Kinematic effects of the velocity fluctuations for dark halos population in LCDM model
aa r X i v : . [ a s t r o - ph . C O ] J a n KINEMATIC EFFECTS OF THE VELOCITY FLUCTUATIONS FOR DARKHALOS POPULATION IN Λ CDM MODEL
E. P. Kurbatov
Institute of astronomy, Russian Academy of Sciences48 Pyatnitskaya st., Moscow, Russian Federation, 119017 [email protected]
Abstract Λ CDM-model predicts an excess of dark halos compared to the observations. Theexcess is seen from the estimates of the virialized mass inside the Local Superclusterand its surroundings. It is shown that account for cosmological velocity fluctuationsin the process of formation of the population of dark halos it is possible to eliminatethis contradiction remaining within the framework of the Λ CDM cosmology. Basedon the formalism of Press and Schechter, we suggest a model of formation of dark halopopulation which takes into account the kinematic effects in the dark matter. The modelallows a quantitative explanation for the observable deficit of the virialized mass in thelocal Universe.
1. Introduction
The inconsistencies between the predictions of the Λ CDM cosmological model and the ob-servations concern the distribution of the matter on the scales smaller than the scale of homo-geneity of the Universe. In particular, note the deficit of virialized matter in the local Uni-verse (Makarov and Karachentsev 2011). Counts of the mass of the galaxies, virialized groups, andclusters of galaxies in the Local Supercluster and its environs performed by many authors revealedthe lack of mass inside the virialized objects, amounting to a half of an order of magnitude (seethe references in Makarov and Karachentsev (2011)). According to the Makarov and Karachentsev(2011) catalog the estimate of the local density parameter inside a sphere of the radius Mpcis about . ± . . As a possible explanation of this deficit Makarov and Karachentsev (2011)suggested that about / of the matter is located outside the virialized areas. Instead, this darkmatter is either concentrated in dark clumps, outside the virial regions, or distributed diffusively.In the favor of the idea of the dark clumps tell observational data on weak lensing (Jee et al. 2005)and on disturbed galaxies (Karachentsev et al. 2006).One of the generally accepted ideas on the Large-Scale Structure formation in the Universe is thehierarchical model. Formation of the virialized halos population in the Λ CDM model is represented 2 –as a continuous process of condensing and clustering of the structures which develop from thedensity perturbations. Stochastic nature of this process is determined by the properties of theinitial cosmological fluctuations. As a result, a hierarchical halos structure forms, which consists ofgalactic clusters, virialized groups, field galaxies and low-mass satellite galaxies. Press and Schechter(1974) suggested a simple model for evolution of the dark halos mass function. Development of thismodel in later studies helped to solve the problem to a good approximation. The extension ofthe PS model known as the Excursion-Set or the Extended PS (EPS) formalism (Bond et al. 1991;Lacey and Cole 1993) is based on two assumptions: (i) the requirement for the perturbation to bevirialized at a given time which can be formulated in the terms of the density perturbation fieldat the linear stage of its evolution; (ii) the mass function of halos is formulated for those objectswhich are at the top of the hierarchical structure, i.e. they do not belong to other halos. In thisformulation, the problem can be solved using the linear perturbation theory only.In the formation process of the dark halos population the environment effects, which can leadto the loss of the halo mass or to the ejection of the sub-halos from the parent clusters, have anessential importance. In the numerical model of Diemand et al. (2007) was shown that the mostintensive loss of sub-halo happens before virialization of the parental halo. An already formed halocan accumulate up to of its mass by the accretion of surrounding dark matter.Despite of the quite obvious role of the environment effects, they are not the only ones whichare able to influence the halo population. For instance, Valageas (2012) have shown that the velocityfluctuations in the dark matter (the author considered the Warm Dark Matter model, WDM) canaffect the density fluctuations statistics on scales . . h − Mpc. Valageas (2012) did not findany considerable effect of the velocity fluctuations for the dark halos population. It should beemphasized however that the calculations were constrained to corrections to the power spectrumonly. More, approach used in the latter study required modification of the Λ CDM-model.An interesting problem is the possibility of direct kinematic effect of the velocity fluctuations onthe process of the formation of the individual halos as well as their population. It is well known thatthe velocity fluctuations grow along to the density fluctuations (Crocce and Scoccimarro 2006). Dueto the moment conservation, the halo during formation process inherits velocity of the dark matteraveraged over its scale. When this halo becomes involved in the formation process of the largerscale gravitational condensation, its velocity may be large enough to leave the parental structure.This effect may change the history of evolution of dark halos, the dark halos population itself, aswell as the history of the chemical evolution of the galaxies.In this paper, a method for accounting for the kinematic effects in the dark halos populationprocess is suggested. The evolutionary model of the population is formulated on the base of theExcursion-Set theory (Bond et al. 1991; Lacey and Cole 1993) in terms of kinetic equation. Also,the problems of the choice of the initial conditions and of the influence of the background structureare considered.In § 2 the EPS model is briefly examined. In § 3 formation model for individual halo with 3 –kinematic effects is suggested and the kinetic equation for the mass function is obtained. In § 4 themodel is applied to the problem of the virialized mass deficit in the local Universe. Conclusions arepresented in § 5.
