Cosmological Perturbations and the Running Cosmological Constant Model
aa r X i v : . [ a s t r o - ph . C O ] M a r Cosmological Perturbations and the Running Cosmological Constant Model
Alan M. Velasquez-Toribio
Departamento de Fisica - ICE,Universidade Federal do Espirito SantoCEP: 29075-910, ES, Brazil (Dated: Received: date / Revised version: date)We study the matter density fluctuations in the running cosmological constant (RCC) model usinglinear perturbations in the longitudinal gauge. Using this observable, we calculate the growth rateof structures and the matter power spectrum, and compare these results to
SDSS data and theavailable data for linear growth rate. The distribution of collapsed structures may also constrainmodels of dark energy. It is shown that the
RCC model enhances departures from the Λ
CDM model for both cluster number and cumulative cluster number predicted. In general, increasing thecharacteristic parameter ν leads to significant growth of the cluster number. We found that thetheory of perturbations provides a useful tool to distinguish between the new model RCC and thestandard cosmological model Λ
CDM . PACS numbers:
I. INTRODUCTION
Recent results from both type Ia supernovae [1, 2] and the Wilkinson Microwave Anisotropy Probe (
W M AP ), inparticular, the five-year data [3, 4], indicate that the present Universe is accelerating and that it has negligible spatialcurvature. In this context, non-relativistic matter contributes about 30% (dark matter plus baryonic matter) of thecritical density of the Universe and the remaining 70% of the energy density is not known and is called dark energy.Dark energy is generally associated with a cosmological constant ( CC ) and can be physically equivalent to the vacuumenergy. This component breaks the strong energy condition, but it is the simplest model that one can build. Thismodel, although satisfactory from an observational point of view, is th,eoretically disfavored because there is a hugedifferent between the predicted and observed values of CC [5, 6]. Other possibilities have been considered, amongthese, the most popular are based on scalar fields, known as quintessence models [7 ? –13], and models based onscalar fields with non-canonical kinetic energy, called k-essence models [14]. Unfortunately, the change in the scalarfield may be extremely slow and there is a degenerescence of the potential of quintessence (and k-essence). Anotherapproach is to consider dark energy as an effect of modified gravitation. Some models of this class are, scalar-tensortheory [15], F ( R ) theories [16], and models that introduce extra-dimensions, such as the DGP model [17, 18]. Arecent review of models of dark energy is given in [19, 20].In practice, dark energy can be seen into of the Friedmann equations through the relation between its energydensity and pressure. This ratio is known as the equation of state (
EOS ) parameter, w ( z ) = P ( z ) /ρ ( z ). The function w ( z ), where z is the redshift, is a key quantity in attempting to understand the dynamics of cosmological expansion.Another important quantity in the cosmological background is the deceleration parameter q(z). These two are themost commonly used functions in studies of observational constraints using background data. In particular, thedeceleration parameter can be used in a model-independent approach (see [21] and [22]).The next step in the study of a cosmological model is to use linear perturbations. In fact, the behavior of linearperturbations in a scalar field and their effect on large scale structure formations has been investigated by manyauthors, e.g. see [23]. Also the behavior of nonlinear gravitational collapse has been investigated [24, 25]. Thesestudies are fundamental to understanding and discriminating among competing models.In this paper, we investigate the cosmological consequences of a model motivated by quantum field theory QF T ,specifically in the renormalization group. The idea of studying renormalization group effects as a way of solving the CC has been explored previously [26–38]. In particular, a model quadratic for the running of the CC was presentedin [39, 40], called the running cosmological constant (RCC). Additionally, this model was extended to the logarithmicrunning of gravitational coupling G [44]. The quadratic model of the running of the CC was solved in reference [41–43].In this model, the energy density of the vacuum ρ Λ ( z ) can be given as a quadratic function of the expansion rate,which allows one to unambiguously define the equation of conservation of energy and determine the hubble parameter H ( z ) as function of the redshift. Reference [45] generalizes this model to run for CC and G . A possible fundamentalrelation with QF T was first proposed in reference [46] for running CC and a logarithmic run of the gravitationalcoupling. Let us also mention the reference [47] where the quadratic evolution law of the CC is also emphasized fromthe point of view of QF T in curved space-time using the effect of zero-point fluctuations in a FLRW background.This model is confronted with the latest accurate observational data in [48]. Additionaly, it is worthwhile to notethat the paper [49] reaches similar conclusions for the evolution of the CC from the point of view of supersymmetrictheories.The model of the running of the CC is equivalent to models of dark energy in the fluid approximation (parameteri-zations of the EOS and quintessence models [50]). The background cosmology of this model has been well investigatedwith data from type Ia supernovae, restricting the values of parameter ν , which represents the ”run” of the CC inthe RCC model. A recent review of the class of these models within QFT in curved space-time is given in [52] and[53]. Additionally, reference [54] determined the matter power spectrum in the synchronous gauge.We investigate matter density perturbations using the longitudinal gauge. This approach was pioneered in [55, 56].Their formalism is applied to our
RCC model to determine the linear growth rate and the matter power spectrum asfunctions of parameter ν . We compared the results to observational data from the SDSS (Sloan Digital Sky Survey).Recently, the running of the CC and of gravitational coupling using the longitudinal gauge was studied [63]. Somecomments between this and our work are given in the conclusion section.On the other hand, it has been recognized in various papers and simulations [57–61] that the evolution of thecluster number counts can determine the properties of dark energy. Therefore, we use our results for linear densityperturbations and the Press-Schechter formalism to determine the mass function, the number counts, and the cumu-lative number counts as functions of redshift, and study the sensitivity of parameter ν . We use the Λ CDM model tocompare our predictions. Additionally, we investigate how the number counts and cumulative number counts dependson the parameters Ω m and h when the value of ν is fixed. Similar quantities were studied in [62]. Some considerationswill be made in the text.Our paper is organized as follows. In section II, we introduce the RCC model and discuss the behavior of thecomoving energy density and deceleration parameter. In section III, we discuss the linear perturbations of matter.Section IV is devoted to the computation of cluster number counts in the Press-Schechter formalism. In section V,we present our conclusions. In the appendix, we display results for the
DGP model and scalar-tensor theory, both ofwhich are useful when calculating the linear growth rate.
