Constraining Dvali-Gabadadze-Porrati gravity from observational data

Jun-Qing Xia*

Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014 Trieste, Italy(Received 26 March 2009; published 27 May 2009)

The accelerating expansion of our Universe at present could be driven by an unknown energy

component (dark energy) or a modification of general relativity (modified gravity). In this paper we

revisit the constraints on a phenomenological model which interpolates between the pure CDM model

and the Dvali-Gabadadze-Porrati braneworld model with an additional parameter . Combining the

cosmic microwave background , baryon acoustic oscillations, and type Ia supernovae, as well as some

high-redshift observations, such as the gamma-ray bursts and the measurements of linear growth factors,

we obtain the tight constraint on the parameter ¼ 0:254 0:153 (68% C.L.), which implies that the flat

Dvali-Gabadadze-Porrati model is incompatible with the current observations, while the pure CDM

model still fits the data very well. Finally, we simulate the future measurements with higher precisions and

find that the constraint on can be improved by a factor of 2 when compared to the present constraints.

DOI: 10.1103/PhysRevD.79.103527 PACS numbers: 98.80.Es, 04.50.h, 95.36.+x

I. INTRODUCTION

Current cosmological observations, such as the CMBmeasurements of temperature anisotropies and polarizationat high redshift z 1090 [1] and the redshift-distancemeasurements of type Ia supernovae (SNIa) at z < 2 [2],have demonstrated that the Universe is now undergoing anaccelerated phase of expansion. The simplest explanationis that this behavior is driven by the cosmological constantor the dynamical dark energy models, which suffer fromthe severe coincidence and fine-tuning problems [3]. Onthe other hand, this observed late-time acceleration of theexpansion on the large scales could also be caused by somemodifications of general relativity, such as the scalar-tensor[4] and fðRÞ theories [5], and gravitational slip [6].

One of the well-known examples is the Dvali-Gabadadze-Porrati (DGP) braneworld model [7], in whichthe gravity leaks off the four dimensional brane into thefive dimensional space-time. On small scales gravity isbound to the four dimensional brane and the general rela-tivity is recovered to a good approximation. In the frame-work of a flat DGP model, the Friedmann equation will bemodified as [8]

H2 H

rc¼ 8G

3m; (1)

where rc ¼ ðH0ð1mÞÞ1 is the crossover scale. Atearly times, Hrc 1, the Friedmann equation of generalrelativity is recovered, while in the future, H ! H1 ¼1=rc, the expansion is asymptotically de Sitter. Recentlythere has been a lot of interest in the phenomenologicalstudies relevant to the DGP model in the literature [9,10].

In this paper we investigate an interesting phenomeno-logical model, first introduced in Ref. [11], which inter-polates between the pure CDM model and the DGP

model with an additional parameter and presents thetight constraints from the current observations and futuremeasurements. The paper is organized as follows: In Sec. IIwe describe the general formalism of the modified gravitymodel. Section III contains the current observations weuse, and Sec. IV includes our main global fitting results. InSec. V we present the forecasts from the future measure-ments, while Sec. VI is dedicated the conclusions.

II. GENERAL FORMALISM

In this phenomenological model, assuming the flatnessof our Universe, the Friedmann equation is modified as[11]

H2 H

r2c

¼ 8G

3m; (2)

where rc ¼ H10 =ð1mÞ2. Thus, we can straightfor-

wardly rewrite the above equation and obtain the expan-sion rate as following:

E2ðzÞ H2

H20

¼ mð1þ zÞ3 þ H2

H20

; (3)

where the last term denotes the modification of theFriedmann equation of general relativity:

H2

H20

ð1mÞH

H0

¼ ð1mÞEðzÞ: (4)

Furthermore, we can obtain the effective equation of state:

weffðzÞ 1þ 1

3

d lnH2

d lnð1þ zÞ ¼ 1þ

3ð1þ zÞE

0ðzÞEðzÞ ;

(5)

where the prime denotes the derivative with respect to theredshift z.*xia@sissa.it

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In Fig. 1 we illustrate the evolutions of the effectiveenergy density ðzÞ 1mðzÞ and weffðzÞ for differ-ent values of parameter . During the matter dominated

era, EðzÞ varies as ð1þ zÞ3=2, which corresponds to theeffective equation of state: weff ¼ 1þ =2. In the futurez ! 0, with the matter density m / ð1þ zÞ3 ! 0, wehave weffðzÞ ! 1 and ðzÞ ! 1.

