Issue 
A&A
Volume 507, Number 1, November III 2009



Page(s)  53  59  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/200911998  
Published online  08 September 2009 
A&A 507, 5359 (2009)
Probing the cosmographic parameters to
distinguish between dark energy and modified gravity models
(Research Note)
F. Y. Wang^{1,2}  Z. G. Dai^{1}  Shi Qi^{3,4}
1  Department of Astronomy, Nanjing University, Nanjing 210093, PR
China
2  Department of Astronomy, University of Texas at Austin, Austin, TX
78712, USA
3  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210008, PR China
4  Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing
University  Purple Mountain Observatory, Nanjing 210093, PR China
Received 6 March 2009 / Accepted 4 August 2009
Abstract
Aims. In this paper we investigate the deceleration,
jerk and
snap parameters to distinguish between the dark energy and modified
gravity models using high redshift gammaray bursts (GRBs) and
supernovae (SNe).
Methods. We first derive the expressions of
deceleration, jerk
and snap parameters in dark energy and modified gravity models. In
order to constrain the cosmographic parameters, we calibrate the GRB
luminosity relations without assuming any cosmological models using SNe
Ia. Then we constrain the model parameters (including dark energy and
modified gravity models) using type Ia supernovae and gammaray bursts.
Finally we calculate the cosmographic parameters. GRBs can extend the
redshiftdistance relation up to high redshifts, because they can be
detected to high redshifts.
Results. We find that the statefinder pair (r,s)
could not be used to distinguish between some dark energy and modified
gravity models, but these models could be differentiated by the snap
parameter. Using the modelindependent constraints on cosmographic
parameters, we conclude that the CDM model is consistent with
current data.
Key words: gamma rays: bursts  cosmology: cosmological parameters  cosmology: distance scale
1 Introduction
Recent observations of the Hubble relation of distant type Ia supernovae (SNe Ia) have provided strong evidence for acceleration of the present universe (Riess et al. 1998; Perlmutter et al. 1999). The observations of the spectrum of cosmic microwave background (CMB) anisotropies (Spergel et al. 2003, 2007), largescale structure (LSS) (Tegmark et al. 2004; Eisenstein et al. 2005) and the distanceredshift relation to Xray galaxy clusters (Allen et al. 2004, 2007) also confirm that the universe is accelerating. Possible explanations for this acceleration have been proposed. A negative pressure term called dark energy is taken into account, such as in the cosmological constant model with equation of state (Weinberg 1989), an evolving scalar field (Peeble & Ratra 1988; Caldwell et al. 1998), phantom energy for which the sum of the pressure and energy density is negative, and the Chaplygin gas (Kamenshchik et al. 2001). All the above models for acceleration are obtained by introducing a new energy component called dark energy. Alternative models, in which gravity is modified, can also drive the universe acceleration, e.g., the DvaliGabadadzePorrati (DGP) model (Dvali et al. 2000; Deffayet et al. 2002), Cardassian expansion model (Freese & Lewis 2002; Wang et al. 2003), and the f(R) gravity model (Vollick 2003; Carroll et al. 2004).
These two families of models, dark energy and modified gravity, are fundamentally different. An important question is whether it is possible to distinguish between the modified gravity and dark energy models that have nearly the same cosmic expansion history. Much work has been done on this topic. A usuallydiscussed quantity is the growth rate of cosmological density perturbations, which should be different in the models depending on different gravity theory even if they have an identical cosmic expansion history. Recently, there have been extensive discussions on discriminating dark energy and modified gravity models using the matter density perturbation growth factor (Linder 2005). But Kunz & Sapone (2007) demonstrated that the growth factor is not sufficient to distinguish between modified gravity and dark energy (Kunz & Sapone 2007). They found that a generalized dark energy model can match the growth rate of the DvaliGabadadzePorrati model and reproduce the 3+1 dimensional metric perturbations.
