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Register a new userCosmology with massive neutrinos II: on the universality of the halo mass function and bias
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We use a large suite of N-body simulations to study departures from universality in halo abundances and clustering in cosmologies with non-vanishing neutrino masses. To this end, we study how the halo mass function and halo bias factors depend on the scaling variable $sigma^2(M,z)$, the variance of the initial matter fluctuation field, rather than on halo mass $M$ and redshift $z$ themselves. We show that using the variance of the cold dark matter rather than the total mass field, i.e., $sigma^2_{cdm}(M,z)$ rather than $sigma^2_{m}(M,z)$, yields more universal results. Analysis of halo bias yields similar conclusions: When large-scale halo bias is defined with respect to the cold dark matter power spectrum, the result is both more universal, and less scale- or $k$-dependent. These results are used extensively in Papers I and III of this series.
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We use a suite of N-body simulations that incorporate massive neutrinos as an extra-set of particles to investigate their effect on the halo mass function. We show that for cosmologies with massive neutrinos the mass function of dark matter haloes selected using the spherical overdensity (SO) criterion is well reproduced by the fitting formula of Tinker et al. (2008) once the cold dark matter power spectrum is considered instead of the total matter power, as it is usually done. The differences between the two implementations, i.e. using $P_{rm cdm}(k)$ instead of $P_{rm m}(k)$, are more pronounced for large values of the neutrino masses and in the high end of the halo mass function: in particular, the number of massive haloes is higher when $P_{rm cdm}(k)$ is considered rather than $P_{rm m}(k)$. As a quantitative application of our findings we consider a Planck-like SZ-clusters survey and show that the differences in predicted number counts can be as large as $30%$ for $sum m_ u = 0.4$ eV. Finally, we use the Planck-SZ clusters sample, with an approximate likelihood calculation, to derive Planck-like constraints on cosmological parameters. We find that, in a massive neutrino cosmology, our correction to the halo mass function produces a shift in the $sigma_8(Omega_{rm m}/0.27)^gamma$ relation which can be quantified as $Delta gamma sim 0.05$ and $Delta gamma sim 0.14$ assuming one ($N_ u=1$) or three ($N_ u=3$) degenerate massive neutrino, respectively. The shift results in a lower mean value of $sigma_8$ with $Delta sigma_8 = 0.01$ for $N_ u=1$ and $Delta sigma_8 = 0.02$ for $N_ u=3$, respectively. Such difference, in a cosmology with massive neutrinos, would increase the tension between cluster abundance and Planck CMB measurements.
Luminous matter produces very energetic events, such as active galactic nuclei and supernova explosions, that significantly affect the internal regions of galaxy clusters. Although the current uncertainty in the effect of baryonic physics on cluster statistics is subdominant as compared to other systematics, the picture is likely to change soon as the amount of high-quality data is growing fast, urging the community to keep theoretical systematic uncertainties below the ever-growing statistical precision. In this paper, we study the effect of baryons on galaxy clusters, and their impact on the cosmological applications of clusters, using the Magneticum suite of cosmological hydrodynamical simulations. We show that the impact of baryons on the halo mass function can be recast in terms on a variation of the mass of the halos simulated with pure N-body, when baryonic effects are included. The halo mass function and halo bias are only indirectly affected. Finally, we demonstrate that neglecting baryonic effects on halos mass function and bias would significantly alter the inference of cosmological parameters from high-sensitivity next-generations surveys of galaxy clusters.
We study the impact of theoretical uncertainty in the dark matter halo mass function and halo bias on dark energy constraints from imminent galaxy cluster surveys. We find that for an optical cluster survey like the Dark Energy Survey, the accuracy required on the predicted halo mass function to make it an insignificant source of error on dark energy parameters is ~ 1%. The analogous requirement on the predicted halo bias is less stringent (~ 5%), particularly if the observable-mass distribution can be well constrained by other means. These requirements depend upon survey area but are relatively insensitive to survey depth. The most stringent requirements are likely to come from a survey over a significant fraction of the sky that aims to observe clusters down to relatively low mass, Mth ~ 10^13.7 Msun/h; for such a survey, the mass function and halo bias must be predicted to accuracies of ~ 0.5% and ~ 1%, respectively. These accuracies represent a limit on the practical need to calibrate ever more accurate halo mass and bias functions. We find that improving predictions for the mass function in the low-redshift and low-mass regimes is the most effective way to improve dark energy constraints.
Understanding the biasing between the clustering properties of halos and the underlying dark matter distribution is important for extracting cosmological information from ongoing and upcoming galaxy surveys. While on sufficiently larges scales the halo overdensity is a local function of the mass density fluctuations, on smaller scales the gravitational evolution generates non-local terms in the halo density field. We characterize the magnitude of these contributions at third-order in perturbation theory by identifying the coefficients of the non-local invariant operators, and extend our calculation to include non-local (Lagrangian) terms induced by a peak constraint. We apply our results to describe the scale-dependence of halo bias in cosmologies with massive neutrinos. The inclusion of gravity-induced non-local terms and, especially, a Lagrangian $k^2$-contribution is essential to reproduce the numerical data accurately. We use the peak-background split to derive the numerical values of the various bias coefficients from the excursion set peak mass function. For neutrino masses in the range $0leq sum_i m_{ u_i} leq 0.6$ eV, we are able to fit the data with a precision of a few percents up to $k=0.3, h {rm ,Mpc^{-1}}$ without any free parameter.
We generalize the stochastic theory of hierarchical clustering presented in paper I by Lapi & Danese (2020) to derive the (conditional) halo progenitor mass function and the related large-scale bias. Specifically, we present a stochastic differential equation that describes fluctuations in the mass growth of progenitor halos of given descendant mass and redshift, as driven by a multiplicative Gaussian white noise involving the power spectrum and the spherical collapse threshold of density perturbations. We demonstrate that, as cosmic time passes, the noise yields an average drift of the progenitors toward larger masses, that quantitatively renders the expectation from the standard extended Press & Schechter (EPS) theory. We solve the Fokker-Planck equation associated to the stochastic dynamics, and obtain as an exact, stationary solution the EPS progenitor mass function. Then we introduce a modification of the stochastic equation in terms of a mass-dependent collapse threshold modulating the noise, and solve analytically the associated Fokker-Planck equation for the progenitor mass function. The latter is found to be in excellent agreement with the outcomes of $N-$body simulations; even more remarkably, this is achieved with the same shape of the collapse threshold used in paper I to reproduce the halo mass function. Finally, we exploit the above results to compute the large-scale halo bias, and find it in pleasing agreement with the $N-$body outcomes. All in all, the present paper illustrates that the stochastic theory of hierarchical clustering introduced in paper I can describe effectively not only halos abundance, but also their progenitor distribution and their correlation with the large-scale environment across cosmic times.
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