Constraining DGP Gravity from Observational Data
aa r X i v : . [ a s t r o - ph . C O ] J u l Constraining DGP Gravity from Observational Data
Jun-Qing Xia ∗ Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014 Trieste, Italy
The accelerating expansion of our universe at present could be driven by an unknown energycomponent (Dark Energy) or a modification of general relativity (Modified Gravity). In this notewe revisit the constraints on a phenomenological model which interpolates between the pure ΛCDMmodel and the Dvali-Gabadadze-Porrati (DGP) braneworld model with an additional parameter α .Combining the cosmic microwave background (CMB), baryon acoustic oscillations (BAO) and typeIa supernovae (SNIa), as well as some high-redshift observations, such as the gamma-ray bursts(GRB) and the measurements of linear growth factors (LGF), we obtain the tight constraint on theparameter α = 0 . ± .
153 (68% C.L.), which implies that the flat DGP model is incompatiblewith the current observations, while the pure ΛCDM model still fits the data very well. Finally, wesimulate the future measurements with higher precisions and find that the constraint on α can beimproved by a factor two, when compared to the present constraints. I. INTRODUCTION
Current cosmological observations, such as the CMBmeasurements of temperature anisotropies and polariza-tion at high redshift z ∼ z < f ( R ) theories [5], and gravitationalslip [6].One of the well-known examples is the DGPbraneworld model [7], in which the gravity leaks off thefour dimensional brane into the five dimensional space-time. On small scales gravity is bound to the four di-mensional brane and the general relativity is recoveredto a good approximation. In the framework of flat DGPmodel, the Friedmann equation will be modified as [8]: H − Hr c = 8 πG ρ m , (1)where r c = ( H (1 − Ω m )) − is the crossover scale. Atearly times, Hr c ≫
1, the Friedmann equation of generalrelativity is recovered, while in the future, H → H ∞ =1 /r c , the expansion is asymptotically de Sitter. Recentlythere have been a lot of interests in the phenomenologicalstudies relevant to the DGP model in the literature [9,10].In this note we investigate an interesting phenomeno-logical model, first introduced in Ref.[11], which inter-polates between the pure ΛCDM model and the DGPmodel with an additional parameter α and presents the ∗ Electronic address: [email protected] tight constraints from the current observations and fu-ture measurements. The paper is organized as follows:In Sec. II we describe the general formalism of the mod-ified gravity model. Sec. III contains the current obser-vations we use, and Sec. IV includes our main globalfitting results. In Sec. V we present the forecasts fromthe future measurements, while Sec. VI is dedicated theconclusions.
II. GENERAL FORMALISM
In this phenomenological model, assuming the flatnessof our universe, the Friedmann equation is modified as[11]: H − H α r − αc = 8 πG ρ m , (2)where r c = H − / (1 − Ω m ) α − . Thus, we can straight-forwardly rewrite the above equation and obtain the ex-pansion rate as following: E ( z ) ≡ H H = Ω m (1 + z ) + δH H , (3)where the last term denotes the modification of the Fried-mann equation of general relativity: δH H ≡ (1 − Ω m ) H α H α = (1 − Ω m ) E α ( z ) . (4)Furthermore, we can obtain the effective equation ofstate: w eff ( z ) ≡ −
1+ 13 d ln δH d ln(1 + z ) = − α z ) E ′ ( z ) E ( z ) , (5)where the prime denotes the derivative with respect tothe redshift z .In Fig.1 we illustrate the evolutions of the effective en-ergy density Ω α ( z ) ≡ − Ω m ( z ) and w eff ( z ) for differentvalues of parameter α . During the matter dominated era, w e ff ( z ) ( z ) FIG. 1: The evolutions of effective energy density Ω α ( z ) andeffective equation of state w eff ( z ) for different values of α .The black solid lines are for α = 0 (ΛCDM), the red dashedlines for α = 0 .