Besides the modification of the Friedmann equation, theflat DGP model also changes the growth function of den-sity perturbation ðaÞ. Under assumptions of a quasistaticregime and subhorizon scales, the correct evolution ofperturbation was found [9,12]

€þ 2H _ 4G

1þ 1

3

m ¼ 0; (6)

where the dot denotes the derivative with respect to thecosmic time t, and the factor is given by

¼ 1 2rcH

1þ _H

3H2

: (7)

However, this phenomenological model Eq. (2) is a pa-rametrization, so the situation is more complicated. One ofthe possible methods was found by Ref. [13]. In order toobtain the growth function of density perturbation within acovariant theory, the authors introduced a correction termand assumed the structure of a modified theory of gravity todetermine this term. Based on those assumptions, it wasconsequently found that the factor was

¼ 1 2

ðHrcÞ2

1þ ð2 Þ _H

3H2

: (8)

In the following analysis, we will use Eqs. (6) and (8) tocalculate the growth of density perturbation.

Defined as the normalized growth gðaÞ ðaÞ=a, thegrowth function equation (6) can be rewritten as

d2g

da2þ

7

2 3

2

weffðaÞ1þXðaÞ

dg

ada

þ 3

2

1 weffðaÞ

1þXðaÞ XðaÞ

1þXðaÞ1þ 1

3

g

a2¼ 0; (9)

where the variable XðaÞ is the ratio of the matter density tothe effective energy density XðaÞ ¼ mðaÞ=ðaÞ. InFig. 2 we plot the linear growth factor gðaÞ as a functionof scale factor a for different values of . One can see thatthe linear growth factor has been suppressed obviously aslong as 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 parameterized as

[15]

f d ln

d lna¼

m; (10)

where is the growth index. And then the growth functionbecomes

df

d lnaþ

1

2 3

2

weffðaÞ1þ XðaÞ

fþ f2

3

2

XðaÞ1þ XðaÞ

1þ 1

3

¼ 0: (11)

For the pure CDM model, the theoretical value of is6=11 0:545, while ¼ 11=16 ¼ 0:6875 in the flat DGPmodel [16].In the framework of this phenomenological model, we

can easily see that the pure CDM model and flat DGPmodel can be recovered when ¼ 0 and ¼ 1, respec-

FIG. 2 (color online). The evolutions of linear growth gðaÞ ðaÞ=a for different values of . The black solid lines are for ¼ 0 (CDM), the red dashed lines are for ¼ 0:25, and theblue dotted lines are for ¼ 1 (DGP).

FIG. 1 (color online). The evolutions of effective energy den-sity ðzÞ and effective equation of state weffðzÞ for differentvalues of . The black solid lines are for ¼ 0 (CDM), thered dashed lines are for ¼ 0:25 and the blue dotted lines arefor ¼ 1 (DGP). And the vertical line denotes today (z ¼ 0).

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tively. In order to be consistent with the cosmologicalobservations, the term should be very small in the earlytimes, such as the big bang nucleosynthesis era (z 109).This limit corresponds to the upper bound: < 2 [11]. Onthe other hand, when < 0, the effective equation of statewill become smaller than1, which leads to the instabilityof linear growth of density perturbation due to the appear-ance of a term equation (8).

At early times, such as the matter dominated era, we

have EðzÞ / ð1þ zÞ3=2 and _H=H2 ’ 3=2. Thus, Eq. (8)becomes

’ 1 ðmð1þ zÞ3Þ1=2

1m

0; (12)

since ð1þ zÞ3 1 at 2< z < 1000. By contrast, at latetimes the matter energy density m / ð1þ zÞ3 ! 0 and theexpansion is asymptotically de Sitter, _H ! 0. And then wehave

’ 1 2= )< 0; for 0<< 2> 0; for < 0

: (13)

Based on Eqs. (12) and (13), we can straightforwardly seethat as long as < 0, during the evolution of the Universethe value of should change the sign at one pivot redshiftzt, which leads to jzt ¼ 0 and ð3Þ1jzt ! 1. In Fig. 3

we have shown the evolution of ð3Þ1 when fixing ¼2. There is an obvious singularity at a 0:595.Therefore, based on these discussions above, we use atophat prior on as 0 < 2 in our calculations.