2. Formation of the halo population in the EPS model2.1. Growth of the cosmological perturbations
In the Λ CDM model the Large-Scale Structure of the Universe is formed from the growingfluctuations of density and velocity in the cold dark matter. Evolution law of the structures withcharacteristic mass m can be linked to the properties of the overdensity field δ ≡ ( ρ − ρ cr , ) /ρ cr , averaged over the volume containing mass m . The averaging is defined as a convolution of the fieldwith a filter W : δ ( a, x , m ) = Z d x ′ W ( x ′ − x , m ) δ ( a, x ′ ) , (1)where a is the scale factor, which is related to the redshift z as a = 1 / ( z + 1) ; m is the halo mass.The same in the terms of the Fourier transform is : ˜ δ ( a, k , m ) = (2 π ) / ˜ W ( k , m ) ˜ δ ( a, k ) . (2)The ” top-hat “ filter will be adopted hereinafter: W ( x , m ) = 34 πX θ ( X − x ) , (3) (2 π ) / ˜ W ( k , m ) = 3 sin kX − kX cos kXk X , (4)where X = [3 m/ (4 π Ω m , ρ cr , )] / .The random fluctuations field is assumed to be Gaussian with zero mean and delta-correlatedover Fourier modes, h ˜ δ ( a, k ) ˜ δ ∗ ( a, k ′ ) i = δ D ( k − k ′ ) P ( a, k ) , (5)where δ D is the Dirac’s delta-function, P ( a, k ) is the overdensity power spectrum at the momentcorresponding to the scale factor a . Thereby, over spatial scale associated with the given mass m the overdensity field can be characterized by only the variance value, which is independent of spatialposition: S ( a, m ) ≡ h δ ( a, x , m ) i = Z d k | ˜ W ( k , m ) | P ( a, k ) . (6)Velocity divergence fluctuations field θ ≡ ∇ x ( a ˙ x ) has the same properties: h ˜ θ ( a, k ) ˜ θ ∗ ( a, k ′ ) i = δ D ( k − k ′ ) P θ ( a, k ) , (7) The Fourier transform is defined as ˜ f ( k ) = (2 π ) − / R d x e i kx f ( x ) . P θ ( a, k ) is the power spectrum of the field θ ( a, x ) . Let denote as v ≡ a ˙ x the field of thephysical velocity determined relatively to the expanding Universe. We neglect the vortical part ofthe velocity fluctuations. In this case, the velocity field is determined by the divergence field only: h ˜ θ ( a, k ) ˜ θ ∗ ( a, k ′ ) i = X j,j ′ k j k ′ j ′ h ˜ v j ( a, k ) ˜ v ∗ j ′ ( a, k ′ ) i = δ D ( k − k ′ ) X j k j h| ˜ v j ( a, k ) | i . (8)Let define as P v ( a, k ) ≡ h| ˜ v j ( a, k ) | i the power spectrum of the velocity in the given spatial direction(the velocity distribution is assumed to be isotropic). Then the power spectrum of the field v willbe P v ( a, k ) = P θ ( a, k ) k , (9)where k = | k | .At large redshifts the overdensity and velocity divergence amplitudes evolve according to thelinear law (Peebles 1980; Crocce and Scoccimarro 2006): ˜ δ ( a, k ) = DD i ˜ δ ( a i , k ) ≡ D ˜ δ L ( k ) , (10) ˜ θ ( a, k ) = aHf Da i H i f i D i ˜ θ ( a i , k ) ≡ aHf D ˜ θ L ( k ) , (11)where H = H ( a ) is the Hubble parameter, D = D ( a ) is the linear growth factor, f ≡ d ln D/d ln a ,index ”i“ marks the initial time. Hereinafter, by the index ”L“ we will denote the values reducedto the unit growth factor, i.e. not depending on the redshift. For linear approximation for theamplitudes, the variances are subject to the quadratic growth law: S ( a, m ) = D S L ( m ) (12) S v ( a, m ) = D v S v, L ( m ) , (13)where D v ≡ aHf D . Physically, interesting is the situation when the initial overdensity and velocitydivergence perturbations are proportional to each other. It can be shown (Crocce and Scoccimarro2006) that in this case ˜ δ ( a i , k ) = − ˜ θ ( a i , k ) a i H i f i , (14)i. e., the power spectra of the fields ˜ δ L ( k ) and ˜ θ L ( k ) coincide.Due to the momentum conservation, the forming halo as a whole inherits the velocity of theproto-halo matter, namely, velocity, averaged over proto-halo mass scale v ( a, x , m ) = Z d x ′ W ( x ′ − x , m ) v ( a, x ′ ) . (15)Let assume, that a proto-halo with mass M had velocity V ( a, X , M ) . Then the variance of aparticular spatial component of the velocity v j ( a, x , m ) of a sub-halo with the mass m , to the first 5 –approximation will depend neither on position of the sub-halo inside the proto-halo nor the velocityof the proto-halo, but only on the variance of the velocity field on the scale M : S v ( a, m, M ) ≡ D v [ S v, L ( m ) − S v, L ( M )] . (16)In the present paper the Λ CDM model parameters obtained by the Planck mission Planck Collaboration et al.(2013) were adopted: h = 0 . , Ω m , = 0 . , Ω Λ , = 1 − Ω m , = 0 . , σ = 0 . , n S = 0 . .The power spectrum of the overdensity fluctuations was calculated for these parameters by meansof the code CAMB (Lewis et al. 2000) for wavenumbers from − Mpc − to Mpc − (on-lineinterface to the CAMB is presented in the LAMBDA project 2013). Outside of this range, the the-oretical power spectrum was used with the power index n S and the transfer function (Bardeen et al.1986, p. 60) with the shape parameter Γ ≡ h Ω m , = 0 . Mpc − . Plots of the overdensity andvelocity variances calculated in such a model are shown in the Fig. 1. The linear growth factor wasadopted from Bildhauer et al. (1992) using normalization D ( a ≈
