Probing the cosmographic parameters to distinguish between dark energy and modified gravity models
aa r X i v : . [ a s t r o - ph . C O ] J a n Astronomy&Astrophysicsmanuscript no. ms˙aa c (cid:13)
ESO 2018October 23, 2018
Probing the cosmographic parameters to distinguish betweendark energy and modified gravity models
F. Y. Wang , , Z. G. Dai and Shi Qi , Department of Astronomy, Nanjing University, Nanjing 210093, China Department of Astronomy, University of Texas at Austin, Austin, TX 78712 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing University - Purple Mountain Observatory, Nanjing 210093, China.Preprint online version: October 23, 2018
ABSTRACT
Aims.
In this paper we investigate the deceleration, jerk and snap parameters to distinguish between the dark energy and modified gravitymodels by using high redshift gamma-ray 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 toconstrain the cosmographic parameters, we calibrate the GRB luminosity relations without assuming any cosmological models using SNeIa. Then we constrain the models (including dark energy and modified gravity models) parameters using type Ia supernovae and gamma-raybursts. Finally we calculate the cosmographic parameters. GRBs can extend the redshift - distance relation up to high redshifts, because theycan 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 di ff erentiated by the snap parameter. Using the model-independent constraints on cosmographic parameters, weconclude that the Λ CDM model is consistent with the current data.
Key words.
Gamma rays : bursts - Cosmology : cosmological parameters - Cosmology : distance scale
1. Introduction
Recent observations of the Hubble relation of distant Type Iasupernovae (SNe Ia) have provided strong evidence for acceler-ation of the present universe (Riess et al. 1998; Perlmutter et al.1999). The observations of the spectrum of cosmic microwavebackground (CMB) anisotropies (Spergel et al. 2003;2007),large-scale structure (LSS) (Tegmark et al. 2004; Eisensteinet al. 2005) and the distance-redshift relation to X-ray galaxyclusters (Allen et al. 2004; 2007) also confirm that the uni-verse is accelerating. Possible explanations for the accelera-tion have been proposed. A negative pressure term called darkenergy is taken into account, such as the cosmological con-stant model with equation of state w = p /ρ = − Send o ff print requests to : F. Y. Wange-mail: [email protected] ity is modified, can also drive the universe acceleration, e.g.,the Dvali-Gabadadze-Porrati (DGP) model (Dvali et al. 2000;De ff ayet 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 modifiedgravity, are fundamentally di ff erent. An important question iswhether it is possible to distinguish between the modified grav-ity and dark energy models that have nearly the same cosmicexpansion history. Many works have been done on this topic.A usually-discussed quantity is the growth rate of cosmologicaldensity perturbations, which should be di ff erent in the modelsdepending on di ff erent gravity theory even if they have an iden-tical cosmic expansion history. Recently, there have been ex-tensive discussions on discriminating dark energy and modifiedgravity models using the matter density perturbations growthfactor (Linder 2005). But Kunz and Sapone (2007) demon-strated that the growth factor is not su ffi cient to distinguishbetween modified gravity and dark energy (Kunz & Sapone2007). They found that a generalized dark energy model canmatch the growth rate of the Dvali-Gabadadze-Porrati modeland reproduce the 3 + F. Y. Wang et al.: Probing the cosmographic parameters
On the other hand, the statefinder pair ( r , s ) has also beenproposed to distinguish between the models, where r ≡ ˙¨ a / aH and s ≡ ( r − / q − / ff ectively discriminatedi ff erent 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 di-agnostic, and found that the statefinder pair could di ff erentiatethese models (Alam et al. 2003). Di ff erent cosmological mod-els exhibit qualitatively di ff erent trajectories of evolution in the r − s plane. The statefinder diagnostic has been extensively usedin many models (Gorini et al. 2003). But the statefinder pairis di ffi cult to measure by cosmological observations (Visser2004; Catto¨en & Visser 2007). The present values of cosmo-graphic parameters can be determined from observations (Riesset al. 2004; Visser 2004). Caldwell & Kamionkowski (2004)showed the jerk parameter could probe the spatial curvatureof the universe (Caldwell & Kamionkowski 2004). The de-celeration, jerk and snap parameters are related to the sec-ond, third and fourth derivative of the scale factor respectively.Visser (2004) expanded the Hubble law to fourth order in red-shift including the snap parameter and put constraints on thedeceleration and jerk parameters using SNe Ia (Visser 2004).Rapetti et al. (2007) constrained the deceleration and jerk pa-rameters from SNe Ia and X-ray cluster gas mass fraction mea-surements. For a redshift range of SNe Ia the terms beyondthe cubic power of Hubble law can be neglected. In order toput a narrow constraint on the snap parameter, we need high-redshift objects. GRBs may be a useful tool. GRBs can bedetectable out to very high redshifts (Ciardi & Loeb 2000).The farthest burst detected so far is GRB 090423, which is at z = . GRB cosmology has been published (Dai, Liang & Xu 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, Dai & Liang 2005; Wang & Dai 2006;Schaefer 2007; Wright 2007; Wang, Dai & Zhu 2007; Gong &Chen 2007; Li et al. 2008; Liang et al. 2008; Qi, Wang & Lu2008a,b; Basilakos & Perivolaropoulos 2008; Kodama et al.2008; Wang, Dai & Qi 2009). Very recently, Schaefer (2007)used 69 GRBs and five relations to build the Hubble diagramout to z = .