25 and the blue dotted lines for α = 1 (DGP).And the vertical line denotes today ( z = 0). E ( z ) varies as (1 + z ) / , which corresponds to the effec-tive equation of state: w eff = − α/
2. In the future z →
0, with the matter density ρ m ∝ (1 + z ) →
0, wehave w eff ( z ) → − α ( z ) → δ ( a ). Under assumptions of aquasi-static regime and sub-horizon scales, the correctevolution of perturbation was found [9, 12]:¨ δ + 2 H ˙ δ − πG (cid:18) β (cid:19) ρ m δ = 0 , (6)where the dot denotes the derivative with respect to thecosmic time t , and the β factor is given by: β = 1 − r c H H H ! . (7)However, this phenomenological model Eq.(2) is aparametrization, so the situation is more complicated.One of the possible methods was found by Ref.[13]. Inorder to obtain the growth function of density perturba-tion within a covariant theory, the authors introduced acorrection term and assumed the structure of modifiedtheory of gravity to determine this term. Based on thoseassumptions, it was consequently found that the β factorwas: β = 1 − α ( Hr c ) − α − α ) ˙ H H ! . (8)In the following analysis, we will use Eq.(6) and Eq.(8)to calculate the growth of density perturbation. g ( a ) = ( a ) / a Scale Factor a =0 ( CDM) =0.25 =1 (DGP)
FIG. 2: The evolutions of linear growth g ( a ) ≡ δ ( a ) /a fordifferent values of α . The black solid lines are for α = 0(ΛCDM), the red dashed lines for α = 0 .
25 and the bluedotted lines for α = 1 (DGP). Defined the normalized growth g ( a ) ≡ δ ( a ) /a , thegrowth function Eq.(6) can be rewritten as: d gda + (cid:18) − w eff ( a )1 + X ( a ) (cid:19) dgada + 32 (cid:20) − w eff ( a )1 + X ( a ) − X ( a )1 + X ( a ) (cid:18) β (cid:19)(cid:21) ga = 0 (9)where the variable X ( a ) is the ratio of the matter densityto the effective energy density X ( a ) = Ω m ( a ) / Ω α ( a ). InFig.2 we plot the linear growth factor g ( a ) as function ofscale factor a for different values of α . One can see thatthe linear growth factor has been suppressed obviously aslong as α is larger than zero. Thus, in the literature thelinear growth has been widely used to study the modifiedgravity models, especially the DGP model [14].Furthermore, the growth factor can be parameterizedas [15]: f ≡ d ln δd ln a = Ω γm , (10)where γ is the growth index. And then the growth func-tion becomes: dfd ln a + (cid:18) − w eff ( a )1 + X ( a ) (cid:19) f + f − X ( a )1 + X ( a ) (cid:18) β (cid:19) = 0 . (11)For the pure ΛCDM model, the theoretical value of γ is6 / ≈ . γ = 11 /
16 = 0 . ( ) - Scale Factor a
FIG. 3: The evolution of the β term (3 β ) − when fixing α = − model can be recovered when α = 0 and α = 1, respec-tively. In order to be consistent with the cosmologicalobservations, the α term should be very small in the earlytimes, such as the Big Bang Nucleosynthesis (BBN) era( z ∼ ). This limit corresponds to the upper bound: α < α <
0, the ef-fective equation of state will become smaller than − β term Eq.(8).At early times, such as the matter dominated era, wehave E ( z ) ∝ (1 + z ) / and ˙ H/H ≃ − /
2. Thus, Eq.(8)becomes β ≃ − (Ω m (1 + z ) ) − α/ − Ω m ≪ , (12)since (1 + z ) ≫ < z < ρ m ∝ (1 + z ) → H →
0. Andthen we have β ≃ − /α ⇒ (cid:26) β < , for 0 < α < β > , for α < . (13)Based on Eq.(12) and Eq.(13), we can straightforwardlysee that as long as α <
0, during the evolution of universethe value of β should change the sign at one pivot redshift z t , which leads to β | z t = 0 and (3 β ) − | z t → ∞ . InFig.3 we have shown the evolution of (3 β ) − when fixing α = −
2. There is an obvious singularity at a ∼ . α as 0 ≤ α < III. METHOD AND DATA
In our calculations we assume a flat space and use auniform prior on the present matter density fraction of the universe: 0 . < Ω m < .
