Faint Blue Galaxies and Merging the Evolution of the Luminosity Function

a r X i v :a s t r o -p h /9612162v 1 17 D e c 1996F AINT BLUE GALAXIES AND MERGING:

THE EVOLUTION OF THE LUMINOSITY FUNCTION

A.CAVALIERE 1,and N.MENCI

21Astro?sica,Dipartimento di Fisica II Universit`a di Roma,

via della Ricerca Scienti?ca 1,I-00133Roma,Italy.

2

Osservatorio Astronomico di Roma,

via dell’Osservatorio,I-00040Monteporzio (Roma),Italy.ABSTRACT.We probe to what extent not only the counts of the faint blue galaxies and their redshift distribution but also their z -resolved luminosity functions can be explained in terms of binary merging (aggregations).We present a dynamical theory of such interactions.On this basis we ?nd that “minimal aggregations”taking place within large scale structures and triggering star-bursts yield rates and timing such as to explain the observations of the local,?at lumi-nosity function and of its progressive rising and steepening for redshifts out to z ～1.Correspondingly,we predict faint blue counts still rising up to m B ～28?29,redshift distributions shifting toward larger z with increasing m B ;in addition,we predict an upturn of the faint end of the luminosity function more pronounced in clusters than in the ?eld.We propose that our picture provides the di?erential dynamics missing in the canon-ical hierarchical clustering theories.This reconciles with the observations the steep luminosity functions predicted at high z by such theories.

Subject headings:cosmology –galaxies:evolution –galaxies:formation.

1.INTRODUCTION

The observations of faint blue galaxies(FBGs)provided statistically signi?cant counts

in works by Koo(1986),Tyson(1988),and Cowie et al.(1988).Then an excess became

apparent,namely,by a factor≈5at magnitudes B～<24and≈7at fainter ones,over the values expected from normal local galaxies?lling a critical Friedmann-Robertson-

Walker universe.Over the last decade the growing data base concerning FBGs kept challenging the interpretations.

Various contributions to that excess have been discussed,the simplest being based

on improved measurements of the local luminosity function(hereafter local LF).These include the correction of the incompleteness by a factor～<1/2in the previous samples of local galaxies,as pointed out by Lilly et al.(1995)and by Ellis et al.(1996), and the consideration of the upturn at M B≥?16from a?atter Schechter shape pinpointed by Marzke,Huchra,&Geller(1994).Both these improvements in the local baseline increase,albeit moderately,the expected counts,since the latter are given by the convolution along the line of sight of the LFs weighted with the cosmological volumes.

Meanwhile,it became apparent that toward faint magnitudes blue galaxies make

up a fraction of all galaxies which increases both with decreasing luminosity and with increasing redshift z(Koo&Kron1992;Colless et al.1993).Thus,some sort of galaxy evolution must be at work.

Passive luminosity evolution driven by the shift with age of stellar populations would provide the observed blue counts only if helped by the larger volume contributed by widely open cosmologies with?o?1(see Pozzetti,Bruzual,&Zamorani1996).A similar result also holds in the red band down to K=24(Cowie et al.1994;Djorgovski et al.1996),though the excess is considerably smaller here.Such cosmologies,however, are out of tune with other evidence(summarized,e.g.,by Ostriker&Steinhardt1995) favoring a?at universe of higher density,i.e.,?o～>0.3with a cosmological constant λo=1??o.

On the other hand,a constraint was added with the advent of large spectroscopic

samples(Cowie,Songaila,&Hu1991;Lilly1993;Colless et al.1993;Glazebrook et al.1995)that provided redshift distributions N(z)with a median value as low as z ≈0.5-0.6for the identi?ed objects with B～<24.The low value of the redshift where N(z)is peaked rules out strong luminosity evolution in a?at universe;rather,for z～<0.5,the distribution resembles the behavior provided by passive-evolution models with sophisticated color mixes(Koo1990).The latter,on the other hand,would imply a very substantial high-z tail unless very speci?c recipes are implemented(see Gronwall &Koo1995),which imply mismatches at lower z(see Cowie et al.1996).

However,recently Lilly et al.(1995)and Ellis et al.(1996)have provided direct evidence of considerable evolution in the resolved luminosity functions over the range z≈0.1?1.This concerns only the blue objects,while the red ones appear to be already “in place”by z≈1,and quite similar to typical L?galaxies of the local kind mainly undergoing passive change.The observed evolution appears to be strongly di?erential,

being faster for the lower luminosities measured.

Two main interpretations have been proposed for the onset of such a blue,later lineage of galaxies.One(Babul&Rees1992;Babul&Ferguson1996)holds these to be very small and numerous lumps in which gas collapse and star formation have been delayed by the high level of the UV background prevailing before z≈1;shortly thereafter,the photoionizing UV?ux might decline sharply enough for this constraint to be lifted.Following the initial short starburst,these objects would rapidly fade away.

