arXiv:1103.1464v2 [astro-ph.CO] 12 Apr 2011
Dynamics and Constraints of the Massive Gravitons Dark Matter Flat Cosmologies
Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece
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
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: firstname.lastname@example.org
†Electronic address: email@example.com
‡Electronic address: firstname.lastname@example.org
§Electronic address: email@example.com
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-
The simplest dark energy candidate corresponds to a
cosmological constant, Λ (see  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 , 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
the natural value estimated using quantum field theory
(ii) Coincidence problem - the fact that the matter and
the vacuum energy densities are of the same order just
prior to the present epoch .
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  (see, however,
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 .
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  and Refs.
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 . 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
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 . 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 . 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
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
In this section we briefly present the main points of
Visser’s massive gravity approach . The full action is
given by (in what follows ? = c = 1)
16πG+ Lmassg(g,g0) + Lmatter(g)
where besides the Einstein-Hilbert Lagrangian and the
Lagrangian of the matter fields, we have the bi-metric
×(g − g0)ρν−1
0)µν(g − g0)µσ(g−1
0)µν(g − g0)µν
where mg is the graviton mass and (g0)µν is a general
The field equations, which are obtained by variation of
(2.1), can be written as:
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:
(g − g0)σρ−1
×(g − g0)αβ
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 . 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 ).
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
In the above framework, the global dynamics of a
flat MGCDM cosmology is driven by the following equa-
8πGρ = 3
8πGp = −2¨ a
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 . 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) +
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:
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:
= Ωma−3+ δH2,(2.10)
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-
In general, using the FLRW equations, one can express
the effective dark energy EoS parameter in terms of the
normalized Hubble parameter 
H0)2is the present contribu-
1 − Ωma−3E−2(a).
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
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
?7 − 10a2
7 − 5a2
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].
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. , 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
?µth(ai,Ωm) − µobs(ai)
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
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.
.In particular, we utilize the following estimator
SDSS data to be A = 0.469 ± 0.017, where zs= 0.35 [or
as = (1 + zs)−1≃ 0.75]. Therefore, the corresponding
BAOfunction can be written as:
y2H(y). Now, from the
, measured from the
BAO(Ωm) =[A(Ωm) − 0.469]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
from the WMAP 7-years data  is R = 1.726±0.019 at
zls= 1091.36 [or als= (1 + zls)−1≃ 9.154 × 10−4]. In
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
cmb(Ωm) =[R(Ωm) − 1.726]2
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
.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 . 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-
tot(Ωm) = χ2
• MGCDM model:
tot(Ωm)/dof ≃ 448.5/367. Such results should
be compared to those found by Alves et al.
tot(Ωm)/dof ≃ 565.06/558. This difference must
be probably attributed to the use of the Union2
supernovae sample  by the latter authors.
=0.276 ± 0.012 with
0.273 ± 0.015 with a
• ΛCDM model:
tot(Ωm)/dof ≃ 439.5/367, which is in good agree-
ment with recent studies [1–8].
=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.
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
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. , does not change the previously
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
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 .)
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 . 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 ,].
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 , which has the following func-
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 
D(a) = ae
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 , 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  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
TABLE I: Data of the growth rate of clustering . The
correspondence of the columns is as follows: redshift, observed
growth rate and references.
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; ) and that
expected in the MGCDM cosmological model, according
?fobs(zi) − fmodel(zi,γ)
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, (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
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
−0.14(χ2/dof ≃ 0.69), while for the ΛCDM
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  obtained γ = 0.60+0.40
while Nesseris & Perivolaropoulos , 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
−0.17. If such a sys-
The difference between
5. COMPARE THE CLUSTER HALO
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  (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 ; here we only
present the basic ideas. The number density of halos,
n(M,z), with masses within the range (M,M +δM) are
where fPSc(σ) =
early extrapolated density threshold above which struc-
tures collapse , 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 , and the values of the baryon density
parameter, the spectral slope and Hubble constant ac-
cording to the recent WMAP7 results . 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. .
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  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 . Now, in order to estimate the
correct model σ8power spectrum normalization, we use
the formulation developed in  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 ) which are based on the WMAP7 re-
sults , on a recent cluster abundances analysis , on
weak-lensing results  and on peculiar velocities based
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
c/2σ2), δcis the lin-
differential halo mass function, n(M,z), according to:
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 , ie:
L(M,z) = 3.087 × 1044
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 :
fν0,lim=2.592 × 108mJy
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 ) 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
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
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.
