A&A 481, 279-294 (2008)
DOI: 10.1051/0004-6361:20067045
M. Demianski^{1,2} - E. Piedipalumbo^{3,4} - C. Rubano^{3,4} - P. Scudellaro^{3,4}
1 - Institute for Theoretical Physics, University of
Warsaw, Hoza 69, 00-681 Warsaw, Poland
2 -
Department of
Astronomy,
Williams College, Williamstown, MA 01267, USA
3 -
Dipartimento di
Scienze Fisiche, Università di Napoli Federico II, Compl.
Univ. Monte S. Angelo, 80126 Naples, Italy
4 -
Istituto Nazionale
di Fisica Nucleare, Sez. Napoli, Via Cinthia, Compl. Univ. Monte
S. Angelo, 80126 Naples, Italy
Received 29 December 2006 / Accepted 6 November 2007
Abstract
Aims. We study cosmological models in scalar tensor theories of gravity with power-law potentials as models of an accelerating universe.
Methods. We consider cosmological models in scalar tensor theories of gravity that describe an accelerating universe and study a family of inverse power-law potentials, for which exact solutions of the Einstein equations are known. We also compare theoretical predictions of our models with observations. For this we use the following data: the publicly available catalogs of type Ia supernovae and high redshift gamma ray bursts, the parameters of large-scale structure determined by the 2-degree Field Galaxy Redshift Survey (2dFGRS), and measurements of cosmological distances based on the Sunyaev-Zel'dovich effect, among others.
Results. We present a class of cosmological models that describe the evolution of a homogeneous and isotropic universe filled with dust-like matter and a scalar field that is non minimally-coupled to gravity. We show that this class of models depends on three parameters: V_{0} - the amplitude of the scalar field potential,
- the present value of the Hubble constant, and a real parameter s that determines the overall evolution of the universe. It turns out that these models have a very interesting feature naturally producing an epoch of accelerated expansion. We fix the values of these parameters by comparing predictions of our model with observational data. It turns out that our model is compatible with the presently available observational data.
Key words: cosmology: theory - cosmology: cosmological parameters - cosmology: observations
Recent observations of the type Ia supernovae and CMB anisotropy indicate that the total matter-energy density of the universe is now dominated by some kind of dark energy causing an accelerated expansion of the Universe (Perlmutter 1997; Riess et al. 1998,2004; Spergel et al. 2006). The origin and nature of this dark energy remains unknown (Zeldovich 1967; Weinberg 1989).
Prompted by this discovery, a new class of cosmological models has recently been proposed. In these models the standard cosmological constant -term is replaced by a dynamical, time-dependent component - quintessence or dark energy - that is added to baryons, cold dark matter (CDM), photons, and neutrinos. The equation of state of the dark energy is assumed to be of a hydrodynamical type , where and are, respectively, the energy density and pressure, and , which implies a negative contribution to the total pressure of the cosmic fluid. When , we recover the standard cosmological constant term. One of the possible physical realization of quintessence is a cosmic scalar field (Caldwell et al. 1998), which induces dynamically a repulsive gravitational force that is responsible for the observed now accelerated expansion of the universe.
The existence of dark energy, which now dominates the overall energy density in the universe, is posing several theoretical problems. First, it is natural to ask why we observe the universe at exactly the time when the dark energy dominates matter (cosmic coincidence problem). The second issue, a fine-tuning problem, arises from the fact that if the dark energy is constant, such as in the standard cosmological constant scenario, then at the beginning of the radiation era its energy density should have been vanishingly small in comparison with the radiation and matter component. This poses a problem, since to explain the inflationary behavior of the early universe and the late time dark energy dominated regime, the dark energy should evolve and cannot simply be a constant. All these circumstances stimulated a renewed interest in the generalized gravity theories, and prompted consideration of a variable term in more general classes of theories, such as the scalar tensor theories of gravity.