The other interpretation stems from the widespread local evidence of starbursts associated with galaxy-galaxy interactions(among the vast literature see the recent papers by Zabludo?et al.1996;Hutchings1996and references therein);this maintains that many more,if smaller,blue starbursts are triggered at increasing depths by tidal interactions taking place mainly with or between numerous dwarfs(Lacey&Silk1991). Some such interactions are likely to end up in galaxy-galaxy aggregations,i.e.,binary merging,also causing considerable number evolution.A phenomenological treatment of these e?ects has been outlined by Guiderdoni&Rocca-Volmerange(1991)and by Broadhurst,Ellis,&Glazebrook(1992).

A dynamical approach to the kinetics of aggregations has been given by Cavaliere& Menci(1993),motivated to complement the standard hierarchical clustering theory by its intrinsic inadequacy in two speci?c respects:it predicts too many small objects,and it envisages too little dynamics to erase them;so by itself it would set and preserve a steep luminosity function.A reconciliation between excess counts and a locally?at LF may be sought in terms of star formation processes(see White1994;Frenk,Baugh,&Cole 1995),but it still falls short of the full goal even on tuning several parameters.Cavaliere &Menci(1993)instead,on the basis of the dynamics of aggregations,predicted that N(M,z)and hence N(L,z)should?atten out with z decreasing in the range z～<1.

We are encouraged to pursue our dynamical approach further not only by the recent data concerning the z-resolved luminosity functions(which we now compare with)but also by some very recent lines of additional evidence.These include the following:a sizable fraction of FBGs at z～1show dynamical signs of ongoing or recent interactions,or the presence of blue satellites(Koo et al.1996;Van den Bergh et al.1996);some FBGs look like spirals with a red bulge(Koo et al.1996)and peripheral starbursts;widespread post-starburst galaxies mark the action of pervasive galaxy-galaxy interactions(Zabludo?et al.1996);measurements of clustered redshifts have shown the presence of at least some galaxy concentrations(Le F`e vre et al.1994; Koo et al.1996).

Here we will start from?rst principles and build up a dynamical theory of in-teractions containing no free parameters once one speci?es the environment where the interactions take place.Our main point will be that many such interactions can occur even at moderate redshifts(z～1)within large scale structures that o?er the best com-bination of volume and density contrast.Such interactions,with their mass-dependent cross sectionΣ(M),naturally yield a di?erential evolution.This a?ects the dwarf galaxy numbers yielding considerable density evolution,while the associated starbursts a?ect all blue luminosities yielding a milder luminosity evolution.In terms of observables,

we?nd counts still rising at very faint magnitudes,redshift distributions peaked at a value shifting from z≈0.5at B～<24toward z≈1for fainter magnitudes,and resolved luminosity distributions which that?atten out from z≈2toward a local Schechter-like shape.

The plan of the paper is as follows.In§2we present our dynamical framework to compute the mass functions evolving by aggregations.In§3we discuss the strong dependence of such evolution on the galaxy environment.We compare three kinds of environments:the homogeneous“?eld”;the bound,virialized structures;and the large-scale structures,which in fact will dominate the observables.In§4we describe how from the evolutionary mass distribution we compute the observables:the resolved luminosity distribution N(L,z),the blue counts N(m B)and the redshift distribution N(z).In§5 we give our results and predictions for these observables.In the?nal section we give and discuss our conclusions.

2.THE MASS FUNCTION FROM BINARY AGGREGATIONS

2.1The dynamics

A dynamical theory for galaxy evolution directly concerns the mass function N(M,t). The governing equation will be of the general form

?N

,(2.1)

τ

whereτ?1is an average evolutionary rate.

In the simplest case when the evolution of a galaxy is not a?ected by surrounding companions,the rate will not depend on the mass function itself.But this produces self-similar N(M,t)with faint-end slope unchanged in time.A well known instance is constituted by the Press&Schechter(1974)formula,corresponding to the choice ofτdiscussed by Cavaliere,Colafrancesco,&Scaramella(1991)and by Kitayama&Suto (1996).

Hence,we consider the case when the evolutionary rateτ?1does depend on the number of surrounding masses that interact and eventually aggregate.For a total density n g(t)of galaxies,this corresponds to consideringτ?1∝n g ΣV ,whereΣis the cross section for aggregation and V is the relative velocity of the two aggregating masses;the average runs over the galaxy velocity distribution.The requirement of total mass conservation(see Lucchin1988)requires a positive and a negative term on the right hand side of equation(2.1),and easily leads to the following Smoluchowski form for the kinetic equation:

?N

2 m0dm′K(m′,m?m′,t)N(m′,t)N(m?m′,t)

?N(m,t) ∞0dm′K(m,m′,t)N(m′,t).(2.2)

Here the kernel K(m,m′,t)=n g(t) ΣV corresponds to the interaction rate for each pair of masses m,m′,and contains the physics driving the evolution of the distribution N(m,t).The latter is written in its comoving form;the masses m≡M/M?are nor-malized in terms of M?,the mass corresponding locally to the standard characteristic luminosity L?(see Peebles1993).

