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arXiv:1103.1464v2 [astro-ph.CO] 12 Apr 2011

Dynamics and Constraints of the Massive Gravitons Dark Matter Flat Cosmologies

S. Basilakos∗

Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece

M. Plionis†

Institute of Astronomy & Astrophysics, Nationals Observatory of Athens, Thessio 11810, Athens, Greece, and

Instituto Nacional de Astrof´ ısica,´Optica y Electr´ onica, 72000 Puebla, Mexico

M. E. S. Alves‡

Instituto de Ciˆ encias Exatas, Universidade Federal de Itajub´ a

Av.BPS, 1303, 37500-903, Itajub´ a, MG, Brazil

J. A. S. Lima§

Departamento de Astronomia (IAGUSP), Universidade de S˜ ao Paulo

Rua do Mat˜ ao, 1226, 05508-900, S. Paulo, Brazil

We discuss the dynamics of the universe within the framework of Massive Graviton Dark Matter

scenario (MGCDM) in which gravitons are geometrically treated as massive particles. In this modi-

fied gravity theory, the main effect of the gravitons is to alter the density evolution of the cold dark

matter component in such a way that the Universe evolves to an accelerating expanding regime, as

presently observed. Tight constraints on the main cosmological parameters of the MGCDM model

are derived by performing a joint likelihood analysis involving the recent supernovae type Ia data,

the Cosmic Microwave Background (CMB) shift parameter and the Baryonic Acoustic Oscillations

(BAOs) as traced by the Sloan Digital Sky Survey (SDSS) red luminous galaxies. The linear evo-

lution of small density fluctuations is also analysed in detail. It is found that the growth factor of

the MGCDM model is slightly different (∼ 1 − 4%) from the one provided by the conventional flat

ΛCDM cosmology. The growth rate of clustering predicted by MGCDM and ΛCDM models are

confronted to the observations and the corresponding best fit values of the growth index (γ) are

also determined. By using the expectations of realistic future X-ray and Sunyaev-Zeldovich cluster

surveys we derive the dark-matter halo mass function and the corresponding redshift distribution

of cluster-size halos for the MGCDM model. Finally, we also show that the Hubble flow differences

between the MGCDM and the ΛCDM models provide a halo redshift distribution departing signifi-

cantly from the ones predicted by other DE models. These results suggest that the MGCDM model

can observationally be distinguished from ΛCDM and also from a large number of dark energy

models recently proposed in the literature.

PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

1.INTRODUCTION

The high-quality cosmological observational data (e.g.

supernovae type Ia, CMB, galaxy clustering, etc), accu-

mulated during the last two decades, have enabled cos-

mologists to gain substantial confidence that modern cos-

mology is capable of quantitatively reproducing the de-

tails of many observed cosmic phenomena, including the

late time accelerating stage of the Universe. Studies by

many authors have converged to a cosmic expansion his-

tory involving a spatially flat geometry and a cosmic dark

sector formed by cold dark matter and some sort of dark

energy, endowed with large negative pressure, in order to

explain the observed accelerating expansion of the Uni-

∗Electronic address: svasil@academyofathens.gr

†Electronic address: mplionis@astro.noa.gr

‡Electronic address: alvesmes@unifei.edu.br

§Electronic address: limajas@astro.iag.usp.br

verse [1–8].

In spite of that, the absence of a fundamental phys-

ical theory, regarding the mechanism inducing the cos-

mic acceleration, have given rise to a plethora of alter-

native cosmological scenarios. Most are based either on

the existence of new fields in nature (dark energy) or in

some modification of Einstein’s general relativity, with

the present accelerating stage appearing as a sort of ge-

ometric effect.

The simplest dark energy candidate corresponds to a

cosmological constant, Λ (see [9] for reviews).

standard concordance cosmological (ΛCDM) model, the

overall cosmic fluid contains baryons, cold dark matter

plus a vacuum energy. This model fits accurately the cur-

rent observational data and it therefore provides an excel-

lent scenario to describe the observed universe. However,

it is well known that the concordance model suffers from,

among others [10], two fundamental problems:

(i) Fine tuning problem - the fact that the observed

value of the vacuum energy density (ρΛ = Λc2/8πG ≃

10−47GeV4) is more than 120 orders of magnitude below

In the

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the natural value estimated using quantum field theory

[11].

(ii) Coincidence problem - the fact that the matter and

the vacuum energy densities are of the same order just

prior to the present epoch [12].

Such problems have inspired many authors to pro-

pose alternative dark energy candidates such as Λ(t) cos-

mologies, quintessence, k−essence, vector fields, phan-

tom dark energy, tachyons, Chaplygin gas and the list

goes on (see [13–28] and references therein). Naturally, in

order to establish the evolution of the dark energy equa-

tion of state (EoS), a realistic form of H(a) is required

which should be constrained through a combination of

independent dark energy probes.