On the other hand, the statefinder pair (r,s) has also been proposed to distinguish between the models, where and . Sahni et al. (2003) demonstrated that the statefinder diagnostic could effectively discriminate different forms of dark energy (Sahni et al. 2003). Alam et al. (2003) investigated the cosmological constant, quintessence, Chaplygin gas, and braneworld models using the statefinder diagnostic, and found that the statefinder pair could differentiate between these models (Alam et al. 2003). Different cosmological models exhibit qualitatively different trajectories of evolution in the rs plane. The statefinder diagnostic has been extensively used in many models (Gorini et al. 2003). But the statefinder pair is difficult to measure with cosmological observations (Visser 2004; Cattoën & Visser 2007).
The present values of cosmographic parameters can be determined from observations (Riess et al. 2004; Visser 2004). Caldwell & Kamionkowski (2004) showed that the jerk parameter could probe the spatial curvature of the universe (Caldwell & Kamionkowski 2004). The deceleration, jerk and snap parameters are related to the second, third and fourth derivative of the scale factor respectively. Visser (2004) expanded the Hubble law to fourth order in redshift including the snap parameter and put constraints on the deceleration and jerk parameters using SNe Ia (Visser 2004). Rapetti et al. (2007) constrained the deceleration and jerk parameters from SNe Ia and Xray cluster gas mass fraction measurements. For a redshift range of SNe Ia, the terms beyond the cubic power of the Hubble law can be neglected. In order to put a narrow constraint on the snap parameter, we need highredshift objects. GRBs may be a useful tool. GRBs can be detectable out to very high redshifts (Ciardi & Loeb 2000). The farthest burst detected so far is GRB 090423, which is at z=8.2 (Olivares et al. 2009). A lot of work in this socalled GRB cosmology has been published (Dai et al. 2004; Ghirlanda et al. 2004; Di Girolamo et al. 2005; Firmani et al. 2005; Friedman & Bloom 2005; Lamb et al. 2005; Liang & Zhang 2005, 2006; Xu et al. 2005; Wang & Dai 2006; Schaefer 2007; Wright 2007; Wang et al. 2007; Gong & Chen 2007; Li et al. 2008; Liang et al. 2008; Qi et al. 2008a,b; Basilakos & Perivolaropoulos 2008; Kodama et al. 2008; Wang et al. 2009). Very recently, Schaefer (2007) used 69 GRBs and five relations to build the Hubble diagram out to z=6.60 and discussed the properties of dark energy in several dark energy models (Schaefer 2007). He found that the GRB Hubble diagram is consistent with the concordance cosmology. Liang et al. (2008) calibrated the luminosity relations of GRBs by interpolating from the Hubble diagram of SNe Ia at z<1.4 with the assumption that objects at the same redshift should have the same luminosity distance (Liang et al. 2008). This method is modelindependent. More recently, Capozziello & Izzo (2008) used the Liang et al. (2008) results to constrain the cosmographic parameters and found that the results calibrated by SNe Ia data agree with the CDM model. Cardone et al. (2009) used 83 GRBs and six correlations to build the Hubble diagram.
Riess et al. (2004) found that the jerk j_{0} is positive at the 92% confidence level based on their ``gold'' dataset and is positive at the 95% confidence level based on their ``gold+silver'' dataset. Neither explicit upper bounds are given for the jerk nor are any constraints placed on the snap s_{0}. Rapetti et al. (2007) measured and in a flat model with constant jerk (Rapetti et al. 2007). Capozziello & Izzo (2008) used 27 GRBs to derive the values of the cosmographic parameters. They found , and . In this paper, we use more GRB data to constrain the cosmography parameters in several dark energy and modified gravity models.
We calibrate the luminosity relations of GRBs using SNe Ia and calculate the deceleration, jerk and snap parameters of several dark energy and modified gravity models using SNe Ia and GRBs. We also use a modelindependent method to constrain the cosmographic parameters. We find that in some models the jerk parameters are almost equal to each other. So this parameter is not used to distinguish between the models. However, the snap parameters in all the models are different, so we can distinguish between the models using the snap parameter.