III. METHOD AND DATA

In our calculations we assume a flat space and use auniform prior on the present matter density fraction of theUniverse: 0:1<m < 0:5. Furthermore, we constrain theHubble parameter to be uniformly in 4 Hubble Space

Telescope region: 0:4< h< 1:0. The resulting plots areproduced with CosmoloGUI.1

In this section we will list the cosmological observationsused in our calculations: CMB, baryon acoustic oscilla-tions (BAO), and SNIa measurements, as well as somehigh-redshift observations, such as the gamma-ray bursts(GRB) and linear growth factors (LGF) data. We havetaken the total likelihood to be the products of the separatelikelihoods (Li) of these cosmological probes. In otherwords, defining 2

L;i ¼ 2 logLi, we get

2L;total ¼ 2

L;CMB þ 2L;BAO þ 2

L;SNIa þ 2L;GRB þ 2

L;LGF:

(14)

If the likelihood function is Gaussian, 2L coincides with

the usual definition of 2 up to an additive constant corre-sponding to the logarithm of the normalization factor ofL.

A. CMB data

CMB measurement is sensitive to the distance to thedecoupling epoch via the locations of peaks and troughs ofthe acoustic oscillations. Here we use the ‘‘WMAP dis-tance information’’ obtained by the WMAP group [1],which includes the ‘‘shift parameter’’ R, the ‘‘acousticscale’’ lA, and the photon decoupling epoch z.

2 R and lAcorrespond to the ratio of angular diameter distance to thedecoupling era over the Hubble horizon and the soundhorizon at decoupling, respectively, given by

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffimH

20

qc

ðzÞ; (15)

lA ¼ ðzÞsðzÞ ; (16)

where ðzÞ and sðzÞ denote the comoving distance to zand the comoving sound horizon at z, respectively. Thedecoupling epoch z is given by [17]

z ¼ 1048½1þ 0:00124ðbh2Þ0:738½1þ g1ðmh

2Þg2;(17)

where

g1 ¼ 0:0783ðbh2Þ0:238

1þ 39:5ðbh2Þ0:763 ;

g2 ¼ 0:560

1þ 21:1ðbh2Þ1:81 :

(18)

We calculate the likelihood of the WMAP distance infor-

FIG. 3. The evolution of the term ð3Þ1 when fixing ¼2.

1http://www.sarahbridle.net/cosmologui/.2In the revised version of the WMAP5 paper [1], they also

extend the baryon density bh2 into the WMAP distance

information. But our calculations are not sensitive to bh2

and they also claim that this extension does not affect theconstraints. Thus, we fix bh

2 ¼ 0:022765 to be the best fitvalue obtained by the WMAP group.

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mation as follows:

2 ¼ ðxthi xdatai ÞðC1Þijðxthj xdataj Þ; (19)

where x ¼ ðR; lA; zÞ is the parameter vector and ðC1Þij isthe 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 directly mea-sure not only the angular diameter distance,DAðzÞ, but alsothe expansion rate of the Universe, HðzÞ. But current BAOdata are not accurate enough for extracting the informationof DAðzÞ and HðzÞ separately [18]. Therefore, one can onlydetermine the following effective distance [19]:

DvðzÞ ð1þ zÞ2D2

AðzÞcz

HðzÞ1=3

: (20)

In this paper we use the Gaussian priors on the distanceratios rsðzdÞ=DvðzÞ:

rsðzdÞ=Dvðz ¼ 0:20Þ ¼ 0:1980 0:0058;

rsðzdÞ=Dvðz ¼ 0:35Þ ¼ 0:1094 0:0033;(21)

with a correlation coefficient of 0.39, extracted from theSDSS and 2dFGRS surveys [20], where rsðzdÞ is the co-moving sound horizon size and zd is the drag epoch atwhich baryons were released from photons given by [21]

zd ¼ 1291ðmh2Þ0:251

1þ 0:659ðmh2Þ0:828 ½1þ b1ðbh

2Þb2; (22)

where

b1 ¼ 0:313ðmh2Þ0:419½1þ 0:607ðmh

2Þ0:674;b2 ¼ 0:238ðmh

2Þ0:223: (23)

C. SNIa data

The SNIa data give the luminosity distance as a functionof redshift

dL ¼ ð1þ zÞZ z

0

cdz0

Hðz0Þ : (24)

The supernovae data we use in this paper are the recentlyreleased Union compilation (307 sample) from the

Supernova Cosmology Project [2], which include the re-cent samples of SNIa from the SNLS and ESSENCEsurveys, as well as some older data sets, and span theredshift range 0 & z & 1:55. In the calculation of the like-lihood from SNIa we have marginalized over the nuisanceparameter, the absolute magnitudeM, as done in Ref. [22]:

2 ¼ A B2

Cþ ln

C

2

; (25)

where

A ¼ Xi

ðdata thÞ22

i

; B ¼ Xi

data th

2i

;