Nevertheless, there are other possibilities to explain

the present accelerating stage. For instance, one may

consider that the dynamical effects attributed to dark

energy can be mimicked by a nonstandard gravity the-

ory. In other words, the present accelerating stage of

the universe can also be driven only by cold dark matter

under a modification of the nature of gravity. Such a re-

duction of the so-called dark sector is naturally obtained

in the so-called f(R) gravity theories [29] (see, however,

[30]).

On the other hand, general relativity predicts that

gravitational waves are non-dispersive and propagate

with the same vacuum light speed. These results lead

to the common believe that the graviton (the “boson”

for general relativity), must be a massless particle. How-

ever, massive gravitons are features of some alternatives

to general relativity as the one proposed by Visser [31].

Such theories have motivated many experiments and ob-

servations in order to detect a possible dispersive behav-

ior due to a non-zero graviton mass (see [34] and Refs.

there in).

More recently, it was shown that the massive graviton

approach proposed by Visser can be used to build realis-

tic cosmological models that can then be tested against

the available cosmological data [35]. One of the main ad-

vantages of such massive graviton cosmology is the fact

that it contains the same number of free parameters as

the concordance ΛCDM model, and, therefore, it does

not require the introduction of any extra fields in its dy-

namics. In this way, since the astronomical community

is planning a variety of large observational projects in-

tended to test and constrain the standard ΛCDM concor-

dance model, as well as many of the proposed alternative

models, it is timely and important to identify and explore

a variety of physical mechanisms (or substances) which

could also be responsible for the late-time acceleration of

the Universe.

In what follows we focus our attention to a cosmolog-

ical model within Visser’s massive graviton theory. In

particular we discuss how to differentiate the massive

graviton model from the concordance ΛCDM model. Ini-

tially, a joint statistical analysis, involving the latest ob-

servational data (SNIa, CMB shift parameter and BAO)

is implemented. Secondly, we attempt to discriminate

the MGCDM and ΛCDM models by computing the halo

mass function and the corresponding redshift distribution

of the cluster-size halos. Finally, by using future X-ray

and SZ surveys we show that the evolution of the clus-

ter abundances is a potential discriminator between the

MGCDM and ΛCDM models. We would like to stress

here that the abundance of collapsed structures, as a

function of mass and redshift, is a key statistical test

for studies of the matter distribution in the universe,

and, more importantly, it can be accessed through ob-

servations [36]. Indeed, the mass function of galaxy clus-

ters has been measured based on X-ray surveys [37–39],

via weak and strong lensing studies [40–42], using opti-

cal surveys, like the SDSS [43, 44], as well as, through

Sunayev-Zeldovich (SZ) effect [45]. In the last decade

many authors have been involved in this kind of stud-

ies and have found that the abundance of the collapsed

structures is affected by the presence of a dark energy

component [46–58].

The paper is planned as follows. The basic elements

of Visser’s theory are presented in section 2, where

we also introduce the cosmological equations for a flat

Friedmann-Lemaitre-Robertson-Walker (FLRW) geome-

try with massive gravitons. In section 3, a joint statisti-

cal analysis based on SNe Ia, CMB and BAO is used to

constraint the massive graviton cosmological model free

parameter. The linear growth factor of matter pertur-

bations is discussed in section 4, while in 5, we discuss

and compare the corresponding theoretical predictions

regarding the evolution of the cluster abundances. Fi-

nally, the main conclusions are summarized in section 6.

2.MASSIVE GRAVITONS COLD DARK

MATTER (MGCDM) COSMOLOGY: BASIC

EQUATIONS

In this section we briefly present the main points of

Visser’s massive gravity approach [31]. The full action is

given by (in what follows ? = c = 1)

S =

?

d4x

?√−gR(g)

16πG+ Lmassg(g,g0) + Lmatter(g)

?

(2.1)

where besides the Einstein-Hilbert Lagrangian and the

Lagrangian of the matter fields, we have the bi-metric

Lagrangian:

Lmass(g,g0) =1

2mg2√−g0

×(g − g0)ρν−1

?

(g−1

0)µν(g − g0)µσ(g−1

0)σρ

(2.2)

2

?(g−1

0)µν(g − g0)µν

?2?

,

where mg is the graviton mass and (g0)µν is a general

flat metric.

The field equations, which are obtained by variation of

(2.1), can be written as:

Gµν−1

2mg2Mµν= −8πGTµν,(2.3)

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where Gµνis the Einstein tensor, Tµνis the energy-

momentum tensor for perfect fluid, and the contribution

of the massive tensor to the field equations reads:

Mµν= (g−1

0)µσ

?

(g − g0)σρ−1

2(g0)σρ(g−1

0)αβ

(2.4)

×(g − g0)αβ

?

(g−1

0)ρν.

Note that if one takes the limit mg→ 0 the standard

Einstein field equations are recovered.