II. RUNNING COSMOLOGICAL CONSTANT
In this section, we will introduce our cosmological model by considering a
F LRW metric and a Universe of matter(baryonic + dark) and dark energy or vacuum energy; thus, the cosmological evolution is governed by the followingFriedmann equation: H = 8 πG ρ m + ρ Λ ) − ka , (1)where ρ m and ρ Λ are the densities of matter and energy, respectively, a is the scale factor, H = ˙ aa is the Hubbleparameter, and k is a constant that can take the values +1 , , −
1. In this investigation, we restrict ourselves to anevaluation of the flat case. Using the Bianchi identity, we can write the conservation law: dρ m dz + dρ Λ dz = 3 ρ m z . (2)In the above equation, we can see that there is an exchange of energy between the two components of the Universe,so both are completely linked. The dark matter can be subject to an energy exchange resulting in both a time-dependent mass and a modification of its equation of state. Therefore, in this model, the dynamic of the Universe isdominated both by the evolution of the CC and by its interaction with dark matter.On the other hand, an important issue to be noted is that equations (1-2) do not form a complete set of equationsbecause there are three free variables, ρ m ( z ) , ρ Λ ( z ) , H ( z ) and two equations. We need a third equation to havea complete system. In this paper, we introduce an additional equation by considering the effects of applying therenormalization group to the CC . In this framework, the energy density in the RCC model is [39, 41, 44]: dρ Λ d ln H = σH M (4 π ) . (3)The above equation was proposed based on the assumption that the renormalization group scale µ is identified with H ( z ). This scale was originally proposed in [39] and can be considered as purely phenomenological ansatzs. M is aneffective mass parameter representing the average mass of the heavy particles in the grand unified theory (GUT) nearthe Planck scale, after taking into account their multiplicities. The coefficient σ can be positive or negative, the signdepends on whether bosons ( σ = +1) or fermions ( σ = −
1) dominate in the loop contributon, this is, it depends onwhether fermions or bosons dominate at the highest energies. Recall that, in the renormalization group framework,equation (3) is interpreted as a ” β -function” of QF T in curved space-time, and that it determines the running of P H z L FIG. 1: Comoving background matter density as a function of redshift in the RCC model. We can see that the decrease indensity is an indicator of the coupling between dark matter and vacuum energy. Note that, in this plot, Λ
CDM correspondsto a constant line equal to one. From bottom to top: ν = 0 .
25 (orange), = 0 .
065 (red), = 10 − (blue). In all models we usedΩ m = 0 .
24 and h = 0 . the CC , for other details see [51]. Therefore, in the background (cosmology of zeroth order), the three equationsabove allow a full description of the evolution of the RCC cosmological model. Using equation (3), we can obtain oneexplicit solution for ρ Λ : ρ Λ = ρ Λ0 + 3 ν π M p (cid:0) H − H (cid:1) , (4)where ρ Λ0 and H are the current values of these parameters. Additionally, in this model, we find a new parameter ν (dimensionless) that is given by ν ≡ σ π M M p , (5)thus, from equation (4), if ν is zero, the effects of a run are canceled and we recover the standard model, Λ CDM .Based on the above equations, equation (1) can be rewritten as H H = 1 + (cid:18) Ω m − ν Ω k − ν (cid:19) (cid:18) (1 + z ) − ν − − ν (cid:19) + Ω k ( z + 2 z )1 − ν , (6)where H Ω k (1 + z ) = − ka and Ω m = Ω MD + Ω B , where Ω DM is the parameter for dark matter and Ω B isthe parameter for baryonic matter. This expression for the Hubble parameter is valid for Universes with positive ornegative curvature. Therefore, equations (1), (4), and (6) define our cosmological model. One of the first quantitiesthat we can calculate is the matter density. We define the comoving matter density function as Π( z ) = ρ m ( z )(1+ z ) . InFigure 1, we displayed it. There is a decrease in the density caused by the coupling between dark matter and darkenergy. Increasing the coupling leads to a faster decrease in the density. In Figure 1, the constant line corresponds toΛ CDM ( ν = 0). Note that this behavior is similar to that of coupled quintessence models [64]. We used only positivevalues for parameter ν . For negative values, the density Π is greater than one.Another important parameter in the cosmology of the background is the deceleration parameter q ( z ) = − − H (1 + z ) ddz ( H ). In Figure 2, we show this parameter for our RCC model. When z >
1, in all cases (negative or positive)the deceleration parameter is positive and tends rapidly toward a matter dominated phase. For values close to z = 0,the deceleration parameter can not distinguish between positive and negative values of ν . In the next section, weconsider linear perturbations and structure formation. - - q H z L FIG. 2: The plot of deceleration parameter q ( z ) in the RCC model. We can see that the deceleration parameter is slightlysensitive to the change of sign of the parameter. We used Ω m = 0 .
