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Register a new userAn Improved Calculation of the Non-Gaussian Halo Mass Function
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The abundance of collapsed objects in the universe, or halo mass function, is an important theoretical tool in studying the effects of primordially generated non-Gaussianities on the large scale structure. The non-Gaussian mass function has been calculated by several authors in different ways, typically by exploiting the smallness of certain parameters which naturally appear in the calculation, to set up a perturbative expansion. We improve upon the existing results for the mass function by combining path integral methods and saddle point techniques (which have been separately applied in previous approaches). Additionally, we carefully account for the various scale dependent combinations of small parameters which appear. Some of these combinations in fact become of order unity for large mass scales and at high redshifts, and must therefore be treated non-perturbatively. Our approach allows us to do this, and to also account for multi-scale density correlations which appear in the calculation. We thus derive an accurate expression for the mass function which is based on approximations that are valid over a larger range of mass scales and redshifts than those of other authors. By tracking the terms ignored in the analysis, we estimate theoretical errors for our result and also for the results of others. We also discuss the complications introduced by the choice of smoothing filter function, which we take to be a top-hat in real space, and which leads to the dominant errors in our expression. Finally, we present a detailed comparison between the various expressions for the mass functions, exploring the accuracy and range of validity of each.
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We compute the effect of primordial non-Gaussianity on the halo mass function, using excursion set theory. In the presence of non-Gaussianity the stochastic evolution of the smoothed density field, as a function of the smoothing scale, is non-markovian and beside local terms that generalize Press-Schechter (PS) theory, there are also memory terms, whose effect on the mass function can be computed using the formalism developed in the first paper of this series. We find that, when computing the effect of the three-point correlator on the mass function, a PS-like approach which consists in neglecting the cloud-in-cloud problem and in multiplying the final result by a fudge factor close to 2, is in principle not justified. When computed correctly in the framework of excursion set theory, in fact, the local contribution vanishes (for all odd-point correlators the contribution of the image gaussian cancels the Press-Schechter contribution rather than adding up), and the result comes entirely from non-trivial memory terms which are absent in PS theory. However it turns out that, in the limit of large halo masses, where the effect of non-Gaussianity is more relevant, these memory terms give a contribution which is the the same as that computed naively with PS theory, plus subleading terms depending on derivatives of the three-point correlator. We finally combine these results with the diffusive barrier model developed in the second paper of this series, and we find that the resulting mass function reproduces recent N-body simulations with non-Gaussian initial conditions, without the introduction of any ad hoc parameter.
A sizeable level of non-Gaussianity in the primordial cosmological perturbations may be induced by a large trispectrum, i.e. by a large connected four-point correlation function. We compute the effect of a primordial non-Gaussian trispectrum on the halo mass function, within excursion set theory. We use the formalism that we have developed in a previous series of papers and which allows us to take into account the fact that, in the presence of non-Gaussianity, the stochastic evolution of the smoothed density field, as a function of the smoothing scale, is non-markovian. In the large mass limit, the leading-order term that we find agrees with the leading-order term of the results found in the literature using a more heuristic Press-Schecther (PS)-type approach. Our approach however also allows us to evaluate consistently the subleading terms, which depend not only on the four-point cumulant but also on derivatives of the four-point correlator, and which cannot be obtained within non-Gaussian extensions of PS theory. We perform explicitly the computation up to next-to-leading order.
The description of the abundance and clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys, which can potentially yield constraints of order unity on the non-Gaussianity parameter f_{NL}. We present tests on N-body simulations of analytical formulae describing the halo abundance and clustering for non-Gaussian initial conditions. We calibrate the analytic non-Gaussian mass function of Matarrese et al.(2000) and LoVerde et al.(2008) and the analytic description of clustering of halos for non-Gaussian initial conditions on N-body simulations. We find excellent agreement between the simulations and the analytic predictions if we make the corrections delta_c --> delta_c X sqrt{q} and delta_c --> delta_c X q where q ~ 0.75, in the density threshold for gravitational collapse and in the non-Gaussian fractional correction to the halo bias, respectively. We discuss the implications of this correction on present and forecasted primordial non-Gaussianity constraints. We confirm that the non-Gaussian halo bias offers a robust and highly competitive test of primordial non-Gaussianity.
The claimed detection of large amounts of substructure in lensing flux anomalies, and in Milky Way stellar stream gaps statistics, has lead to a step change in constraints on simple warm dark matter models. In this study we compute predictions for the halo mass function both for these simple models and also for comprehensive particle physics models of sterile neutrinos and dark acoustic oscillations. We show that the mass function fit of Lovell et al. underestimates the number of haloes less massive than the half-mode mass, $M_mathrm{hm}$ by a factor of 2, relative to the extended Press-Schechter (EPS) method. The alternative approach of applying EPS to the Viel et al. matter power spectrum fit instead suggests good agreement at $M_mathrm{hm}$ relative to the comprehensive model matter power spectra results, although the number of haloes with mass $<M_mathrm{hm}$ is still suppressed due to the absence of small scale power in the fitting function. Overall, we find that the number of dark matter haloes with masses $<10^{8}M_{odot}$ predicted by competitive particle physics models is underestimated by a factor of $sim2$ when applying popular fitting functions, although careful studies that follow the stripping and destruction of subhaloes will be required in order to draw robust conclusions.
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.
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