Thus, from the construction of the Visser’s theory, it

can be classified as a bimetric theory of gravitation. This

kind of theory was first studied by N. Rosen [32]. In the

Rosen’s concept the metric gµνdescribes the geometry of

the spacetime in the same way as in the context of the

general relativity theory, and the second metric (g0)µν

(that Rosen denoted by γµν) refers to the flat spacetime

and describes the inertial forces. It is worth to mention

that Rosen has shown that a bimetric theory satisfies the

covariance and the equivalence principles, a fact that was

also pointed out by Visser (for more discussion see [33]).

In this way, in order to follow the Rosen’s approach we

have constrained the background metric to respect the

Riemann-flat condition, that is, Rλ

way that we have no ambiguity on the choice of (g0)µν, it

will always be chosen to be a flat metric, depending only

on the particular coordinates we are dealing, of course.

Regarding the energy-momentum conservation we will

follow the same approach of Refs. [59, 60]. Since the

Einstein tensor satisfies the Bianchi identities ∇νGµν=

0, the energy conservation law is expressed as:

µνκ(g0) ≡ 0 in such a

∇νTµν=

mg2

16πG∇νMµν.(2.5)

In the above framework, the global dynamics of a

flat MGCDM cosmology is driven by the following equa-

tions1:

8πGρ = 3

?˙ a

a

?2

+3

4mg2(a2− 1),(2.6)

8πGp = −2¨ a

a−

?˙ a

a

?2

−1

4mg2a2(a2− 1),(2.7)

where ρ is the energy density, p is the pressure and a(t)

is the scale factor.

1In the present article we restrict our analysis to the flat cosmolo-

gies in order to compare our results with those of the flat ΛCDM

model that is the most accepted cosmological model as shown,

e.g., by the WMAP7 data [5]. A generalization of the model for a

non spatially flat cosmology will appear in a forthcoming article.

From Eq. (2.5) we get the evolution equation for the

energy density, namely:

˙ ρ + 3H

?

(ρ + p) +

mg2

32πG(a4− 6a2+ 3)

?

= 0,(2.8)

where H = ˙ a/a. By integrating the above equation for a

matter dominated universe (p = 0) one obtains:

ρ(a) =ρ0

a3−3mg2

32πG

?a4

7

−6a2

5

+ 1

?

,(2.9)

where ρ0 is the present value of the energy density. As

expected, in the limiting case mg → 0 all the standard

FLRW expressions are recovered.

Now, inserting (2.9) in the modified Friedmann equa-

tion (2.6) we obtain the normalized Hubble parameter:

E2(a) =H2(a)

H2

0

= Ωma−3+ δH2,(2.10)

with

δH2=1

2Ωg

?7a2− 5a4?,(2.11)

where H0 is the Hubble constant, Ωm is the matter

density parameter (for baryons and dark matter Ωi =

ρi0/ρc0, where ρc0 = 3H02/8πG is the critical density

parameter), and Ωg =

tion of the massive gravitons. It should be stressed that

the last term of the above normalized Hubble function

(2.10) encodes the correction to the standard FLRW ex-

pression.

In general, using the FLRW equations, one can express

the effective dark energy EoS parameter in terms of the

normalized Hubble parameter [61]

1

70(mg

H0)2is the present contribu-

wDE(a) =

−1 −2

1 − Ωma−3E−2(a).

3adlnE

da

(2.12)

After some simple algebra, it is also readily seen that

the effective (“geometrical” in our case) dark energy EoS

parameter is given by (see [25, 62]):

wDE(a) = −1 −1

3

dlnδH2

dlna

.(2.13)

In our case, inserting Eq. (2.11) into Eq. (2.13) it is

straightforward to obtain a simple analytical expression

for the geometrical dark energy EoS parameter:

wDE(a) = −1 −2

3

?7 − 10a2

7 − 5a2

?

.(2.14)

It thus follows that in the cosmological context, the

modified gravity theory as proposed by Visser can be

treated as an additional effective fluid with EoS param-

eter defined by (2.14). Note also that the current Hub-

ble function has only two free parameters (H0and Ωm),

exactly the same number of free parameters as the con-

ventional flat ΛCDM model. Naturally, the value of H0

is not predicted by any of the models and it is set to its

observational value of H0= 70.4 km s−1Mpc−1[5, 63].

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3.LIKELIHOOD ANALYSIS

Let us now discuss the statistical treatment of the ob-

servational data used to constrain the MGCDM model

presented in the previous section.

To begin with, we consider the Constitution super-

novae Ia set of Hicken et al. [6], but in order to avoid

possible problems related to the local bulk flow, we use a

subset of this sample containing 366 SNe Ia all with red-

shifts z > 0.02. The likelihood estimator is determined

by a χ2

SNIastatistics:

χ2

SNIa(Ωm) =

366

?

i=1

?µth(ai,Ωm) − µobs(ai)

σi

?2

, (3.1)

where ai= (1 + zi)−1is the scale factor of the Universe

at the observed redshift zi, µ is the distance modulus

µ = m − M = 5logdL+ 25 and dL is the luminosity

distance2, dL(a,Ωm) = ca−1?1

likelihood analysis we find that Ωm= 0.266±0.016 with

χ2

tot(Ωm)/dof ≃ 446.5/365.