60 and discussed the properties of dark energy inseveral dark energy models (Schaefer 2007). He found that theGRB Hubble diagram is consistent with the concordance cos-mology. Liang et al.(2008) calibrated the luminosity relationsof GRBs by interpolating from the Hubble diagram of SNe Iaat z < . Λ CDM model. Cardone et al.(2009) used 83 GRBs and six correlations to build the Hubblediagram. Butler et la. (2009) found a real, intrinsic correlationbetween E iso and E peak using latest Swift GRB sample.Riess et al. (2004) found that the jerk j is positive at the92% confidence level based on their “gold” dataset and is pos-itive at the 95% confidence level based on their “gold + silver” dataset. Neither explicit upper bounds are given for the jerk norare any constraints placed on the snap s . Rapetti et al. (2007)measured q = − . ± .
14 and j = . + . − . in a flat modelwith constant jerk (Rapetti et al. 2007). Capozziello & Izzo(2008) used 27 GRBs to derive the values of the cosmographicparameters. They found q = − . ± . j = . ± . s = . ± .
16. In this paper, we use more GRB data toconstrain the cosmography parameters in several dark energyand modified gravity models.In this paper, we calibrate the luminosity relations of GRBsusing SNe Ia and calculate the deceleration, jerk and snap pa-rameters of several dark energy and modified gravity modelsusing SNe Ia and GRBs. We also use a model-independentmethod to constrain the cosmographic parameters. We find thatin some models the jerk parameter is almost equal to each other.So this parameter is not used to distinguish between the mod-els. However, the snap parameter in all the models is di ff erent,so we can distinguish between the models using the snap pa-rameter.The structure of this paper is organized as follows. In sec-tion 2 we introduce the hubble, deceleration, jerk and snap pa-rameters. In section 3 we derive expressions of cosmographicparameters of the Hubble law in several dark energy models.In section 4 we present expressions of cosmographic parame-ters of the Hubble law in modified gravity models. The con-straints on models parameters and cosmographic parameters ofthe Hubble law are given in section 5. Finally, section 6 con-tains conclusions and discussions.
2. Hubble, deceleration, jerk and snap parameters
The expansion rate of the Universe can be written in terms ofthe Hubble parameter, H = ˙ a / a , where a is the scale factor and˙ a is its first derivative with respect to time. As we known that q is the deceleration parameter, related to the second derivativeof the scale factor, j is the so-called “jerk” or statefinder pa-rameter, related to the third derivative of the scale factor, and s is the so-called “snap” parameter, which is related to the fourthderivative of the scale factor. These quantities are defined as q = − H ¨ aa ; (1) j = H ˙¨ aa ; (2) s = H ¨¨ aa . (3)The deceleration, jerk and snap parameters are dimensionless,and a Taylor expansion of the scale factor around t provides a ( t ) = a ( + H ( t − t ) − q H ( t − t ) + j H ( t − t ) + s H ( t − t ) + O [( t − t ) ] } , (4)and so the luminosity distance d L = cH ( z +
12 (1 − q ) z − (cid:16) − q − q + j (cid:17) z + h − q − q − q + j + q j + s i z + O ( z ) } , (5) . Y. Wang et al.: Probing the cosmographic parameters 3 (Visser 2004). For the redshift range of SNe Ia the terms be-yond the cubic power in Eq. (5) can be neglected. If modelshave the same deceleration and jerk parameters, we can seedegeneracy of these models from Eq. (5). Therefore we mustmeasure the snap parameters to distinguish between the mod-els. This needs high-redshift objects. The relations among the q ( z ) , j ( z ) and s ( z ) are j ( z ) = q ( z ) + q ( z ) + (1 + z ) dqdz ( z ); (6) s ( z ) = − (1 + z ) d jdz ( z ) − j ( z ) − j ( z ) q ( z ) . (7)The Friedmann equation is H = ( ˙ aa ) = π G X i ρ i . (8)From Einstein’s equations, we can obtain the dynamical equa-tion of universe¨ aa = − π G X i ( ρ i + P i ) . (9)The conservation equation is˙ ρ i + H ( ρ i + P i ) = . (10)In order to derive the jerk and snap parameter, we di ff erentiateEq.