5. Furthermore, we con-strain the Hubble parameter to be uniformly in 4 σ Hub-ble Space Telescope (HST) region: 0 . < h < .
0. Theresulting plots are produced with CosmoloGUI .In this section we will list the cosmological observationsused in our calculations: CMB, BAO and SNIa measure-ments, as well as some high-redshift observations, such asthe GRB and LGF data. We have taken the total likeli-hood to be the products of the separate likelihoods ( L i )of these cosmological probes. In other words, defining χ L,i = − L i , we get: χ L, total = χ L, CMB + χ L, BAO + χ L, SNIa + χ L, GRB + χ L, LGF . (14)If the likelihood function is Gaussian, χ L coincides withthe usual definition of χ up to an additive constant cor-responding to the logarithm of the normalization factorof L . A. CMB Data
CMB measurement is sensitive to the distance to thedecoupling epoch via the locations of peaks and troughsof the acoustic oscillations. Here we use the “WMAPdistance information” obtained by the WMAP group [1],which includes the “shift parameter” R , the “acousticscale” l A , and the photon decoupling epoch z ∗ . R and l A correspond to the ratio of angular diameter distanceto the decoupling era over the Hubble horizon and thesound horizon at decoupling, respectively, given by: R = p Ω m H c χ ( z ∗ ) , (15) l A = πχ ( z ∗ ) χ s ( z ∗ ) , (16)where χ ( z ∗ ) and χ s ( z ∗ ) denote the comoving distance to z ∗ and the comoving sound horizon at z ∗ , respectively.The decoupling epoch z ∗ is given by [17]: z ∗ = 1048[1 + 0 . b h ) − . ][1 + g (Ω m h ) g ] , (17)where g = 0 . b h ) − . . b h ) . , g = 0 . . b h ) . . (18) In the revised version of WMAP5 paper [1], they also extend thebaryon density Ω b h into the WMAP distance information. Butour calculations are not sensitive to Ω b h and they also claimthat this extension does not affect the constraints. Thus, we fixΩ b h = 0 . Table I. Inverse covariance matrix for the WMAP distanceinformation l A , R and z ∗ . The maximum likelihood valuesare R = 1 . l A = 302 .
10 and z ∗ = 1090 .
04, respectively. l A ( z ∗ ) R ( z ∗ ) z ∗ l A ( z ∗ ) 1 .
800 27 . − . R ( z ∗ ) 5667 . − . z ∗ . We calculate the likelihood of the WMAP distance infor-mation as follows: χ = ( x th i − x data i )( C − ) ij ( x th j − x data j ) , (19)where x = ( R, l A , z ∗ ) is the parameter vector and ( C − ) ij is the inverse covariance matrix for the WMAP distanceinformation shown in Table I. B. BAO Data
The BAO information has been already detected in thecurrent galaxy redshift survey. The BAO can directlymeasure not only the angular diameter distance, D A ( z ),but also the expansion rate of the universe, H ( z ). Butcurrent BAO data are not accurate enough for extract-ing the information of D A ( z ) and H ( z ) separately [18].Therefore, one can only determine the following effectivedistance [19]: D v ( z ) ≡ (cid:20) (1 + z ) D A ( z ) czH ( z ) (cid:21) / . (20)In this note we use the gaussian priors on the distanceratios r s ( z d ) /D v ( z ): r s ( z d ) /D v ( z = 0 .
20) = 0 . ± . ,r s ( z d ) /D v ( z = 0 .
35) = 0 . ± . , (21)with a correlation coefficient of 0 .
