Challenges to the DGP Model from Horizon-Scale Growth and Geometry
Wenjuan Fang, Sheng Wang, Wayne Hu, Zoltan Haiman, Lam Hui, Morgan May
aa r X i v : . [ a s t r o - ph ] N ov Challenges to the DGP Model from Horizon-Scale Growth and Geometry
Wenjuan Fang, Sheng Wang, Wayne Hu,
2, 3
Zolt´an Haiman, Lam Hui, and Morgan May Department of Physics, Columbia University, New York, NY 10027 Kavli Institute for Cosmological Physics, Enrico Fermi Institute, University of Chicago, Chicago, IL 60637 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637 Department of Astronomy, Columbia University, New York, NY 10027 Brookhaven National Laboratory, Upton, NY 11973 (Dated: October 23, 2018)We conduct a Markov Chain Monte Carlo study of the Dvali-Gabadadze-Porrati (DGP) self-accelerating braneworld scenario given the cosmic microwave background (CMB) anisotropy, super-novae and Hubble constant data by implementing an effective dark energy prescription for modifiedgravity into a standard Einstein-Boltzmann code. We find no way to alleviate the tension betweendistance measures and horizon scale growth in this model. Growth alterations due to perturbationspropagating into the bulk appear as excess CMB anisotropy at the lowest multipoles. In a flatcosmology, the maximum likelihood DGP model is nominally a 5 . σ poorer fit than ΛCDM. Cur-vature can reduce the tension between distance measures but only at the expense of exacerbatingthe problem with growth leading to a 4 . σ result that is dominated by the low multipole CMBtemperature spectrum. While changing the initial conditions to reduce large scale power can flattenthe temperature spectrum, this also suppresses the large angle polarization spectrum in violation ofrecent results from WMAP5. The failure of this model highlights the power of combining growthand distance measures in cosmology as a test of gravity on the largest scales. I. INTRODUCTION
On the self-accelerating branch of the Dvali-Gabadadze-Porrati (DGP) braneworld model [1], cosmicacceleration arises from a modification to gravity at largescales rather than by introducing a form of mysteriousdark energy with negative pressure [2]. In this model ouruniverse is a (3 + 1)-dimensional brane embedded in aninfinite Minkowski bulk with differing effective strengthsof gravity. The relative strengths define a crossover scaleon the brane beyond which (4 + 1)-dimensional gravityand bulk phenomena become important.A number of theoretical and observational problemsof the self-accelerating branch of DGP have been re-cently uncovered. This branch suffers from pathologiesrelated to the appearance of ghost degrees of freedom[3, 4, 5, 6, 7, 8, 9, 10]. Ghosts can lead to runaway excita-tions when coupled to normal modes and their existencecan invalidate the self-accelerating background solutionas well as linear perturbations around it.Observational problems also arise if one posits thatghosts and strong coupling do not invalidate the idea ofself-acceleration itself, i.e. that our Hubble volume be-haves in a manner that is perturbatively close on scalesnear the horizon to the modified Friedmann equationspecified on this branch. These problems fall into twoclasses, those due to the background expansion historyand those due to the growth of structure during the ac-celeration epoch.In a flat spatial geometry the DGP model adds onlyone degree of freedom to fit acceleration, the crossoverscale. Like ΛCDM, the extra degree of freedom can bephrased as the matter density relative to the critical den-sity. Remarkably, with this one parameter, the ΛCDMmodel can fit three disparate sets of distance measures: the local Hubble constant and baryon acoustic oscilla-tions, the relative distances to high-redshift supernovae(SNe), and the acoustic peaks in the cosmic microwavebackground (CMB). The flat DGP model cannot fit theseobservations simultaneously and the significance of thediscrepancy continues to grow. The addition of spatialcurvature can help alleviate some, but not all, of thistension [11, 12, 13].On intermediate scales that encompass measurementsof the large scale structure of the Universe, the earlyonset of modifications to the expansion also imply a sub-stantial reduction in the linear growth of structure whichis itself slightly reduced by the modification to gravity[14, 15]. If extended to the mildly nonlinear regime wherestrong coupling effects may alter growth, this reductionis in substantial conflict with weak lensing data [16].On scales approaching the crossover scale, which areprobed by the CMB, the unique modification to grav-ity in the DGP model itself strongly alters the growthrate. Here the propagation of perturbations into the bulkrequires a (4 + 1)-dimensional perturbation framework[17]. The approximate iterative solutions introduced inRfn. [18] have been recently verified to be sufficientlyaccurate by a more direct calculation [19] but still arecomputationally too expensive for Monte Carlo explo-rations of the DGP parameter space. More recently, a(3 + 1)-dimensional effective approach dubbed the pa-rameterized post-Friedmann (PPF) framework [20, 21]has been developed that accurately encapsulates mod-ified gravity effects with a closed effective dark energysystem. The PPF approach enables standard cosmolog-ical tools such as an Einstein-Boltzmann linear theorysolver to be applied to the DGP model.In this
Paper , we implement the PPF approach toDGP and conduct a thorough study of the tension be-tween CMB distance, energy density, and growth mea-sures, along with the SNe and local distance measures,across an extended DGP parameter space. We find thateven adding epicycles to the DGP model does not sig-nificantly improve the agreement with the data. Curva-ture, while able to alleviate the problem with distances,exacerbates the problem with growth. Changing the ini-tial power spectrum to remove excess power in the tem-perature spectrum destroys the agreement with recentpolarization measurements from the five-year WilkinsonMicrowave Anisotropy Probe (WMAP) [22].The outline of the paper is as follows. In § II we re-view the impact of the DGP modifications on distancemeasures and the growth of structure emphasizing aneffective dark energy PPF approach that is detailed inAppendix A and compared with a growth-geometry split-ting approach in Appendix B. We present the results ofthe likelihood analysis in § III and discuss these results in § IV.