Each galaxy aggregation comprises two stages:?rst,the dark matter halos coalesce; second,the baryonic cores may eventually coalesce.The former process requires grazing, weakly hyperbolic encounters for which the cross section is given by(Saslaw1986, Cavaliere,Colafrancesco,&Menci1992,hereafter CCM92)

Σ=?(V

r m V2 with r m=r?m1/3,v m=v?m1/3,(2.3)

where v?and r?are the velocity dispersion and dark-halo radius of the present M?galaxy.We use M?=51011h?1M⊙1;r?is conservatively taken to be40h?1kpc(see Persic,Salucci,&Stel1996),and v?=270km/s(three dimensional).The function ?(V/v m)describes the decreasing e?ciency of the aggregations for increasing relative velocities.Following the results from N-body simulations(see Richstone&Malumuth 1983)we will take?=1for V<3v m,and just?=0for larger values;we shall discuss the e?ect of such a cuto?in§5and§6.The second term(proportional to G)in the cross section describes the focusing e?ect of gravity,an important addition to the purely geometrical cross section proportional toπr2m only for large masses and in the range V<

√

1We parameterize the Hubble constant using H o=100h km/s/Mpc.

we think of aggregations as complementary to the standard hierarchical clustering(for which see Peebles1993),we focus on initial mass functions having the Press&Schechter (1974)form

N(m)dm=2a bδc

πM?

φ(m)dm,(2.4)

whereφ(m)=m?2+a exp(?b2δ2c m2a/2).Hereδc≈1.68,b?1～1?xes the amplitude, and a=(n+3)/6is related to the spectral index n～?2of the power spectrum for the density?uctuations from which galaxies form(see White1994).

In the same spirit,the typical galactic mass at the initial time M?(t in)is identi?ed with the characteristic mass M c(t in)provided by the standard hierarchical theory;when normalized to its present value[e.g.,M c(t o)=0.6×1015h?1M⊙for?=1],this reads (Lupini1995)

M c(t in)?

o

α?2

t2

in

+3λo

√M?(t

in

) φ(m)dm,(2.6) integrated from M?/500(the smallest mass considered);hereρco is the critical cosmo-logical density at the present epoch,and C makes explicit the contrast factor at t in of the environment where interactions take place.

Note that assuming the Press&Schechter expression(2.4)and the above values for the parameters is not mandatory;because of the very negative values of the spectral index n at galactic scales,the exponent of the power-law inφ(m)is always close to-2,a generic feature of the mass distributions also found in other variants of the hierarchical clustering scenario(see,e.g.,White1994).On the other hand,shortly after t in the integro-di?erential nature of equation(2.2)causes the solutions to lose memory of the detailed shape of the initial conditions.

2.3Interaction rates

The numerical value of the interaction kernel K=n g(t) ΣV determines the timescale for the aggregation https://www.doczj.com/doc/1617194488.html,puting n g as described above,and substituting?ducial

values for r?and for the initial relative velocity V in,we obtain an expression for K that we conveniently write in the form

K(m,m′,t)=310?4H o t o(m2/3+m′2/3) 1+v2?

V in

10kpc 2

1+z in 4,(3.3) where z in is the redshift corresponding to the initial time t in.

The corresponding rapid decrease of f(t)with time makes the aggregation rate rapidly negligible,so that the evolution of the mass function after equation(2.2)will

become mild at most.Nevertheless,for comparison purposes we compute it using a value for z in typical of the formation era of galaxies,according to equation(2.5)which in turn depends on?o,λo and on the perturbation spectrum.For a scale-free spectrum with n=?2.2we obtain z in=3.5when?=1andλo=0;and z in=4.5when ?o=0.2andλo=0.8.The initial V in=V o(1+z in)is given in terms of the local value V o for the pairwise galaxy velocity dispersion.

The latter was measured?rst by Davis&Peebles(1983)(see also Tormen et al. 1993)who found V o≈340km/s.Recently Marzke et al.1995combined the data from the CfA Redshift Survey and the Southern Redshift Survey and obtained a larger value ≈540km/s;however,when the Abell clusters with R≥1are removed from the sample, the pairwise“velocity dispersion of the remaining galaxies drops to V o=298km/s”. We adopt the value300km/s.Actually,such a value for V o is conceivably due to the present perturbed gravitational potential;so our scaling back as1+z in this subsection will optimistically maximize the e?ect,which will be shown to be inadequate anyway.