39, extracted from theSDSS and 2dFGRS surveys [20], where r s ( z d ) is the co-moving sound horizon size and z d is the drag epoch atwhich baryons were released from photons given by [21]: z d = 1291(Ω m h ) . . m h ) . [1 + b (Ω b h ) b ] , (22)where b = 0 . m h ) − . [1 + 0 . m h ) . ] ,b = 0 . m h ) . . (23) C. SNIa Data
The SNIa data give the luminosity distance as a func-tion of redshift d L = (1 + z ) Z z cdz ′ H ( z ′ ) . (24) The supernovae data we use in this paper are the recentlyreleased Union compilation (307 sample) from the Su-pernova Cosmology project [2], which include the recentsamples of SNIa from the SNLS and ESSENCE survey, aswell as some older data sets, and span the redshift range0 < ∼ z < ∼ .
55. In the calculation of the likelihood fromSNIa we have marginalized over the nuisance parameter,the absolute magnitude M , as done in Ref.[22]:¯ χ = A − B C + ln (cid:18) C π (cid:19) , (25)where A = X i ( µ data − µ th ) σ i ,B = X i µ data − µ th σ i ,C = X i σ i . (26) D. GRB Data
GRBs can potentially be used to measure the luminos-ity distance out to higher redshift than SNIa. Recently,several empirical correlations between GRB observableswere reported, and these findings have triggered intensivestudies on the possibility of using GRBs as cosmological“standard” candles. However, due to the lack of low-redshift long GRBs data to calibrate these relations, ina cosmology-independent way, the parameters of the re-ported correlations are given, assuming an input cosmol-ogy, and obviously they depend on the same cosmologicalparameters that we would like to constrain. Thus, apply-ing such relations to constrain cosmological parametersleads to biased results. In Ref.[23] the circular problemis naturally eliminated by marginalizing over the free pa-rameters involved in the correlations; in addition, someresults show that these correlations do not change signif-icantly for a wide range of cosmological parameters [24].Therefore, in this paper we use the 69 GRBs sample overa redshift range from z = 0 . − .
60 published in Ref.[25]but we keep in mind the issues related to the “circularproblem” that are more extensively discussed in Ref.[23].
E. LGF Data
As we point out above, the linear growth factor willbe suppressed in the modified gravity model. It will behelpful using the measurements of linear growth factorto constrain the modified gravity models. Therefore, inTable II we list linear growth factors data we use in ouranalysis: the linear growth rate f ≡ Ω γm from galaxypower spectrum at low redshifts [26, 27, 28, 29, 30] andlyman- α growth factor measurement obtained with the TABLE II. The currently available data for linear growthrates f we use in our analysis. z f σ Ref.0.15 0.51 0.11 [26]0.35 0.70 0.18 [27]0.55 0.75 0.18 [28]0.77 0.91 0.36 [29]1.40 0.90 0.24 [30]3.00 1.46 0.29 [31]TABLE III. Constraints on the parameters α , Ω m and γ .Here we have shown the mean values and errors from thecurrent observations and the standard derivations from thefuture measurements. α Ω m γ CMB+BAO+SN 0 . ± .
175 0 . ± . − All Real Data 0 . ± .
153 0 . ± .
017 0 . ± . lyman- α power spectrum at z = 3 [31]. It is worth notingthat the data points in Table II are obtained with assum-ing the ΛCDM model, thus, one should use these datavery carefully, especially for the points obtained fromRefs.[27, 28, 30]. The corresponding χ is simply givenby: χ = X i ( f th i − f data i ) σ i . (27) IV. NUMERICAL RESULTS
In this section we present our main results of con-straints on this phenomenological model from the currentobservational data, as shown in Table III.In Fig.4 we illustrate the posterior distribution of α from the current data. Firstly, we neglect the high red-shift probes. The result shows that the current observa-tions yield a strong constraint on the parameter: α = 0 . ± .