In our earlier paper (Demianski et al. 2006) we analyzed extended quintessence models, for which exact solutions of the Einstein equations are known, and discussed how in these models it is possible to treat the fine tuning problem in an alternative way. We applied our consideration to a special model, based on one of the most commonly used quintessence potentials , corresponding to the coupling (so-called induced gravity). We showed that this model corresponds to a special, and physically significant, case that emerged by requiring the existence of a Noether symmetry in the pointlike Lagrangian. In this paper we analyze a new and wider class of theories derived from the Noether symmetry requirement. One of the main advantages of such models is that they exhibit power-law couplings and potentials and admit a tracker behavior. In some sense we complete and generalize the analysis initiated in Marino & de Ritis (2001) and de Ritis et al. (2000), where the attention was focused on the mechanism of obtaining an effective cosmological constant through the cosmological no-hair theorem, and the analysis of the solution was restricted to the asymptotical regime. Extending our analysis to the whole time evolution, we are not only able to clarify the properties of such solutions, but also to compare predictions of these models with observations. We concentrate on the following data: the publicly available data on type Ia supernovae and gamma ray bursts, the parameters of large scale structure determined by the 2-degree Field Galaxy Redshift Survey (2dFGRS), and the measurements of cosmological distance with the Sunyaev-Zel'dovich effect.
Since the detailed properties of a quintessence model, whose
coupling and potential form are derived by requiring the existence of
a Noether symmetry, are extensively discussed in Demianski et al. (2006, Paper I), here we only summarize the basic results,
referring readers to our previous paper for details. Let us consider
the general action for a scalar field ,
non minimally-coupled
with gravity, but not coupled with matter; in this case, we have
From now on we restrict ourselves to a dust-filled universe with
,
and
.
Using the point-like Lagrangian Eq. (2) in the action and varying it with respect to ,
we
obtain the Euler-Lagrange equations
(8) |
(9) |
(10) |
(17) | |||
(18) | |||
(19) |
(20) | |||
(21) |
As is apparent from Eqs. (15) and (16) there are two additional particular values of s, namely s=0 and s=-3, which should be treated independently.
When s= 0, a Noether symmetry exists if
Figure 1: Diagram of as function of s. It turns out that an attractive gravity requires . | |
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Figure 2: Behavior of the coupling factor (red curve) and the power-law exponent p(s) (blue curve). We see that with an appropriate choice of s in the range (-1.5 , -1) all the values for the exponents are available. | |
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D | = | (22) | |
= | (23) | ||
Using the available observational data, we can further constrain the range of possible values of s. Actually requiring that today , as indicated by observations of supernovae Ia and WMAP, we constrain the range of possible values of s to .
Figure 3: Plot of versus (solid black line). The upper and lower dashed lines indicate the log-log plot of a^{-3} and a^{-4} versus a, respectively. It turns out that scales as a^{-n}, with 3<n<4. In this and subsequent plots, we use the mean values for the parameters obtained through fits (see Table 2). | |
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From Eqs. (15) and (16) it turns out
that for low values of t the scale factor and the scalar field
behave as
(27) | |||
(28) |
Recently, the cosmological relevance of extended gravity theories as scalar tensor or higher order theories has been widely explored. However, in the weak field approximation, all these classes of theories are expected to reproduce the Einstein general relativity that, in any case, is experimentally tested only in this limit. This fact is a matter of debate, since several relativistic theories do not reproduce Einstein results at the Newtonian approximation but, in some sense, generalize them, giving rise, for example, to Yukawa-like corrections to the Newtonian potential, which could have interesting physical consequences. Moreover, in general, any relativistic theory of gravitation can yield corrections to the Newton potential (see for example, Will 1993), which in the post-Newtonian (PPN) formalism could furnish tests for such theory, mainly based on the Solar System experiments. In this section we want to discuss the Newtonian limit of our class of scalar-tensor theories of gravity, the induced gravity theories, and to study the parametrized post Newtonian (PPN) behavior of these theories. In particular, it turns out that the Newtonian limit depends on . Furthermore, we find a quadratic correction to the Newtonian potential strictly depending on the presence of the scalar-field potential, which acts as a cosmological constant.
Figure 4: as a function of , for the averaged mean values provided by our analysis, as shown in Table 2. We observe a transition from a small constant value in the past, , to at present. | |
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Figure 5: Rate of change in the equation of state as measured by versus the parameter. The values of the parameters correspond to the average values provided by our analysis and shown in Table 2. | |
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Figure 6: Plot of versus in the Jordan frame. The vertical bar marks . The solid red straight line indicates the log-log plot of versus a. The matter dominated-era and the transition to the present dark-energy dominated regime are represented. | |
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A satisfactory description of the PPN limit for scalar tensor
theories is developed in Esposito-Farese (2004) and Damour et al. (1993). In
these papers, this limit has been thoroughly discussed leading to
interesting results even in the case of strong gravitational sources
like pulsars and neutron stars where the deviations from general
relativity are considered in a non-perturbative regime
(Damour et al. 1993). The starting point for such an analysis is a
redefinition of the non minimally-coupled Lagrangian action in terms
of a minimally-coupled scalar field model via a conformal
transformation of the form
.