(9)˙¨ a = − π G X i [˙ a ρ i (1 + w i ) + a ˙ ρ i (1 + w i ) + a ρ i × w i ] = − π G X i [ aH ρ i (1 + w i ) − Ha ρ i (1 + w i )(1 + w i ) + a ρ i × w i ] , (11)¨¨ a = − π G X i d [˙ a ρ i (1 + w i ) + a ˙ ρ i (1 + w i ) + a ρ i × w i ] dt = − π G X i [¨ a ρ i (1 + w i ) + a ˙ ρ i (1 + w i ) + a ρ i ˙ w i + a ¨ ρ i (1 + w i ) + a ˙ ρ i ˙ w i + a ρ i × w i ] . (12)
3. Dark energy models w ( z ) parameterization model We first consider the dark energy with a constant equation ofstate. w ( z ) = w (13)For this model, we obtain q XCDM0 =
32 [1 + w (1 − Ω M )] − , (14) dqdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) XCDM0 = w (1 − Ω M ) Ω M , (15) d jdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) XCDM0 = − w (1 + w )( Ω M − Ω M , (16) j XCDM0 =
12 (2 + − Ω M ) w + − Ω M ) w ) , (17) s XCDM0 =
14 ( − − − Ω M ) w − − Ω M + Ω M ) w − − Ω M + Ω M ) w ) . (18)These expressions are consistent with Bertolami & Silva(2006).A more interesting approach to explore dark energy is touse time-dependent dark energy model. The simplest parame-terization including two parameters is (Maor et al. 2001; Weller& Albrecht 2001) w ( z ) = w + w z . (19)In this dark energy model the luminosity distance is (Linder2003) d L = cH − (1 + z ) Z z dz [(1 + z ) Ω M + (1 − Ω M )(1 + z ) + w − w ) e w z ] − / . (20) q WCDM0 = +
32 (1 − Ω M ) w (21) j WCDM0 =
12 (2 + − Ω M ) w + − Ω M ) w + − Ω M ) w )(22) s WCDM0 =
14 ( − − − Ω M + Ω M ) w − − Ω M + Ω M ) w − − Ω M ) w + w (1 − Ω M )( − − ( Ω M − w )) (23)We consider the Chevallier-Polarski-Linder parameteriza-tion (Chevallier & Polarski 2001; Linder 2003) w ( z ) = w + w z + z . (24)The luminosity distance is (Chevallier & Polarski 2001; Linder2003) d L = cH − (1 + z ) Z z dz [(1 + z ) Ω M + (1 − Ω M )(1 + z ) + w + w ) e − w z / (1 + z ) ] − / . (25)The cosmographic parameters are: q CPL0 = +
32 (1 − Ω M ) w , (26) j CPL0 =
12 (2 + − Ω M ) w + − Ω M ) w + − Ω M ) w ) , (27) s CPL0 =
14 ( − − − Ω M + Ω M ) w − − Ω M + Ω M ) w − − Ω M ) w + w (1 − Ω M )( − − ( Ω M − w )) . (28)Capozziello, Cardone & Salzano (2008) and Capozziello &Izzo (2008) also derived cosmographic parameters in thismodel. Our results are consistent with theirs. F. Y. Wang et al.: Probing the cosmographic parameters
We consider the generalized Chaplygin gas (GCG) model,which is characterized by the equation of state p GCG = − A /ρ α GCG . (29)We can integrate the conservation equation for generalizedChaplygin gas, leading to ρ GCG = ρ GCG0 [ A s + (1 − A s ) a − + α ) ] / (1 + α ) (30)where ρ Ch0 is the energy density of GCG today, and A s = A /ρ + α Ch0 . The attractive feature of the model is that it can unifydark energy and dark matter. The reason is that, from Eq. (30),the GCG behaves as dustlike matter at an early epoch and asa cosmological constant at a later epoch (Kamenshchik et al.2001; Bento et al. 2002). The Friedmann equation can be ex-pressed as H ( z , H , A s , α ) = H E ( z , A s , α ) , (31)where E ( z , A s , α ) = Ω b (1 + z ) + (1 − Ω b )[ A s + (1 − A s )(1 + z ) + α ) ] + α , (32) Ω b is the density parameter of the baryonic matter. The lumi-nosity distance is d L = cH − (1 + z ) Z z dz { (1 + z ) Ω b + (1 − Ω b )[ A s + (1 − A s )(1 + z ) + α ) ] + α } − / . (33)For the GCG model we obtain (Bertolami & Silva 2006; Wang,Dai & Qi 2009) q GCG0 =
