpc, $N=1024^3$ particles, initialised at $z_i=49$). In a 6-parameter space spanning ${omega_m,A_s,n_s,w,omega_b,h}$, the output relative power spectra of RelFit are consistent with the predictions of the CosmicEmu emulator to 1% or better for a wide range of cosmologies up to $ksimeq 10$/Mpc. Thus, our approach could provide an inexpensive and democratically accessible route to fulfilling the 1%-level accuracy demands of the upcoming generation of large-scale structure probes, especially in the exploration of non-standard or exotic cosmologies on nonlinear scales.
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We discuss an analytical approximation for the matter power spectrum covariance matrix and its inverse on translinear scales, $k sim 0.1h - 0.8h/textrm{Mpc}$ at $z = 0$. We proceed to give an analytical expression for the Fisher information matrix of the nonlinear density field spectrum, and derive implications for its cosmological information content. We find that the spectrum information is characterized by a pair of upper bounds, plateaux, caused by the trispectrum, and a knee in the presence of white noise. The effective number of Fourier modes, normally growing as a power law, is bounded from above by these plateaux, explaining naturally earlier findings from $N$-body simulations. These plateaux limit best possible measurements of the nonlinear power at the percent level in a $h^{-3}textrm{Gpc}^3$ volume; the extraction of model parameters from the spectrum is limited explicitly by their degeneracy to the nonlinear amplitude. The value of the first, super-survey (SS) plateau depends on the characteristic survey volume and the large scale power; the second, intra-survey (IS) plateau is set by the small scale power. While both have simple interpretations within the hierarchical textit{Ansatz}, the SS plateau can be predicted and generalized to still smaller scales within Takada and Hus spectrum response formalism. Finally, the noise knee is naturally set by the density of tracers.
The power spectrum of density fluctuations is a foundational source of cosmological information. Precision cosmological probes targeted primarily at investigations of dark energy require accurate theoretical determinations of the power spectrum in the nonlinear regime. To exploit the observational power of future cosmological surveys, accuracy demands on the theory are at the one percent level or better. Numerical simulations are currently the only way to produce sufficiently error-controlled predictions for the power spectrum. The very high computational cost of (precision) N-body simulations is a major obstacle to obtaining predictions in the nonlinear regime, while scanning over cosmological parameters. Near-future observations, however, are likely to provide a meaningful constraint only on constant dark energy equation of state wCDM cosmologies. In this paper we demonstrate that a limited set of only 37 cosmological models -- the Coyote Universe suite -- can be used to predict the nonlinear matter power spectrum at the required accuracy over a prior parameter range set by cosmic microwave background observations. This paper is the second in a series of three, with the final aim to provide a high-accuracy prediction scheme for the nonlinear matter power spectrum for wCDM cosmologies.
Context. Weak gravitational lensing is a powerful probe of large-scale structure and cosmology. Most commonly, second-order correlations of observed galaxy ellipticities are expressed as a projection of the matter power spectrum, corresponding to the lowest-order approximation between the projected and 3d power spectrum. Aims. The dominant lensing-only contribution beyond the zero-order approximation is the reduced shear, which takes into account not only lensing-induced distortions but also isotropic magnification of galaxy images. This involves an integral over the matter bispectrum. We provide a fast and general way to calculate this correction term. Methods. Using a model for the matter bispectrum, we fit elementary functions to the reduced-shear contribution and its derivatives with respect to cosmological parameters. The dependence on cosmology is encompassed in a Taylor-expansion around a fiducial model. Results. Within a region in parameter space comprising the WMAP7 68% error ellipsoid, the total reduced-shear power spectrum (shear plus fitted reduced-shear correction) is accurate to 1% (2%) for l<10^4 (l<2x10^5). This corresponds to a factor of four reduction of the bias compared to the case where no correction is used. This precision is necessary to match the accuracy of current non-linear power spectrum predictions from numerical simulations.
We present a general method to compute the nonlinear matter power spectrum for dark energy and modified gravity scenarios with percent-level accuracy. By adopting the halo model and nonlinear perturbation theory, we predict the reaction of a $Lambda$CDM matter power spectrum to the physics of an extended cosmological parameter space. By comparing our predictions to $N$-body simulations we demonstrate that with no-free parameters we can recover the nonlinear matter power spectrum for a wide range of different $w_0$-$w_a$ dark energy models to better than 1% accuracy out to $k approx 1 , h , {rm Mpc}^{-1}$. We obtain a similar performance for both DGP and $f(R)$ gravity, with the nonlinear matter power spectrum predicted to better than 3% accuracy over the same range of scales. When including direct measurements of the halo mass function from the simulations, this accuracy improves to 1%. With a single suite of standard $Lambda$CDM $N$-body simulations, our methodology provides a direct route to constrain a wide range of non-standard extensions to the concordance cosmology in the high signal-to-noise nonlinear regime.
We study the effects of dark energy (DE) anisotropic stress on features of the matter power spectrum (PS). We employ the Parametrized Post-Friedmannian (PPF) formalism to emulate an effective DE, and model its anisotropic stress properties through a two-parameter equation that governs its overall amplitude ($g_0$) and transition scale ($c_g$). For the background cosmology, we consider different equations of state to model DE including a constant $w_0$ parameter, and models that provide thawing (CPL) and freezing (nCPL) behaviors. We first constrain these parameters by using the Pantheon, BAO, $H_0$ and CMB Planck data. Then, we analyze the role played by these parameters in the linear PS. In order for the anisotropic stress not to provoke deviations larger than $10%$ and $5%$ with respect to the $Lambda$CDM PS at $k sim 0.01 ,h/text{Mpc}$, the parameters have to be in the range $-0.30< g_0 < 0.32$, $0 leq c_g^2 < 0.01$ and $-0.15 < g_0 < 0.16$, $0 leq c_g^2 < 0.01$, respectively. Additionally, we compute the leading nonlinear corrections to the PS using standard perturbation theory in real and redshift space, showing that the differences with respect to the $Lambda$CDM are enhanced, especially for the quadrupole and hexadecapole RSD multipoles.
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