In fact, assuming the transformation rules
(42) |
(45) |
Table 1: A brief summary of recent constraints on the PPN-parameters.
We summarize the experimental results in Table 1. These
results have been used by Schimd et al. (2005) to set the following
constrains:
For the sake of completeness, here we even take into account the
shift that the scalar-tensor gravity induces on the theoretical
predictions for the local value of the gravitational constant as
coming from the Cavendish-like experiments. This quantity
represents the gravitational coupling measured when the Newton
force arises between two masses:
Figure 7: Current limits on the PPN parameters restrict the range of the parameter s. We see that the constraint on leads to , as shown in the inner zoom. | |
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Finally, in Fig. 8 we plot the Brans-Dicke parameter as a function of s: actually, for our model . It turns out that just for , satisfies the limits coming both from the Solar System experiments, (Will 1993), and current cosmological observations, including cosmic microwave anisotropy data and the galaxy power spectrum data, give (Acquaviva et al. 2005)^{}.
Figure 8: Behavior of the Brans-Dicke parameter as a function of s. For , satisfies limits placed by the solar system experiments ( ) and by current cosmological observations ( ). | |
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Conformal transformations are often used
to convert the non minimally-coupled scalar field models into the
minimally-coupled ones, to gain mathematical simplification. The
Jordan frame, in which the scalar field is non minimally-coupled to the Ricci curvature, is mapped into the Einstein
frame in which the transformed scalar field is minimally-coupled
but at a price of coupling matter to the scalar field. The two
frames are not physically equivalent, and some care has to be taken
in applying this technique (see for instance Faraoni 2000 for a
critical discussion of this point). In this section we study the
effect of conformal transformations on our models and show that, in
presence of matter, it can mimic a coupling between the quintessence
scalar field and dark matter. We discuss some implications of such a
fictitious interaction on the effective equation of state.
Actually it turns out that, since the interaction alters the
redshift dependence of the matter density, it is possible to obtain
an effective transformed dark-energy equation of state of
.
Let us start from the transformation
rules connected with the conformal transformation
:
Let us consider our nonminimally coupled model characterized by the
functions
and
.
According to the rules in Eqs. (49)-(52) we obtain the following relations
between the transformed and original dynamical quantities:
Figure 9: Time evolution of the transformed scalar field . | |
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Figure 10: Evolution with the redshift of in the Einstein frame. | |
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Concluding this section we present the traditional plot
and compare
it with the
relation (see Fig. 11). Interestingly we see that, just because of the
interaction term,
no longer tracks
the matter during the matter dominated era, as happens in the
Jordan frame (see Fig. 6), but becomes dominant at
earlier times.
It is interesting to write down the effective equation of
state
(see Eq. (57)), which
mimics a CDM model. Actually in our case,
is
In this section we
briefly discuss how the scalar tensor theories of gravity could
be involved in a cosmological model with mass varying
neutrinos that mimic the dark energy, a quite different
theoretical scenario of evolution of the universe that recently
has been suggested by Fardon et al. (2004). Let us recall that the mass differences between neutrino mass eigenstates
(m_{1}, m_{2}, m_{3}) have recently been measured in oscillation
experiments (Lesgourgues & Pastor 2006). Observations of atmospheric
neutrinos suggest a squared mass difference of
,
while solar neutrino observations and
results from the KamLAND neutrino experiment point towards
.
While only weak
constraints on the absolute mass scale (
)
have been obtained from single -decay
experiments, the double-beta decay searches from the
Heidelberg-Moscow experiment have reported a signal for a
neutrino mass at >
level (Klapdor-Kleingrothaus et al. 2004), recently
promoted to > level (Klapdor-Kleingrothaus 2006). This last result
translates into a total neutrino mass of
at
c.l., but this claim is still considered controversial (see Elliott & Engel 2004).