24 and h = 0 . III. THE LINEAR PERTURBATION EQUATIONS
The theory of cosmological perturbations is based on the expansion of Einstein’s equations to linear order aroundthe background metric. The decomposition theorem states that perturbations in the metric can be divided into threetype: scalar, vector, and tensor. In this paper, we will consider scalar perturbations, specifically, density perturbationsin the
RCC model because we are interested in the matter power spectrum. Another fundamental question is thechoice of gauge or coordinate system. General Relativity leads to the question of gauge freedom. This means that ifwe change the coordinate system we use, we would get a metric of a different form. One way of dealing with the gaugeproblem is to eliminate gauge dependence entirely. This approach is referred to as using gauge-invariant variables andwas pioneered by Bardeen [65]. However, in the literature, many other gauges have been used [68]. For example, thedynamics of density perturbations in the
RCC model were investigated in the synchronous gauge [54]. In the presentwork, we are extending the analysis of [54] by computing the matter power spectrum in the longitudinal gauge andwe will study the cluster number counts.The longitudinal gauge is, from the physical point of view, much more intuitive because metric perturbations aresimilar to Newtonian perturbations. This gauge is commonly chosen for work in CMB and gravitational lensing.On the other hand, conceptually, this gauge fixes all spurious degrees of freedom, and the two scalar potentials Ψand Φ that appear in the line element correspond to Bardeens gauge invariants [56, 65]; this is, however, not thecase of the synchronous gauge where there exist residual transformations that lead to the appearance of unphysicalsolutions. However, the use of this gauge is justified because these spurious modes are canceled to some extent whencalculating a physical observable, which, by definition, cannot depend on a given system of coordinates. The metricin the longitudinal gauge is given by [55]: ds = − (1 + 2Ψ( ~x, t )) dt + (1 + 2Φ( ~x, t )) dx j dx j . (7)It should be noted that the longitudinal gauge is restricted to scalar modes; nonetheless, it can be easily generalizedto include the vector and tensor degrees of freedom [66]. Further, in the absence of anisotropic stress, one of Einstein’sequations gives Ψ = − Φ; the two gravitational potentials are equal and opposite [67]. Therefore, there remains onlyone free metric perturbation variable, which is a generalization of the Newtonian gravitational potential. This justifiesthe name of Newtonian gauge.To derive the equations for the density perturbations, we follow the standard formalism [66–68]. We consider theentropy perturbation to be negligible and the energy-momentum tensor to be free of anisotropic stresses. Thus, inthese conditions, the energy-momentum tensor has the form of a perfect fluid, T µν = pg µν + ( ρ + p ) U µ U ν , (8)where U µ = dx µ / ( − ds ) / is the four-velocity of fluid, p is the pressure, and ρ is the energy density of a perfect fluid.For a fluid moving with a small velocity v i ≡ dx i /dτ (peculiar velocity), the v i can be treated as a perturbation ofthe same order either as δρ = ρ − ¯ ρ or as δp = p − ¯ p . The quantities ¯ ρ and ¯ p refer to the background. In linear order,the perturbations of the energy-momentum tensor that we used are given by: T = − (¯ ρ + δρ ) ,T i = (¯ ρ + ¯ p ) v i = − T i , (9) T ij = (¯ p + δ ¯ p ) δ ij . The perturbed four-velocity is U α = ((1 − Ψ) , v i ) . (10)Although one can directly work with the Einstein’s equations, it turns out to be convenient to use the equationsof motion for the matter variables because we are ultimately interested in matter perturbations. We consider theconservation of the energy-momentum tensor as T αβ ; α = ∂T αβ ∂x α + Γ αδα T δβ − Γ γβα T αδ = 0 . (11)This expression gives us two equations, one for β = 0 and the other for β = i . To determine these equations, weconsider the perturbations for the two densities and the metric ρ m → ρ m (1 + δ m ) ,ρ Λ → ρ Λ (1 + δ Λ ) , (12) g µν → g µν + h µν . For the sake of simplicity, we want to make full use of the symmetry under spatial translations; this can best beexploited by working with the Fourier components of perturbations. Therefore, we have f ( ~x ) = Z d k (2 π ) e i~k.