0) = a . − − − − S L M/M ⊙ S v , L / M p c Fig. 1.— Dependence of the variance of overdensity fluctuations (upper panel) and the variance ofvelocity fluctuations (lower panel) on the mass scale. Closer the centers of the volumes M and m are to each other, the better this approximation is (Hoffman and Ribak1991). See also (Kurbatov 2014). The EPS model (Peacock and Heavens 1990; Bond et al. 1991) can be formulated with theexcursion set approach as follows. Consider the overdensity fluctuations field δ ( a, x , m ) with themass scale m . If the perturbation is spherical, it can be argued that its amplitude on the linearstage is determined by the initial amplitude only (Peebles 1980), or, equivalently, by δ L ( x , m ) . Thisfeature can be used to choose such perturbations which collapse and virialize to the halo stage by agiven redshift. It’s necessary to note, however, that some already virialized halos may be absorbedby more massive ones. Therefore, the only initial perturbations accounted in the mass function forcertain redshift should be the most massive ones with required virialization time.These ideas may be represented as a typical problem of theory of the stochastic processes. Astochastic process δ ( S ) starts from the point δ = 0 with the parameter S = 0 , which correspondsto m = ∞ . Its evolution obeys equation dδdS = η , (17)where η = η ( S ) is a Gaussian random process with zero mean, unit variance, and the correlationfunction h η ( S ) η ( S ′ ) i = δ D ( S − S ′ ) (Maggiore and Riotto 2010a). The mass m corresponding tothe value of the parameter S , when a certain threshold δ c /D ( z ) is reached by the process δ ( S ) forthe first ” time “ S , is interpreted as the mass of the halo virialized by the given redshift z . Thedistribution function of these masses is the halo mass function sought for.The value δ c is the linear amplitude of the spherical overdensity perturbation (10) collapsing atthe moment z . In the spherical collapse model (Peebles 1980) this value reduced to the unit growthfactor is approximately . . In our case δ c = 1 . D ( z = 0) . (18)The resulting halo mass function can be represented as a probability density (Bond et al. 1991;Lacey and Cole 1993) f (0) ( z, S ) = ω √ πS exp (cid:18) − ω S (cid:19) (19)or the cumulative distribution function F (0) ( z, < S ) = Z S dS ′ f (0) ( z, S ′ ) = erfc (cid:18) ω √ S (cid:19) , (20)where notation ω ≡ δ c /D ( z ) is introduced. Hereafter the index ”L“ will be omitted as the redshift dependency will be denoted explicitly. In models of the EPS class the linear growth factor D is usually normalized to take unit value at z =0 (Lacey and Cole 1993). In the present paper the alternative normalization is used: D ( z ) ≈ / ( z + 1) at z ≫ . Forcosmological parameters adopted here (see previous Sec.) we have D ( z = 0) ≈ . . S or the mass of a halo with a largest formation rate at the given redshiftcorrespond to the maximum of the function ∂f (0) /∂ω and thus obeys equation ω = (3 + √ S .Most of the halos formed per unit time has the variance in the interval log S = log[ ω / (3+ √ ± . .At large redshifts, the interval can be estimated as S ≈ (0 . . . . .
5) ( z + 1) . (21)It should be noted that the process δ ( S ) defined by Eq. (1) for an arbitrary filter strictlyspeaking does not comply to Eq. (17) where the r.h.s. is delta-correlated. That is, the processis not Markovian. Moreover, the spherical collapse model is quite a rough approximation, neitherany possible deviations from the spherical symmetry in the initial conditions nor tidal action ofenvironment are considered. These issues can be resolved by corrections to the excursion set for-malism (Bond et al. 1991; Maggiore and Riotto 2010a,b). However in the present paper we base onthe simple case of the excursion set theory.
3. Kinematic effects in the dark matter3.1. Formation of a single halo
Transition of a spherical perturbation into a halo proceeds in four stages: initial expansion,detachment from the Hubble flow, compression, virialization (Peebles 1980). The moment of de-tachment or turnaround point is located approximately in the middle between the start of theexpansion and the virialization, when the formation is completed (Peebles 1980). In the spheri-cal collapse model, the halo formation is described by motion of disjoint layers or test particles inthe gravitational field of a point mass. At that, the mass enclosed inside the layer is conserved.For the perturbation to become collapsed at a given redshift z f , its linear overdensity must growas (Lacey and Cole 1993) ∆( z ) = D ( z ) D ( z f ) δ c D ( z = 0) . (22)It was mentioned above that sub-halos have random velocities relatively to the parent halo.Let consider the possibility that the velocity of the sub-halo is large enough to leave the proto-halo.Let’s write an expression for mechanical energy of a test particle inside the parent proto-halo usingphysical coordinates, and neglecting the Λ -term: E = ˙ r . (23)At large redshifts, the overdensity amplitude of the parent proto-halo is low, thus