Revising the Halofit Model for the Nonlinear Matter Power Spectrum

TL;DR

This work revisits the Halofit model by calibrating it against high-resolution N-body simulations for 16 cosmologies around the WMAP best-fit, including variations in the dark-energy equation of state. The authors introduce a 35-parameter revised Halofit, with coefficients that depend on the effective spectral index and curvature and include small w-dependent corrections, achieving ~5% accuracy for $k \leq 1\,h\,{ m Mpc}^{-1}$ and ~10% for $1 \le k \le 10\,h\,{ m Mpc}^{-1}$ up to $z \le 3$, extending to $k \le 30\,h\,{ m Mpc}^{-1}$ and $z \le 10$. They validate the fit against additional simulations (Coyote emulator and ray-tracing results) and show improved weak-lensing and CMB lensing predictions relative to the original Halofit, with practical impact for upcoming surveys. The revised model is implemented in CAMB, enabling users to compute nonlinear $P(k)$ and subsequent lensing spectra with higher fidelity, while acknowledging caveats from baryons and massive neutrinos that require further modeling.

Abstract

Based on a suite of state-of-the-art high-resolution $N$-body simulations, we revisit the so-called halofit model (Smith et al. 2003) as an accurate fitting formula for the nonlinear matter power spectrum. While the halofit model has been frequently used as a standard cosmological tool to predict the nonlinear matter power spectrum in a universe dominated by cold dark matter, its precision has been limited by the low-resolution of $N$-body simulations used to determine the fitting parameters, suggesting the necessity of improved fitting formula at small scales for future cosmological studies. We run high-resolution $N$-body simulations for 16 cosmological models around the Wilkinson Microwave Anisotropy Probe (WMAP) best-fit cosmological parameters (1, 3, 5, and 7 year results), including dark energy models with a constant equation of state. The simulation results are used to re-calibrate the fitting parameters of the halofit model so as to reproduce small-scale power spectra of the $N$-body simulations, while keeping the precision at large scales. The revised fitting formula provides an accurate prediction of the nonlinear matter power spectrum in a wide range of wavenumber ($k \leq 30h$\,Mpc$^{-1}$) at redshifts $0 \leq z \leq 10$, with 5% precision for $k\leq1 h$ Mpc$^{-1}$ at $0 \leq z \leq 10$ and 10% for $1 \leq k\leq 10 h$ Mpc$^{-1} $ at $0 \leq z \leq 3$. We discuss the impact of the improved halofit model on weak lensing power spectra and correlation functions, and show that the improved model better reproduces ray-tracing simulation results.

Figures (7)

• Figure 1: Power spectra $P(k)$ from the different simulation box sizes for the WMAP5 model at redshifts $z=0.35$, $1$, $3$, $5$, and $10$. The vertical axis shows power spectra in our simulations normalized by the theoretical nonlinear matter power spectrum model from the original halofit model in S03. Green, blue, and orange symbols are simulation results from the box sizes of $L=2000$, $800$, and $320h^{-1}$Mpc, respectively. Red symbols are the same as the oranges, but including the folding method with $n=2$ (see Section \ref{['sec:ps']}). Gray symbols show simulation results from vn11a. Arrows denote wavenumbers where $P(k)$ with the different box sizes are connected. Dashed curves denote the shot noise. Vertical solid lines indicate the maximum wavenumber $k_{\rm max}$ for deriving our fitting formula.
• Figure 2: Power spectra $P(k)$ from our simulations of $L=1000h^{-1}$Mpc divided by those of the Cosmic emulator for the ten Coyote models at $z=0$ ( top) and $1$ ( bottom). Red symbols with error bars are for the fiducial model m00. Horizontal dotted lines indicate the fractional error of $3 \%$.
• Figure 3: Power spectra $P(k)$ for the WMAP cosmological models at $z=0$, $0.35$, $1$, and $3$. The WMAP7a,b are similar to the WMAP7 but changing the equation of state of dark energy ($w=-0.8,-1.2$). In the vertical axis, the power spectrum $P(k)$ is multiplied by the factor $k^{1.5}$ in order to show the differences between the simulation results and the theoretical models clearly. Black filled circles with the error bars plot our simulation results, and gray symbols are the results from vn11a. Red solid curves show our revised halofit model (see Appendix), whereas black solid curves show the original halofit model in S03. Black dashed curves plot the linear power spectra.
• Figure 4: Similar to Figure \ref{['pk']}, but focusing on the BAO scale of $0.05h$ Mpc$^{-1}<k<0.3h$ Mpc$^{-1}$. The vertical axis show $P(k)$ normalized by the smooth nonlinear power spectrum $P_{\rm nw}^{\rm S03}(k)$ which is calculated by using a no-wiggle fitting formula of eh98 and the original halofit model in S03 for the nonlinear correction. Again, red solid curves show our revised halofit model, whereas black solid curves show the original halofit model in S03. Green curves show the prediction by the closure theory which is a higher-order perturbation theory by th08. Dashed curves show the linear power spectra.
• Figure 5: Our simulation results of the power spectrum $P_{\rm sim}(k)$ divided by our improved fitting formula $P_{\rm model}(k)$ (see Appendix) for all the cosmological models at $z=0$, $1$, $3$, $5$, and $10$ (from top to bottom). Filled circles show results for the six WMAP models, whereas plus and cross symbols are for the ten Coyote models. Blue and red big circles are WMAP5 and WMAP7. Black, green, gray and yellow circles are results for the WMAP1, 3, 7a and 7b models, respectively. Horizontal dotted lines indicates the relative error of $5 \%$.