3.2Virialized structures

Following the standard hierarchical clustering,virialized structures like groups or clus-ters as environments for galaxy interactions may be characterized simply in terms of the redshift z at virialization.Then,the density contrast is～180,the galaxy density scales as n g(z)/n g(0)=(1+z)3,and the virial velocity dispersion as

V(z)/V o=(1+z)n?1

2t in

1+z in

2n+1

1+z in 2n+1

We start our computation at a redshift typical for formation of groups of galaxies,

z in=2.2for an index n=?1.3at the cluster scale.With this value,the initial V in is evaluated from equation(3.4)on the basis of a(three-dimensional)value V o=1270

km/s for clusters of richness1,to obtain V in=480km/s.The corresponding e?ective contrast from equation(3.7)is C≈45for?=1.

3.3Large scale structures

The large scale structures(LSSs;among the vast literature see De Lapparent,Geller &Huchra1992;Sahni&Coles1995;Vettolani et al.1996)constitute an interesting

environment for interactions to take place in;over the?eld they have the advantage of a larger density,and over groups and clusters the advantage of lower velocity dispersions.

However,they are quantitatively less understood as yet from both the observational and the theoretical points of view.

In the following,we shall use the semi-analytical approach proposed by Do-roshkevich et al.(1995)based on the Zel’dovich approximation and supported by nu-

merical N-body simulations(see again Doroshkevich et al.1995,and references therein). In such a picture,caustics form?rst in the matter?eld at z f≈4(for?=1),then LSSs build up by accretion of matter falling onto them.In this stage aggregations are

not e?ective,because most galaxies have either closely parallel motions or large relative velocities.

Aggregation can be e?ective in a subsequent stage when the thickness of the pan-cakes approaches half of their typical separation.This is the epoch z in when we start our calculations.We evaluate it taking up the results by Doroshkevich et al.(1995),for ?=1.For z～<3the mean comoving distance L between LSSs increases for decreasing z after

L1≈r c 8152)exp(?y2

1+z

1?

1+[(1+z t)/(1+z)]2?1 1/2 (Doroshkevich et al.1995),with z t≈0.2.From such an expression we?nd the comoving thickness of the structures to scale approximately as(1+z)1/2so that the contrast

behaves like C(z)∝(1+z)(D?3)/2,where D is the dimensionality of the LSS(D=2 for sheet-like structures,and D=1for?laments).In terms of the function f(z)de?ned in equation(2.7)this translates to

f(z)= (1+z)/(1+z in) (3+D)/2.(3.10) If the local contrast is≈5,consistent with observations(Geller&Huchra1989; Ramella,private communication)and with numerical simulations(Brainerd and Vil-lumnsen1991;Menci&Valdarnini1993),then from the scaling of C(z)we infer an initial contrast C=5(1+z in)(D?3)/2,with the value C≈3for D=2.

In this section we have de?ned the quantities which enter equation(2.2)for the computation of the evolving mass function in di?erent environments.A summary of the resulting numerical values is given in Table1.Next we describe how,from the mass function,we compute the observables like the luminosity function,the galaxy number counts and redshift distributions.

4.FROM N(M,t)TO OBSERVABLES

Our basic assumption in deriving the observables is that interactions also trigger star formation(see,e.g.,Zabludo?et al.1996,Hutchings1996,and references therein). Speci?cally,we conservatively assume that in the range of parameters where aggrega-tions are e?ective,they also induce and sustain star formation at a rate s(z)equal to the rate of dynamical interactionsτ?1(z)acting on the available gas(see Lacey&Silk 1991;Broadhurst,Ellis,&Glazebrook1992).

The associated luminosity for the dwarf galaxies is given by the relation L∝M gas/τ,considering thatτexceeds the dynamical time of a single galaxy and con-stitutes the limiting factor.Moreover,the deposition of energy from Supernovae will be initially balanced by cooling,but then for shallow potential wells it will go predom-inantly into gas expulsion.In this regime,even more directly than in White&Frenk (1991),the relation M gas∝M v2m is obtained.Sinceτ?1has the same M and t(or z) dependencies as given for K by equation(2.7),we obtain

L/L?= M/M? ηand L?(z)∝f(z,λo,?o),(4.1) whereη=4/3.Such a value applies when the cross section(2.3)is purely geometrical and is consistent with the observational results by Kormendy(1990).From equation (4.1)the luminosity function N(L)=N[M(L)]dM/dL is obtained.

Since our aim is to isolate the e?ects of dynamical evolution,we deliberately simplify colors and k-corrections to the bones.So we identify the prompt luminosity L with that in the blue band(see Broadhurst et al.1992),and in the blue spectral region at moderate z we assume a neutral k-correction(see Lilly,Cowie,&Gardner1991and Koo&Kron 1992).We take an absolute characteristic magnitude M B?=?20(see Lilly et al.1995; Ellis et al.1996).