Figure 11: Plot of versus in the Einstein frame. The vertical bar marks . | |
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It is known in the literature (Lesgourgues & Pastor 2006) that massive neutrinos can be extremely relevant for cosmology as they leave key signatures in several cosmological data sets. More specifically, massive neutrinos suppress the growth of fluctuations on scales below the horizon scale when they become non relativistic. Current cosmological data have been able to indirectly constrain the absolute neutrino mass to eV at c.l. (Spergel et al. 2006), and are challenging the Heidelberg-Moscow claim. However, as first noticed by (Hannestad 2005), there is some form of anticorrelation between the equation of state parameter w and . The cosmological bound on neutrino masses can therefore be relaxed by considering a dark energy component with a more negative value of than a cosmological constant. Actually it has been proved that the Heidelberg-Moscow result is compatible with the cosmological data only if the equation of state (with w being constant) is at .
This result suggests an interesting link between neutrinos and dark energy (see for instance Brookfield et al. 2006b,a; Kaplan et al. 2004; Amendola 2004; Bi et al. 2004). According to this scenario the late-time accelerated expansion of the universe is driven by the coupling between the quintessential scalar field and neutrinos. Because of this coupling the mass of neutrinos becomes a function of this scalar field. Since the scalar field evolves with time, the mass of neutrinos is not constant (mass-varying neutrinos): the main theoretical motivation for this connection relies on the fact that the energy scale of the dark energy is close to the neutrinos mass scale. Moreover, as discussed above, in interacting dark energy models, the net effect of the interaction is to change the apparent equation of state of the dark energy, allowing a so-called superquintessence regime, with . Interestingly enough, if the Heidelberg-Moscow results are combined with the WMAP 3-years data and other independent cosmological observations, such as the ones connected with the large-scale structure - coming from galaxy redshift surveys and Lyman- forests - or with the SNIa surveys, it is possible to constrain the equation of state to at c.l., ruling out a cosmological constant greater than c.l. (see De La Macorra et al. 2007).
In the following we discuss the coupling between neutrinos
and dark energy from the point of view of the conformal
transformation, according to the arguments outlined in the
previous section; i.e. we show that the neutrino mass and
scalar field coupling can be interpreted as an effect of
conformal transformations from the Jordan to the Einstein frames,
as it happens for the coupling between the dark energy and
the dark matter. For our purpose the neutrinos can be either
Dirac or Majorana particles, the only necessary ingredient is
that, according to Fardon et al. (2004), the neutrino mass is a
function of the scalar field. In the cosmological context,
neutrinos should be treated as a gas (Brookfield et al. 2006b) and
described by the collisionless distribution function
in the phase space (where
is the
conformal time) that satisfies the Boltzmann equation. When
neutrinos are collisionless, the distribution function f does
not depend explicitly on time. We can then solve the Boltzmann
equation and calculate the energy density stored in neutrinos
(f_{0} is the background neutrino distribution function):
(66) |
(67) |
(71) |
Figure 12: Redshift dependence of the second derivative of the scale factor. The transition from a decelerating to an accelerating expansion occurs close to , as predicted by recent observations of SNIa (Riess et al. 2004, 2007). | |
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Figure 13: Observational data of the SNIa sample compiled by Riess et al. (2007) fitted to our model. The solid curve is the best fit curve, compared with a standard CDM model with (red dashed line). | |
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After having explored the Hubble diagram
of SNIa, let us now follow the more general approach, as suggested by
Daly & Djorgovski (2004) and already tested in Paper I.
Consider as a cosmological observable the dimensionless coordinate
distance defined as
Figure 14: Updated Daly & Djorgovski database (Daly & Djorgovski 2005) fitted to our model. The solid curve is the best fit curve with for 248 data points, and the best fit values are , s=-1.49^{+0.02}_{-0.04}. | |
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In this section we discuss a possible observational determination of
H(z) based on the method developed by Jimenez et al. (2003), which
involves differential age measurements. We present some constraints
that can be placed on the evolution of our quintessence model by
this data. First, it is worth pointing out some aspects connected with
the sensitivity of the cosmology to the t(z) and
relations. Actually, it is well known that in scalar tensor theories
of gravity, as well as in general relativity, the expansion history
of the universe is determined by the function H(z). This implies
that observational quantities, such as the luminosity distance, the
angular diameter distance, and the lookback time, all depend on
H(z). It turns out that the most appropriate mathematical tool for
studing the sensitivity of the cosmological model to such observables
is the functional derivative of the corresponding relations with
respect to the cosmological parameters (see Saini et al. 2003, for a
discussion about this point in relation to distance measurements).