MP acknowledges funding by Mexican CONACyT
grant 2005-49878, and JASL is partially supported by
CNPq and FAPESP under grants 304792/2003-9 and
 A. G. Riess, et al., Astrophys. J., 659, 98, (2007).
 W. M. Wood-Vasey et al., Astrophys. J., 666, 694,
(2007); T. M. Davis et al., Astrophys. J., 666, 716 (2007).
 D. N. Spergel, et al., Astrophys. J. Suplem., 170, 377
 M. Kowalski, et al., Astrophys. J., 686, 749 (2008).
 E. Komatsu, et al., Astrophys. J. Suplem., 180, 330
(2009); E. Komatsu, et al., Astrophys. J. Suplem., 192,
 M. Hicken et al., Astroplys. J., 700, 1097 (2009).
 J. A. S. Lima and J. S. Alcaniz, Mon. Not. Roy. As-
tron. Soc. 317, 893 (2000), astro-ph/0005441; J. F. Jesus
and J. V. Cunha, Astrophys. J. Lett. 690, L85 (2009),
 S. Basilakos, and M. Plionis, Astrophys. J. Lett, 714,
 P. J. Peebles and B. Ratra, Rev. Mod. Phys., 75, 559
(2003); T. Padmanabhan, Phys. Rept., 380, 235 (2003);
J. A. S. Lima, Braz. Journ. Phys., 34, 194 (2004),
[astro-ph/0402109]; E. J. Copeland, M. Sami and S. Tsu-
jikawa, Int. J. Mod. Phys. D15, 1753 (2006); M. S.
Turner and D. Huterer, Ann. Rev. Astron. & Astrophys.,
46, 385 (2008).
 L. Perivolaropoulos, [arxXiv.0811.4684], (2008).
 S. Weinberg, Rev. Mod. Phys., 61, 1 (1989).
 P.J. Steinhardt, in: Critical Problems in Physics, edited
by V.L. Fitch, D.R. Marlow and M.A.E. Dementi
(Princeton Univ. Pr., Princeton, 1997); P.J. Steinhardt,
Phil. Trans. Roy. Soc. Lond. A361, 2497 (2003).
 B. Ratra and P. J. E. Peebles, Phys. Rev D., 37, 3406
 M. Ozer and O. Taha, Nucl. Phys. B 287, 776 (1987).
 W. Chen and Y-S. Wu, Phys. Rev. D 41, 695 (1990);
J. C. Carvalho, J. A. S. Lima and I. Waga, Phys. Rev.
D 46, 2404 (1992); J. A. S. Lima and J. M. F. Maia,
Phys. Rev D 49, 5597 (1994); J. A. S. Lima, Phys. Rev.
D 54, 2571 (1996), [gr-qc/9605055]; A. I. Arbab and A.
M. M. Abdel-Rahman, Phys. Rev. D 50, 7725 (1994); J.
M. Overduin and F. I. Cooperstock, Phys. Rev. D 58,
 S. Basilakos, M. Plionis and S. Sol` a, Phys. Rev. D. 80,
 C. Wetterich, Astron. Astrophys. 301, 321 (1995)
 R. R. Caldwell, R. Dave, and P.J. Steinhardt, Phys. Rev.
Lett., 80, 1582 (1998).
 P. Brax, and J. Martin, Phys. Lett. B468, 40 (1999).
 A. Kamenshchik, U. Moschella, and V. Pasquier, Phys.
Lett. B. 511, 265, (2001); M. Makler, S. Q. de Oliveira
and I. Waga, Phys. Lett. B 68, 123521 (2003); J. S.
Alcaniz and J. A. S. Lima, Astrophys. J. 618 (2005); J.
A. S. Lima, J. V. Cunha and J. S. Alcaniz, Astrop. Phys.
31 233 (2009).
 A. Feinstein, Phys. Rev. D., 66, 063511 (2002).
 R. R. Caldwell, Phys. Rev. Lett. B., 545, 23 (2002).
 M. C. Bento, O. Bertolami, and A.A. Sen, Phys. Rev. D.,
70, 083519 (2004).
 L. P. Chimento, and A. Feinstein, Mod. Phys. Lett. A,
19, 761 (2004).
 E. V. Linder, Phys. Rev. D. 70, 023511, (2004); E. V.
Linder, Rep. Prog. Phys., 71, 056901 (2008).
 A. W. Brookfield, C. van de Bruck, D.F. Mota, and D.
Tocchini-Valentini, Phys. Rev. Lett. 96, 061301 (2006).