In addition to the SNe Ia data, we also consider

the BAO scale produced in the last scattering surface

by the competition between the pressure of the cou-

pled baryon-photon fluid and gravity.

acoustic waves leave (in the course of the evolution)

an overdensity signature at certain length scales of the

matter distribution.Evidence of this excess was re-

cently found in the clustering properties of SDSS galax-

ies (see [64–66]) and it provides a suitable “standard

ruler” for constraining dark energy models. In this work

we use the measurement derived by Eisenstein et al.

[64].In particular, we utilize the following estimator

√Ωm

[z2

as

a2E(a)

SDSS data to be A = 0.469 ± 0.017, where zs= 0.35 [or

as = (1 + zs)−1≃ 0.75]. Therefore, the corresponding

χ2

BAOfunction can be written as:

a

dy

y2H(y). Now, from the

The resulting

A(Ωm) =

sE(as)]1/3

??1

da

?2/3

, measured from the

χ2

BAO(Ωm) =[A(Ωm) − 0.469]2

0.0172

.(3.2)

The likelihood function peaks at Ωm= 0.306+0.026

Finally, a very interesting geometrical probe of dark

energy is provided by the angular scale of the sound hori-

zon at the last scattering surface. It is encoded in the

location of the first peak of the angular (CMB) power

spectrum [67, 68], and may be defined by the quantity

R =

from the WMAP 7-years data [5] is R = 1.726±0.019 at

zls= 1091.36 [or als= (1 + zls)−1≃ 9.154 × 10−4]. In

−0.025.

√Ωm

?1

als

da

a2E(a).The shift parameter measured

2Since only the relative distances of the SNIa are accurate and

not their absolute local calibration, we always marginalize with

respect to the internally derived Hubble constant.

this case, the χ2

cmbfunction reads

χ2

cmb(Ωm) =[R(Ωm) − 1.726]2

0.0182

.(3.3)

It should be stressed that for CMB shift parameter, the

contribution of the radiative component, (ΩRa−4, where

ΩR ≃ 4.174 × 10−5h−2) needs also to be considered

[5]. Note also that the measured CMB shift parame-

ter is somewhat model dependent but such details of the

models were not included in our analysis. For example,

such is the case when massive neutrinos are included.

The robustness of the shift parameter has been tested

and discussed in [69]. In this case the best fit value is:

Ωm= 0.263± 0.03.

The derived Ωmvalues from each individual probe ap-

pear to be quite different, although within their mutual

2σ uncertainty range. Therefore, in order to put tighter

constraints on the corresponding parameter space of

any cosmological model, the above probes are combined

through a joint likelihood analysis3, given by the product

of the individual likelihoods according to: Ltot(Ωm) =

LSNIa× LBAO× Lcmb, which translates in the joint χ2

function in an addition: χ2

Now, by applying our joint statistical procedure for

both cosmologies, we obtain the following best fit pa-

rameters:

tot(Ωm) = χ2

SNIa+χ2

BAO+χ2

cmb.

• MGCDM model:

χ2

tot(Ωm)/dof ≃ 448.5/367. Such results should

be compared to those found by Alves et al.

[35], namely:Ωm

=

χ2

tot(Ωm)/dof ≃ 565.06/558. This difference must

be probably attributed to the use of the Union2

supernovae sample [70] by the latter authors.

Ωm

=0.276 ± 0.012 with

0.273 ± 0.015 with a

• ΛCDM model:

χ2

tot(Ωm)/dof ≃ 439.5/367, which is in good agree-

ment with recent studies [1–8].

Ωm

=0.280 ± 0.010 with

3Likelihoods are normalized to their maximum values.

present analysis we always report 1σ uncertainties on the fitted

parameters. Note also that the total number of data points used

here is Ntot= 368, while the associated degrees of freedom are:

dof= 367. Note that we sample Ωm∈ [0.1,1] in steps of 0.001.

In the

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FIG. 1: Normalized Hubble parameter as a function of red-

shift. The solid line is the prediction of the MGCDM model.

For comparison, the dashed line corresponds to the traditional

ΛCDM model.

FIG. 2: Expansion history. In the upper panel we display the

evolution of the dark energy effective EoS parameter. In the

lower panel we compare the deceleration parameters of the

MGCDM (solid line) and the concordance ΛCDM (dashed

line) models. In the insert we show the relative deviation

∆(q − qΛ) of the two deceleration parameters.

It should be mentioned here that using the BAO results

of Percival et al. [65], does not change the previously

presented constraints.

4.MGCDM VERSUS ΛCDM COSMOLOGY

A.The cosmic expansion history

In Figure 1 we plot the normalized MGCDM Hubble

function (solid line) as a function of redshift, which ap-

pears quite different both in amplitude and shape with

respect to the corresponding ΛCDM model expectations

(dashed-line).