However, from an empirical point of view, it is also possible to
show that the lookback time is much more sensitive to the
cosmological model than other observables, such as the luminosity
distance and the distance modulus. This circumstance encourages us
to use, together with the other more standard techniques discussed
above, the age of cosmic clocks to test alternative
cosmological scenarios. Apart from the advantage of providing an
alternative investigation instrument, the age-based methods use
the higher sensitivity to the cosmological parameters of the
relation, as shown in Figs. 15 and 16. Moreover, as we discuss in the following,
such a method reveals its full strength when applied to old objects
at very high z. Actually it turns out that this kind of analysis
could remove, or at least reduce, the degeneracy that we observe at
lower redshifts, for example the one in the Hubble diagram for SNIa
observations, which can be fitted by different cosmological models
with a similar statistical significance.
Since the Hubble parameter can be related to the differential age of
the universe as a function of redshift by the equation
Figure 15: The sensitivity of the relation compared to the values of the parameters in our model. The red line shows , and the blue line shows . | |
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Figure 16: The sensitivity of the relation compared to the values of the parameters in our model in. The red line shows , and the blue line shows . | |
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Here we use the dz/dt data from
(Simon et al. 2005) to determine H(z) in the redshift range
0.1 < z <
1.8. To follow the procedure described in Simon et al. (2005), first we
group all galaxies together that are within
of each
other. This gives an estimate of the age of the universe at a given
redshift. We then compute age differences only for those bins in
redshift that are separated by more than
but less
than
.
The first limit is imposed so that the age
evolution between the two bins is larger than the error in the age
determination. We note here that differential ages are less
sensitive to systematic errors than absolute ages (see Jimenez et al. 2003). The observational value of H(z) is then directly computed by
using Eq. (77), and after that the ages data have been
scaled according to our choice of the unit of time (the unknown
scaling factor has been provided by the
procedure). To
determine the best-fit parameters, we define the following merit
function:
Figure 17: The best-fit curve of the measured values of H(z) corresponding to , s=-1.49^{+0.03}_{-0.09}. | |
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Figure 18: The best-fit curve to the H(z) data for our nmc model (dark blue line) and for the quintessence QCDM fitted to the new released WMAP- three years + SNLS data (WMAP New Three Year Results 2006) , , w=-1.06^{+0.13}_{-0.08} (blue line). | |
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In this section we discuss how the parameters of our model can be
constrained by the angular diameter distance
as measured using
the Sunyaev-Zeldovich effect (SZE) and the thermal bremsstrahlung
(X-ray brightness data) for galaxy clusters. The distance
measurements using Sunyaev-Zeldovich effect and X-ray emission from
the intracluster medium have to take into account that these
processes depend on different combinations of some of the parameters
of the clusters (see Birkinshaw 1999, and references therein). The SZE is
a result of the inverse Compton scattering of the CMB photons on hot
electrons of the intercluster gas, which preserves the number of
photons but allows photons to gain energy thereby generating a
decrement of the temperature in the Rayleigh-Jeans part of the
black-body spectrum while an increment appears in the Wien region.
We limit our analysis to the so-called thermal or static
SZE. The kinematic effect, present only in clusters with a
nonzero peculiar velocity with respect to the Hubble flow along the
line of sight, will be neglected since the thermal SZE is typically
an order of magnitude larger than the kinematic one. As in Paper I,
we introduce the so-called Compton parameter, y, defined as the
optical depth
times the energy gain
per scattering:
Recently distances to 18 clusters with redshift ranging from
to
have been determined from a likelihood joint
analysis of SZE and X-ray observations (see Table 7 in
Reese et al. 2002). Our analysis used angular diameter
distance measurements for a sample of 83 clusters, containing the 18
above-mentioned clusters, another 24 known previously (see
Birkinshaw 1999), and a recently released sample with the measurement of
the angular diameter distances from the Chandra X-ray imaging
and Sunyaev-Zel'dovich effect mapping of 39 high-redshift clusters
of galaxies (
)
(Bonamente et al. 2005). The
unprecedented spatial resolution of Chandra, combined with
its simultaneous spectral resolution, allows a more accurate
determination of distances. Let us consider the merit function of
the form