the densitydistribution inside the parent proto-halo may be considered as uniform. Gravitational potential inthis case is Φ = − GMR (cid:18) − r R (cid:19) , (24) 8 –where M is the proto-halo mass; the scale R is defined as a physical radius of the proto-halo at theturnaround, containing the mass M : M = 4 π ρ cr , Ω m , (1 + ∆) R a . (25)We formulate escape condition for the particle as the constraints for the energy value, E > : ( a ˙ x + aHx ) > H Ω m , a X (1 + ∆) (cid:18) − x X (cid:19) , (26)where X = R/a , x = r/a , x X , a < a f . Condition (26) defines the lower bound of the velocitysufficient to leave the proto-halo: ˙ x > − Hx + (cid:20) H Ω m , a X (1 + ∆) (cid:18) − x X (cid:19)(cid:21) / . (27)Let’s estimate the escape probability. Assume the following: (i) the test particle’s location isnear the edge of the proto-halo; (ii) the proto-halo collapses at a high redshift ( H ≈ H Ω / , a − / );(iii) the amplitude of the proto-halo is low ( ∆ ≪ ). Then the escape condition (27) takes the form ˙ x > HX ∆ . (28)Sub-halos served as the test particles in our formulation of the problem. Let’s multiply the lastinequality by the scale factor a , then substitute into the l.h.s. the most probable value of the relativevelocity modulus of the sub-halo inside the proto-halo, assuming the Gaussian distribution with zeromean and the variance (16). Also, let express X via the proto-halo mass. In the high- z limit, thecondition (27) takes the form [ S v ( m ) − S v ( M )] / & z f + 1) (cid:18) M M ⊙ (cid:19) / Mpc . (29)It can be shown (see Appendix) that in not very strict conditions ( m → , M < M ⊙ ) the l.h.s.of the last inequality can be estimated as [ S v (0) − S v ( M )] / ≈ . { . . . } (cid:18) M M ⊙ (cid:19) / Mpc , (30)where expression in braces is slowly changing decreasing function of M (it becomes close to when M = 10 M ⊙ , and when M = 10 M ⊙ ). Adopting (29), we get (27) in the form { . . . } & z f + 1 . (31)This inequality formally shows that for any z f there are perturbations which masses are small enoughto obey the escape condition for its sub-halos. It is necessary to remember, however, that for anyredshift there is a certain mass interval in which most of the halos form, see Eq. (21). For sub-halos 9 –escape to have an effect upon the population formation process, the proto-halo mass must satisfyboth Eqs. (21) and (29). This requirement can be represented as S v (0) − S v ( M )( M/ M ⊙ ) / Mpc & S ( M ) . (32)Let’s estimate both sides of this inequality in the low-mass limit. For simplicity let use the ” k-sharp “ filter for which the Fourier image is ˜ W ( k , M ) = θ (1 − kX ) . Then the overdensity and the velocityvariances are S ( M ) ∝ Z X − ( M )0 dk k P ( k ) (33)and S v (0) − S v ( M ) ∝ Z ∞ X − ( M ) dk P ( k ) . (34)Calculating the differentials, it’s easy to show that ddS S v (0) − S v ( M ) M / ∝ − XP ( X − ) Z ∞ X − dk P ( k ) , (35)where the proportionality factor is a positive constant. We assumed also k ≡ X − . Defining thepower spectrum as P ( k ) ≡ Π( k ) /k , by formal integration of the r.h.s we obtain ddS S v (0) − S v ( M ) M / ∝ X Π( X − ) Z d Π ′ X ′ > . (36)If Π is a constant (this is true in a rough approximation) then the r.h.s. of (36) is zero. Ina more strict approximation we have Π ∝ ln ( k/ Γ) i.e. slowly growing function of the wavenum-ber (Bardeen et al. 1986). The conclusion is: the lesser proto-halo mass is, the lesser the probabilityfor its sub-halos to escape. Given that the formation of the structures with time goes from the lowmasses to the larger ones, it can be argued that in the high redshift limit the sub-halo escape doesnot affect the formation of the dark halo population.Let’s denote as σ v ≡ [ S v ( m ) − S v ( M )] / the standard deviation of the 1-D velocity (reduced tothe unit growth factor) of a sub-halo m relatively to a proto-halo M . The probability for modulusof the Gaussian distributed sub-halo velocity to exceed some v is P v ( > v ) = erfc (cid:18) v √ σ v (cid:19) + 2 √ π v √ σ v exp (cid:18) − v σ v (cid:19) . (37)Given the uniform spatial distribution of the sub-halos at the early stage of the proto-halo evolution,the probability for sub-halo m to escape proto-halo M formed at the redshift z f is χ ej ( z f , M, m ) ≡ X Z X dx x P v ( > v ej ) , (38)where a threshold velocity value v ej is the r.h.s. of (27) reduced to the unit growth factor: v ej = − aHD v x + (cid:20) H Ω m , aD v X (1 + ∆) (cid:18) − x X (cid:19)(cid:21) / . (39) 10 –Distribution of χ ej for the sub-halos of low masses is shown in Fig. 2. Comparing the position ofcontour lines and the mass interval of proto-halos formed by a given era (the mass interval is boundedby thick dashed lines), one can see that for the formation redshifts z f . the mass fraction ofescaping sub-halos of low masses is approximately constant for all z f and can be estimated as ,roughly. The same is true in the case when the sub-halo mass is one-third of that of the proto-halo,but the escaping mass fraction is about (see Fig. 3). Thereby, a significant fraction of thesub-halos can escape the parent proto-halo, thus avoiding the absorption. − − − − −
