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Register a new userCosmology with Galaxy Clusters: Systematic Effects in the Halo Mass Function
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Aseem Paranjape
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We investigate potential systematic effects in constraining the amplitude of primordial fluctuations sigma_8 arising from the choice of halo mass function in the likelihood analysis of current and upcoming galaxy cluster surveys. We study the widely used N-body simulation fit of Tinker et al. (T08) and, as an alternative, the recently proposed analytical model of Excursion Set Peaks (ESP). We first assess the relative bias between these prescriptions when constraining sigma_8 by sampling the ESP mass function to generate mock catalogs and using the T08 fit to analyse them, for various choices of survey selection threshold, mass definition and statistical priors. To assess the level of absolute bias in each prescription, we then repeat the analysis on dark matter halo catalogs in N-body simulations designed to mimic the mass distribution in the current data release of Planck SZ clusters. This N-body analysis shows that using the T08 fit without accounting for the scatter introduced when converting between mass definitions (alternatively, the scatter induced by errors on the parameters of the fit) can systematically over-estimate the value of sigma_8 by as much as 2sigma for current data, while analyses that account for this scatter should be close to unbiased in sigma_8. With an increased number of objects as expected in upcoming data releases, regardless of accounting for scatter, the T08 fit could over-estimate the value of sigma_8 by ~1.5sigma. The ESP mass function leads to systematically more biased but comparable results. A strength of the ESP model is its natural prediction of a weak non-universality in the mass function which closely tracks the one measured in simulations and described by the T08 fit. We suggest that it might now be prudent to build new unbiased ESP-based fitting functions for use with the larger datasets of the near future.
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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.
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.
The halo mass function (HMF) is a critical element in cosmological analyses of galaxy cluster catalogs. We quantify the impact of uncertainties in HMF parameters on cosmological constraints from cluster catalogs similar to those from Planck, those expected from the Euclid, Roman and Rubin surveys, and from a hypothetical larger future survey. We analyse simulated catalogs in each case, gradually loosening priors on HMF parameters to evaluate the degradation in cosmological constraints. While current uncertainties on HMF parameters do not substantially impact Planck-like surveys, we find that they can significantly degrade the cosmological constraints for a Euclid-like survey. Consequently, the current precision on the HMF will not be sufficient for Euclid (or Roman or Rubin) and possible larger surveys. Future experiments will have to properly account for uncertainties in HMF parameters, and it will be necessary to improve precision of HMF fits to avoid weakening constraints on cosmological parameters.
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.
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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