32 (1 − A s ) − dqdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) GCG0 = A s (1 − A s )(1 + α ) , (35) d jdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) GCG0 = − α (1 + α )(2 A s −
1) ( A s − A s , (36) j GCG0 =
34 (1 − A s )(1 + (3 + α ) A s ) , (37) s GCG0 =
38 ( A s − + + α − α ) A s + − + α + α ) A s ) . (38)
4. Modified gravity models
The original Cardassian model was first introduced in (Freese& Lewis 2002) as a possible alternative to explain the accel-eration of the universe. They modified the Friedmann equationas H = π G ρ m + B ρ nm . (39)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 ac-celeration requires n < /
3. If n =
0, it is the same as thecosmological constant universe. We can obtain H ( z ) by usingEq. (39) and ρ m = ρ m (1 + z ) = Ω m ρ c (1 + z ) , H ( z ) = H [ Ω m (1 + z ) + (1 − Ω m )(1 + z ) n ] , (40)where ρ c = H / π G is the critical density of the universe. Theluminosity distance in this model is d L = cH − (1 + z ) Z z dz [(1 + z ) Ω m + (1 − Ω m )(1 + z ) n ] − / . (41)For the Cardassian expansion model, we obtain q Card0 = +
32 (1 − n )( Ω M − , (42) dqdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Card0 =
92 ( n − (1 − Ω M ) Ω M , (43) d jdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Card0 =
272 ( n − (1 − Ω M ) Ω M n , (44) j Card0 =
12 (2 + n ( Ω M − + n (1 − Ω M )) (45) s Card0 =
14 (4 − Ω M − n (3 − Ω M + Ω M ) − n (4 − Ω M + Ω M ) + n (11 − Ω M + Ω M )) . (46) In the DGP model the modified Friedmann equation due to thepresence of an infinite-volume extra dimension is (De ff ayet etal. 2002) H = H " Ω k (1 + z ) + (cid:18) p Ω r c + q Ω r c + Ω m (1 + z ) (cid:19) , (47)where the bulk-induced term, Ω r c , is defined as Ω r c ≡ / r c H . (48)For a flat universe, Ω k =
0. In the above equation, r c is thecrossover scale beyond which the gravitational force followsthe 5-dimensional 1 / r behavior. Note that on short lengthscales r ≪ r c (at early times) the gravitational force followsthe usual four-dimensional 1 / r behavior. For a spatially flatuniverse, Ω r c = (1 − Ω m ) /
4. We obtain q DGP0 = + Ω M − + Ω M , (49) dqdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) DGP0 = Ω M (1 − Ω M )(1 + Ω M ) , (50) d jdz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) DGP0 = Ω M (1 − Ω M )(1 + Ω M ) , (51) j DGP0 = + Ω M − Ω M + Ω M (1 + Ω M ) , (52) s DGP0 = − Ω M − Ω M − Ω M + Ω M − Ω M (1 + Ω M ) . (53) . Y. Wang et al.: Probing the cosmographic parameters 5 f ( R ) gravity f ( R ) gravity models, in which the gravitational Lagrangian is afunction of the curvature scalar R , also can explain current cos-mic acceleration (Vollick 2003; Carroll et al. 2004; Capozzielloet al. 2009). Poplawski (2006) derived a quite complicated ex-pression of jerk parameter in f ( R ) = R − α R (Poplawski 2006): j ′ = [( φ f ′ − f )(2 f ′ + φ f ′ f ′′ − f f ′′ )(30 φ f ′ f ′′ + φ f ′ f ′′′ − φ f f ′ f ′′ − φ f ′ f ′′ − φ f f ′ f ′′′ − φ f ′ + f f ′ + φ f f ′′ + φ f f ′ f ′′′ + φ f f ′ f ′′ − f f ′′′ − f f ′ f ′′ ) − (2 φ f ′ + φ f ′ f ′′ − φ f ′ f f ′′ − φ f f ′ + φ f f ′ f ′′ − f f ′′ ) × (3 φ f ′ f ′′ − φ f f ′′ − f ′ + f f ′ + f f ′ f ′′ − φ f ′ f ′′ + φ f ′ f ′′′ − φ f f ′ f ′′′ + f f ′′′ )] × [( φ f ′ − f ) (2 f ′ + φ f ′ f ′′ − f f ′′ ) ] − . (54)The snap parameter in this model is (Poplawski 2007) s = j ′ f ′ ( φ f ′ − f )(2 f ′ + φ f ′ f ′′ − f f ′′ ) − j φ f ′ − f φ f ′ − f . (55)Poplawski (2007) calculated q = − . + . − . , j = . + . − . and s = − . + . − . . These expressions of the jerk and snapparameters are only valid in Palatini variational principle. Ageneric formulae of cosmographic parameter are derived byCapozziello, Cardone & Salzano (2008) (for more details, seeEquations (23)-(33) in their paper). They also gave the bestfitted value: q = − . ± . j = . ± . s = − . ± . f ( R ) gravity.
