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This content was downloaded from IP address 186.217.236.55 on 22/07/2019 at 15:01 JCAP03(2017)032 . ⊙ M nd in 1 − h 16 hysics 10 P − a high precision 12 and redshift range le ⊙ l to Klypin’s Bolshoi ic M t 1 , ina − h ar 14 Astronomy, ass range 10 10.1088/1475-7516/2017/03/032 , 10 proach and an improved barrier stant, the angular momentum es. zhou University, doi: × and mical friction. In the case of the o Paulo (IFT-UNESP), do Rio Grande do Norte, d strop — 5 A ⊙ M 1 − h 9 10 × = 5 Francesco Pace [email protected] vir , M a,b,c e,f,g osmology and and osmology C = 0 we also compared our MF to several fitting formulae, and fou z galaxy formation, dark matter theory
In this paper, extending past works of Del Popolo, we show how [email protected] 10. For rnal of rnal .
ou z An IOP and SISSA journal An IOP and 2017 IOP Publishing Ltd and Sissa Medialab srl . [email protected] Instituto de e Astrof´ısica do Ciˆencias Espa¸co, Universidade de Lisboa, FaculdadeEd. de C8, Ciˆencias, Campo Grande,E-mail: 1769-016 Lisboa, Portugal Dipartimento di Fisica e Astronomia, University of Catania Instituto de Teorica, F´ısica Universidade Estadual de S˜a Viale Andrea Doria 6,INFN I-95125 sezione Catania, di Italy Catania, Via S. Sofia 64,International I-95123 Institute Catania, of Italy Physics,59012-970 Universidade Natal, Federal Brazil Jodrell Bank Centre forThe Astrophysics, University School of of Manchester, PhysicsManchester, and M13 9PL, U.K. Rua Dr. Bento Teobaldo Ferraz01140-070 Paulo, 271, S˜ao SP Bloco Brazil 2 —Institute Barra of Funda, Theoretical Physics,No. Physics 222, Department, South Lan Tianshui Road, Lanzhou, Gansu 730000, P.R. Ch b c e g d a f c
Received August 24, 2016 Revised February 8, 2017 Accepted March 3, 2017 Published March 14, 2017 Abstract.
J A high precision semi-analyticfunction mass Antonino Del Popolo, Morgan Le Delliou mass function (MF) can be obtainedtaking using the implicitly excursion set into ap acquired account by a tidal interaction non-zeroΛCDM of paradigm, cosmological proto-structures we con and findsimulation, dyna that in our the MF is mass in range agreement at the 3% leve 0 Moreover, we discuss our MF validity forKeywords: different cosmologi
particular agreement with Bhattacharya’s within 3% in the m JCAP03(2017)032 – 1 4 7 10 15 16 18 11 and w nsitive ]. ] and the cosmic 19 10 , , 9 18 re spherical, with a curate prediction of rse reionization his- , 2 , the MF is a fundamental n to be very successful in del or standard model of Big Mpc. high precision MF, valid for dark matter halos, or more in 1 val [see ich the Universe is constituted d by the cosmological constant − cision MF is related to ongoing h nd evolution of galaxies through xtraction of cosmological param- rvations, X-rays, or the Sunyaev- e tuning problem [ m the linear phase until collapse 8 n epoch (identified with the collapse ng other problems: the cusp/core problem [ ), the equation of state parameter Λ ]. and Ω – 1 – 31 m 1 ]. , variations in cosmological parameters like the Universe ]. 8 17 8 σ , – 7 1 2, the high mass end of the MF (clusters of galaxies) is very se 2 ≤ ]. z 29 – 20 ] and quasar abundance [e.g. 30 ] (PS) proposed a simple model in which initial fluctuations a represents the linear power spectrum amplitude on a scale of 32 [ At redshifts Apart from its use to determine the cosmological parameters At higher redshifts, the MF is an important probe of the Unive A further fundamental test of the ΛCDM model resides in the ac 8 From a theoretical point of view, the model is afflicted by the fin σ 1 2 ] or the missing satellite problem [ tory [e.g. ingredient to study DMsemi-analytic distribution, and aspects analytic of formation models.and a upcoming Furthermore, surveys a detecting highZel’dovich clusters pre using (SZ) optical effect. obse different cosmologies Thus, and clearly, redshiftseters a and is allowing simple a a and helpful precise and accurate e valuable asset. 16 Bang cosmology, is a “doubleby dark” cold cosmological dark model, matter in plusΛ. wh a vacuum On density energy, largefitting represente a and large intermediate variety scales of data this [ model has been prove A Inclusion of the angular momentum B Expressions for the mass function 1 Introduction The ΛCDM model, often referred to as “cosmic concordance” mo 3 Multiplicity and mass function 4 Results 5 Discussion Contents 1 Introduction 2 ESF choices of barrier matter and dark energy (DE) content (Ω coincidence problem. At kpc-scales, the ΛCDM model is sufferi Gaussian distribution, andusing their a spherical evolution collapse model is (SCM). followed At the fro virializatio the halo mass functiondetail (MF), the namely the number density mass of distribution of dark matter halos per mass inter to cosmological parameters like its evolution [ JCAP03(2017)032 , ] ] . ) ⊙ ⊙ 45 35 ]. M M 15 49 14 10 10 ≤ × ] compared the angular on that the 5 ]. The situa- 50 − 47 ry, the collapse 9 depends on mass = 30 were under- 10 m or most bound = 30, and galaxies c ses [ z t predictions, espe- × δ z logy and redshift, poses ]. Obviously a universal . Unfortunately, in the c δ 56 10% for masses of two different techniques: , or different cosmologies and redicted, and conversely for a result universal” behaviour [ ≃ olution at high redshifts. [ , which is independent from tions have been used to obtain 54 c king length is typically chosen onal to the number of cosmic the MF at all high and medium δ – els for the mass function showed ticles, within a certain distance ent with N-body simulations [ an calculate the probability that ld density, given with respect to ization scenarios and reionization 52 r than nsity of high redshift QSOs) or for nd redshifts. Their overestimation , (SO) algorithms. The FOF method ]. In the case of the Bolshoi simula- , finding discrepancies only at uncom- 50 ⊙ , 48 depends on mass. The mass function ob- , M 48 c ]. Even the extended-PS formalism taking 11 = 0 in the mass range 5 47 δ z 36 10 – × 3 33 5, namely proto-galaxies at – 2 – ]. is defined with respect to the mean interparticle ≃ ] found that the PS MF underestimates the rarest ≃ , calculated within linear perturbation theory, gets 43 b 48 ¯ ρ , ¯ ρ , − 10 times more haloes than simulations. [ ρ 42 46 ≃ = ] with their Friends-of-Friends (FoF) MF (see the following δ = 10. Haloes hosting star populations at 2, where 55 z . – ] does not solve the problem. at 2. The ST MF had a much better performance than the PS MF, 51 40 = 0 , 3 for the rarest haloes [ = 5 and down to ⊙ – ≃ b z 48 M ≃ 37 1 , ] (eq. 28, figure 6), the tidal interaction with neighbours and − 46 h 41 , ] showed that moving from a spherical to an elliptical geomet 686 for an Einstein-de Sitter cosmology. Under the assumpti 11 . 44 1 44 15 and . 10 , ≃ ≃ = 0. Excluding the PS MF, the remaining agree to 32 c = 0 = 10 the ST MF gives z δ ], and even the ST MF overpredicts the halo number at large mas b z 46 ], the discrepancy is smaller than 10% at 49 The SO method first finds the halo centre from potential minimu Similarly, [ As shown in [ The majority of numerical simulations identify halos by one The examples above show how an imprecise MF produces incorrec The universality of the MF, that is its independence on cosmo particle to identify haloes with spheres reaching a thresho while at density field has athe Gaussian overdensity probability on distribution, one a c given scale exceeds the critical value the mass ofstructures the characterized collapsing by object. a This density quantity perturbation is greate proporti redshift), the density contrast, PS theory the numberobjects of in objects the in low the mass high tail mass of tail the is MF underp [e.g. tained with the elliptical collapse was shown to be in agreem momentum acquired modifies the collapse of a given region. As tion [ MF would avoid the needfor to its use time N-body evolution. simulations to study it f haloes in their simulations by a factor of several MF [ at redshift merging into account [ depends on the initial overdensity and shear, and (ST). However, a deeper analysissome of problems: those the semi-analytic PS mod MF,masses as [ already reported, overpredicts tion worsens if one studies thea PS better and ST understanding MF of evolution.tested the the Simula MF ST MF at up low to redshifts and of its ev spacing. The FOF halo mass function scales very close to the “ mon (rare) density enhancements. [ the value with masses either friends-of-friends (FOF) oridentifies spherical halos overdensity by a(the linking percolation length technique, b)between to connecting each par other, in the same halo. The lin and this changes the mass function [ estimated by a factor of but its predictive power decreased withgoes increasing up masses a to a factor of cially for halo numbers atastrophysical high redshift phenomena (e.g., happening the at numberhistory). high de redshift (reion an important issue studied by several authors [e.g. JCAP03(2017)032 . ] . c ]. c . ρ ⊙ δ 48 57 z M , espect 2. As ]. The 10 [ . 15 . 56 rovided ] showed − 51 ]. 10 56 e at higher = 0 54 × es 0 , = 0 ] showed the Their results b 3 z 54 36 − 4 – o FOF mapping, 5. ollapse threshold . It has therefore 11 52 σ ] found a universal , ≤ 10 0%. [ 54 48 z × g length . simulation to obtain a endency of FOF to link sality in the time evolu- ifferent cosmologies they ⊙ he effective spectral index trongly correlated [ M 2, 6 ed halo mass functions are ied. [ 1 edshift range d in the process of merging. − ]. At higher redshifts, the SO shift changes. h ] calculated the MF from their s”, showed the same evolution eature more frequent with the o definitions can be considered 50 5 . 50 respect to the critical density 0% of FOF halos with irregular have an exact universality, one needs a