10 0 10 20lg(
M/M ⊙ )020406080100120140160180200220240260280300 z f . . . . . . Fig. 2.— Isolines of the probability of ejection of sub-halos as a function of proto-halo mass (hor-izontal axis) and redshift (vertical axis). The probability is estimated using Eq. (29) for sub-halomasses m → . The solid curve marks the era of the most intensive formation proto-halo of a givenmass. The primary mass interval of the proto-halos formed at a given epoch is bounded by thedashed lines. In the standard excursion set approach a mass inside each collapsed perturbation is conservedduring the collapse and equals to the subsequent halo mass. This suggestion makes realizations ofthe random process δ ( S ) independent up to some small corrections noted above. As it became clear,proto-halos may loss a significant fraction of the sub-halos resulting in transition of the sub-halosto proto-halo of larger scales which undergo collapse at later times. Consequently, the values of δ ( S ) will correlate both in scales and redshifts. This fact makes it difficult to use the excursion setformalism explicitly.Greater clarity can have an approach based on the kinetic equation, when the evolution ofthe mass function explicitly presented as a result of the absorption process of the sub-halo by the 11 – − − − − −
10 0 10 20lg(
M/M ⊙ )020406080100120140160180200220240260280300 z f . . . . . . Fig. 3.— Same as in the Fig. 2, but the sub-halo masses are one-third of the proto-halo mass.large-scale perturbations. Let’s write the total probability of a halo formation at the moment ω ,the mass of which corresponds to the interval ( S , S + dS ) , and another halo ( S , S + dS ) at themoment ω , given conditions S > S and ω > ω : p (0) ( ω , S ; ω , S ) dS dS = f (0) ( ω , S | ω , S ) f (0) ( ω , S ) dS dS = (40) = f (0) ( ω , S | ω , S ) f (0) ( ω , S ) dS dS , (41)where f (0) ( ω, S ) is the PS mass function (19). Reducing the total probability, one obtains Z ∞ S dS p (0) ( ω , S ; ω , S ) = f (0) ( ω , S ) , (42) Z S dS p (0) ( ω , S ; ω , S ) = f (0) ( ω , S ) . (43)These relations suggest a possible form of the kinetic equation: f (0) ( ω, S ) − f (0) ( ω + ∆ ω, S ) == Z ∞ S dS ′ p (0) ( ω + ∆ ω, S ′ ; ω, S ) − Z S dS ′ p (0) ( ω + ∆ ω, S ; ω, S ′ ) , (44)i.e. − ∂f (0) ( ω, S ) ∂ω == lim ∆ ω → ω (cid:20)Z ∞ S dS ′ p (0) ( ω + ∆ ω, S ′ ; ω, S ) − Z S dS ′ p (0) ( ω + ∆ ω, S ; ω, S ′ ) (cid:21) . (45) 12 –There is a ” minus “ sign in the l.h.s. of Eq. (45) because the ω parameter decreases with the time.Terms in the r.h.s. are the source and the sink respectively.According to the total probability formula (40, 41) the source and the sink can be written indifferent ways. We choose the way which leads to the most familiar form of kinetic equation: − ∂f (0) ( ω, S ) ∂ω = lim ∆ ω → ω (cid:20) Z ∞ S dS ′ f (0) ( ω, S | ω + ∆ ω, S ′ ) f (0) ( ω + ∆ ω, S ′ ) − (46) − f (0) ( ω + ∆ ω, S ) Z S dS ′ f (0) ( ω, S ′ | ω + ∆ ω, S ) (cid:21) . (47) ” Transfer cross section “ here is the conditional PDF f (0) ( ω, S | ω + ∆ ω, S ′ ) , and the unknown func-tion is f (0) ( ω + ∆ ω, S ) . Since the random process δ ( S ) is considered as markovian, it can beargued (Lacey and Cole 1993) that f (0) ( ω , S | ω , S ) = f (0) ( ω − ω , S − S ) . Expression forconditional PDF f (0) ( ω , S | ω , S ) can be obtained via the formula of total probability (40,41).Let’s modify the source and the sink of (46) in a way to allow the mass ejection from the forminghalos. Consider the probability f (0) ( ω, S | ω + ∆ ω, S ′ ) dS , which is a mass fraction of fluctuationshaving variance range from S to S + dS and collapsing by the time ω , after each absorbs halos S ′ , which existed before by the time ω + ∆ ω . As the sub-halos S ′ may escape the proto-halo, thisfraction may decrease.Let denote as f ( ω, S ) the unknown mass function and to assume that the statistics of thefluctuations is not affected by the sub-halos ejection occurring in other proto-halos (those whichcollapsed earlier). In this case, the product [1 − χ ej ( ω, S, S ′ )] f (0) ( ω, S | ω + ∆ ω, S ′ ) f ( ω + ∆ ω, S ′ ) dS gives contribution to the mass of the halo S by the sub-halos S ′ , which could not leave the parent.Then the mass fraction of all the halos that form during interval ∆ ω is P + ( ω + ∆ ω, ω, S ) dS ≡≡ dS Z ∞ S dS ′ [1 − χ ej ( ω, S, S ′ )] f (0) ( ω, S | ω + ∆ ω, S ′ ) f ( ω + ∆ ω, S ′ ) . (48)Here f ( ω + ∆ ω, S ′ ) is the mass function of the halos formed by the time ω + ∆ ω . Expression (48)is the first component of the source. It corresponds to the mass redistribution after growth andvirialization of the perturbations. Besides that, we need to take into account those of the sub-haloswhich leave the proto-halos and return to the population: P + , re ( ω + ∆ ω, ω, S ) ≡ Z S dS ′′ χ ej ( ω, S ′′ , S ) f (0) ( ω, S ′′ | ω + ∆ ω, S ) f ( ω + ∆ ω, S ) . (49)This term must be incorporated into the source with negative sign.The sink also has two components: P − ( ω + ∆ ω, ω, S ) ≡ f ( ω + ∆ ω, S ) Q − ( ω + ∆ ω, ω, S ) , (50) 13 – P − , re ( ω + ∆ ω, ω, S ) ≡ f ( ω + ∆ ω, S ) Q − , re ( ω + ∆ ω, ω, S ) , (51)where Q − ( ω + ∆ ω, ω, S ) ≡ Z S dS ′′ [1 − χ ej ( ω, S ′′ , S )] f (0) ( ω, S ′′ | ω + ∆ ω, S ) , (52) Q − , re ( ω + ∆ ω, ω, S ) ≡ Z S dS ′′ χ ej ( ω, S ′′ , S ) f (0) ( ω, S ′′ | ω + ∆ ω, S ) . (53)After all substitutions, the kinetic equation has the form: − ∂f∂ω = lim ∆ ω → ω [ P + − P + , re − ( P − − P − , re )] . (54)It’s easy to show by evaluation that Z ∞ dS ( P + + P − , re ) = Z ∞ dS ( P − + P + , re ) = 1 . (55)Expressions (54) and (55) reveal that the normalization of the PDF f is conserved with the time.Note also that P + , re = P − , re . Let’s finally rewrite the equation (54) using Taylor expansion in ∆ ω and obtain then a recurrent procedure for calculation of the mass function: f ( ω, S ) = f ( ω + ∆ ω, S ) + P + ( ω + ∆ ω, ω, S ) − f ( ω + ∆ ω, S ) Q − ( ω + ∆ ω, ω, S ) + O ( | ∆ ω | ) . (56)Neglecting the quadratic residue, the recurrent relation (56) can be written as f ( ω, S ) = Z ∞ dS ′ r ( ω, S | ω + ∆ ω, S ′ ) f ( ω + ∆ ω, S ′ ) , (57)where r ( ω, S | ω + ∆ ω, S ′ ) = θ ( S ′ − S ) q ( ω, S | ω + ∆ ω, S ′ ) + (58) + δ D ( S − S ′ ) − δ D ( S − S ′ ) Z S ′ dS ′′ q ( ω, S ′′ | ω + ∆ ω, S ′ ) , (59)and q ( ω, S | ω + ∆ ω, S ′ ) ≡ [1 − χ ej ( ω, S, S ′ )] f (0) ( ω, S | ω + ∆ ω, S ′ ) . (60)The representation (57) is useful for numerical solution of the kinetic equation by Monte-Carlomethod. One step of the solution procedure may look like this:(a) generate the random value S ′ with initial PDF f ( ω, S ′ ) ;(b) generate the random value S according to the transition PDF (58);(c) redefine ω − ∆ ω ω , S S ′ and go to (b). 14 –A cumulative transfer distribution function is more convenient for this algorithm than the PDF: Z S dS ′′ r ( ω, S ′′ | ω + ∆ ω, S ′ ) = Z S dS ′′ q ( ω, S ′′ | ω + ∆ ω, S ′ ) , S S ′ , S > S ′ (61)Examples of how the sub-halo ejection affects the halo mass function are shown in Fig. 4. Inthese calculations, the escape probability was assumed constant. Also the EPS mass function wasused as the initial PDF. It is clearly seen that for larger escape probability χ ej the ” tail “ of the lowmass end (high S values) of the mass function becomes heavier , i.e. the evolution of the populationeffectively slows down as χ ej increases. − − − − − − S f ( S ) . . . . . − S . . . . . . F ( < S ) . . . . . Fig. 4.— The mass function (PDF and the cumulative) for the case χ ej ≡ const . The rightmostdashed line designates the initial distribution at z = 100 for all runs. The leftmost dashed linedesignates the final distribution z = 0 ) for the case χ ej = 0 . The intermediate curves correspond tovarious χ ej (see the legend on the plot). 15 – In the kinetic approach presented above it is possible to use an appropriate arbitrary initialmass function. Thus, the problem of the choice of the initial conditions arises. According to themodern cosmological theory, matter-dominated era started at z ≈ × (Gorbunov and Rubakov2011). According to the formal relation (21), structures with the variances S ∼ . . . or masses log( m/M ⊙ ) ≪ − formed at that era (see Appendix). It is obvious, however, that the masses ofthe structures can not be lower than the mass of a hypothetical dark matter particle. If we take thevalue m DM = 100 GeV ∼ − M ⊙ as the mass of the dark matter particle (this value correspondsto S DM ≈ ), then the earliest structures may form at the redshifts as low as z ∼ (see Eq.(21)), and it is unlikely for structures to form at the higher redshifts. Finally, recalling the sub-haloescape model, we see that the sub-halos can not escape at the redshifts greater than (see Fig. 2).Considering all these restrictions, we assume the redshift value z i ≡ as the initial moment forall calculations below. As the initial mass function we assume the cumulative distribution function F ( ω i , < S ) = ( F (0) ( ω i , < S ) , S S DM , S > S DM , (62)where ω i = δ c /D ( z i ) . Note that F (0) ( ω i , < S DM ) ∼ − .In Fig. 5 the mass functions in the model without sub-halo ejection ( χ ej = 0 ) are plotted forthe different redshifts. The initial mass function (62) peaks at the greater masses relatively to thepeak of the EPS mass function (at z = 300 the EPS peaks at S ∼ ), so the mass function atthe subsequent times is also shifted to the higher masses. However, at z . the massive end ofthe distribution coincides with the EPS model. In general, the low mass end of the mass functionis suppressed at all redshifts, because of absence of the structures of mass m DM and lower.Obvious consequence of using the initial mass function suggested in this section is that theformation of the structures with masses lower than m DM is impossible. This property may be usedfor the analysis of the resolution effects of cosmological codes built on top of the N-body or gridmethods. The lowest possible mass scale in such applications should correspond to the spatialresolution of the numerical method. Effect of the low density background structure (the supercluster or the void) can be consideredin our model. Such a structure may be set in terms of the statistical constraints for the field of theoverdensity fluctuations (Kurbatov 2014). Applying the constraints leads to considerable change instatistics of modes of the fluctuations. E.g. the statistics is not more spatially uniform, the pertur-bations field ” feels “ the size and shape of the background structure. Thus, the halo mass functionalso changes. In the EPS theory the effects of the background was considered first by Bond et al.