175 (1 σ ) . (28)One can see that the pure ΛCDM model ( α = 0) stillfits data very well at 2 σ uncertainty, which is consistentwith the current status of global fitting results [1, 32].And the 95% upper limit is α < . α = 1) and the current observations, which isconsistent with other works (e.g. Ref.[33, 34]). How-ever, unlike other works [33], in our analysis we use the“WMAP distance information” which includes the “shiftparameter” R , the “acoustic scale” l A , and the photon α P r obab ili t y FIG. 4: One dimensional constraint on the parameter α fromthe different current data combinations: CMB+BAO+SN(blue dashed lines) and all real data (red solid lines). decoupling epoch z ∗ , instead of R only, to constrain thisphenomenological model. Recently, many results showthat the “WMAP distance information” can give the sim-ilar constraints, when compared with the results from thefull CMB power spectrum [35]. By contrast, R could notbe an accurate substitute for the full CMB data and mayin principle give some misleading results [36].And then, we include some high redshift probes, suchas GRB and LGF data sets. From Table III and Fig.4,we can find that the constraint on α becomes slightlytighter: α = 0 . ± .
153 (1 σ ) , (29)and α < .
541 at 2 σ confidence level. As we have men-tioned before, the effective equation of state of this phe-nomenological model will depart from the cosmologicalconstant boundary at high redshifts. Therefore, thesehigh redshift observations are helpful to improve the con-straints on this phenomenological model.These results (Eqs.(28-29)) are not surprising. FromFig.1 we find that the effective equation of state of theflat DGP model, w eff ≈ − α/ − .
5, will departfrom the cosmological constant w = − w isclosed to w = − α to match the current observations. There is a smalldifference that our result slightly favors a non-zero valueof α , but not significantly, which needs more accuratemeasurements to verify it further.In Fig.5 we plot the two dimensional constraint in the(Ω m , α ) panel. Ω m and α are strongly anti-correlated.The reason of this degeneracy is that the constraintmainly comes from the luminosity and angular diameterdistance information. From Eq.(3) and Eq.(4) we can seethat when α is increased, the contribution of last α termto the expansion rate will become large, due to the pos-itive E ( z ). Consequently, Ω m must be decreased corre- Ω m α FIG. 5: 68% and 95% constraints in the (Ω m , α ) plane fromthe current observations. spondingly in order to produce the same expansion rate.When combining those current observational data, thematter energy density has been constrained very strin-gent: Ω m = 0 . ± .
017 (1 σ ), which is also consistentwith the current status of global fitting results [1, 32].Naturally, the constraint on α will also be improved, be-cause of the tight constraint on the matter energy density.Furthermore, we also investigate the limit on thegrowth index γ and obtain γ = 0 . ± .
205 at 68%confidence level. Obviously, the growth index of the pureΛCDM γ = 6 / ≈ .
545 is consistent with this result.However, the theoretical value of growth index in the flatDGP model, γ = 11 /
16 = 0 . V. FUTURE CONSTRAINTS
Since the present data clearly do not give very strin-gent constraint on the parameter α , it is worthwhile todiscuss whether future data could determine α conclu-sively. For that purpose we have performed an analysisand chosen the fiducial model as the mean values of TableIII obtained from the current constraints.The projected satellite SNAP (Supernova / Accelera-tion Probe) would be a space based telescope with a onesquare degree field of view with 10 pixels. It aims toincrease the discovery rate for SNIa to about 2000 peryear in the redshift range 0 . < z < .
7. In this paperwe simulate about 2000 SNIa according to the forecastdistribution of the SNAP [37]. For the error, we followthe Ref.[37] which takes the magnitude dispersion 0 . σ sys = 0 . × z/ .
7. The wholeerror for each data is given by: σ mag ( z i ) = s σ ( z i ) + 0 . n i , (30)where n i is the number of supernovae of the i ′ th redshiftbin. Furthermore, we add as an external data set a mockdataset of 400 GRBs, in the redshift range 0 < z < . Ω m α FIG. 6: 68% and 95% constraints in the (Ω m , α ) plane fromthe future measurements. with an intrinsic dispersion in the distance modulus of σ µ = 0 .