C ¼ Xi

1

2i

:

(26)

D. GRB data

GRBs can potentially be used to measure the luminositydistance out to higher redshift than SNIa. Recently, severalempirical correlations between GRB observables werereported, and these findings have triggered intensive stud-ies 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, in acosmology-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, applyingsuch relations to constrain cosmological parameters leadsto biased results. In Ref. [23] the circular problem isnaturally eliminated by marginalizing over the free pa-rameters involved in the correlations; in addition, someresults show that these correlations do not change signifi-cantly for a wide range of cosmological parameters [24].Therefore, in this paper we use the 69 GRBs sample over aredshift range from z ¼ 0:17–6:60 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 will besuppressed in the modified gravity model. It will be helpfulusing the measurements of linear growth factors to con-strain the modified gravity models. Therefore, in Table IIwe list linear growth factors’ data that we use in ouranalysis: the linear growth rate f

m from the galaxypower spectrum at low redshifts [26–30] and lyman-growth factor measurement obtained with the lyman-power spectrum at z ¼ 3 [31]. It is worth noting that thedata points in Table II are obtained by assuming theCDM model, thus, one should use these data very care-fully, especially for the points obtained fromRefs. [27,28,30]. The corresponding 2 is simply given by

TABLE I. Inverse covariance matrix for the WMAP distanceinformation lA, R, and z. The maximum likelihood values areR ¼ 1:710, lA ¼ 302:10, and z ¼ 1090:04, respectively.

lAðzÞ RðzÞ zlAðzÞ 1.800 27.968 1:103RðzÞ 5667.577 92:263z 2.923

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2 ¼ Xi

ðfthi fdatai Þ22

i

: (27)

IV. NUMERICAL RESULTS

In this section we present our main results of constraintson this phenomenological model from the current obser-vational data, as shown in Table III.

In Fig. 4 we illustrate the posterior distribution of fromthe current data. Firstly, we neglect the high-redshiftprobes. The result shows that the current observations yielda strong constraint on the parameter:

¼ 0:263 0:175ð1Þ: (28)

One can see that the pure CDM model ( ¼ 0) still fitsthe data very well at 2 uncertainty, which is consistentwith the current status of global fitting results [1,32]. Andthe 95% upper limit is < 0:686, which implies that thereis a significant tension between the flat DGP model ( ¼1) and the current observations, which is consistent withother works (e.g. Ref. [33,34]). However, unlike otherworks [33], in our analysis we use the ‘‘WMAP distanceinformation’’ which includes the ‘‘shift parameter’’ R, the‘‘acoustic scale’’ lA, and the photon decoupling epoch z,instead of R only, to constrain this phenomenologicalmodel. Recently, many results show that the ‘‘WMAPdistance information’’ can give the similar constraintswhen compared with the results from the full CMB powerspectrum [35]. By contrast, R could not be an accuratesubstitute for the full CMB data and may in principle givesome misleading results [36].

And then, we include some high-redshift probes, such asGRB and LGF data sets. From Table III and Fig. 4, we canfind that the constraint on becomes slightly tighter:

¼ 0:254 0:153ð1Þ; (29)

and < 0:686 at 2 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, these high-redshift observations are helpful to improve the constraintson this phenomenological model.These results [Eqs. (28) and (29)] are not surprising.

From Fig. 1 we find that the effective equation of state ofthe flat DGP model, weff 1þ =2 ¼ 0:5, will de-part from the cosmological constant w ¼ 1 at the high-redshift Universe significantly. But the current constrainton w is closed to w ¼ 1 [1,32], so we require the smallvalue of to match the current observations. There is asmall difference that our result slightly favors a nonzerovalue of , but not significantly, which needs more accu-rate measurements to verify it further.In Fig. 5 we plot the two dimensional constraint in the

ðm; Þ panel. m and are strongly anticorrelated. Thereason for this degeneracy is that the constraint mainlycomes from the luminosity and angular diameter distanceinformation. From Eqs. (3) and (4) we can see that when is increased, the contribution of the last term to theexpansion rate will become large, due to the positiveEðzÞ. Consequently, m must be decreased correspond-ingly in order to produce the same expansion rate. When

TABLE III. Constraints on the parameters ,m, and . Here we have shown the mean valuesand errors from the current observations and the standard derivations from the future measure-ments.

m

CMBþ BAOþ SN 0:263 0:175 0:276 0:018 -

All Real Data 0:254 0:153 0:277 0:017 0:570 0:205Future 0.07 0.005 0.050

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α

Pro

babi

lity

FIG. 4 (color online). One dimensional constraint on the pa-rameter from the different current data combinations: CMBþBAOþ SN (blue dashed lines) and all real data (red solid lines).