~x F ( ~k ) . (13)With these considerations, we use the line element (equation 7), equations (9) and (12) put into equation (11), andwe get ˙ δρ m + ˙ δρ Λ + ρ m ( θ + 3Ψ , ) + 3 Hδρ m = 0 , (14)˙ ρ m θ + ρ m ˙ θ − k Ψ2 + 3 Hρ m θ = − ( ka ) δρ Λ , (15)where the dot is the derivative with respect to cosmic time and θ = ∂ i v i . The energy density in the RCC model, ρ Λ ,can be written as [54] ρ Λ = A + B ( ∇ µ U µ ) , (16)where we have used the fact that ∇ µ U µ = 3 H and defined A = ρ Λ0 − ν π M p H and B = νM p π . It is important tonote that the expression above is not unique; although, it has a compact form and is useful for investigating linearperturbations. The forms of the terms A and B have specific forms as functions of the ν parameter. Additionally, theform of (16) is the simplest quadratic form, that is, a constant plus constant by quadratic form. It is important tonote that other papers have used different methods for constructing the perturbation equations (see [62],[90]). Usingthe perturbed four-velocity, equation (10), the perturbed Christoffel symbols, and keeping only the linear order term,we find δ Λ = 2 Hν [ θ −
3( ˙Ψ + H Ψ)] ρ Λ ( z ) . (17)By contrast, the Einstein equations in the longitudinal gauge are k Ψ + 3 H ( z )( ˙Ψ + H ( z )Ψ) = − πGa ( δ m ρ m ( z ) + δ Λ ρ Λ ( z )) , (18)˙Ψ + H ( z )Ψ = − πGaρ m θ. (19)Substituting equation (20) into equation (19), we obtain a new equation for the scalar potential as a function of thevariables of matter k Ψ = 3 Hρ m θ k (1 + z ) − ( δ m ρ m + δ Λ ρ Λ )2(1 + z ) . (20)It is convenient for our numerical calculations to write equations (15), (16), (18), and (21) in terms of the redshift,i.e., using dt = dzH (1+ z ) . Further, it is advantageous to use the following two ratios [54], f ( z ) = ρ m ρ t = (1 + z )2 H ( H ′ ) − H Ω k (1 + z ) H − H Ω k (1 + z ) ) , (21) f ( z ) = ρ Λ ρ t = 3 H − H (1 + z ) H ′ − H Ω k (1 + z ) H − H Ω k (1 + z ) ) , and ˙Ψ − H Ψ = ρ t v k (1 + z ) , (22) θ − vf + 3 H (cid:18) Ψ + ρ t v z ) Hk (cid:19) , (23)where ρ t ( z ) = 3 H ( z ) − H Ω k (1 + z ) . Using equations (22-24), the system of equations for the perturbations canbe written as δ Λ ( z ) = νvHρ t f [ 1 f − ρ t k (1 + z ) ] , (24a) δ ′ m ( z ) + δ m ( z ) " f ′ f + 3 f z − z ) + 3 f (1 + z ) δ Λ + 3Ψ(1 + z ) + vH (1 + z ) (cid:18) f + 3 ρ t k (1 + z ) (cid:19) + 2 νf ( vf ′ + f v ′ ) ( K ( z ) + M ( z )2 ) + 2 νf vf (cid:18) K ′ ( z ) + M ′ ( z )2 (cid:19) = 0 ,v ′ ( z ) + 3( f − v z = k (1 + z ) H (cid:18) δ Λ f − Ψ9 H (cid:19) , (24b) k Ψ = ρ t z ) (cid:20) vk (1 + z ) − ( δ m ( z ) f + δ Λ ( z ) f ) (cid:21) , (24c)where the prime refers to the derivative with respect to redshift, v = f θ , and the M ( z ) and K ( z ) terms of theequation (24b) are given by M ( z ) = 1 f k (1 + z ) , (25) K ( z ) = 13 Hf f . (26)In these equations, ν is the parameter defined in (5) and is very important because when ν = 0, the perturbation inthe vacuum energy is canceled. In this way, we recover the Λ CDM scenario as a particular case. In equation (24),is important to consider some aspects of gauge dependence. In reference [54], the linear perturbation equations inthe synchronous gauge were determined, equations in which there is no scale dependence explicitly in the equationfor δ m (see equation (3.19) of reference [54]). On the other hand, in equation (24), there are terms that explicitlycontain k ; however, the two gauges clearly agree at small scales, where 1 /k →
0. At large scales, these equationsare not equivalent. The Newtonian gauge includes the expansion of the Universe and is, therefore, more appropriatefor a description of large scale perturbations because it corresponds to a time slicing of isotropic expansion [56].The synchronous gauge corresponds to a time slicing obtained for a free falling observer frame. Additionally, asmentioned at the beginning of the section, the Newtonian gauge is directly related to the gauge invariant quantitiesin the approximation of Bardeen [65]. Therefore, the Newtonian gauge is more important from the observationalpoint of view; however, strictly speaking, it is necessary to make some changes when we intend to use the theoreticalquantity δ m together with observations. Recently, in reference [69], the matter power spectrum was investigated usinggauge-dependent quantities which could introduce gauge modes that can cause artificial large scale enhancement ofthe power spectrum. For an accurate determination of the matter power spectrum, it is essential to consider theeffects of redshift space distortions (caused by the peculiar velocities), the scale dependence of the galaxy bias, andmagnification by gravitational lensing [70–72]. In the following sections we used the theoretical quantity δ m in theNewtonian gauge and SDSS data for the matter power spectrum, ignoring the above aspects. ∆ H a L Ν= Ν= Ν= L CDM g H a L Ν= Ν= L CDM Ν= Ν= L CDM0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.60.70.80.91.0
FIG. 3: The growth factor (left) and the linear growth rate (right) for the
RCC model. We also show the linear growth ratefor the
DGP and scalar-tensor models (see appendix for details). In all cases, we use Ω m = 0 .