(1991); Bower (1991), and investigated further by Mo and White (1996); Sheth and Tormen (1999) 16 – − − − − − − − − − − m f ( m ) − − − − − m/M ⊙ . . . . . . F ( > m ) Fig. 5.— Mass function (PDF and the cumulative) for the case χ ej = 0 and the initial conditionsEq. (62). The dotted lines designates the PS mass functions. Five epochs are represented in theplot (from left to right on both panels): z = 250 , z = 150 , z = 50 , z = 15 , and z = 0 .and others. When deviation of density of the background structure from uniform distribution islow, and the structure is large enough in all directions, the effect of the background presence canbe accounted by a simple approach. Let’s denote S B ≡ S ( m B ) the variance of the overdensityfluctuations averaged over the background structure’s mass scale m B , then the variance of thefield over the mass scale m ≪ m B placed deep inside the background structure is approximately S ( m ) − S B (Hoffman and Ribak 1991; Kurbatov 2014). The same considerations are true for thevelocity fluctuations field also (this fact was already utilized when the escape criterion (16) wasobtained).Bond et al. (1991); Bower (1991) have shown that the large scale background structure can beeasily applied to the PS mass function Eq. (19) by formal substitutions of the kind ω ω − ω B , S S − S B , (63) 17 –where ω B is the background overdensity calculated via the linear law (10), and reduced to the unitgrowth factor: ω B = D ( z = 0) D ( z i ) ∆( z i ) = δ c D ( z f ) . (64)The second equality in this expression allows to define an overdense structure which collapses bythe moment z f . The equivalent description can be reached in the kinetic approach proposed in theprevious section, if the substitutions (63) are made in the transfer PDF f (0) ( ω, S | ω + ∆ ω, S ′ ) .The same substitutions may be utilized in the model with kinematic effects. It should benoted however, that the substitutions (63) must not be applied to escape probability χ ej ( ω, S, S ′ ) ,because in fact the probability depends on the redshift and the halo mass, but not the overdensitythreshold and the overdensity variance. Moreover, the velocity variance already enters Eq. (38), asthe difference, hence its value, does not change as the velocity field varies, if these changes have theform (63). Finally, the transfer cross section is q ( ω, S | ω + ∆ ω, S ′ ) = [1 − χ ej ( ω, S, S ′ )] f (0) ( ω − ω B , S − S B | ω − ω B + ∆ ω, S ′ − S B ) , (65)where ω B ≡ δ B /D . Note that the initial conditions (62) must not be transformed by (63).
4. Virialized matter in the local Universe
Mass counts of the galaxies, virialized groups of galaxies, and clusters, made by many authorsreveal the deficiency of the mass inside virialized objects, up to the factor three (see references inMakarov and Karachentsev (2011)) compared to predictions of Λ CDM model. According to thecatalog Makarov and Karachentsev (2011), local density parameter of the matter Ω m ≡ M m , tot πρ cr , D , (66)may be estimated as . ± . . Here M m , tot is the total mass of the matter inside a sphere of radius D . The authors of the catalog proposed several explanations of the deficiency. The most plausibleof them is the suggestion about possibility that the essential part of the dark matter in the Universe(about / ) is scattered outside the virial or collapsing regions, being distributed diffusively orconcentrated in dark ” clumps “ (Makarov and Karachentsev 2011). Certain evidence for existence ofthe dark clumps may be provided by the observations of weak lensing events (Natarajan and Springel2004; Jee et al. 2005), and the properties of the disturbed dwarf galaxies (Karachentsev et al. 2006).The estimate Ω m = 0 . ± . was obtained by Makarov and Karachentsev (2011) using thetotal mass of all virialized groups of galaxies having velocities up to km / s. To estimate thedistances and masses the authors used Hubble parameter’s value km / s Mpc. Thus the distance D can be estimated as / ≈ Mpc, and the estimate of the total mass of the matter (given Ω m , = 0 . ) is . × M ⊙ . The value of the Hubble parameter adopted in the present paperis H = 67 . km / s Mpc, whence D = 52 Mpc, and the total mass is thus M m , tot ≈ . × M ⊙ . 18 –The variation of the Hubble parameter causes just a minor correction to the estimated galacticmasses, about . The estimate of the local matter density parameter changes more significantlyand gives (when the masses of galaxies are corrected also) Ω m = 0 . ± .