TABLE II. The currently available data for the linear growthrates f we use in our analysis.

z f 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]

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combining those current observational data, the matterenergy density has been very stringently constrained :m ¼ 0:277 0:017ð1Þ, which is also consistent withthe current status of global fitting results [1,32]. Naturally,the constraint on will also be improved, because of thetight constraint on the matter energy density.

Furthermore, we also investigate the limit on the growthindex and obtain ¼ 0:570 0:205 at a 68% confi-dence level. Obviously, the growth index of the pureCDM ¼ 6=11 0:545 is consistent with this result.However, the theoretical value of the growth index in theflat DGP model, ¼ 11=16 ¼ 0:6875, is disfavored.

V. FUTURE CONSTRAINTS

Since the present data clearly do not give very stringentconstraint on the parameter , it is worthwhile to discusswhether future data could determine conclusively. Forthat purpose we have performed an analysis and chosen thefiducial model as the mean values of Table III obtainedfrom the current constraints.

The projected satellite, Supernova/Acceleration Probe,would be a space based telescope with a one square degreefield of view with 109 pixels. It aims to increase thediscovery rate for SNIa to about 2000 per year in theredshift range 0:2< z < 1:7. In this paper we simulateabout 2000 SNIa according to the forecast distribution ofthe Supernova/Acceleration Probe [37]. For the error, wefollow Ref. [37] which takes the magnitude dispersion 0.15and the systematic error sys ¼ 0:02 z=1:7. The whole

error for each data is given by

magðziÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

sysðziÞ þ 0:152

ni

s; (30)

where ni is the number of supernovae of the i0th redshiftbin. Furthermore, we add as an external data set a mockdataset of 400 GRBs, in the redshift range 0< z < 6:4with

an intrinsic dispersion in the distance modulus of ¼0:16 and with a redshift distribution very similar to that ofFigure 1 of Ref. [38].For the linear growth factors’ data, we simulate the mock

data from the fiducial model with the error bars reduced bya factor of 2. This is probably reasonable given the largeramounts of galaxy power spectrum and lyman- forestpower spectrum data that will become available soon alongwith a better control of systematic errors in the nextgenerated large scale structure survey. In addition wealso assume a Gaussian prior on the matter energy densitym as ¼ 0:007, which is close to future Planck con-straints [39].From Table III it is clear that the future measurements

with higher precision could improve the constraints dra-matically. The standard derivation of is reduced by afactor of 2. Assuming the mean value remains unchangedin the future, the nonzero value of will be confirmedaround a 3 confidence level by the future measurements.In addition we also illustrate the two dimensional contourof parameters m and in Fig. 6. Compared with thecontour in Fig. 5, the allowed parameter region has beenshrunk significantly. The future measurements could haveenough ability to distinguish between the modified gravitymodel and the pure CDM 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 interests inthe phenomenological studies recently. In this paper weinvestigate an interesting phenomenological model whichinterpolates between the pure CDM model and the flatDGP braneworld model with an additional parameter .First, we find that when is less than zero, the growth

function of density perturbation ðaÞ will appear as anapparent singularity. This is because the variable will

Ωm

α

0.22 0.24 0.26 0.28 0.3 0.32

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FIG. 5 (color online). 68% and 95% constraints in the ðm; Þplane from the current observations.

Ωm

α

0.22 0.24 0.26 0.28 0.3 0.32

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FIG. 6 (color online). 68% and 95% constraints in the ðm; Þplane from the future measurements.

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change the sign during the evolution of our Universe. Andthen the term caused by the modified gravity model willbe divergent at some redshift zt.

From the current CMB, BAO, and SNIa data, we obtaina tight constraint on the parameter ¼ 0:2630:175ð1Þ, which implies that the flat DGP model ( ¼1) is incompatible with the current observations, while thepure CDM model still fits the data very well. Whenadding the high-redshift GRB and LGF data, the constraint

is more stringent ¼ 0:254 0:153ð1Þ, which meansthat these high-redshift observations are helpful to improvethe constraints on this phenomenological model.Finally, we simulate the future measurements with

higher precisions to limit this phenomenological model.And we find that these accurate probes will be helpful toimprove the constraints on the parameters of the model andcould distinguish between the modified gravity model andthe pure CDM model.

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