In figure 2 (upper panel), we present the evolution of

the MGCDM effective dark energy EoS parameter. One

can divide the evolution of the cosmic expansion history

in different phases on the basis of the varying behavior

of the MGCDM and ΛCDM models. We will investigate

such variations in terms of the deceleration parameter,

q(a) = −(1 + dlnH/dlna), which is plotted in the lower

panel of figure 2. In the inset plot we display the rela-

tive deviation of the deceleration parameter, ∆(q − qΛ),

between the two cosmological models. We can divide the

cosmic expansion history in the following phases:

• at early enough times a∼< 0.1 the deceleration pa-

rameters of both models are positive with q ≃ qΛ,

which means that the two cosmological models pro-

vide a similar expansion rate of the universe. Note

that by taking the limit lim

a→0wDE(a) = −5/3 for the

MGCDM model, while we always have wDE= −1

for the ΛCDM model;

• for 0.1 ≤ a ≤ 0.44 the deceleration parameters are

both positive with q > qΛ, which means that the

cosmic expansion in the MGCDM model is more

rapidly “decelerating” than in the ΛCDM case;

• between 0.44 < a < 0.52 the deceleration parame-

ters remain positive but q < qΛ;

• for 0.52 ≤ a ≤ 0.57 the traditional Λ model re-

mains in the decelerated regime (qΛ > 0) but the

MGCDM is starting to accelerate (q < 0);

• for 0.57 < a ≤ 0.94 the deceleration parameters

are both negative and since q < qΛ, the MGCDM

model provides a stronger acceleration than in

the ΛCDM model (the opposite situation holds at

0.85 ≤ a ≤ 0.94).

Interestingly, prior to the present epoch (a > 0.94) the

the deceleration parameter of the MGCDM model be-

comes positive and when a = 1 we have wDE = 0, ie.,

the universe becomes again matter dominated, implying

that the late time acceleration of the universe was a tran-

sient phase which has already finished.

From the inset panel of figure 2 it becomes clear that

the MGCDM model reaches a maximum deviation from

the ΛCDM cosmology prior to a ≃ 0.75 and again at

a ≃ 1. Finally, the deceleration parameters at the present

time are q0≃ 0.50 and q0Λ≃ −0.58. If we go further to

the future we find from Eq. (2.14) that the state param-

eter as well as the deceleration parameter diverges for

a =?7/5. This value sets the turning point after which

the universe begins to contract in the MGCDM model

(for more details see [71].)

B.The growth factor and the rate of clustering

It is well known that for small scales (smaller than the

horizon) the dark energy component (or ”geometrical”

dark energy) is expected to be smooth and thus it is fair

to consider perturbations only on the matter component

of the cosmic fluid [72]. This assumption leads to the

usual equation for matter perturbations

¨δm+ 2H˙δm− 4πGeffρmδm= 0,

where the effect of ”geometrical” dark energy is intro-

duced via the expression of Geff= Geff(t) [see [73],[74]].

(4.1)

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In the context of general relativity Geff coincides with

the Newton’s gravitational constant. Now, for any type

of dark energy an efficient parametrization of the mat-

ter perturbations (δm∝ D) is based on the growth rate

f(a) ≡ dlnD/dlna [76], which has the following func-

tional form:

f(a) =dlnD

dlna= Ωγ

m(a) ,(4.2)

where D(a) is the linear growth factor, Ωm(a)

Ωma−3/E2(a) and γ is the so called growth index (see

Refs. [25, 62, 73, 77, 78]). Since the growth factor of

a pure matter universe (Einstein de-Sitter) has the form

DEdS= a, one has to normalize the different cosmologi-

cal models such that D ≃ a at large redshifts due to the

dominance of the non-relativistic matter component. Us-

ing the latter condition we can easily integrate Eq. (4.2)

to derive the growth factor [25]

=

D(a) = ae

?a

0(dx/x)[Ωγ

m(x)−1].(4.3)

In the present case we are working with a modification

of Einstein’s gravity instead of an extra fluid, in such a

way the usual Poisson equation for the gravitational po-

tential is modified due the presence of the mass term.

In the simple case of the non-relativistic limit we have

a Yukawa-like potential which accomplishes corrections

to the Newtonian potential to scales of the order of the

Compton wavelength of the graviton, λ = m−1

this kind of potential, the classic limit for the graviton

mass obtained from solar system dynamics observations

is mg< 7.68×10−55g [75], but one of the most stringent

constraints is obtained by requiring the derived dynam-

ical properties of a galactic disk to be consistent with

observations [60] thereby yielding mg < 10−59g. Now,

by considering the best fit value obtained here for Ωgwe

have mg ∼ 10−65g, which is nearly 6 orders of magni-

tude below to the previous bound. This value gives a

Compton wavelength of the order of the horizon. The

Compton wavelength can be seen as the physical length

of graviton’s perturbations which implies that these per-

turbations play some role only close to the Hubble radius

and thus they will be negligible at sub-horizon scales. In

other words, Eqs. (4.1), (4.3) are both valid also in the

MGCDM model.

g. Using

TABLE I: Data of the growth rate of clustering [79]. The

correspondence of the columns is as follows: redshift, observed

growth rate and references.

zfobs

Refs.