017 = (0 . ± . m , .The claims of Makarov and Karachentsev (2011) were based on observational data of the virializedgroups having masses above ≈ × M ⊙ , and containing at least two galaxies. Completenessof the sample was . The mass distribution function of the virialized groups (see Fig. 8 inMakarov and Karachentsev (2011)) showed decrease of the number of objects with masses lowerthan M ⊙ . This can be caused by the absence of the field galaxies in the sample. In the furtheranalysis we will use just a fraction of the sample composed by the objects as massive as M ⊙ and above. The reason is that this fraction should more closely match the mass distribution of thedark matter gravitational condensations.According to EPS distribution, the total mass fraction of the dark matter halos more massivethan M ⊙ is M m ( > M ⊙ ) /M m , tot ≈ . , i.e. more than / of all the dark matter settles inthe halos with masses below M ⊙ (see Fig. 6). Calculation of the model presented in Section 3.2,performed neglecting the sub-halo escape but using the initial conditions (62) leads to approximatelythe same distribution as the EPS predicts (dashed line on Fig. 6). The observational data (thicksolid line on Fig. 6) show the deficit of the virialized halos, up to the factor two and a half comparingto the EPS prediction: M m ( > M ⊙ ) /M m , tot ≈ . .Accounting for the effect of sub-halos escape in the presented model, gives the cumulativefraction of all the massive halos about . , i.e. slightly lower than observed. The most part ofthe matter is not incorporated into the structures as massive as M ⊙ and above but becomesdistributed nearly uniformly in structures with masses greater than − M ⊙ (see Fig. 7). Quitea good agreement between the observed and the theoretical distribution functions of the massivehalos is mainly due to the fact that the differential distribution functions for both cases nearlycoincide at the masses from × M ⊙ and above (see the upper plot in Fig. 6), whereas twoleft bins on this plot show twice as large deficit of the halos predicted the theoretical model sug-gested in this study. While the uncertainties in the estimates of the groups’ masses can be as highas (Makarov and Karachentsev 2011), the average values of the observed mass function aresystematically higher than those of the theoretical one. This may be explained in two ways:(i) Average matter density in the vicinities of the local Supercluster seems to be higher thancosmological value (Makarov and Karachentsev 2011). As a consequence of this, the massivetail of the halo mass PDF should be heavier (Mo and White 1996; Kurbatov 2014).(ii) The proposed effect of the mass decrease of proto-halos may be overestimated as we did not ac-count for, e.g., accretion of the matter on the proto-halo from its environment. Diemand et al.(2007) have shown that halo may accrete up to of mass after virialization has ended, i,e.,accretion may partially compensate the escape process leading to effective decrease of theescape probability χ ej . 19 – ∆ N m/M ⊙ . . . . . . . F ( > m ) Fig. 6.— Mass function of the virialized structures with masses > M ⊙ :tMakarov and Karachentsev (2011) (thick solid line), the model with sub-halo ejection (thinsolid line), the model without ejection, but with the initial conditions Eq. (62) (dashed line), thePS model (dotted line).In general, a good agreement between the observed mass function (Makarov and Karachentsev2011) and predictions of the model proposed in this paper should be noted.
5. Conclusions
In the present paper the influence of the dark matter random velocities on formation of thedark halos population is considered. It is shown that this kinematic effect leads to the escape of thesignificant fraction of sub-halos (up to ). However, at the redshifts greater than the escapeis negligible. At the moderate and small redshifts the escape probability is higher for sub-halos oflower masses.We suggested a model of the dark halo population formation, based on the EPS theory. Themodel was built in the frame of kinetic approach where the source and the sink explicitly describe 20 – − − − − − − − − − − m f ( m ) − − − − − m/M ⊙ . . . . . . F ( > m ) Fig. 7.— Mass function (PDF and the cumulative) calculated with Eq. (38). Lines indicate thesame as in Fig. 5.the hierarchical merging process and the kinematic effects, and potentially may be used to accountenvironmental effects. Special initial conditions were proposed to use with the kinetic equation wherethe lowest possible halo mass exists corresponding to the dark halo particle’s mass. The consequencesof these initial conditions are systematic shift of the low-mass end of the mass distribution functiontowards high masses. However, this shift is almost invisible in the high-mass end at lower redshifts.The consequences of the sub-halo escape is the redistribution of the massive halos toward lowermasses. If no escape is included, the EPS mass function is the invariant solution of the kineticequation.The model developed in this paper allowed to explain quantitatively the observable deficit ofthe virialized objects in the Local Universe, at least for groups of galaxies with masses greater than M ⊙ . The missing matter is distributed over the low-mass halos down to the lowest limit.We need to note that the kinetic approach proposed in this paper, with the initial conditionslimiting the lowest possible halo mass, may be used for analysis of the resolution effects in thecosmological numerical codes. 21 –
6. Appendix A
Let us obtain an approximate expression for the overdensity variance reduced to unity scalefactor: S ( M ) = Z d k | ˜ W ( k , M ) | P ( k ) . (67)Assume ” k-sharp “ filter with the Fourier image ˜ W ( k , M ) = θ (1 − kX ) , where X = (3 M/ π Ω m , ρ cr , ) / .Then the variance is S ( M ) = 12 π Z X − M dk k P ( k ) . (68)The overdensity power spectrum for large wavenumbers may be written as (Bardeen et al. 1986) P ≈ Ak n S ln (2 . k/ Γ) k , (69)where constant A is defined by normalization σ ; Γ = h Ω m , Mpc − . Assume M ≪ M ∗ , then S ( M ) ≈ S ( M ∗ ) + A π (1 − n S ) Γ − n S (cid:20) (1 − n S ) ln κ + 2(1 − n S ) ln κ + 2 κ − n S (cid:21) (Γ X ∗ ) − (Γ X ) − . (70)Making all substitutions, we obtain S ( M ) ≈ .
29 + 10 (cid:2) . − (cid:0) .
001 lg M − .
062 lg M + 2 (cid:1) M . (cid:3) , (71)where M ≡ M/ M ⊙ .The variance of the relative velocity of a low mass sub-halo inside the parent halo can beestimated in a similar way. The variance on a mass scale M reduced to the unit growth factor is S v ( M ) = Z d k | ˜ W ( k , M ) | P ( k ) k = S v (0) − π Z ∞ X − dk P ( k ) . (72)The relative velocity variance for low-mass sub-halos is then S v (0) − S v ( M ) ≈ A π Γ − n S (3 − n S ) ln κ + 2(3 − n S ) ln κ + 2(3 − n S ) κ − n S (cid:12)(cid:12)(cid:12)(cid:12) κ =(Γ X ) − . (73)Assuming that the power spectrum has unit spectral index n S = 1 , after all substitutions we have: [ S v (0) − S v ( M )] / ≈ . (cid:2) .
178 log M − .
55 log M + 3 . (cid:3) / M / , (74)The expression in the square root term is a slowly changing monotonic decreasing function of theproto-halo mass. Approximately, it is for M = 10 M ⊙ and unity for M = 10 M ⊙ . 22 – REFERENCES
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