The counts and the redshift distributions are found by convolving the comoving N(L,z)with the cosmological volume V,to obtain

N(m B)dm B= z max0dz d V dm B dm B,(4.2)

where the relation between L and m B is given,e.g.,by Guiderdoni&Rocca-Volmerange (1991),and the cosmological volume for a?at universe with?o+λo=1is given,e.g., by Peebles(1993).The range of integration over z extends out to the maximum z from which the magnitude m B can be observed.

5.RESULTS

In this section we present our results for the mass distribution,the z-depen-

dent luminosity function N(L,z),the counts N(m B)and the redshift distributions N(z) in di?erent magnitude ranges.These observables are derived from galaxy evolution driven by aggregations acting in the di?erent environments described in§3.

The plots we present are obtained through the following steps:

i)For a given galaxy environment we compute the aggregation rate given by equation

(2.7);the parameters C,t in(or z in),V in and the function f(?o,λo,t)are discussed in §3,and summarized in Table1.

ii)We numerically integrate equation(2.2)from the initial time t in to the present time t o.The time step is set at?t=(t?t o)/500while the mass step is M?/500.

iii)Following the equations recalled in§4we compute N(L,z),N(m B)and N(z). The results we obtain are compared with the observations concerning the faint blue counts,the z-distributions,and the z-resolved LFs.

5.1The homogeneous“?eld”

For comparison,we put on record the corresponding results,although this is not ex-pected to be an environment conducive to many late binary aggregations due to the rapid decrease of density and velocities.

Figure1a shows that such is indeed the case for?=1(λo=0).Because of the fast decrease of f(?o,λo,t)with time(see Table1),the interaction rate rapidly becomes small,so that N(M,t)hardly evolves from the initial shape.The corresponding LFs are shown in Figure1b.Any small evolution that occurs takes place at high redshifts, so that no appreciable change can be observed in the range z～<1.The faint counts in Figure1c are well?tted,which is not not surprising considering the steep LFs at all redshifts.But the z-distributions(Figure1d)are peaked at z≈1even at relatively bright magnitudes.

Thus,aggregations of galaxies uniformly distributed in a FRW universe are un-interesting,and clearly fail to account for the observed evolution in the LFs out to z≈1.

One might expect that the plateau(actually the in?ection)characteristic of the cosmic scale factor in a Lema?itre universe(λ=0)could provide an era of roughly con-stant density when the binary aggregation processes could proceed for a while.However, in a?at universe such processes are suppressed after the in?ection by the accelerated expansion.The net e?ect is still close to the case?=1,as can be seen from Figures 2a?2d computed for?o=0.2andλo=0.8.

5.2Virialized structures

A more favorable environment for galaxy aggregations is constituted by the bound, virialized structures like groups and clusters of galaxies.Here the large density contrast C(though convoluted with the limited survival time discussed in§3.2),and the milder time decrease of the interaction kernel f(?o,λo,t)yield a sustained evolution of N(M,t) (see Figure3a)compared to that found for the“?eld”.In Figures3b?3d we show the LFs,the counts,and the z-distributions expected if all galaxies resided within virialized structures for a considerable fraction of their lifetime.It is found that the evolution of N(L,z)consists of a rapid?attening in the sub-L?range,with relatively little change in the range from z≈1to the present.

It must be noted that in clusters,in spite of the larger value of the contrast C,the aggregation rate is suppressed not only by the limited survival time(see§3.2),but also by the increase with decreasing z of the velocity dispersion entering the cuto??(V/v m) in equation(2.3).The latter circumstance has the e?ect of decreasing the gravitational term inΣ,and even of shrinking to zero the whole cross section of the smaller galaxies with the lower internal velocities;so at the faint end the mass distribution will hardly be changed from the initial steep shape.The net result will be a LF with an upturn toward the faint end(see Figure3e).The balance between the e?ect of the sharp, mass-dependent decrease of the cross section and the general speeding up driven by the large e?ective contrast is a delicate one,and deserves an aimed study motivated by the LFs recently found(see Driver et al.1994)to be appreciably steeper in local clusters than in the“?eld”.

5.3Large scale structures

A result in a way intermediate between the two extremes above is provided by the environment constituted by the LSSs,where observations show most galaxies to reside not only locally(Geller&Huchra1989),but also at increasingly larger redshifts(see Broadhurst et al.1990;Schectman et al.1995;Vettolani et al.1995;Carlberg et al. 1996).For galaxies inside sheet-like structures(our case D=2)the parameters entering the aggregation rate are summarized in Table1,and yield the evolution of N(M,t) shown in Figure4a.The corresponding counts and redshift distributions(Figures4c, 4d)agree with the observations.In particular,they agree with the spectroscopic redshift distribution by Glazebrook at al.(1995)if L∝M4/3holds,and yield an enhanced high-z tail for a stronger dependence of L on M,e.g.,L∝M1.5.The luminosity functions

(Figure4b)show a clear?attening from z≈1toward a Schechter-like,local shape with α≈?1.1in agreement with the data.