[81, 82]

[83]

[84]

[85]

[86]

0.15

0.35

0.55

1.40

3.00

0.51 ± 0.11

0.70 ± 0.18

0.75 ± 0.18

0.90 ± 0.24

1.46 ± 0.29

Clearly in order to quantify the evolution of the growth

factor we need to know the growth index. Since for the

current graviton model there is yet no theoretically pre-

dicted value of growth index, we attempt to provide a

relevant value by performing a standard χ2minimiza-

tion procedure (described previously) between the ob-

servationally measured growth rate (based on the 2dF

and SDSS galaxy catalogs; see Table I; [79]) and that

expected in the MGCDM cosmological model, according

to:

χ2(γ) =

5

?

i=1

?fobs(zi) − fmodel(zi,γ)

σi

?2

, (4.4)

where σi is the observed growth rate uncertainty. Note

that for comparison we perform the same analysis also

for the ΛCDM model.

FIG. 3: Upper Panel: Comparison of the observed (solid cir-

cles[79], (see Table I) and theoretical evolution of the growth

rate of clustering f(z). The lines correspond to the MGCDM

(solid curve) and the ΛCDM (dashed curve) models. Bot-

tom Panel: The evolution of the growth factor, with that

corresponding to the MGCDM model (γ = 0.56) showing a

∼ 1−4% difference with respect to that of the ΛCDM model

(γΛ = 0.62), especially at large redshifts (z ≥ 1). Errorbars

are plotted only for the MGCDM model in order to avoid

confusion.

In Figure 3 (upper panel), we present the measured

fobs(z) (filled symbols) with the estimated growth rate

function, f(z) = Ωγ

m(z), for the two considered cosmolog-

ical models. Notice, that for the MGCDM cosmological

model (solid line) we use Ωm= 0.276 and for the Λ case

(dashed line) Ωm= 0.280, which are the values provided

by our likelihood analysis of section 3. In the inset panel

of figure 3 we plot the variation of ∆χ2= χ2(γ)−χ2

around the best γ fit value. For the MGCDM model we

find γ = 0.56+0.15

min(γ)

−0.14(χ2/dof ≃ 0.69), while for the ΛCDM

Page 7

7

model we obtain γΛ= 0.62+0.18

is somewhat greater, but within 1σ, of the theoretically

predicted value of γΛ ≃ 6/11. Such a discrepancy be-

tween the theoretical ΛCDM and observationally fitted

value of γ has also been found by other authors. For ex-

ample, Di Porto & Amendola [80] obtained γ = 0.60+0.40

while Nesseris & Perivolaropoulos [79], based on mass

fluctuations inferred from independent observations at

different redshifts, found γ = 0.67+0.20

tematic difference between the measured and the the-

oretical γ ΛCDM values is due to observational uncer-

tainties or the method used to estimate the observed

γ, then one may expect a similar systematic difference

to affect the measured γ value for the MGCDM model,

pointing to a probably more realistic value for this model

of γ ≃ 0.49. Since however this value is within the 1σ

observational uncertainty, we will consider the originally

fitted MGCDM γ value as the nominal one.

Using the above best fit γ values we present, in the

lower panel of figure 3, the growth factor evolution de-

rived by integrating Eq. (4.3) for the two cosmologi-

cal models (MGCDM-solid and ΛCDM-dashed). The er-

ror bars correspond to the 1σ uncertainty of the fitted

γ values. Note that the growth factors are normalized

to unity at the present time.

the fitted growth factors lies, at redshifts z ≥ 1, in the

interval ∼ 1 − 4%, while when using the theoretically

predicted ΛCDM value of γΛ≃ 6/11 the difference is less

than 1.5%. For a consistent treatment of the two models

and for the corresponding comparison of their respective

mass functions and halo redshift distributions we will use,

throughout the rest of the paper, the observationally de-

rived γ values, ie., γΛ≃ 0.62 and γMGCDM≃ 0.56.

−0.15(χ2/dof ≃ 0.75), which

−0.30,

−0.17. If such a sys-

The difference between

5.COMPARE THE CLUSTER HALO

ABUNDANCES

It is important to define observational criteria that will

enable us to distinguish between the MGCDM model and

the concordance ΛCDM cosmology. An obvious choice,

that has been extensively used, is to compare the theo-

retically predicted cluster-size halo redshift distributions

and to use observational cluster data to distinguish the

models. Recently, the halo abundances predicted by a

large variety of DE models have been compared with

those corresponding to the ΛCDM model [16, 57]. As

a result, such analyses suggest that many DE models ex-

plored in this study (including some of modified gravity)

are clearly distinguishable from the ΛCDM cosmology.