24 and h = 0 . A. The Linear Growth Rate
The solution to system (25) allows us to determine the density contrast of matter, δ m , which is necessary todetermine the growth factor, defined as D ( a ) = δ m ( a ) δ m ( a = 1) . (27)In Figure 3 (left) we show the growth factor in the RCC model for different values of ν ; we can see that for ν = 10 − there is concordance with a Λ CDM model. However, observationally, it is more important to know the linear growthrate, which measures how rapidly structure is being assembled in the Universe as a function of cosmic time (scalefactor or redshift). Linear growth rate is defined by g ( a ) = d ln D ( a ) d ln a . (28)This quantity has been measured using different catalogs. In general, the redshift maps of galaxies are distorted bythe peculiar velocities of galaxies along the line of sight. At large scales, this distortion can be expressed through the TABLE I: In this table we compile the different values of the linear growth rate found in the literature.Survey β ± δβ b L ± δb L g ± δg z2dFGRS [74] 0 . ± .
09 1 . ± .
10 0 . ± .
14 0.152dF-SDSS LRG and QSO [76] 0 . ± .
05 1 . ± .
35 0 . ± .
35 0.55VVDS [75] 0 . ± .
26 1 . ± .
10 0 . ± .
36 0.80 redshift distortion parameter β and can be shown to be related to the linear growth rate as [73] β = g ( z ) b L , (29)where b L is the linear bias value ( b L = σ galR /σ massR ), that is, the ratio between the root-mean-squared ( rms ) densitycontrasts in the galaxy and mass distributions on scale R where linear theory applies. Therefore, a measure of g ( z )can be obtained using these two parameters. The β parameter may be measured from redshift surveys by measuringthe power spectrum of the galaxies [73] and the b L can be obtained from the skewness induced in the bispectrum ofa given survey. Using this technique, values of β and b L have been measured using the 2dFGRS sample of 220,000galaxies [74]. Recently, another measure was made using the spectroscopic data from the VIMOS-LT Deep Survey[75]. Finally, another measure of growth rate was made using the 2dF-SDSS LRG and QSO survey [76]. However,in this last case, the value of β and b L are not fully independent, because they have been obtained by imposingsimultaneous consistency with the clustering measured at z = 0. All these data are compiled in table 1. In Figure 3(right) we show these estimates of the growth rate and compare them to predictions from various theoretical models.We plot the linear growth rate for the DGP and scalar-tensor models. Here, we use the results shown in the appendix.Despite the large error bars, the measurements indicate the need for a small value for parameter ν , and hence, arevery close to the Λ CDM model.
B. The Matter Power Spectrum
Another important amount that we can determine from the set of equations in (25) is the matter power spectrum,defined as P ( k, z ) = δ ( k, z ) . (30)To set the initial conditions, we can use the BBKS approximation for the transfer function [77], T ( k ) = ln(1 + 2 . q ( k ))(2 . q ( k ))[1 + 3 . q + (16 . q ) + (5 . q ) + (6 . q ) ] / , (31)in the presence of CC , we know that q = k/ (Σ hM pc − ), where the shape parameter is Σ = Ω m he − Ω b and h = H /
100 because observable wavenumbers are in units of hM pc − . Further, we assume that only 4% of the cosmicdensity is provided by conventional baryonic matter. The matter power spectrum can be written in the form P ( k ) = Ak n T ( k ) , (32)where n measures the slope of the primordial power spectrum (we will assume n = 1 [3, 87]), and A is a normalizationconstant. To obtain power spectra for our RCC model, we have performed the numerical analysis using the equationsin (25) from z = 500 to z = 0. At z = 500, matter is dominant. In normalizing the power spectrum, we followed themethodology presented in [54].In Figure 4, we present the matter power spectrum for our model, where Ω m = 0 .
24 and we use the power spectrumestimated by Percival et al. based on data from the SDSS Project. The impact that the
RCC model has on the linearpower spectrum is dominated by equation (24a) and depends on the value of the parameter ν . If parameter ν is zero,we recover the Λ CDM model, because δ Λ = 0. Consequently, a large value of ν entails a greater damping of thepower spectrum in the case of ν being positive. In the case of ν being negative, the deviation of the power spectrumwith respect to the observational data is even stronger. In principle, this feature can also be seen in Figure 4. Wecan interpret this behavior theoretically because in equation (15), the parameter ν appears as a factor in the term k is proportional which is derived from the pressure gradient. When ν is negative, this term changes sign. Thus, it is - - - - - l og P @ k D Ν= L CDM Ν= Ν=-
Ν=-
FIG. 4: The matter power spectrum for the
RCC model. The blue curve is the Λ
CDM and is very close to the
RCC modelwith the best fit ν = (1 . ± . × − . We used Ω m = 0 .
24 and h = 0 . expected that for small scales ( k > .
15) and ν <
0, the pressure terms become important. In general, if we considerlarge scales ( k < .
