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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, 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 (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

2 S ], one can obtain an analytical approxi- ∂S S n / ( 83 2 ) is the so-called “multiplicity function”, 37 3 ν ∂ M 93 ν ), defined as the average comoving number B − ( n , ) is [ ) f − log ! log 68 S ]. n d  r, ν d M,z − (

( 92 ( dM δ ]. The latter found out that larger shear and an- n 2 – 7 – exp ρ + 5 | =0 ∝ 82 M ) X n , S M ta proportionally to its turn-around time ( t 81 j − )= T )= , | ∝ S 78 ( M j = , T M,z ( 77 dS n , and thus need higher density contrast to collapse and form ) j S ( f peaks tend to resist more to gravitational collapse than hig ν ) is lower in overdense than underdense regions. M,z ( c ) can be obtained using a Taylor expansion of δ ]. Low- S is the background density. The quantity ( 80 T ρ – Those results agree with [ Conversely, as high peaks are more probable in denser region This reflects the different aspects of tides depending on the s Angular momentum possesses similar effects as a non-zero cos In the case of constant and linear barriers [ 75 than more massive ones. , c structures. This issmall why scales, structures where need, shear on is average, more higher important. because they acquire larger acquires specific angular momentum particular, it especially slowscosmological down large constant mass vanishes structures at high redshift (see also s correlated with the peak height: density of haloes in a mass range more sensitive to external tides, thus fixed time collapse le 71 massive “peaks” (in the initialδ random field) form structure threshold where mation for the first crossing.generating In a other large cases, ensemble thebarriers, of first one random crossi can walks. approximate As the shown first by crossing [ distributi 3 Multiplicity and massIn function the ESF, the unconditional mass function dark energy. gular momentum slow down theangular collapse. momentum, As collapse smaller scales ofThose sta structures results at have been those extended scales more re recently by [ the angular momentum andbarrier behaviour the that cosmological reduces small constantwith haloes’ effects, respect abundance, an to amodel flat of barrier ellipsoidal (PS collapse,haloes, mass caused leading function). to in A a this similar larger case be collapse by time the [see lar the distribution of the first crossing. where JCAP03(2017)032 ) , 3.2 ) (3.7) (3.8) (3.5) (3.6) (3.4) (3.9) 2 (3.10)  4 ) derive . 0 a ) 02 , . 0  . Although aν 2 ( ) ] A , 2 1 α  + ) 2 − , 2 ) α  , ) 2 ν 585 the next section. ) . 1

β 0 2 45 a aν . )  5526 ( ( 0 4) ] give ), the use of eqs. ( . 07 6 ) 1 ological constant, given . , . l Friction, it is given by 0 aν 0 β − + 0 ( 68 5 ) nstants (except 2.4 aν . 2) ( α α 0 ( ) aν papers, the effects of angular ( ion of the ellipsoidal barrier β 1+ + [1 + aν/ aν  ν ··· 4 5! ( ndent random walks and that . 1 −  0 a 1+ , eq. (G3)]. 1) aν ) 02 5 45 .  . . 0 1+ 94 − 0 0 aν 1 . )  ( exp( aν − α 0 4019 . ( . 2 1 aν + aν 0 aν π ( α − 2 1 2 − = 1 + − ( 585 ) presents our mass function, calculated in − . ( r 0 4 )]exp . ( ) 1 dν 0  5526 ... . . The multiplicity function is now given by exp ) ) α 3 3.10 ] exp ∗ . 0 aν ν ( 0079 − 0 π ( . ( S – 8 – exp g aν ) π 68 aν 2 0 1 1 ( f aν 2 ) gives the normalization constant √ π 1) α aν aν 2 r ∞ − + ( 1+ r − 0 ) 3.9 =  2!  r Z ν  α 6 1 ∗ . ( 585 12 4 .  . 1+ . a 0 σ 0 α 2 0 ( ) ) )  2 ) 094 1 2 1218 974 must satisfy the constraint . . α . β + 0079 aν have been given previously. Then [ α aν 0 = 5, ) A 0 . aν aν ( ( ( 0 1 ( α g n ≃ aν = 0 β 2 ( − + ) β [1 + 4019 1 . 1+ , being ν 1+ 1 =0.707. Thus eq. ( π 0 A ∗ (

 + 2 ν a  and S 585 − . ) α νf 0 1 ≃ ν/ π ( ) α ) 1 )= aν 2 1218 α = ) ( . a α ν , aν 0 aν ( 2 ( r 1 ( βg ( exp ) p g 2 a ∗ σ νf A σ × ). ( 1+ 1+ )= 93702 and ∗ ≃ . 322. ν   S S,t . ) ( ( , whose validity shall now be confronted with simulations in 1 1 ν ), gives ( = 0 B ≡ A A νf = 0 Sf 2 ) gives, at fifth order 2.5 νf ≃ S A A )= 3.3 The same method, for the barrier taking into account the cosm At this point, it is important to stress that all numerical co In the following we will use the CDM spectrum of [ Applying the previous methods to the barrier given by eq. ( Finally, in the case of the barrier including, as in previous )= ν ( ν ( νf momentum and Λ, and now introducing the new effect of Dynamica where by eq. ( where νf and the values of appendix and ( from barrier calculations: condition ( which is aobtained good through approximation the tofitting simulations the the GIF of first simulations unconstrained, crossing [see figure distribut indepe 2 of The normalization factor where where JCAP03(2017)032 2. 