Note that an upturn at faint luminosities should also be expected here;the mecha-nism is similar to that discussed at the end of§5.2,but the e?ect is considerably milder (and shifted toward fainter luminosities)because the relative velocities inside LSSs are and remain smaller than those in clusters.

De?ning a characteristic luminosity by the ratio of the second to the?rst moment of N(L,z),we?nd this evolves by about1from z=1to z=0,as also illustrated in Figure4a.Actually,the intrinsic luminosity change L?(z)∝(1+z)(3+D)/2(see eqs.

[4.1]and[3.10])is nearly balanced by the change following approximately(1+z)?1.5, due to characteristic mass after the relation L∝M4/3.The net result is a moderate change(about1Mag)of all luminosities,competing with the large change in number for setting the shape of N(L,z).The corresponding M/L ratio evolves(for an L?galaxy) as M?(z)?1/3(1+z)?(3+D)/2which yields M?/L?～(1+z)?(2+D)/2.

6.CONCLUSIONS AND DISCUSSION

We have derived for FBGs a luminosity function that changes considerably for increasing z as a result of number evolution and a milder luminosity evolution.In particular,in the observed range out to z≈1the luminosity function N(L,z)rises and steepens from a local logarithmic slope about?1.1to a value close to?1.6at redshift ≈0.8,as observed(see Figure4b,and below for a discussion).

This is derived from a dynamical theory of the galaxy mass function based on mass-conserving binary aggregations.The resulting evolution of N(M,t)is far from being self-similar;indeed,it is strongly di?erential with mass,causing for smaller galaxies stronger number changes and shift of the mass scale.This constitutes an important complement to the canonical hierarchical clustering,which in fact contains too little dynamics to account for the observed evolution of the luminosity function,as recalled in the Introduction.

The actual strength of the evolution induced by aggregations depends on the envi-ronment where these take place.In CCM92we stressed that in long-lived groups strong, gravitationally focused interactions of large galaxies occur,leading to the formation of cD-like mergers.Here,we?nd that in the large-scale structures—which comprise most local galaxies,are being observed out to z～1,and in a critical universe are expected to loom out at redshifts z～2—small-small and small-large aggregations are favored. Their strength is such as to match–with no special e?ort at optimization–the existing data for the resolved N(L,z)and for the integrated but deeper observables like counts and magnitude distributions(see Figures4a to4d).

A simple way of summarizing the interaction(and starburst)rates derived in§3.3is to note that they scale–di?erently than in the“maximal merging”model,see Carlberg

(1996)–likeτ?1=τ?1

o (M?/M)1/3(1+z)2.5.This is because in the rateτ?1=n gΣV

the scaling of the background density(1+z)3is partly o?set by the LSS contrast C(z)∝(1+z)?0.5;with mass,on the other hand,the scaling results from n g∝M?1

andΣ∝M2/3.Toomre(1977)has?xed the observed interaction probability t o/τo at

a few percent for interactions of bright local galaxies;we?ndτ?1

o =C n g(t o)ΣV o≈

1/500Gyr?1.At z≈1our probability t/τfor galaxies of5×109M⊙exceeds30 %.The bright career of this dwarf population activated by interactions is e?ectively started at z≈2when the large scale structures begin to loom out;they reshu?e the conditions set by the canonical hierarchical clustering including the initial correlations of linear perturbations,sustain contrasts larger than unity and produce non-linear relative velocities con?ned to one or two dimensions.This career terminates,due to the Hubble expansion,with a progressive fading for z～<0.5and with the number of sub-L*units reduced by～10?1.

The LSS features used in our computation–?lamentary(D=1)or sheet-like (D=2as considered above)structures produce similar results–come down to three key points:formation era around z in≈2,local density contrast≈5,and slow growth of the contrast from z in to the present.These features,which appear in the relatively local observations and also in all N-body experiments,are also incorporated in analytic interpretations di?erent from those of Zel’dovich’s(see Bond,Kofman,&Pogosyan 1996).We stress that all such structures are ultimately traced back to long-distance correlations,not considered in the homogeneous renditions of the hierarchical cluster-ing scenario,but actually intrinsic,and e?ective in bringing to interact galaxies that otherwise would not have met.

The detailed value of the initial contrast C is the only“free”parameter at present. In fact,its magnitude is constrained by its very de?nition to be of the order of a few,as we have used.As to?ner adjustments,its local distribution is currently being measured (Ramella,private communication);the analytic theory of its evolution is being re?ned, beyond the approximation used in§3.3(see Sahni&Coles1995);observations are reaching beyond z≈1(see Willinger et al.1996).

Our results are conserved when the shape of the initial mass function is changed from the detailed Press&Schechter form but is still steep as predicated by all hierar-chical clustering scenarios(cf.White1994).