The structure of this paper is as follows. In Sect. 2 we introduce the Hubble, deceleration, jerk and snap parameters. In Sect. 3 we derive expressions of cosmographic parameters of the Hubble law in several dark energy models. In Sect. 4 we present expressions of cosmographic parameters of the Hubble law in modified gravity models. The constraints on model parameters and cosmographic parameters of the Hubble law are given in Sect. 5. Finally, Sect. 6 contains conclusions and discussions.
2 Hubble, deceleration, jerk and snap parameters
The expansion rate of the Universe can be written in terms of the
Hubble parameter, ,
where a is the scale factor and
is its first derivative with respect to time. As we known that q
is the deceleration parameter, related to the second derivative of the
scale factor, j is the socalled ``jerk'' or
statefinder parameter, related to the third derivative of the scale
factor, and s
is the socalled ``snap'' parameter, which is related to the fourth
derivative of the scale factor. These quantities are
defined as
(1) 
(2) 
(3) 
The deceleration, jerk and snap parameters are dimensionless, and a Taylor expansion of the scale factor around t_{0} provides
a(t)  =  
(4) 
and so the luminosity distance
=  
(5) 
(Visser 2004). For the redshift range of SNe Ia the terms beyond the cubic power in Eq. (5) can be neglected. If models have the same deceleration and jerk parameters, we can see degeneracy of these models from Eq. (5). Therefore we must measure the snap parameters to distinguish between the models. This needs highredshift objects. The relations among the q(z),j(z) and s(z) are
(6) 
(7) 
The Friedmann equation is
(8) 
From Einstein's equations, we can obtain the dynamical equation of the universe
The conservation equation is
(10) 
In order to derive the jerk and snap parameter, we differentiate Eq. (9)
=  
=  
(11) 
=  
=  
(12) 
3 Dark energy models
3.1 w(z) parameterization model
We first consider the dark energy with a constant equation of state
w(z)=w_{0}.  (13) 
For this model, we obtain
(14) 
(15) 
(16) 
(17) 
=  
(18) 
These expressions are consistent with Bertolami & Silva (2006).
A more interesting approach to explore dark energy is to use a
timedependent dark energy model. The simplest parameterization
including two parameters is (Maor et al. 2001; Weller
& Albrecht 2001)
w(z)=w_{0}+w_{1}z.  (19) 
In this dark energy model the luminosity distance is (Linder 2003)
=  
(20) 
The cosmographic parameters are:
(21) 
=  
(22) 
=  
(23) 
We consider the ChevallierPolarskiLinder parameterization (Chevallier & Polarski 2001; Linder 2003)
(24) 
The luminosity distance is (Chevallier & Polarski 2001; Linder 2003)
=  
(25) 
The cosmographic parameters are:
(26) 
=  
(27) 
=  
(28) 
Capozziello et al. (2008) and Capozziello & Izzo (2008) also derived cosmographic parameters in this model. Our results are consistent with theirs.
3.2 Generalized Chaplygin gas model
We consider the generalized Chaplygin gas (GCG) model, which is
characterized by the equation of state
(29) 
We can integrate the conservation equation for generalized Chaplygin gas, leading to
where is the energy density of GCG today, and . An attractive feature of the model is that it can unify dark energy and dark matter. The reason is that, from Eq. (30), the GCG behaves like dustlike matter at an early epoch and as a cosmological constant at a later epoch (Kamenshchik et al. 2001; Bento et al. 2002). The Friedmann equation can be expressed as
(31) 
where
=  
(32) 
is the density parameter of the baryonic matter. The luminosity distance is
=  
(33) 
For the GCG model we obtain (Bertolami & Silva 2006; Wang et al. 2009)
(34) 
(35) 
(36) 
(37) 
(38) 
4 Modified gravity models
4.1 Cardassian expansion model
The original Cardassian model was first introduced in Freese &
Lewis (2002)
as a possible alternative to explain the acceleration of the universe.