3. . / sity 156. . vir (3.13) (3.14) (3.12) (3.11) = 0 ∆ b ρ = 0 r an EdS b = ¯ ained by the 75, vir . ]. The authors R owing expression ρ 96 ], this issue needs = 0 n be minimised by as a function of the 96 ] also quantified this more affected by the Λ a 54 malism of the spherical adius is ned by [ . . Ref. [ 25 and Ω d will also affect the definition b provided the mass resolution of ume that using the appropriate sed for ΛCDM cosmologies. As . retical derivation of the linking he results of [ d by that of two particles in a ty an isothermal profile for the 68 ], it was determined as a fit to the . f the mass function on the linking , . pendence of ) = 0 95 x , + 0 ∆ m c x 2 (  ) 1+ ψ c ], is a FoF with a linking length of 2 is significantly larger than 178 and that vir . − 86 178 ∆ 53 2 unaffected. This could probably not be the . ) ) can be obtained with the following relation: c x  ], and thus could be interpreted as depending = 0 a )(1 + ∆ + 1 b 244 c is largely unaffected by the choice of the specific 44 24 univ ( . b – 9 – a µ = much different from the optimal one are used, due 3 = 0 − b )= 3  c − ( 2 (called )=ln(1+ . b  ψ 0 x b , hence a correct value for ( 2  . µ vir univ 0 b  . Given a background cosmological model, the virial overden b . , one can relate it to a mean separation between particles. Fo b b = 0, the FoF overdensity for , which gives the number of high mass haloes, could also be obt and on the halo concentration. Their analysis led to the foll z a b 2, while for a ΛCDM cosmology with Ω . ], deviations from universality for the FoF mass function ca 54 to ∆ (the enclosed FoF overdensity) = 0 b b can be evaluated, under certain assumptions, within the for In the light of these considerations, it is reasonable to ass Since the number of halos, defined via FoF or SO techniques, is vir where it depends on This empirical correlation can be explained in the light of t relating excursion set theory with a diffusing barrier, as shown by [ the parameter to the bridging problem affecting FoF methods. As also explai and to be investigated more deeply and quantitatively. showed that at massive haloes number in the simulations of [ using the appropriate value of ∆ It is clear thatof the the way linking the length. concentrationlength This parameter implies parameter is therefore a define dependence o on the halo finder, which, in our case, similarly to [ and showed that the best choice for case if, for a fixed cosmology, values of linking length for a given cosmology, will leave algorithm. Different considerations hold instead for the de ∆ change of technique at low massesthe rather simulation than is at good high enough), masses the ( value of linking length parameter collapse model. Havingdark this value matter and halos, assuming the for simplici mean density of the halo at the virial r Assuming that the densitysphere at of the radius virial radius is represente While the previously presentedlength, values in proceed practice from the the valuenoted theo of by an [ EdS model is also usually u model, JCAP03(2017)032 ] , ] ] . - ⊙ )) = ⊙ 56 68 46 46 3%, M , M ], its 3.7 10%, vir 1 1 . 35 68 − − M − is much and the h h z 15 ], based on 3% level in Apart from 16 ]’s appendix 10 55 ≃ 8 49 ), while on the mparison with the = 0 corresponds —10 -dependence: [ z or [ 12 ]dosoto5 M > z 3.10 B dependence encoded 54 rical collapse model, ] give a universal mass ], this behaviour reveals in the Bolshoi simulation ] overpredicts simulation 68 56 , range 10 ] proceeds from the multi- = 0, the rightmost dashed 68 by eq. ( tiplicity function (eq. ( = 6, and for masses 54 z 45 on of the mass function of [ lation with deviations ment with simulations by [ accuracy, while [ 31 which at en compare its evolution with , ). . ulations, [ , rly than our MF, as seen below. z intersecting all the error bars. = 0 with several mass functions z , a function both of 1 ( c ions showing steepening of the MF with mass 51 c z 44 δ logy), as the MF dependence with δ sed by [ , < At ). At 1 44 z − ( , σ c δ 35 log . [ ⊙ < M 10 but over a more restricted mass range. Our 55 1 . value has been long kept. It has also been shown , we plot the ratio between the Bhattacharya MF − 0 ) (see definition in appendix c – 10 – 2 h δ . At higher redshifts, [ − M ⊙

......

