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Register a new userA Stochastic Theory of the Hierarchical Clustering I. Halo Mass Function
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We present a new theory for the hierarchical clustering of dark matter (DM) halos based on stochastic differential equations, that constitutes a change of perspective with respect to existing frameworks (e.g., the excursion set approach); this work is specifically focused on the halo mass function. First, we present a stochastic differential equation that describes fluctuations in the mass growth of DM halos, as driven by a multiplicative white (Gaussian) noise dependent on the spherical collapse threshold and on the power spectrum of DM perturbations. We demonstrate that such a noise yields an average drift of the halo population toward larger masses, that quantitatively renders the standard hierarchical clustering. Then, we solve the Fokker-Planck equation associated to the stochastic dynamics, and obtain the Press & Schechter mass function as a (stationary) solution. Moreover, generalizing our treatment to a mass-dependent collapse threshold, we obtain an exact analytic solution capable of fitting remarkably well the N-body mass function over a wide range in mass and redshift. All in all, the new perspective offered by the theory presented here can contribute to better understand the gravitational dynamics leading to the formation, evolution and statistics of DM halos across cosmic times.
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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.
We use a set of N-body simulations employing a modified gravity (MG) model with Vainshtein screening to study matter and halo hierarchical clustering. As test-case scenarios we consider two normal branch Dvali-Gabadadze-Porrati (nDGP) gravity models with mild and strong growth rate enhancement. We study higher-order correlation functions $xi_n(R)$ up to $n=9$ and associated hierarchical amplitudes $S_n(R)equivxi_n(R)/sigma(R)^{2n-2}$. We find that the matter PDFs are strongly affected by the fifth-force on scales up to $50h^{-1}$Mpc, and the deviations from GR are maximised at $z=0$. For reduced cumulants $S_n$, we find that at small scales $Rleq10h^{-1}$Mpc the MG is characterised by lower values, with the deviation growing from $7%$ in the reduced skewness up to even $40%$ in $S_5$. To study the halo clustering we use a simple abundance matching and divide haloes into thee fixed number density samples. The halo two-point functions are weakly affected, with a relative boost of the order of a few percent appearing only at the smallest pair separations ($rleq 5h^{-1}$Mpc). In contrast, we find a strong MG signal in $S_n(R)$s, which are enhanced compared to GR. The strong model exhibits a $>3sigma$ level signal at various scales for all halo samples and in all cumulants. In this context, we find that the reduced kurtosis to be an especially promising cosmological probe of MG. Even the mild nDGP model leaves a $3sigma$ imprint at small scales $Rleq3h^{-1}$Mpc, while the stronger model deviates from a GR-signature at nearly all scales with a significance of $>5sigma$. Since the signal is persistent in all halo samples and over a range of scales, we advocate that the reduced kurtosis estimated from galaxy catalogues can potentially constitute a strong MG-model discriminatory as well as GR self-consistency test.
Studies of flux anomalies statistics and perturbations in stellar streams have the potential to constrain models of warm dark matter (WDM), including sterile neutrinos. Producing these constraints requires a parametrization of the WDM mass function relative to that of the cold dark matter (CDM) equivalent. We use five WDM models with half-mode masses, $M_mathrm{hm}=[1.3,35]times10^{8}$~$M_{odot}$, spread across simulations of the Local Group, lensing ellipticals and the $z=2$ universe, to generate such a parametrization: we fit parameters to a functional form for the WDM-to-CDM halo mass function ratio, $n_mathrm{WDM}(M_{X})/n_mathrm{CDM}(M_{X})$, of ($1+(alpha M_mathrm{hm}/M_{X})^{beta})^{gamma}$. For $M_{X}equiv$ virial mass of central halos we obtain $alpha=2.3$, $beta=0.8$, and $gamma=-1.0$, and this fit is steeper than the extended Press-Schechter formalism predicts. For $M_{X}equiv$ mass of subhalos we instead obtain $alpha=4.2$, $beta=2.5$ and $gamma=-0.2$; in both mass definitions the scatter is $sim20$~per~cent. The second fit typically underestimates the relative abundance of $z=2$ WDM subhaloes at the tens of per cent level. We caution that robust constraints will require bespoke simulations and a careful definition of halo mass, particularly for subhalos of mass $<10^{8}M_{odot}$.
We study halo mass functions with high-resolution $N$-body simulations under a $Lambda$CDM cosmology. Our simulations adopt the cosmological model that is consistent with recent measurements of the cosmic microwave backgrounds with the ${it Planck}$ satellite. We calibrate the halo mass functions for $10^{8.5} lower.5exhbox{$; buildrel < over sim ;$} M_mathrm{vir} / (h^{-1}M_odot) lower.5exhbox{$; buildrel < over sim ;$} 10^{15.0 - 0.45 , z}$, where $M_mathrm{vir}$ is the virial spherical overdensity mass and redshift $z$ ranges from $0$ to $7$. The halo mass function in our simulations can be fitted by a four-parameter model over a wide range of halo masses and redshifts, while we require some redshift evolution of the fitting parameters. Our new fitting formula of the mass function has a 5%-level precision except for the highest masses at $zle 7$. Our model predicts that the analytic prediction in Sheth $&$ Tormen would overestimate the halo abundance at $z=6$ with $M_mathrm{vir} = 10^{8.5-10}, h^{-1}M_odot$ by $20-30%$. Our calibrated halo mass function provides a baseline model to constrain warm dark matter (WDM) by high-$z$ galaxy number counts. We compare a cumulative luminosity function of galaxies at $z=6$ with the total halo abundance based on our model and a recently proposed WDM correction. We find that WDM with its mass lighter than $2.71, mathrm{keV}$ is incompatible with the observed galaxy number density at a $2sigma$ confidence level.
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.
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