5. Constraints from SNe Ia and GRBs
Davis et al. (2007) fitted the SNe Ia dataset that include 60ESSENCE SNe Ia (WoodVasey et al. 2007), 57 SNe Ia fromSuper-Nova Legacy Survey (SNLS) (Astier et al. 2006), 45nearby SNe Ia and 30 SNe Ia detected by HST (Riess et al.2007) with the MCLS2K2 method. With the luminosity dis-tance d L in units of megaparsecs, the predicted distance modu-lus is µ = d L ) + . (56)The likelihood functions can be determined from χ statistic, χ = N X i = [ µ i ( z i ) − µ , i ] σ µ , i + σ ν , (57)where σ ν is the dispersion in the supernova redshift (trans-formed to distance modulus) due to peculiar velocities, µ , i isthe observational distance modulus, and σ µ , i is the uncertaintyin the individual distance moduli. The confidence regions canbe found through marginalizing the likelihood functions over H (i.e., integrating the probability density p ∝ exp − χ / for allvalues of H ).We use the calibration results obtained by using the inter-polation methods directly from SNe Ia data (Liang et al. 2008).The calibrated luminosity relations are completely cosmologyindependent. We assume these relations do not evolve with red-shift and are valid in z > .
40. The luminosity or energy of GRB can be calculated. So the luminosity distances and dis-tance modulus can be obtained. After obtaining the distancemodulus of each burst using one of these relations, we use thesame method as Schaefer (2007) to calculate the real distancemodulus, µ fit = ( X i µ i /σ µ i ) / ( X i σ − µ i ) , (58)where the summation runs from 1 − µ i is the best estimated distance modulus fromthe i -th relation, and σ µ i is the corresponding uncertainty. Theuncertainty of the distance modulus for each burst is σ µ fit = ( X i σ − µ i ) − / . (59)The χ value is χ = N X i = [ µ i ( z i ) − µ fit , i ] σ µ fit , i , (60)where µ fit , i and σ µ fit , i are the fitted distance modulus and its er-ror. We combine SNe Ia and GRBs by multiplying the likeli-hood functions. The total χ value is χ = χ + χ . Thebest fitted value is obtained by minimizing χ . In our analysis, we consider the flat cosmology. We use h = . ± .
08 from the
Hubble Space Telescope key projects (Freedman et al. 2001). Riess et al. (2009) used the old dis-tance ladder and observed Cepheids in the near-infrared wherethey are less sensitive to dust and found h = . ± . Ω M in the flat Λ CDM model from SNe Ia andGRBs. From this figure, we have Ω M = . ± .
04. The cos-mographic parameters in Λ CDM model are q = − + Ω M , j = . s = − Ω M . We can obtain q = − . ± . j = . s = − . ± . Ω M and w from 1 σ to3 σ using 192 SNe Ia and 69 GRBs in the w = w model. Wemeasure Ω M = . + . − . and w = − . + . − . . The cosmo-graphic parameters in the w = w model are q = − . + . − . , j = . + . − . and s = − . + . − . .In Fig.3 we present constraints on w and w from 1 σ to 3 σ using 192 SNe Ia and 69 GRBs in the w = w + w z model. Thevalues of parameters are w = − . ± .
19 and w = . ± .