0 0 0 0 0 1 1 1 1 1

son Wat / MFs short-dashed line the Sheth-Tormen mass function [ those obtained in Bhattacharya’s paper at MF at Figure 2 separation of the curves,lines in show the analogy ratio to at their figure 5. The red curve represents the Courtin massfunction function and [ that of Bhattacharya [ measured in numerical simulations hasing only been on directly fitted b and redshift evolution, whatshift is independence commonly of understood the relation as between “uni the linear and no mass function, multiplying it by a correction factor function [ the pink dashed-dotted curves thedotted two Reed curve the mass functions Angulo [ massand function the [ Bhattacharya [ at a given redshift with respect to the one proposed by Bhatta haloes virialization, which dependsit. upon cosmology These and results red demonstrate the importance of nonlinearprocess e creates deviationsspherical from collapse threshold, universality, responsible fromaccounting for for cosmol deviations it reduces MF discrepancy between models. a good agreement between analytic and numerical MF [ that taking into account the cosmology dependence of JCAP03(2017)032 × 5 − . Over 9 ⊙ 10 M e relatively 1 × − h n with respect 14 = 0 MF. 10 z t there that our MF × shows the discrepancy 75 3 . l detail in the following. 1 ] claimed that the corrected rques. Such improvements ≈ he mass range 5 igher masses. This is easily gure 49 ower is the number of objects. erical collapse based PS model, pancies with simulations. Our M stant Λ, the effects of dynamical onsidering the effect of Λ in the s. Diamonds represent the Bolshoi MF. = 10 yields 38% for the correction collapse without taking into account z ] introduced the effects of asphericity ]. 45 49 = 0 and with their redshift evolution with z ] and this work with the Bolshoi MF. Left panel: show that the MF generated from our barrier 68 , showing the comparison of the Bolshoi data = 0. The points and the error-bars correspond 4 4 – 12 – z and ], are characterised by the facts that all walks cross ] at ] (dashed line), and the result of our model (solid line). 3 , like ours, allow mergers and fragmentation, whereas 74 S 49 , 49 73 ] and simulations. [ 68 [e.g., S ]. Note how all the error-bars intersect our curve. Errors ar 49 receives the values 10, 6, 2.5, 0. Right panel: zoom of the z we show the fractional accuracy of our proposed mass functio 5 . This is presented in figure ⊙ M . Comparison of the mass functions of [ is the linear growth factor normalized to unity today. [ 1 δ − ] instead of the claimed 10%, as opposed to our 3%. It is eviden h The results displayed in figures In figure 49 14 where Figure 3 barriers decreasing with to the values from [ to the numerical mass function of [ Note however thatby calculating [ the averagegives error a on much better result than the correction by [ them and fragmentation is not allowed. small for low-massexplained objects taking and into account graduallyThe that point increase the most higher towards differing the from h mass, our the mass l function has a mass (diamonds) with the correction by [ is in good agreement both with simulations at considering an intuitive parametrization of an elliptical the dashed line representsFrom the left ST to MF, while right the solid line our between the predictions by [ the interaction withbarrier. neighbours Thus, (isolated while spheroid), their modelit nor is remains an c improvement partial, on themodel, leading sph in to contrast, the takes aforementionedfriction into MF account and the discre of cosmologicalgive con angular rise momentum, to acquiredMoreover, a barriers MF through increasing in tidal with good to agreement with simulations, as we wil mass function deviates by less than 10% from simulations in t a precision of the order of 3%. By contrast, the left panel of fi 10 JCAP03(2017)032 ] 49 y [ ]. Curves and symbols 49 ’s right panel where we 2 r proposed mass function s of [ re . n this work with respect to the ] and our proposed mass function at 49 . 