16 and with a redshift distribution very similarto that of Figure 1 of Ref.[38].For the linear growth factors data, we simulate themock data from the fiducial model with the error barsreduced by a factor of two. This is probably reasonablegiven the larger amounts of galaxy power spectrum andlyman- α forest power spectrum data that will becomeavailable soon as long with a better control of systematicerrors in the next generated large scale structure survey.In addition we also assume a Gaussian prior on the mat-ter energy density Ω m as σ = 0 . α is reduced bya factor two. Assuming the mean value remains un-changed in the future, the non-zero value of α will beconfirmed around 3 σ confidence level by the future mea-surements. In addition we also illustrate the two dimen-sional contour of parameters Ω m and α in Fig.6. Com-paring with the contour in Fig.5, the allowed parameterregion has been shrunk significantly. The future measure-ments could have enough ability to distinguish betweenthe modified gravity model and the pure ΛCDM model. VI. CONCLUSIONS
As an alternative approach to generate the late-timeacceleration of the expansion of our universe, models ofmodifications of gravity have attracted a lot of interestsin the phenomenological studies recently. In this note weinvestigate an interesting phenomenological model whichinterpolates between the pure ΛCDM model and the flatDGP braneworld model with an additional parameter α .Firstly, we find that when α is less than zero, thegrowth function of density perturbation δ ( a ) will appearan apparent singularity. This is because the variable β will change the sign during the evolution of our universe.And then the β term caused by the modified gravitymodel will be divergent at some redshift z t .From the current CMB, BAO and SNIa data, we ob-tain a tight constraint on the parameter α = 0 . ± .
175 (1 σ ), which implies that the flat DGP model ( α =1) is incompatible with the current observations, whilethe pure ΛCDM model still fits the data very well. Whenadding the high-redshift GRB and LGF data, the con-straint is more stringent α = 0 . ± .
153 (1 σ ), which means that these high redshift observations are helpful toimprove the constraints on this phenomenological model.Finally, we simulate the future measurements withhigher precisions to limit this phenomenological model.And we find that these accurate probes will be helpful toimprove the constraints on the parameters of the modeland could distinguish between the modified gravity modeland the pure ΛCDM model. [1] E. Komatsu, et al. , arXiv:0803.0547.[2] M. Kowalski et al. , arXiv:0804.4142.[3] S. Weinberg, Rev. Mod. Phys. , 1 (1989); I. Zlatev,L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett. ,896 (1999).[4] S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner,Phys. Rev. D , 043528 (2004); C. Schimd, J. P. Uzanand A. Riazuelo, Phys. Rev. D , 083512 (2005).[5] See e.g. V. Acquaviva, C. Baccigalupi and F. Perrotta,Phys. Rev. D , 023515 (2004); P. Zhang, Phys. Rev.D , 123504 (2006); Y. S. Song, H. Peiris and W. Hu,Phys. Rev. D , 063517 (2007); E. Bertschinger andP. Zukin, Phys. Rev. D , 024015 (2008); and referencestherein.