06, for example) the k terms may be negligible.We can determine the mass in terms of the Planck mass for ν ≈ − as M ≈ . × − M p , that is, approximatelyto the GUT particle spectrum. Therefore, the RCC model is viable for masses smaller than the Planck mass.To get a more accurate value of ν (without using the rigorous theory of parameter estimation), we use the χ statistic. We determined the quality of the fit between the theoretical estimate and the observational data, which isdefined as: χ = Σ i ( P ob − P the ) σ ob , where P ob is the observational value of the power spectrum, P the is the correspondingtheoretical result, and σ ob denotes the observational error bars. Therefore, the probability distribution function canbe defined as P ( ν ) = F e − χ / , where F is a normalization constant. Minimizing χ , we can determine the mostprobable value of parameter ν . In our case, this value is ν = (1 . ± . × − , where we have assumed thatΩ m = 0 .
24 and h = 0 . ν >
0, and a detailed analysis of the case of ν <
RCC model, rather than determinethe observational constraints, which requires a rigorous application of the theory of parameter estimation [79].
IV. THE NUMBER COUNTS
It has long been recognized that modeling a cluster of galaxies provides a useful test of the fundamental cosmologicalparameters. The total abundance of cluster N and its distribution in redshift dNdz should be determined by the geometryof the Universe and the power spectrum of initial density fluctuations. One of the first cosmological parameters tobe constrained was σ , the amplitude of mass density fluctuations on a scale of 8 h − M pc . For example, recently,Komatsu et al. determined a value of σ = 0 . ± .
026 using the data from WMAP. However, in general, the valueof σ is inaccurate (for example, there is an implicit uncertainty in the value of σ as a function of w ; see [80, 81]).Our objective in the present section is to determine the clusters number count and its evolution with redshift andinvestigate whether these quantities depend significantly on ν . We used the Press-Schechter ( P S ) [82, 83] formalismto give a prescription for estimating the mass function for a hierarchical gaussian density field.In the
P S model, the comoving number density of collapsed dark matter haloes of mass M in the interval dM is0 d V (cid:144) d z d W L CDM Ν= Ν= Ν= d V (cid:144) d V E d S L CDM Ν= Ν= Ν= FIG. 5: The figure shows the evolution of the comoving volume element with redshift for values different of ν . We show (below)the volume element with respect to the de-Sitter volume. given by dndM = − r π δ c Π σ ( M, z ) M d ln( σ ( M, z )) dM Exp ( − δ c σ ( M, z )) ) , (33)where Π is the comoving matter mean density of the Universe and δ c is the linearly extrapolated density thresholdabove which structures collapse, i.e., δ c = δ L ( z = z col ). In an Einstein-de Sitter (EdS) model, an overdensityregion collapses with a linear contrast δ c = 1 .
686 and is the value that we adopt for our calculations. As a firstapproximation, we use this value in the EdS (we postponed for future work the use of spherical collapse modelto calculate δ c = δ L ( z col )). In principle, this election is not far from reality because for homogeneus quintessencemodels coupled δ c does not separate too far from the EdS value (see [96]).Recall that the RCC model is equivalentto quintessence models [50]. Additionally, using the newtonian formalism for the spherical collapse, in appendix ofthe reference [62], were found a δ c as function of the redshift, showing a slight deviation from the EdS value, beingthe best-fit δ c = 1 .
685 very closed to our value used. The quantity σ ( M, z ) = D ( z ) σ M is the rms linear fluctuationof density in spheres of radius R containing a mass M and with growth factor D ( z ). In our analysis, the rms of thesmoothed overdensity is given by σ M = σ ( MM ) − γ/ , (34)where M = 6 × Ω m h − M ⊙ , the mass inside a sphere of radius R = 8 h − M pc , where M ⊙ is the solar mass. Theindex γ is a function of the mass scale and the shape parameter Γ [86] γ = (0 .
3Γ + 0 . .
92 + 13 log( MM )] . (35)We use Γ = 0 .
167 [87]. We associate galaxy clusters with dark matter haloes of the same mass. Our analysis ofthe effects of a running cosmological constant on the number of dark matter haloes is carried out by computing twoquantities. The first quantity is the number of haloes per unit of redshift in a given range of mass dNdz = Z π d Ω Z M Sup M inf dndM dVdzd Ω dM , (36)where dVdzd Ω is the comoving volume element, and is given by r ( z ) /H ( z ), where r ( z ) = R z dz/H ( z ). In Figure 5, wedisplay the comoving volume element as a function of the redshift in the RCC model. We can see that there is a1 Ν= Ν= Ν= L CDM d N (cid:144) d z (cid:144) Ν= Ν= Ν= L CDM d N (cid:144) d z (cid:144) FIG. 6: The figure shows the evolution of the number counts with redshift and the effect of the value of the parameter ν forobjects with a mass within the range 10 < M/ ( h − M ⊙ ) < . The evolution of number counts for objects with mass withinthe range 10 < M/ ( h − M ⊙ ) < (below). In both cases the curve ν = 10 − is indistinguishable from the Λ CDM model.We used a fixed value of Ω m = 0 .