0 – 13 – , except that now the dashed line represents the correction b 3 . Fractional ratio between the numerical MF by [ . Comparison of our analytic mass function with the Bolshoi’ It is also interesting to evaluate how the ratio between of ou = 0. Figure 5 Figure 4 all the points, the accuracy of the mass function presented i z simulation used as comparison is, on average, better than 4% are as in theto left the panel ST of MF. figure and that of Bhattacharya changes in time. This is shown in figu JCAP03(2017)032 n of physically , where we 2 ) where the σ ]. These fits have 68 , n section 35 d precise MF, especially )), are produced by high ccurately predict the dark rrier, whose shape depends r masses (low formation physics motivated ular momentum acquisition, 3.10 10 shifts considered, the overall s the presence of a non-zero ing to rely on numerical results: mbedding it in the barrier. q. ( han high resolution simulations. ecially at small scales. om solid physical and theoretical cal multiplicity function: improv- ant and angular momentum. taining a realistic analytical form ]. Multiplicity functions presented 8 ximation accuracy. Moreover, this orms to [e.g. 53 6 z 4 when angular momentum and Λ is taken into account – 14 – 2. Points and corresponding error bars are those z , . The dashed line represents the same quantity when 1 2 , ⊙ M 1 = 0 = 2 demonstrate that redshift evolution is important and − 0 z h z 2

2.2 2.1 1.9 2.5 2.4 2.3 11

c (z) ) obtained in this paper does provide an excellent predictio ]. = 10 = 1 and 53 3.10 z M . Such form is both able to better “describe” simulations and . The collapse threshold in terms of first principles On top of this theoretical advantage, our approach can very a In conclusion, the excursion set approach, with a structure The role of angular momentum in shaping the MF was discussed i The data sets at Thus, a precise MF requires a precise determination of theAt ba this stage it is important to emphasize the need for a new an high resolution simulations, and atarguments. the same time derives fr matter halo distribution atThis much lower is computational because cost we can t it derive follows its up functional directly form without by hav using an improved barrier. barrier, produces an excellent approximationing to the the barrier numeri formnon-zero (with cosmological more constant, and etc.)method more increases physical displays the effects: a appro cosmological ang remarkable constant, versatility: is very any easy to effect, take such into account a by e Figure 6 (solid line) for a mass in this paper, suchresolution as N-body Bhattacharya’s simulations and fits, except similar forno in our’s theoretical functional (e f foundations, revealingfrom the importance of ob showed that it reduces or prevents structure formation, esp motivated. The MF ( must be taken intoagreement account; is in of addition, the over order the of range 3%, of except red for few points at highe agreement is at the 5% level. dynamical friction is taken into account. consider the following redshifts: when it agrees with previous ones [e.g., Bhattacharya’s on the effects of dynamical friction, the cosmological const obtained in figure 5 of [ JCAP03(2017)032 , ] ): 56 82 M , , 42 54 niversal – effect as , 50 41 , haloes, that ar momentum z 48 ) (discussed in , ]. 