We use the Press and Schecther [87] (hereafter PSc)

formalism, based on random Gaussian fields, which de-

termines the fraction of matter that has formed bounded

structures as a function of redshift. Mathematical de-

tails of our treatment can be found in [57]; here we only

present the basic ideas. The number density of halos,

n(M,z), with masses within the range (M,M +δM) are

given by:

n(M,z)dM =

¯ ρ

M

dlnσ−1

dM

fPSc(σ)dM ,(5.1)

where fPSc(σ) =

early extrapolated density threshold above which struc-

tures collapse [88], while σ2(M,z) is the mass variance

of the smoothed linear density field, extrapolated to red-

shift z at which the halos are identified. It depends on

the power-spectrum of density perturbations in Fourier

space, P(k), for which we use here the CDM form ac-

cording to [89], and the values of the baryon density

parameter, the spectral slope and Hubble constant ac-

cording to the recent WMAP7 results [5]. Although the

Press-Schecther formalism was shown to provide a good

first approximation to the halo mass function provided

by numerical simulations, it was later found to over-

predict/under-predict the number of low/high mass ha-

los at the present epoch [90, 91]. More recently, a large

number of works have provided better fitting functions

of f(σ), some of them based on a phenomenological ap-

proach. In the present treatment, we adopt the one pro-

posed by Reed et al. [92].

We remind the reader that it is traditional to

parametrize the mass variance in terms of σ8, the rms

mass fluctuations on scales of 8 h−1Mpc at redshift

z = 0.

In order to compare the mass function predictions of

the different cosmological models, it is imperative to use

for each model the corresponding value of δcand σ8. It

is well known that for the usual Λ cosmology δc≃ 1.675,

while Weinberg & Kamionkowski [46] provide an accu-

rate fitting formula to estimate δc for any DE model

with a constant equation of state parameter. Since for

the current graviton cosmological vacuum model the ef-

fective dark energy EoS parameter at the present time

is w ≃ 0 which implies that the Hubble parameter is

matter dominated it is fair to use the Einstein de-Sitter

value δc ≃ 1.685 [46]. Now, in order to estimate the

correct model σ8power spectrum normalization, we use

the formulation developed in [57] which scales the ob-

servationally determined σ8,Λ value to that of any cos-

mological model. The corresponding MGCDM value is

σ8,MGCDM= 0.828 and it is based on σ8,Λ= 0.804 (as in-

dicated also in Table 1), derived from an average of a va-

riety of recent measurements (see also the corresponding

discussion in [57]) which are based on the WMAP7 re-

sults [5], on a recent cluster abundances analysis [93], on

weak-lensing results [94] and on peculiar velocities based

analyses [95].

Given the halo mass function from Eq.(5.1) we can

now derive an observable quantity which is the redshift

distribution of clusters, N(z), within some determined

mass range, say M1 ≤ M/h−1M⊙ ≤ M2 = 1016. This

can be estimated by integrating, in mass, the expected

?2/π(δc/σ)exp(−δ2

c/2σ2), δcis the lin-

Page 8

8

differential halo mass function, n(M,z), according to:

N(z) =dV

dz

?M2

M1

n(M,z)dM,(5.2)

where dV/dz is the comoving volume element. In order

to derive observationally relevant cluster redshift distri-

butions and therefore test the possibility of discriminat-

ing between the MGCDM and the ΛCDM cosmological

models, we will use the expectations of two realistic fu-

ture cluster surveys:

(a) the eROSITA satellite X-ray survey, with a flux limit

of: flim= 3.3×10−14ergs s−1cm−2, at the energy band

0.5-5 keV and covering ∼ 20000 deg2of the sky,

(b) the South Pole Telescope SZ survey, with a limiting

flux density at ν0= 150 GHz of fν0,lim= 5 mJy and a

sky coverage of ∼ 4000 deg2.

FIG. 4: The expected cluster redshift distribution of the

MGCDM (solid curve) and ΛCDM (dashed curve) models for

the case of two future cluster surveys (upper panels), and the

corresponding fractional difference with respect to the refer-

ence ΛCDM model (lower panels). Errorbars are 2σ Poisson

uncertainties, while the dashed lines in the lower panel bracket

the range due to the uncertainty of the observationally fitted

value of γ.

To realize the predictions of the first survey we use the

relation between halo mass and bolometric X-ray lumi-

nosity, as a function of redshift, provided in [96], ie:

L(M,z) = 3.087 × 1044

?

ME(z)

1015h−1M⊙

?1.554

h−2ergs−1.

(5.3)

The limiting halo mass that can be observed at redshift z

is then found by inserting in the above equation the lim-

iting luminosity, given by: L = 4πd2

Lflimcb, with dLthe

luminosity distance correspondingto the redshift z and cb

the band correction, necessary to convert the bolometric

luminosity of Eq.(5.3) to the 0.5-5 keV band of eROSITA.

We estimate this correction by assuming a Raymond-

Smith (1977) plasma model with a metallicity of 0.4Z⊙,

a typical cluster temperature of ∼ 4 keV and a Galactic

absorption column density of nH= 1021cm−2.

The predictions of the second survey can be realized

using again the relation between limiting flux and halo

mass from [96]:

fν0,lim=2.592 × 108mJy

d2

A(z)

?