The main trends are also robust with respect to variations of the M/L ratio from the value M/L∝M?1/3used in§§4and5,to include M/L∝const.and M/L∝M?2/3 as shown in Figure5.The di?erent M/L ratios tilt,but moderately,the shape of the distant LFs;correspondingly,the tail of N(z)for z>1rises to the values indicated by the data by Cowie et al.(1996)if the dependence of M/L goes toward M?2/3.

We also computed the counts in the K-band.These,as expected,increase consid-erably less than in the blue due to the long evolutionary times of the red stars and to the stronger k-correction.

One recurrent objection to the aggregation scenario concerns the correlation func-tion,as discussed by Efstathiou(1995).This is a delicate issue,as is shown in Figure 6.At the statistical level of the correlation function,aggregations with cross section less than linear in mass(likeΣ∝M2/3for nearly geometrical cross sections that ap-ply here,see§2)do not cause the slope of the initial correlation function to change.

A steepening instead takes place for the stronger,gravitationally focussed interactions

with non-linear cross sectionΣ∝M4/3.The whole issue may peter out,however,with the growing evidence for clustered redshifts at z≈0.8?1as observed and discussed by Le F`e vre et al.(1994)and by Koo et al.(1996).

The other recurrent objection to aggregations is that the spiral disks may be dam-aged by merging,as discussed by Ostriker(1990).In our framework,however,at the level of M?the mass growth is due to minor episodes of merging with the numerous but much smaller(and often more di?use)galaxies.On the other hand,the overall change of the characteristic mass seen in Figure4a is mainly due to the changing shape of the mass function,i.e.,to its?attening corresponding to the disappearance of small galaxies.This results in a mild statistical change in time of the second moment of the distribution,which does not imply a corresponding change in the masses of individual galaxies.For similar reasons,the population of ellipticals does not evolve appreciably by aggregations in the redshift range considered here.

To conclude,excess counts at B～<24may result from the combination of several e?ects:local shape and normalization of the luminosity function,long line-of-sight as-sociated with?o<1,passive evolution,and interactions as stressed here.The shape of N(z)is more telling,and at relatively bright magnitudes m B～<24requires some evo-lution,yet it rules out the pure luminosity kind in a?at universe.The resolved N(L,z) with its evolving shape restricts the issue to the following alternatives:luminosity-dependent luminosity evolution or a combination of(di?erential)density plus luminos-ity evolution,as naturally provided by aggregations.The steepening associated with the latter drives a speci?c di?erential shift of N(z),characterized by the peak marching on toward higher z at fainter magnitudes(see Figure4d);it also sustains the rise of the counts at very faint magnitudes(see Figure4c)even beyond b j=27measured by Metcalfe et al.(1995),probing in an integrated fashion the driver’s action at redshifts not yet reached by spectroscopy.

The above predictions stem from simple dynamics in which interactions produce aggregations and starbursts together.On the other hand,Cavaliere&Menci(1993) and Moore et al.(1996)point out that in clusters higher densities but larger relative velocities cause more frequent but weaker encounters(“harassment”)resulting mainly in pure starbursting and in blueing of the Butcher-Oemler(1984)type.In fact,such ?ybys and the true aggregations constitute two facets of a single process,di?ering as for the detailed form of the cross section(see equation2.3)and for the ratio s(z)/τ?1(z) of the star formation to the interaction rates.

These two facets stem from the same dynamics,and concur to stress the importance of interactions as drivers of galaxy evolution.Actually,the intermediate regime should also be observable in clusters,in the form of a local LF frozen with a steeper faint end or a more pronounced upturn,as shown by Figure3e.

We have bene?ted from many informative discussions with G.De Zotti,M.D’Onofrio, E.Giallongo,M.Ramella,R.Windhorst and G.Zamorani.We thank the referee for his helpful comments which improved our presentation.This work was supported by partial grants from MURST and ASI.

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FIGURE CAPTIONS

Figure1:Results for galaxies with M/L∝M?1/3interacting in the homogeneous “?eld”with?=1andλ=0.

Panel a):The solid lines show the comoving mass function at10equally spaced time intervals,from the initial time t in(dashed line)to the present.

Panel b):The corresponding luminosity functions in the B band for three redshift bins:z=0?0.2,0.2?0.5and z>0.5;the data are taken from Ellis et al.1996.

Panel c):The computed galaxy counts(solid line),compared with data from various authors:Maddox et al.(1990)solid circles;Jones et al.(1991),open triangles;

Tyson(1988),open circles;Lilly,Cowie&Gardner(1991),open stars;Metcalfe et al.(1995),solid triangles.

Panel d):The computed redshift distribution in the magnitude range22.5

Figure2:Same as Figure1but for?o=0.2andλo=0.8.

Figure3:Same as Figure1but for galaxies within virialized structures.