Figure 19: Observational SZE data fitted to our model with the best-fit values , s=-1.49^{+0.03}_{-0.09}, and . The empty boxes indicate distance measurements for a sample of 44 mentioned clusters (see Birkinshaw 1999; Reese et al. 2002),while the filled diamonds indicate the measurement of the angular diameter distances from Chandra X-ray imaging and Sunyaev-Zel'dovich effect mapping of 39 high-redshift clusters (Bonamente et al. 2005). | |
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Gamma-ray bursts (GRBs) are bright explosions visible across most of the Universe, certainly out to redshifts of and likely out to . Recent studies have pointed out that GRBs may be used as standard cosmological candles (Friedman & Bloom 2005; Ghirlanda et al. 2004). It turns out that the energy released during bursts spans nearly three orders of magnitude, and the distribution of the opening angles of the emission, as deduced from the timing of the achromatic steepening of the afterglow emission, spans a similar wide range of values. However, when the apparently isotropic energy release and the conic opening of the emission are combined to infer the intrinsic, true energy release, the resulting distribution does not widen, as is expected for uncorrelated data, but shrinks to a very well-determined value (Frail & Kulkarni 2003), with a remarkably small (one-sided) scattering, corresponding to about a factor of 2 in total energy. Similar studies in the X-ray band have reproduced the same results. It is thus very tempting to study to what extent this property of GRBs makes them suitable cosmological standard candles. Schaefer (2003) proposed to use the two well-known correlations of the GRBs luminosity (with variability and with time delay), while others have exploited the recently reported relationship between the beaming-corrected -ray energy and the locally observed peak energy of GRBs (see for instance Dai et al. 2004). As for the possible variation of ambient density from burst to burst, which may widen the distribution of bursts energies, Frail & Kulkarni (2003) note that this spread is already contained in their data sample, and yet the distribution of energy released is still very narrow. There are at least two reasons GRBs are better than type Ia supernovae as cosmological candles. On the one hand, GRBs are easy to find and locate: even 1980s technology allowed BATSE to locate 1 GRB per day, making the build-up of a 300-object database a one-year enterprise. The Swift satellite launched on 20 November 2004, detects GRBs at about the same rate as BATSE, but with a nearly perfect capacity for identifying their redshifts simultaneously with the afterglow observations^{}. Second, GRBs have been detected out to very high redshifts: even the current sample contains several events with z> 3, with one (GRB 000131) at z = 4.5 and another at z=6.3. This should be contrasted with the difficulty of locating SN at z > 1 and absence of any SN with z > 2. On the other hand, the distribution of luminosities of SNIa is narrower than the distribution of energy released by GRBs, corresponding to a magnitude dispersion rather than . Therefore GRBs may provide a complementary standard candle, out to distances that cannot be probed by SNIa, their major limitation being the larger intrinsic scatter of the energy released, as compared to the small scatter in peak luminosities of SNIa. There currently exists enough information to calibrate luminosity distances and independent redshifts for nine bursts (Schaefer 2003). These bursts were all detected by BATSE with redshifts measured from optical spectra of either the afterglow or the host galaxy. The highly unusual GRB980425 (associated with supernova SN1998bw) is not included because it is likely to be qualitatively different from the classical GRBs. Bursts with red shifts that were not recorded by BATSE still cannot have their observed parameters converted to energies and fluxes that are comparable with BATSE data. For the present analysis we use a sample of GRBs that had their redshifts estimated (), as represented in Fig. 20, with the distance modulus , given by Eq. (72).
To this aim, the only difference with respect to the SNIa is that
we slightly modify the correction term of Eq. (73) and take
Figure 20: Observational Hubble diagram for the recent SNIa sample compiled by Riess et al. (2006) (empty lozenges), and the GRBs data by () (empty boxes) fitted to our model. The solid curve is the best-fit curve with, , s=-1.43^{+0.02}_{-0.04}. The red dashed line corresponds to the standard CDM model with . | |
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Measurements of the gas mass fraction in galaxy clusters have been
proposed as a test of cosmological models (Allen et al. 2002). Both
theoretical arguments and numerical simulations predict that the
baryonic mass fraction in the largest relaxed galaxy clusters should
not depend on the redshift and should provide an estimate of the
cosmological baryonic density parameter
(Eke et al. 1998).