M

1015h−1M⊙

?1.876

E2/3(z)

(5.4)

where dA(z) ≡ dL/(1 + z)2is the angular diameter dis-

tance out to redshift z.

In figure 4 (upper panels) we present the expected red-

shift distributions above a limiting halo mass, which is

M1 ≡ Mlimit = max[1014h−1M⊙,Mf], with Mf corre-

sponding to the mass related to the flux-limit at the

different redshifts, estimated by solving Eq.(5.3) and

Eq.(5.4) for M. In the lower panels we present the

fractional difference between the MGCDM and ΛCDM.

The error-bars shown correspond to 2σ Poisson uncer-

tainties, which however do not include cosmic variance

and possible observational systematic uncertainties, that

would further increase the relevant variance. A further

source of uncertainty that should be taken into account

is related to the uncertainty of the observationally de-

rived value of γ (see section 4).

the lower panels of Fig.3 bracket the corresponding num-

ber count relative model differences due to the 1σ un-

certainty in the value of γ, with the lower curve cor-

responding to (γ,σ8) = (0.71,0.787) and the upper to

(γ,σ8) = (0.42,0.876).

The results (see also Table II) indicate that significant

model differences should be expected to be measured

up to z∼< 1 for the case of the eROSITA X-ray survey,

and to much higher redshifts for the case of the South

Pole Telescope SZ survey. What is particularly interest-

ing is the differential difference between the ΛCDM and

MGCDM models, which is negative locally (z∼< 0.3), pos-

itive at intermediate redshifts (0.4∼< z∼< 1) and negative

again for z∼> 1. This appears to be a unique signature

of the MGCDM model, which differentiates it from the

behaviour of a large class of DE models (see [57]) and

makes it relatively easier to be distinguished. In Table

II, one may see a more compact presentation of our re-

sults including the relative fractional difference between

the MGCDM model and the ΛCDM model, in character-

istic redshift bins and for both future surveys.

The dashed lines in

6.CONCLUSIONS

In this work, the large and small scale dynamical prop-

erties of a flat FLRW cold dark matter cosmology, en-

dowed with massive gravitons (MGCDM), were discussed

Page 9

9

from an analytical and a numerical viewpoints. We find

that the MGCDM can accommodate a “dynamic phase

transition” from an early decelerating phase (driven only

by cold dark matter) to a late time accelerating expan-

sion and a subsequent recent re-deceleration phase.

Interestingly, the Hubble function of the MGCDM

model contains only two free parameters, namely H0and

Ωm, which is the same number of free parameters as the

ΛCDM model. Performing, a joint likelihood analysis us-

ing the current observational data (SNIa, CMB shift pa-

rameter and BAOs), we have provided tight constraints

on the main cosmological parameter of the MGCDM

model, i.e., Ωm = 0.276 ± 0.012. We then compared

the MGCDM scenario with the conventional flat Λ cos-

mology regarding the rate of clustering as well as the

predicted halo redshift distribution.

The main conclusions of such a comparison are:

• At large redshifts the amplitude of the linear per-

turbation growth factor of the MGCDM model is

slightly different to the Λ solution (at a 1 − 4%

level), while the observationally determined growth

index of clustering (γ ≃ 0.56) is smaller than the

corresponding fit for the Λ model (γΛ≃ 0.62), al-

though within their respective 1σ uncertainties.

• The shape and amplitude for the redshift distribu-

tion of cluster-size halos predicted by the MGCDM

model is quite different from the one of a flat

ΛCDM cosmology. Such a difference depends on

redshift and has a characteristic signature that can

discriminate the current graviton model from other

contender DE models in the future cluster surveys.

Acknowledgments

MP acknowledges funding by Mexican CONACyT

grant 2005-49878, and JASL is partially supported by

CNPq and FAPESP under grants 304792/2003-9 and

04/13668-0, respectively.

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Modelσ8

γ(δN/NΛ)eROSITA

z < 0.3

0.00

-0.09

0.00

-0.19

(δN/NΛ)SPT

0.6 ≤ z < 0.9

0.00

0.11

0.18

0.05

0.6 ≤ z < 0.9

0.00

0.15±0.01

0.25±0.01

0.06±0.01

z < 0.3

0.00

-0.09

0.00

-0.19

1.3 ≤ z < 2

0.00

-0.03

-0.01

-0.05

ΛCDM

MGCDM

MGCDM

MGCDM

0.804

0.831

0.875

0.789

0.62

0.56

0.42

0.71

TABLE II: Numerical results. The 1stcolumn indicates the cosmological model. The 2rdand 3rdcolumns lists the corresponding

σ8and γ values, respectively. The remaining columns present the fractional relative difference of the abundance of halos between

the MGCDM and the ΛCDM cosmology for two future cluster surveys discussed in the text. The lower two rows show results

corresponding to the upper and lower 1σ range of the observational γ value uncertainty. Errorbars are 2σ Poisson uncertainties

and are shown only if they are larger than 10−2).