46 54 A.1 , 35 observed MF, and why tween our MF and the es the collapse threshold imations in several cases, m eq. ( t with simulations. In this rtion of high- wn the collapse [ cts are taken into account, ass tail, (Feyereisen private he strongest in slowing down to the cosmological constant. in the shaping of the MF. In portant role in the shaping of entum are taken into account. ion, reducing their abundance a slight redshift dependence of se model, but in parallel with damental importance. After the ]. ns and gives a fit to their results. pite the effect of Λ reducing with mulas [e.g., 106 ]. At the same time the cosmological – ], further advances came from N-body nd angular momentum for the ΛCDM and dark 68 while it already is the case for mass . At the same time, the approach leaves 101 102 , z , z ] for a ΛCDM model [for a generalization due to other factors [see 45 95 83 , z 44 11 – 15 – . Thus structure formation is “suppressed” at high z ] and [ 42 ), following [ , (figure 3)]. z ( c 83 δ . This explains why our MF predicts a smaller abundance than ]. dependence on mass, one needs to take into account its time z 90 c ], taking into account the angular momentum and Λ for a mass 10 – δ ], since at high redshifts the effect of Λ decreases, the non-u at all values of 88 86 we plot – 53 co ). The term involving the cosmological constant has the same 6 δ 84 A . The dashed line adds the effect of dynamical friction. Angul ⊙ M 1 − ] with increasing h to be a monotonic decreasing function of , as done in many of the papers cited in the Introduction. 68 c a 11 δ The paper shows that the introduction of a moving barrier mak The effect of the cosmological constant can be understood fro Although MF obtained through simulations yield good approx Before concluding, we want to point out that the agreement be Dynamical friction also slows down the collapse, similarly In addition to the As discussed by [ ) is larger than A similar result was obtained in [ For an alternative description of the effects of tidal shear a and ]. This gives rise to a delay in large-scale structure format = 10 z by angular momentum. ( 10 11 2 , the non-universal behaviour persists at high c mass-dependent, contrary toextended the models where standard shear, spherical tidal fields collap and/or angular mom us with a semi-analytical form of the MF in very good agreemen sense our result is much more physical than that of simulatio would be smaller than the resolution of simulations [ the simulations blackmakes box it nature, difficult in toour which disentangle approach, the we many selected role physical physicalthe of effe MF effects those and known mechanisms explained to with playthe them an ST why im MF the has PS MF problems gives in bad reproducing fits it to at the high and steepening the MF. This would also produce a larger propo z 100 δ 5 Discussion The determination of a high precision mass function is of fun first improvements of the PS MF by [ to DE models, see detail in appendix those involving angular momentum and DF, namely, slowing do constant clearly decreasescommunication the and number paper in of preparation). halos in the high-m Of the three effectsthe taken collapse, into account, followed angular by dynamical momentum is friction. t Bolshoi simulation data couldA be further improved assuming simulations, used as tools to calibrate proposed fitting for that of [ evolution with redshift should bez suppressed. However, des and more recently from a new diffusing barrier [ M causes evolution. In figure energy models, we refer to [ JCAP03(2017)032 em: (A.2) (A.3) (A.1) ]. A similar 108 = 0 and during , z M.Le D. has been 62 pproach problems, in d modify its functional nowledge IFT/UNESP. ) the angular momentum cosmological constant and per radius of a shell can be minor compared with other n of the angular momentum, ts that helped improving the r, ν y to introduce effects such as ( mass function yields results in r, ture, such as fits to numerical L (high) mass [ 3 Λ ical constant, can be obtained by s the linear extrapolation of the ns predictions at high redshifts. , on collapses at a given time, when + cal effects such as the cosmological 3 i r , ) dt dr ], within 3% level at ) ,t η i ]. ] fitted their mass function with an EdS 53 ,t r i − ( , r 44 3 ) ( 107 a the initial radius, and that the mass is given r 49 a ) i ( ] found a moving barrier, taking into account , i r g r ,t i 41 44 r − – 16 – ( ] the barrier was extended to account for the role )= ) the acceleration, ) ρ r 3 ,t 62 π ( r i 3 4 g , 2 r r, ν ( ( ]: 2 r 61 = M L 114 M = – ). The general behaviour of our proposed mass function is r 3 109 dt dv , 70 ) and shares with them, albeit at a lower level, the same probl , 2 61 ) is the expansion parameter and ,t i r ( a the coefficient of dynamical friction. Recalling that the pro The positive consequence of these aspects is to solve the PS a The barrier for the first crossing shapes the mass function an An interesting feature of a moving barrier is the possibilit The delay of collapse of a perturbation due to the acquisitio FP is funded by an STFC post-doctoral fellowship. The work of The effect of introducing the cosmological constant remains η effects such as tidalthe fields angular and momentum angular slow momentum, down the but collapse. both the particular to reduce (increase) the number of objects at low where written as by and of the cosmological constant and of DF. result has been found for the ellipsoidal collapseform [ with respect to thevery simple good PS formulation. agreement with The improved N-body simulations [ an excess of structures with respect to numerical simulatio Acknowledgments The authors thank an anonymousscientific referee content for of the this useful work. commen angular momentum acquisition. In [ model). in agreement with other functional forms proposed in litera mergers, tidal torques, dynamicalconstant friction, (note and that, cosmologi as also pointed out by [ simulations (see figure where Λ is the cosmological constant, its time evolution (see figure supported by PNPD/CAPES20132029. M.Le D. also wishes to ack A Inclusion of theAs angular already momentum discussed, inits hierarchical models, overdensity exceeds a perturbati athreshold to critical the threshold. present time. The We saw barrier that i [ the presence of dynamical friction,solving and the a equation non-zero [ cosmolog JCAP03(2017)032 ] 113 (A.7) (A.4) (A.9) (A.6) (A.5) (A.8) (A.11) (A.10) . adius, which is , ) may be written ) i ] ) 2 max A.1 C , 2 ta a 2 108 r a. − 3 − Λ − 2 2 a. C , 2 , eq. ( a + a r 2 2 i 3 for the growing mode [e.g., ( Λ  ( . Linear theory can be used 3 Λ 6 H i πG Λ ]. Solving numerically [ i 2 6 − Λ dt 3 8 t δ da + r - 2 = 0. The binding energy of a + η + v 3 ta r + GM 112 = 3 3 r a 6 3 − Λ r, a a ci dr = 0, we get [ 10 i 2 3 )) it is possible to obtain the linear ρ 3 3 Λ ) 10 r i a 2 r δ 2 2 r dr/dt +Λ 2 )+ 2 2 at time ) 2 L L . The connection between the binding 10 i r , ) L δ 2 r c L i M r ( (1+ )+ 2 and A.10 δ 3 2 δ 2 δ d 2 g r G da/dt  r G 10 ) i ) · ta L ( ) 2 r G ). 8 4 3 r δ 2 ] and can be obtained using the conditions r δ dr 2 g 4 ,t 2 8 i 2 = R i β (1 + − ) with α r G H L 4 (1 + Λ da ( − (1+ 4 A.8 ν 112 + 4 3 i ta GM ci Ω ) H a ) and ( dr (1 + r ρ  A.7 H 3 ), similarly to [ ) 4 max a c r + – 17 – a ta + dr/dt 1 ta 3 = δ 2 r, ν r R H r A.8 i Z r ( ) α r r, ν ) ) β 2 2 δ ( ν M + Z − ,t + ,t + i 2 i L r i r r 1 M 2 ) ( ) L ( i δ 3 a δ ) we can write δ  2 = ρ 1+ 1+ a 2 (1 + max   =0 for eq. ( 2 1+ 2 r a Z a A.6 a dt co co H 2 dv (1 + GM (1 + )= − δ δ ) (or eqs. ( = 2 h 2 i − δ ,t 2 i 2 = ≃ H a H da/dt r =  A.9 1+ c ( r − δ h ) and ( ρ = 2 a dt dr 2 i 2 ta = 2 r dt d  H 0 A.5  2 a ) and Z ], the threshold becomes 2 ) and ( r dt dt d da = 62 A.7 can be obtained by means of the relation  A.7 i max ta t a δ 0 Z = 0 the two equations become = , and η is the binding energy of the shell [see ta ) for a given mass and turn-around time gives the turn-around r ]. C t ] C =0 for eq. ( For Integrating once more, recalling that