The baryonic content in galaxy clusters is dominated by the hot
X-ray emitting intra-cluster gas so that what is actually
measured is the gas mass fraction
,
and it is this quantity
that should not depend on the redshift. Moreover, it is expected
that the baryonic mass fraction in clusters is equal to the
universal ratio
so that
should indeed
be given by
,
where the
multiplicative factor b is motivated by simulations that suggest
that the gas fraction is lower than the universal ratio. Following
the procedure described in Allen et al. (2002, 2004), and already
used in Paper I we adopt the standard CDM model (i.e., a flat
universe with
and h = 0.5) as a reference cosmology
in making the measurements, so that the theoretical expectation for
the apparent variation of
with the redshift is
Figure 21: The best-fit curve to the data for our nmc model (red thick line) and for the quintessence model (black thick line) considered in Lima et al. (2003). It is interesting to note that, as pointed out also for the model described in Demianski et al. (2006), even if the statistical significance of the best-fit procedure for these two models is comparable, the best fit relative to our nmc model seems to be dominated by smaller redshift data, and the one relative to the Lima et al. model by higher redshift data. | |
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In this section we consider the evolution of scalar density
perturbations in the longitudinal gauge d
.
It is well-known that in the
framework of the minimally-coupled theory, where we have to deal
with a fully relativistic component, that becomes homogeneous on smaller
scales than the horizon, the standard quintessence
cannot cluster on such scales. In the non minimally-coupled
quintessence theories it is instead possible to separate a pure
gravitational term both in the stress-energy tensor
and in the energy density ,
so the situation changes, and
it is necessary to consider also fluctuations of the scalar field.
However, it turns out (Boisseau et al. 2000; Riazuelo & Uzan 2002) that the equation for
dust-like matter density perturbations inside the horizon can be
written as follows
In this paper we have extended the analysis that we performed in Paper I (where we have analyzed a special extended quintessence model, based on one of the most commonly used quintessence potentials , corresponding to the coupling ), considering a new and wider class of theories for which exact solutions of the Einstein equations are known. We also discussed how it is possible in such models to treat the fine-tuning problem in an alternative way. We have shown that an epoch of accelerated expansion appears in a natural way in the family of such models selected by requiring that their corresponding point like Lagrangian admits a Noether symmetry. In the non minimally-coupled scalar tensor theory of gravity, it is possible to perform an appropriate conformal transformation and to move from the Jordan picture to the standard Einstein one, but then matter becomes coupled to the scalar field. We explored both descriptions and also considered the neutrino mass varying model as a possible example of non minimally-coupled scalar tensor theory.
Figure 22: The growth index f in different cosmological models. The thick dashed red line corresponds to our non minimally-coupled model. The blue thin dashed curve corresponds to the standard CDM model with , and the black solid line corresponds to another quintessence model with an exponential potential (described in Demianski et al. 2005). | |
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Table 2: The basic cosmological parameters derived from our model are compared with observational data.
It turns out that the imposed requirement of a Noether
symmetry is quite restrictive, so we obtained a family of models
that is fully specified by 3 parameters: a parameter s that
determines the strength of the non-minimal coupling and the
potential of the scalar field, H_{0} as the Hubble constant, and a
parameter V_{0} that determines the scale of the potential. To
determine the values of these parameters, we compared predictions of
our model with several independent observational data. The results
of this parameter determination procedure are presented in Table 2.
We see that with our average value of s= - 1.46 the scale factor,
for small t, is changing as
,
and for
large t, as
while
and
in corresponding asymptotic
regimes. The potential V decays to zero for large t, after
reaching a maximum value (see Fig. 23). Similarly,
the effective gravitational coupling
decreases for large t, until it becomes zero for
(we have a sort of asymptotic freedom at
). It turns out that in our model the
observational constraints on the variation of the effective
gravitational constant are respected. Actually a new analysis of the
Big Bang Nucleosynthesis (Copi et al. 2004) restricts the variations in G to
(98) |
(99) |
Figure 23: The potential V as a function of the redshift z. | |
Open with DEXTER |
Initially, for small t, the matter energy density is higher than the energy density of the scalar field. In Table 2 we present results of our analysis, and they show that predictions of our model are fully compatible with the recent observational data. Comparing results of this paper with our previous analysis of minimally coupled scalar field models (see Paper I), we conclude that the present-day observational data connected with the post recombination evolution of the universe can be fitted by several different models of quintessence. More data on high-redshift supernovae of type Ia and GRBs are needed, as is more information on the early phase of structure formation in order to place stronger restrictions on the allowed type of dark energy.
Acknowledgements
This work was supported in part by the grant of Polish Ministry of Science and Higher Education 1-P03D-014-26, and by INFN Na12. The authors are very grateful to Professor Djorgovski for providing the data that we used in Sect. 3.1.1, and to Professors Verde and Simon, for providing the data used in Sect. 3.2.2. Of course we take full responsibility for the fitting procedure.