Figure3e:The e?ect of varying the position of the cuto?for the e?ciency?(V/v m)in the cross section(2.3).In this?gure?=0is taken for V>2.5v m rather than V>3v m as adopted throughout the paper.

Figure4:Same as Figure1but for galaxies within large scale structures with D=2.

In panel b)we show also our prediction for the luminosity function at z=1(dot-dashed line).In panel c)we show also our predictions for24

Figure5:The evolution of the luminosity function of interacting galaxies within large scale structures,computed for M/L=constant(left panel)and for M/L∝M?2/3 (right panel).

Figure6:The time evolution of the slopeγof the two-point correlation function for aggregating galaxies,as computed from a typical Montecarlo simulation(see Menci, Colafrancesco,&Biferale1993for details);γin is the initial value.The dotted line shows the result for a cross section scaling asΣ～M4/3,and the solid line that for Σ～M2/3.

-22-20-18-16-5-4-3-2-1

101214024681618202224d -1-0.500.5

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小学奥数之容斥原理

五．容斥原理问题 1．有100种赤贫.其中含钙的有68种,含铁的有43种,那么,同时含钙和铁的食品种类的最大值和最小值分别是( ) A 43,25 B 32,25 C32,15 D 43,11 解：根据容斥原理最小值68+43-100＝11 最大值就是含铁的有43种 2．在多元智能大赛的决赛中只有三道题.已知:(1)某校25名学生参加竞赛,每个学生至少解出一道题;(2)在所有没有解出第一题的学生中,解出第二题的人数是 解出第三题的人数的2倍:(3)只解出第一题的学生比余下的学生中解出第一题的人数多1人;(4)只解出一道题的学生中,有一半没有解出第一题,那么只解出第二题的学生人数是( ) A，5 B，6 C，7 D，8 解：根据“每个人至少答出三题中的一道题”可知答题情况分为7类：只答第1题，只答第2题，只答第3题，只答第1、2题，只答第1、3题，只答2、3题，答1、2、3题。 分别设各类的人数为a1、a2、a3、a12、a13、a23、a123 由（1）知：a1+a2+a3+a12+a13+a23+a123＝25…① 由（2）知：a2+a23＝（a3+ a23）×2……② 由（3）知：a12+a13+a123＝a1－1……③ 由（4）知：a1＝a2+a3……④ 再由②得a23＝a2－a3×2……⑤ 再由③④得a12+a13+a123＝a2+a3－1⑥ 然后将④⑤⑥代入①中，整理得到 a2×4+a3＝26 由于a2、a3均表示人数，可以求出它们的整数解： 当a2＝6、5、4、3、2、1时，a3＝2、6、10、14、18、22 又根据a23＝a2－a3×2……⑤可知：a2>a3 因此，符合条件的只有a2＝6，a3＝2。 然后可以推出a1＝8，a12+a13+a123＝7，a23＝2，总人数＝8+6+2+7+2＝25，检验所有条件均符。 故只解出第二题的学生人数a2＝6人。 3．一次考试共有5道试题。做对第1、2、3、、4、5题的分别占参加考试人数的95%、80%、79%、74%、85%。如果做对三道或三道以上为合格，那么这次考试的合格率至少是多少？ 答案：及格率至少为71％。 假设一共有100人考试 100-95＝5 100-80＝20 100-79＝21 100-74＝26 100-85＝15 5+20+21+26+15＝87（表示5题中有1题做错的最多人数）

实验四 血涂片的制备与染色

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高中生物实验所有染色剂及其性质详解(完整版)

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应用：用于DNA粗提取实验的鉴定试剂. 6 吡罗红使RNA呈现红色 原理：RNA+吡罗红→红色 应用：可以显示RNA在细胞中的分布. 注意：在观察DNA和RNA在细胞中的分布时用的是甲基绿和吡罗红混合染色剂,而不是单独染色. 7 台盼蓝使死细胞染成蓝色 原理：正常的活细胞,细胞膜结构完整具有选择透过性能够排斥台盼蓝,使之不能够进入胞内；死细胞或细胞膜不完整的细胞,胞膜的通透性增加,可被台盼蓝染成蓝色. 应用：区分活细胞和死细胞；检测细胞膜的完整性. 8 线粒体的染色 原理：健那绿染液是专一性染线粒体的活细胞染料,可以使活细胞中的线粒体呈现蓝绿色,而细胞质接近无色. 应用：可以用高倍镜观察细胞中线粒体的存在. 9 酒精的检测 原理：橙色的重铬酸钾溶液在酸性条件下与酒精发生化学反应,变成灰绿色. 应用：探究酵母菌细胞呼吸的方式；制作果酒时检验是否产生了酒精；检查司机是否酒后驾驶. 10 CO2的检测 原理：CO2可以使澄清的石灰水变混浊,也可使溴麝香草酚蓝水溶液由蓝变绿在变黄.

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