Integrating eqs. ( As shown in [ Using eqs. ( 42 A.9 115 , related to the binding energy through eq. ( energy, to get the overdensity at turn-around and collapse, growing mode is uniquely given by the overdensity where and 112 being the average density and [ overdensity at turnaroundeq. and collapse ( ( dr/dt as and JCAP03(2017)032 , ) d ]. co 71 T A.1 o their (B.4) (B.2) (B.1) (B.3) ǫ ]. The (A.14) (A.13) (A.12) (A.15) 3 a = 49 ], [see o × λ a 76 s generating n is the collapse = co T ac , dϑ , n and using the mass ) o o ,  δ δ σ ] (eq. 29); m ( ) ) 2 , f r ϑ σ Ω 69 ( (  ) ta cr GM 2 3 = 0 starting from eq. ( ) are given by [ f r / ρ 6 ϑ f approximations of [ ϑ 2 ( dσ η ). ( ) 2 dk , σdM − 2 ) = N f dM f ) ⊙ +Λ ) M 2.5 . ϑ ( 12 ρ is the halo virial mass and ) . ( ) M . M δ 1 1 2 ν k,M ( dσ ( f (  2 ( h o M σ µ / 2 0 ) and 3 is given in [ , λ 3 0 , 3 / , , appendix D], while ϑ νf ) 4 − M )] log ( 2 W )), cr ) for eq. (  / ϑ 1 ) ν 42 1 Ω their total number, 2 ρ 3 M ϑ f 3 k / 3 2 0 π 3 , 3.1 r, ν  ( α 1 a + ], [ − ] β ( cos ν log m n P sin δ log r ac [ 2 70 d i 3 + 1 − d d + n k 3 sys − r · a 2 δ 1 H 3 2 R , eq. (C5)], and then integrating the torque over Mpc) ϑ (1 ∞ ρ 2 ) can be written as N 2  – 18 – ( β ) = Ω 1 in eq. ( π 0 3 M α L ) 4 Λ 71 − σ Z π π Gm δ ν ρ ( )= 2 ( ta h GM 0 B.1 Ω ) c ( f r ], [ √ 2 = Z 3 a )= 44[ ( π . 2 11 + r, ν ta ) 2 76 / ( 2 4 r i N 5 δ 0 α dM r 10 β c ) M o − , eq. ( ] ν Z ) = = M,z ( T δ 1 × ( δ 0 , due to the tidal interaction with neighbours, is calculate 2 M , and the functions ( − t )= n / ( dσ o σ 116 L µ o 3 t 75 1+ 1+ σ τ . dM M   ρ the system radius [ log 3 ( ηa , that the background density / 2 d co co 45, 2 . = 2 = σ δ δ = sys ρ  M o R 3 4 ≃ = )) is given by ǫ ndM = 0 dn c dM  δ )= 3.1 3 1 3 = σ β M ( , the mass and the number density of field particles (particle f a dn )= n 07, ], . 35 r, ν , eq. (35)] obtaining is the critical density today ( ( 6= 0, the barrier can be obtained similarly to the case = 0 and 0 76 L , η is the tidal torque at 3 a , )). cr o α ρ τ m 71 The angular momentum For In this case, the threshold becomes A.4 the fluctuating field), respectively, getting the r.m.s. of the torque [see where the constants were already given in section where time [ (or ( given B Expressions for theIn mass the following, function we writemass the function MF (eq. using ( the same notations and Recalling that time of a pure top hat model [ function of [ where where comoving number, and where JCAP03(2017)032 ) the (B.6) (B.5) (B.7) (B.8) ). The z k,M ] il from the ( , (2007) 169 W (2013) 2 ) (1 + 2 (2011) 18 , /  51 1 45 − . 1548 192 0  = 1 07 ) . ⊙ ation, we assume the 0 , a aν M (  ]. 1 ν + , arXiv:1210.7231 − × [ 1 2 M ugh 4 h a . π Astron. Rep. 0 aν 2 , ) − 02 eters 12 SPIRE .  r 0 IN aν 1, using the expression proposed 10 . ( n. The linear growth-rate function ] [   0 AIP Conf. Proc. exp , + . (2013) 86 45 ≡ . , > ν ⊙ 0 1 Astrophys. J. Suppl. . ) √ m , 585 0 M .  (1) 779 0 aν 3 1 . ) , y ( 5526 0 − . /D − 0 aν h ) + ) ( 333 a . . 7 First year Wilkinson Microwave Anisotropy Probe ν 4 4 ( . 0 Seven-year Wilkinson Microwave Anisotropy Probe 1 0 − y 10 a D ) 707. – 19 – . 1+ 10 0079 22  . . aν × astro-ph/0302209 0 ( [ )= 12 = 0 1+( Astrophys. J. 41 . a . M >  2 + 6 + ( , 1 0 δ a 1 y aν 20 9 a π . . 585 2 0 . ], we have 0 y ]. ]. 16 ) r 1218 68 4019 . . A (2003) 175 0 aν 102 0 . ( 322 and − . SPIRE SPIRE ) the primordial power spectrum of perturbations, )= ( k 148 IN IN ( σ 1+ ), the r.m.s. density fluctuation 1 + 1 = 0 ( ) can be approximated, for Ω ] [ ] [ A measurement of the cosmic microwave background damping ta  P a f exp ( M 2 A ( Non- in cosmology Dark matter and structure formation a review , A D )= × σ 2  ≃ M ) ( ) 27 the error is smaller than 7 . σ M collaboration, E. Komatsu et al., collaboration, D.N. 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