 J. Grande, J. Sol` a and H.ˇStefanˇ ci´ c, JCAP 08, (2006),
011; Phys. Lett. B645, 236 (2007).
 C. G. Boehmer, and T. Harko, Eur. Phys. J. C50, 423
 G. Allemandi, A. Borowiec, M. Francaviglia and S. D.
Odintsov, Phys. Rev. D 72, 063505 (2005); L. Amen-
dola, D. Polarski and S. Tsujikawa, Phys. Rev. Lett. 98,
131302 (2007); J. Santos, J. S. Alcaniz, F. C. Carvalho
and N. Pires, Phys. Lett. B 669, 14 (2008); J. Santos
and M. J. Reboucas, Phys. Rev. D 80, 063009 (2009);
V. Miranda, S. E. Joras, I. Waga and M. Quartin, Phys.
Rev. Lett. 102, 221101 (2009); S. H. Pereira, C. H. G.
Bessa and J. A. S. Lima, Phys. Lett. B 690 103 (2010),
[arXiv:0911.0622]. For a review see, T. P. Sotiriou and V.
Faraoni, Rev. Mod. Phys. 82 451, (2010).
 R. Reyes et al., Nature 464, 256 (2010).
 M. Visser, Gen. Rel. Grav. 21, 1717, (1998).
 N. Rosen, Gen. Rel. Grav. 4, 435, (1973)
 C.de Rham,G.
 K. Lee et al., Astrophys. J. Lett. 722, 1589 (2010).
 M. E. S. Alves, et al., Phys. Rev. D. 82, 3505 (2010).
 A. E. Evrard et al., Astrophys. J., 573, 7 (2002).
 S. Borgani et al., Astrophys. J., 561, 13 (2001).
 T.H. Reiprich, H. B¨ ohringer, Astrophys. J., 567, 716
 A. Vikhlinin et al., Astrophys. J., 692, 1060, (2009).
 M. Bartelmann, A. Huss, J. M. Colberg, A. Jenkins, and
F. R. Pearce, Astron. Astrophys. 330, 1, (1998).
 H. Dahle, Astrophys. J., 653, 954 (2006).
 V. L. Corless, L.J. King, Mon. Not. Roy. Astron. Soc,
396, 315 (2009).
 N. A. Bahcall, et al., Astrophys. J., 585, 182 (2003).
 Z. L. Wen, J.L. Han, F.S. Liu, Mon. Not. Roy. Astron.
Soc, 407, 553, (2010)
 J. A. Tauber, New Cosmological Data and the values of
the Fundamental Parameters, 201, 86, (2005).
 N. N. Weinberg and M. Kamionkowski, Mon. Not. Roy.
Astron. Soc, 341, 251 (2003).
 L. Liberato and R. Rosenfeld, JCAP 0607, 009 (2006).
 M. Manera and D. F. Mota, Mon. Not. Roy. Astron. Soc.
371, 1373 (2006).
 L. R. Abramo, R. C. Batista, L. Liberato, and R. Rosen-
feld, JCAP 11, 012 (2007).
 M. J. Francis, G. F. Lewis, and E. V. Linder, Mon. Not.
Roy. Astron. Soc, 393, L31, (2009); M. J. Francis, G. F.
Lewis, and E. V. Linder, Mon. Not. Roy. Astron. Soc,
 F. Schmidt, A. Vikhlinin and Wayne Hu, Phys. Rev. D.,
80, 083505, (2009)
 M. J. Mortonson, Phys. Rev. D., 80, 123504 (2009).
 D. Rapetti, S. W. Allen, A. Mantz and H. Ebeling, Mon.
Not. Roy. Soc., 406, 1796, (2010)
 F. Pace, J-C Waizmannm and M. Bartelman, Mon. Not.
Roy. Astron. Soc, 406, 1865, (2010).
 U. Alam, Z. Lukic´ and S. Bhattacharya, Astrophys. J.,
727, 87, (2011)
 S. Khedekar, and S. Majumdar, Phys. Rev. D., 82,
081301, (2010); S. Khedekar, S. Majumdar and S. Das,
Phys. Rev. D., 82, 041301, (2010).
 S. Basilakos, M. Plionis, A. Lima, Phys. Rev. D., 82,
 L. Lombriser, A. Slosar, U. Seljak and Wayne Hu,
 P. Rastall, Phys. Rev. D 6, 3357 (1972); J. Narlikar, J.
Astrophys. Astr. 5, 67 (1984);
 M. E. S. Alves, O. D. Miranda and J. C. N. de Araujo,
Gen. Relat. Grav., 39, 777 (2007); J. C. N. de Araujo
and O. D. Miranda, Gen. Relativ. Gravit. 39, 777 (2007)
 T. D. Saini, S. Raychaudhury,V. Sahni, and A. A.
Starobinsky, Phys. Rev. Lett., 85, 1162, (2000); D.