44. The cosmographic parameters are q = − . ± . j = . ± .
93 and s = − . ± . w and w from 1 σ to3 σ using 192 SNe Ia and 69 GRBs in the w = w + w z / (1 + z )model. We measure w = − . ± .
30 and w = . + . − . . Thecosmographic parameters are q = − . ± . j = . + . − . and s = − . + . − . .Fig.5 shows constraints on A s and α from 1 σ to 3 σ usingSNe Ia and GRBs in the GCG model. The parameters are A s = F. Y. Wang et al.: Probing the cosmographic parameters p r obab ili t y M Fig. 1.
Luminosity distance - redshift diagram. The circles arethe GRBs. The solid line is the result of our fitting M w Fig. 2.
Constraints on Ω M and w from 1 σ to 3 σ using 192 SNeIa in the w = w model.0 . ± .
12 and α = . + . − . . The cosmographic parametersare q = − . ± . j = . + . − . and s = − . + . − . .In Fig.6 we present constraints on Ω M and n from 1 σ to 3 σ using 192 SNe Ia and 69 GRBs in the Cardassian expansionmodel. We measure Ω M = . ± .
11 and n = − . + . − . . Thecosmographic parameters are q = − . + . − . , j = . + . − . and s = . + . − . .Fig.7 shows constraints on Ω M using SNe Ia and GRBs inthe DGP model. The value of Ω M is Ω M = . ± .
02. Thecosmographic parameters are q = − . ± . j = . ± .
02 and s = − . ± . -2.0 -1.5 -1.0 -0.5 0.0-202 w w Fig. 3.
The same as Fig.2 but in the w = w + w z model. -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0-4-20246 w w Fig. 4.
The same as Fig.2 but in the w = w + w z / (1 + z ) model.tance only depends on redshift z and cosmographic parame-ters. So this method is fully model independent. We use the192 SNe Ia and 69 GRBs and find the best fit parameters are q = − . ± . j = . ± .
80 and s = . ± .
50. Theresults are consistent with the flat Λ CDM model.In Table 1 we summarize the constraints on cosmographicparameters. The deceleration and jerk parameters in the w = w , GCG, Cardassian expansion and f(R) models are almostthe same in the 1 σ confidence level. These values are consistentwith the deceleration and jerk parameters of the Λ CDM modelin the 1 σ confidence level. So these models can not be discrim-inated using the present value of the statefinder pair. Howeverthe snap parameter in all the models is di ff erent and thus canbe used to discriminate the cosmological models. In the future, . Y. Wang et al.: Probing the cosmographic parameters 7 As Fig. 5.
The same as Fig.2 but in the GCG model. M n Fig. 6.
The same as Fig.2 but in the Cardassian expansionmodel.more data will give a precise snap parameter in di ff erent mod-els.
6. conclusions and discussions
The cosmic acceleration could be due to a mysterious dark en-ergy, or a modification of general relativity (modified gravity).In this paper we investigate the deceleration, jerk and snap pa-rameters in modified gravity models and dark energy models.We calibrate the GRB luminosity relations without assumingany cosmological models using SNe Ia. Because gamma-raybursts can be detected in high redshifts, we calculate the de-celeration, jerk and snap parameters using type Ia supernovae p r obab ili t y M Fig. 7.
The same as Fig.1 but in the DGP model.and gamma-ray bursts. GRBs can extend the redshift - distancerelation up to high redshifts. We find that the deceleration andjerk parameters in the w = w , GCG, Cardassian expansion andf(R) models are almost the same in the 1 σ confidence level. Sothese models can not be discriminated using the present valueof the statefinder pair. We find that the dark energy models andmodified gravity models could be distinguished between by thesnap parameter. Using the model-independent constraints oncosmographic parameters, we find the Λ CDM model is consis-tent with the current data.
Acknowledgements
This work is supported by the National Natural ScienceFoundation of China (grants 10233010, 10221001 and10873009) and the National Basic Research Program of China(973 program) No. 2007CB815404. F. Y. Wang was also sup-ported by the Jiangsu Project Innovation for PhD Candidates(CX07B-039z).
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The cosmographic parameters value model q j s Λ CDM − . ± .
06 1 . ± . − . ± . w = w − . + . − . . + . − . − . + . − . w = w + w z − . ± .
21 2 . ± . − . ± . w = w + w z + z − . ± .
33 3 . + . − . − . + . − . GCG − . ± .
18 1 . + . − . − . + . − . Cardassian − . + . − . . + . − . . + . − . DGP − . ± .
04 0 . ± . − . ± . − . + . − . . + . − . − . + . − ..