24 for all curves. Ν= Ν= Ν= L CDM N H M > h - M Ÿ L (cid:144) Ν= Ν= L CDM Ν= N H M > h - M Ÿ L (cid:144) FIG. 7: The figures show the integrated number counts up to redshift z = 5 for objects with mass M > h − M ⊙ (top) and M > h − M ⊙ (below). strong dependence on the value of ν . The Λ CDM model is plotted for comparison. In the right panel, we plot thecomoving volume element compared to the Einstein-de Sitter volume for all cases.The other quantity that we compute is the all sky integrated number counts above a given mass threshold, M inf ,and up to redshift z N ( z, M > M inf ) = Z π d Ω Z ∞ M inf Z z dndM dVdzd Ω dzd Ω . (37)This result is also called a cumulative mass function. To compute the two quantities, we must choose a normalizationof the number density of haloes n ( M ). This is commonly expressed in terms of σ . We choose to normalize all2models by fixing the number density of haloes at redshift zero. At zero redshift, all models have the same comovingbackground density Π and growth factor D . Our model fiducial is Λ CDM (Ω m = 0 . h = 0 .
72) with σ = 0 . ν . In Figure 6, we display the numbercounts as a function of redshift. It is clear that there is a strong dependence on the value assumed by the parameter ν . An increase in the value of ν produces an increase in the value of dNdz . Comparing the two panels, we can seethat there is a larger variation for greater values of mass. The results for the total number of collapsed structures aredisplayed in Figure 7. In the top panel, we show the integration in the range M > M ⊙ and in the bottom panelfor M > M ⊙ ; in both cases, M sup = 10 . We do not use strictly infinite, as N ( z, M > M inf ) is dominated bythe contribution of the lower bound of the mass integration range.The parameter ν can be considered a coupling parameter between dark energy (vacuum energy) and dark matter.Models with more coupling have higher values of dN/dz . This can be understood by the behavior of other observables.Equations (37) and (38) have a dependences on the growth factor D ( z ), comoving energy density, and the comovingvolume element. An increase in the comoving volume translates into an increase in the number counts; however,average density and D ( z ) decrease. Both effects produce the observed results shown in Figures 6 and 7. Thisbehavior is similar to models of quintessence homogeneous [96]. < M (cid:144)H h - M Ÿ L < W m0 = W m0 = W m0 = d N (cid:144) d z (cid:144) W m0 = W m0 = W m0 = N H M > h - M Ÿ L (cid:144) < M (cid:144)H h - M Ÿ L < W m0 = W m0 = W m0 = d N (cid:144) d z (cid:144) W m0 = W m0 = W m0 = N H M > h - M Ÿ L (cid:144) FIG. 8: The panels show the sensitivity on the parameter Ω m . The right column shows the expected redshift distribution of10 < M/ ( h − M ⊙ ) < (upper) and 10 < M/ ( h − M ⊙ ) < (lower) clusters. The left column shows the integratednumber counts of M/ ( h − M ⊙ ) < (upper) and M/ ( h − M ⊙ ) < (lower) clusters. A. Cosmological Sensitivity on Ω m and h The cluster number counts depend on the cosmological parameters via the energy density, growth factor, andcomoving volume element. The cosmological dependence is implicit, but is very strong. First, we consider theeffects of changing Ω m , which are displayed in Figure 8. In the left column results for dN/dz , with a mass between10 < M/ ( h − M ⊙ ) < (top left) and 10 < M/ ( h − M ⊙ ) < (bottom left) are shown. The curves are fora flat RCC universe with h = 0 . ν = 10 − in all cases, and Ω m = 0 .
20 (dashed line), Ω m = 0 .
24 (solid line)and Ω m = 0 .
30 (dotted line). The right column shows the total number of clusters N ( z ) for the same values of allparameters. Several conclusions can be drawn from Figure 8. Overall, a decrease in Ω m increases the number ofclusters at all redshifts (and vice versa). The curve closest to Λ CDM is the central curve. Note that the dependenceon Ω m is strong, for instance, a 16 ,
7% decrease in Ω m increases the total number of clusters N ( z ) by 21% for moremassive structures (see the bottom right panel). < M (cid:144)H h - M Ÿ L < h = = = d N (cid:144) d z (cid:144) h = = = N H M > h - M Ÿ L (cid:144) < M (cid:144)H h - M Ÿ L < h = = = d N (cid:144) d z (cid:144) h = = = N H M > h - M Ÿ L (cid:144) FIG. 9: The same as in Figure 8 for the sensitivity on the parameter h . Figure 9 demonstrates the effects of changing h . Comparing Figure 8 to Figure 9, the quantitative behavior ofthe observables ( N ( z ) and dN/dz ) under changes in h and Ω m are similar: decreasing h increases the total numberof clusters, but does not significantly change their redshift distribution for objects with mass within the range of10 < M/ ( h − M ⊙ ) < ; however, for masses between 10 < M/ ( h − M ⊙ ) < , the change is greater.4 V. SUMMARY AND DISCUSSIONS
In this paper we have determined the cosmological implications of the