[6] S. F. Daniel, R. R. Caldwell, A. Cooray and A. Mel-chiorri, Phys. Rev. D , 103513 (2008); S. F. Daniel,R. R. Caldwell, A. Cooray, P. Serra and A. Melchiorri,arXiv:0901.0919.[7] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett.B , 208 (2000).[8] C. Deffayet, Phys. Lett. B , 199 (2001).[9] A. Lue, Phys. Rept. , 1 (2006).[10] C. Deffayet, Int. J. Mod. Phys. D (2008) 2023; R. Dur-rer and R. Maartens, arXiv:0811.4132; and referencestherein.[11] G. Dvali and M. S. Turner, arXiv:astro-ph/0301510.[12] K. Koyama and R. Maartens, JCAP , 16 (2006).[13] K. Koyama, JCAP , 17 (2006).[14] E. V. Linder, Phys. Rev. D , 043529 (2005); D. Hutererand E. V. Linder, Phys. Rev. D , 023519 (2007); D. Po-larski, arXiv:astro-ph/0605532; S. A. Thomas, F. B. Ab-dalla and J. Weller, arXiv:0810.4863.[15] P. J. E. Peebles, Large-Scale Structure of the Universe,Princeton Univ. Press, 1980; P. J. E. Peebles, Astrophys.J. , 439 (1984); O. Lahav, P. B. Lilje, J. R. Primackand M. J. Rees, Mon. Not. R. Astron. Soc. , 128(1991).[16] E. V. Linder and R. N. Cahn, Astropart. Phys. , 481(2007).[17] W. Hu and N. Sugiyama, Astrophys. J. , 542 (1996).[18] T. Okumura, T. Matsubara, D. J. Eisenstein, I. Kayo,C. Hikage, A. S. Szalay and D. P. Schneider, Astrophys.J. , 889 (2008).[19] D. J. Eisenstein et al. Astrophys. J. , 560 (2005).[20] W. J. Percival, et al. , Mon. Not. Roy. Astron. Soc. , 1053 (2007).[21] D. Eisenstein and W. Hu, Astrophys. J. , 605 (1998).[22] E. Di Pietro and J. F. Claeskens, Mon. Not. Roy. Astron.Soc. , 1299 (2003).[23] H. Li, J. Q. Xia, J. Liu, G. B. Zhao, Z. H. Fan andX. Zhang, Astrophys. J. , 92 (2008).[24] C. Firmani, V. Avila-Reese, G. Ghisellini andG. Ghirlanda, Rev. Mex. Astron. Astrofis. , 203(2007).[25] B. E. Schaefer, Astrophys. J. , 16 (2007).[26] E. Hawkins et al. , Mon. Not. Roy. Astron. Soc. , 78(2003); L. Verde et al. , Mon. Not. Roy. Astron. Soc. ,432 (2002).[27] M. Tegmark et al. , Phys. Rev. D , 123507 (2006).[28] N. P. Ross et al. , arXiv:astro-ph/0612400.[29] L. Guzzo et al. , Nature , 541 (2008).[30] J. da Angela et al. , arXiv:astro-ph/0612401.[31] P. McDonald et al. Astrophys. J. , 761 (2005).[32] J. Q. Xia, H. Li, G. B. Zhao and X. Zhang, Phys. Rev.D , 083524 (2008).[33] M. Fairbairn and A. Goobar, Phys. Lett. B , 432(2006); Z. K. Guo, Z. H. Zhu, J. S. Alcaniz andY. Z. Zhang, Astrophys. J. , 1 (2006); S. Ryd-beck, M. Fairbairn and A. Goobar, JCAP , 003(2007); T. M. Davis et al. , Astrophys. J. , 716 (2007);M. S. Movahed, M. Farhang and S. Rahvar, Int. J. Theor.Phys. , 1203 (2009).[34] Y. S. Song, Phys. Rev. D , 024026 (2005); Y. S. Song,I. Sawicki and W. Hu, Phys. Rev. D , 064003 (2007);W. Fang, S. Wang, W. Hu, Z. Haiman, L. Hui andM. May, Phys. Rev. D , 103509 (2008); Z. H. Zhu andM. Sereno, arXiv:0804.2917; S. A. Thomas, F. B. Abdallaand J. Weller, arXiv:0810.4863; and references therein.[35] P. S. Corasaniti and A. Melchiorri, Phys. Rev. D ,103507 (2008); H. Li, J. Q. Xia, G. B. Zhao, Z. H. Fanand X. Zhang, Astrophys. J. , L1 (2008).[36] O. Elgaroy and T. Multamaki, Astron. Astrophys. ,65 (2007).[37] A. G. Kim, E. V. Linder, R. Miquel and N. Mostek, Mon.Not. Roy. Astron. Soc. , 909 (2004).[38] D. Hooper and S. Dodelson, Astropart. Phys.27