They modified the Friedmann equation as
This model has no energy component besides ordinary matter. If we consider a spatially flat FRW universe, the Friedmann equation is modified as Eq. (39). The universe undergoes acceleration requiring n < 2/3. If n=0, it is the same as the cosmological constant universe. We can obtain H(z) by using Eq. (39) and ,
where is the critical density of the universe. The luminosity distance in this model is
(41) 
For the Cardassian expansion model, we obtain
(42) 
(43) 
(44) 
(45) 
=  
(46) 
4.2 DvaliGabadadzePorrati model
In the DGP model the modified Friedmann equation due to the
presence
of an infinitevolume extra dimension is (Deffayet et al. 2002)
where the bulkinduced term, , is defined as
For a flat universe, . In the above equation, is the crossover scale beyond which the gravitational force follows the 5dimensional 1/r^{3} behavior. Note that on short length scales (at early times) the gravitational force follows the usual fourdimensional 1/r^{2} behavior. For a spatially flat universe, . We obtain
(49) 
(50) 
(51) 
(52) 
(53) 
4.3 f(R) gravity
f(R) gravity models, in which the
gravitational Lagrangian is a function of the curvature scalar R,
also can explain the current cosmic acceleration (Vollick 2003;
Carroll et al. 2004;
Capozziello et al. 2009).
Poplawski (2006)
derived a quite complicated expression for the jerk parameter in (Poplawski
2006):
The snap parameter in this model is (Poplawski 2007)
Poplawski (2007) calculated , and . These expressions of the jerk and snap parameters are only valid in Palatini variational principle. A generic formula for the cosmographic parameter is derived by Capozziello, Cardone & Salzano (2008; for more details, see Eqs. (23)(33) in their paper). They also gave the best fitted value: , and using SNe Ia. In this paper, we only use Poplawski (2006) as an example for the f(R) gravity.
5 Constraints from SNe Ia and GRBs
Davis et al. (2007)
fitted the SNe Ia dataset that includes 60 ESSENCE SNe Ia (WoodVasey
et al. 2007),
57 SNe Ia from SuperNova Legacy Survey (SNLS) (Astier
et al. 2006), 45
nearby SNe Ia and 30 SNe Ia detected by HST (Riess
et al. 2007) with
the MCLS2K2 method. With the luminosity distance
in units of megaparsecs, the
predicted distance modulus is
(56) 
The likelihood functions can be determined from the statistic,
(57) 
where is the dispersion in the supernova redshift (transformed to distance modulus) due to peculiar velocities, is the observational distance modulus, and is the uncertainty in the individual distance moduli. The confidence regions can be found by marginalizing the likelihood functions over H_{0} (i.e., integrating the probability density for all values of H_{0}).
We use the calibration results obtained by using the
interpolation methods directly from SNe Ia data (Liang et al. 2008). The
calibrated luminosity relations are completely cosmology
independent. We assume that these relations do not evolve with redshift
and are valid in z>1.40.
The luminosity or energy of GRB can be calculated. Thus, the luminosity
distances and distance
modulus can be obtained. After obtaining the distance modulus of each
burst using one of these relations, we use the same method as Schaefer (2007) to
calculate the real distance modulus,
(58) 
where the summation runs from 15 over the relations with available data, is the best estimated distance modulus from the ith relation, and is the corresponding uncertainty. The uncertainty of the distance modulus for each burst is
(59) 
The value is
(60) 
where and are the fitted distance modulus and its error.
We combine SNe Ia and GRBs by multiplying the likelihood functions. The total value is . The best fitted value is obtained by minimizing .
5.1 Constraints on cosmographic parameters
In our analysis, we consider a flat cosmology. We use from the Hubble Space Telescope key projects (Freedman et al. 2001).
Let us first consider observational constraints on dark energy models. In Fig. 1, we show the distribution probabilities as a function of in the flat CDM model from SNe Ia and GRBs. From this figure, we have . The cosmographic parameters in CDM model are , j_{0}=1.0 and . We can obtain , j_{0}=1.0 and .