RCC model using linear perturbations. Weused the longitudinal gauge to determine the density perturbations of matter. In analyzing these perturbations wefound, similar to reference [54], that the perturbation of vacuum energy density is proportional at the ν parameter(equation (25a)); thus, when ν is zero, the standard scenario Λ CDM is recovered.Linear perturbation theory allows us to calculate the linear growth rate g ( z ) and compare our results both to thoseother models frequently studied in the literature and to data for growth rate. In Figure 3, we have seen that for valuesof ν ≤ − , our model predicts an overdensity similar to that of other models (Λ CDM , DGP , and scalar-tensortheory). In Figure 3 (bottom), we observed that for large values of ν the predicted g ( z ) is far from the data. Thesethree data of g ( z ) are not sufficient to rule out a given model. In Figure 4, we compared the matter power spectrumto the SDSS data obtained by Percival et al. [78] and showed that a RCC model with ν ≈ − is compatible withΛ CDM . The best-fit for our free parameter is ν = (1 . ± . × − .We investigated the expected evolution of cluster number counts in the RCC cosmology. In the present paper ithas been studied using the
P S mass function. We have shown that there is a significant dependence on the clusternumber counts in the
RCC model via the amount of coupling between dark matter and vacuum energy. Increasingthe coupling, that is to say, increasing the value of ν (see Figures 6 and 7), increase the cluster number counts. Thisfeature is compatible with the clustering properties in quintessence models [64].In general, our results for power spectrum and for linear growth rate are compatible with the majority of othermodels. For example, in the coupled quintessence model presented in reference [88], the interaction must be veryweak, on the order of 10 − , for the model to be compatible with data from 2dFGRS. Recently, in reference [90] theperturbations of a model called Λ XCDM have been studied; this model includes two free parameters, parameter ν ,which has the same meaning as in our case, and an equation of state parameter w X , the cosmon component. Thismodels was proposed and studied in reference [91, 92], which showed that both components can interact. Reference[93] performed an analysis of the linear perturbation of these models; their results were consistent with data from2dFGRS.In reference [94] a holographic model with infrared decay in CDM is considered. They use three types of cut-offand, in all case, the model has modes of growth for the density contrast when the effective equation of state is between − < w eff < − /
3. The authors have used a Newtonian approximation, therefore, it is only for the cut-off of theHubble horizon that the model implies a dark energy density proportional to the square of the Hubble parameter(similar to our case). We feel that it is necessary to have a fully relativistic approach to the density contrast.In reference [95], a model with a cosmological term that decays linearly with the Hubble parameter is considered.In that paper, the authors consider a relativistic treatment of perturbations in the synchronous gauge; the darkcomponent is also perturbed. They calculate the matter power spectrum and show that their results are inconsistentwith 2dFGRS data (the matter parameter is very large Ω m ≈ . CDM in the inhomogeneouscase, and that for both types of model, the largest deviations are observed for massive structures
M > . Theseresults are consistent with our calculations.Finally, it would be interesting to consider the other astrophysical implications of the RCC model with a modifica-tion of the spherical collapse model, that is, the δ c function of the redshift of collapse. Additionally, one could evaluatethe profile of density contrast around the cluster, and supercluster of void matter, using as a first approximation, a N F W (Navarro, Frenk, and While) profile [98]. Another important issue is the study of the concentration of haloes.For example, following the prescription given in reference [99, 100], we can investigate whether the concentrationof haloes in the
RCC model decreases with an increase in the mass, as in the case of the Λ
CDM model. Thisinvestigation would also determine the limits of the model.
Appendix A: Equations used to model DGP and Scalar-Tensor Theory
In this appendix we write the equations used to determine the growth factor in the DGP model and scalar-tensortheory. The growth factor is defined in equation (28) and obeys equation [101, 102] D ′′ ( k, a ) + ( 3 a + H ′ ( a ) H ( a ) ) D ′ ( k, a ) − m a H ( a ) f ( k, a ) D ( k, a ) = 0 , (A1)5with the initial condition D ( a ) ≈ a for a ≈ f ( k, a ) expresses the connection between themetric perturbations and the matter density perturbations. Therefore, the connection depends on the particulargravity theory.
1. DGP Model
In the DGP model, in the case of flatness and matter being only found on the brane, H ( z ) is given by [17] H DGP = p Ω r c + p Ω m a − + Ω r c , (A2)where Ω r c = (1 − Ω m ) and, in this theory, f ( k, a ) is given by [101] f ( k, a ) = (cid:18) β (cid:19) , (A3)with β = 1 − H DGP ( a ) H p Ω r c (1 + aH ′ DGP ( a )3 H DGP ( a ) ) . (A4)
2. Scalar-Tensor Theory
Scalar-tensor theory is the simplest generalization of General Relativity in which the fundamental constants arevariable. The function f ( k, a ) in this case is given by [101] f ( k, a ) = G eff ( a ) G eff ( a = 1) (cid:18) k/ma ) (cid:19) , (A5)where G eff is the effective Newton constant, a is the scale factor and a = 1 is the present value, and m is the massof the scalar field Φ inducing a Yukawa cut-off in the gravitational field. We used a simple ansatz given by [102] G eff ( a ) G eff ( a = 1) = 1 + ξ (1 − a ) . (A6)In the case of the DGP model and scalar-tensor theory, the detection of a value of f ( k, a ) = 1 would be a signatureof alternative theories of gravity. For the numerical calculations we used ξ = − . Acknowledgement
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