Huterer, and M. S. Turner, Phys. Rev. D., 64, 123527,
 E. V. Linder and A. Jenkins, Mon. Not. Roy. Astron.
Soc., 346, 573 (2003).
 W. Freedman et al., Astrophys. J., 553, 47 (2001)
 D. J. Eisenstein et al., Astrophys. J., 633, 560, (2005);
N. Padmanabhan, et al., Mon. Not. Roy. Astron. Soc.,
378, 852 (2007).
 W. Percival et al., Mon. Not. Roy. Astron. Soc., 401,
 E. A. Kazin, Astrophys. J., 710, 1444 (2010).
 J. R. Bond, G. Efstathiou and M. Tegmark, Mon. Not.
Roy. Astron. Soc. 291, L33 (1997).
 S. Nesseris and L. Perivolaropoulos, JCAP 0701, 018,
 O. Elgaroy & T. Multamaki, Astron. Astrophys., 471,
65 (2007); P.S. Corasaniti & A. Melchiorri Phys.Rev.D,
77, 103507 (2008).
 R. Amanullah et al., Astrophys. J., 716, 712 (2010)
 M. E. S. Alves, O. D. Miranda and J. C. N. de Araujo,
 R. Dave, R. R. Caldwell and P. J. Steinhardt, Phys. Rev.
D., 66, 023516 (2002)
 A. Lue, R. Scossimarro, and G. D. Starkman, Phys. Rev.
D., 69, 124015, (2004)
 S. Tsujikawa, K. Uddin and R. Tavakol, Phys. Rev. D.,
77, 043007, (2008)
 C. Talmadge, J. P. Berthias, R. W. Hellings and E. M.
Standish, Phys. Rev. Lett., 61 (1988)
 P. J. E. Peebles, “Principles of Physical Cosmology”,
Princeton University Press, Princeton New Jersey (1993).
 L. Wang, and J.P. Steinhardt, Astrophys. J., 508, 483
 E. V. Linder, and R. N. Cahn, Astrop. Phys., 28, 481
 S. Nesseris and L. Perivolaropoulos, Phys. Rev. D., 77,
 C. Di Porto and L. Amendola, Phys. Rev. D., 77, 083508,
 L. Verde, et al., Mon. Not. Roy. Astron. Soc., 335, 432,
 E. Hawkins, et al., Mon. Not. Roy. Astron. Soc., 346, 78,
 M. Tegmark, et al., Phys. Rev. D., 74, 123507, (2006)
 N. P. Ross, et al., Mon. Not. Roy. Astron. Soc., 381, 573,
 N. P. Ross, et al., Mon. Not. Roy. Astron. Soc., 383, 656,
 P. McDonald, et al., Astrophys.. J., 635, 761, (2005)
 W. H. Press and P. Schechter, Astrophys. J. 187, 425
 V. Eke, S. Cole &, C. S. Frenk, Mon. Not. Roy. Astron.
Soc., 282, 263, (1996)
 J. M. Bardeen, J. R. Bond, N. Kaiser, and, A. S. Szalay,
Astrophys. J., 304, 15 (1986); N. Sugiyama, Astrophys.
J. Suplem., 100, 281 (1995)
 A. Jenkins, et al., Mon. Not. Roy. Astron. Soc., 321, 372
 L. Marassi and J. A. S. Lima, Int. J. Mod. Phys. D 13,
1345 (2004); ibdem, IJMPD 16, 445 (2007).
 D. Reed, R. Bower, C. Frenk, A. Jenkins, and T. Theuns,
Mon. Not. Roy. Astron. Soc., 374, 2 (2007).
 E. Rozo, et al., Astrophys. J., 713, 1207 (2010)
 L. Fu, et al., Astron. Astrophys. 479, 9 (2008)
 R. Watkins, H.A. Feldman and, M.J. Hudson, 2009, Mon.
Not. Roy. Astron. Soc., 392, 743
 C. Fedeli, L. Moscardini, and S. Matarrese, Mon. Not.
Roy. Astron. Soc., 397, 1125 (2009)
12 Download full-text
z < 0.3
0.6 ≤ z < 0.9
0.6 ≤ z < 0.9
z < 0.3
1.3 ≤ z < 2
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).