Figure 1: Luminosity distanceredshift diagram. The circles are the GRBs. The solid line is the result of our fitting. 

Open with DEXTER 
Figure 2: Constraints on and w from to using 192 SNe Ia in the w=w_{0} model. 

Open with DEXTER 
In Fig. 2 we present constraints on and w from to using 192 SNe Ia and 69 GRBs in the w=w_{0} model. We measure and . The cosmographic parameters in the w=w_{0} model are , and .
In Fig. 3 we present constraints on w_{0} and w_{1} from to using 192 SNe Ia and 69 GRBs in the w=w_{0}+w_{1} z model. The values of the parameters are and . The cosmographic parameters are , and .
Figure 3: The same as Fig. 2 but in the w=w_{0}+w_{1} z model. 

Open with DEXTER 
In Fig. 4 we present constraints on w_{0} and w_{1} from to using 192 SNe Ia and 69 GRBs in the w=w_{0}+w_{1} z/(1+z) model. We measure and . The cosmographic parameters are , and .
Figure 4: The same as Fig. 2 but in the w=w_{0}+w_{1} z/(1+z) model. 

Open with DEXTER 
Figure 5 shows constraints on A_{s} and from to using SNe Ia and GRBs in the GCG model. The parameters are and . The cosmographic parameters are , and .
Figure 5: The same as Fig. 2 but in the GCG model. 

Open with DEXTER 
In Fig. 6 we present constraints on and n from to using 192 SNe Ia and 69 GRBs in the Cardassian expansion model. We measure and . The cosmographic parameters are , and .
Figure 6: The same as Fig. 2 but in the Cardassian expansion model. 

Open with DEXTER 
Figure 7 shows constraints on using SNe Ia and GRBs in the DGP model. The value of is . The cosmographic parameters are , and .
Figure 7: The same as Fig. 1 but in the DGP model. 

Open with DEXTER 
We directly use Eq. (5) to constrain the cosmographic parameters. This analysis uses the FRW metric only, so we have not specified any gravitational theory yet. The luminosity distance only depends on redshift z and cosmographic parameters. So this method is fully model independent. We use the 192 SNe Ia and 69 GRBs and find the best fit parameters are , and . The results are consistent with the flat CDM model.
In Table 1 we summarize the constraints on cosmographic parameters. The deceleration and jerk parameters in the w=w_{0}, GCG, Cardassian expansion and f(R) models are almost the same in the confidence level. These values are consistent with the deceleration and jerk parameters of the CDM model in the confidence level. So these models cannot be discriminated between by using the present value of the statefinder pair. However the snap parameter in all the models is different and thus can be used to discriminate between the cosmological models. In the future, more data will give a precise snap parameter in different models.
Table 1: The cosmographic parameter values.
6 Discussions and conclusions
The cosmic acceleration could be due to unidentified dark energy, or a modification of general relativity (modified gravity). In this paper we investigate the deceleration, jerk and snap parameters in modified gravity and dark energy models. We calibrated the GRB luminosity relations using SNe Ia without assuming any cosmological models. Because gammaray bursts can be detected at high redshifts, we calculated the deceleration, jerk and snap parameters using type Ia supernovae and gammaray bursts. GRBs can extend the redshiftdistance relation up to high redshifts. We find that the deceleration and jerk parameters in the w=w_{0}, GCG, Cardassian expansion and f(R) models are almost the same at a confidence level. So these models cannot be discriminated between using the present value of the statefinder pair. We found that the dark energy models and modified gravity models could be distinguished between by the snap parameter. Using the modelindependent constraints on cosmographic parameters, we found that the CDM model is consistent with the current data.
AcknowledgementsThis work is supported by the National Natural Science Foundation of China (grants 10233010, 10221001 and 10873009) and the National Basic Research Program of China (973 program) No. 2007CB815404. F. Y. Wang was also supported by the Jiangsu Project Innovation for Ph.D. Candidates (CX07B039z).
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All Tables
Table 1: The cosmographic parameter values.
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