II. DGP DISTANCE AND GROWTHPHENOMENOLOGY
In the DGP model, gravity remains a metric theoryon the brane in a background that is statistically ho-mogeneous and isotropic. Its modifications therefore areconfined to the field equations that relate the metric tothe matter. Once the metric is obtained, all the usualimplications for the propagation of light from distantsources and the motion of matter remain unchanged.In this section, we review the DGP modifications tothe background metric, or expansion history, and thegravitational potentials, or linear metric perturbations.We cast these modifications in the language of an ef-fective dark energy contribution under ordinary gravity[20, 21, 23, 24, 25, 26, 27].
A. Background Evolution
That the DGP model is a metric theory in a statisti-cally homogeneous and isotropic universe imposes a back-ground Friedmann-Robertson-Walker (FRW) metric onthe brane. The FRW metric is specified by two quanti-ties, the evolution of the scale factor a and the spatialcurvature K . The DGP model modifies the field equa-tion, i.e. the Friedmann equation, relating the evolutionof the scale factor H = a − da/dt to the matter-energycontent. On the self-accelerating branch it becomes (see, e.g. , Rfns. [2, 28]) H = s πG X i ρ i + 14 r c + 12 r c ! − Ka , (1)where r c is the crossover scale and subscript i labels thetrue matter-energy components of the universe to be dis- tinguished below from the effective dark energy contri-bution.On the other hand, given the metric the matter evolvesin the same way as in ordinary gravity˙ ρ i = − aH ( ρ i + P i ) . (2)Overdots represent derivatives with respect to conformaltime η = R dt/a . From these relations, one sees that as a → ∞ , H → r − c and the Universe enters a de Sitterphase of accelerated expansion.In the limit that r c → ∞ the ordinary Friedmann equa-tion is recovered. The effect of a finite r c compared withthe Hubble scale can be encapsulated in a dimensionlessparameter Ω r c much like the usual contributions of thedensity Ω i = 8 πGρ i / H and curvature Ω K = − K/H to the expansion rateΩ r c ≡ r c H . (3)The modified Friedmann equation (1) today then be-comes the constraint equation1 = p Ω r c + s Ω r c + X i Ω i + Ω K . (4)It is convenient and instructive to recast the impact of r c as an effective dark energy component. With the samebackground evolution, the effective dark energy will havean energy density of ρ e ≡ πG (cid:18) H + Ka (cid:19) − X i ρ i . (5)Conservation of its energy-momentum tensor is guaran-teed by the Bianchi identities and requires its “equationof state” w e ≡ P e /ρ e to be given by Eq. (2) w e = P i ( ρ i + P i )3( H + a − K ) / πG + P i ρ i − . (6)If we define Ω e in the same way as Ω i , the usual constraintcondition applies P i Ω i + Ω e + Ω K = 1. Comparing it toEq. (4), we obtain the following relationship between Ω e and Ω r c Ω e = 2 p Ω r c (1 − Ω K ) . (7)Given w e and Ω e , we can now describe the backgroundevolution of the DGP cosmology by using the ordinaryFriedmann equation for H . Likewise the Hubble param-eter specifies the comoving radial distance D ( z ) = Z z dz ′ H ( z ′ ) , (8)and the luminosity distance d L ( z ) = (1 + z ) √− Ω K H sin (cid:16)p − Ω K H D ( z ) (cid:17) , (9) -2 -1 -0.8-0.7-0.6-0.5-0.4-0.3 w e z FIG. 1: Equation of state of the effective dark energy w e forthe self-accelerating DGP model with Ω m = 0 .
26 and Ω K = 0. as usual.The effective dark energy for DGP has quite a dif-ferent equation of state from that of a cosmological con-stant. For example, when there is no curvature, w e startsat − / (1 + Ω m ) at the present, approaches − / − / w e is always larger than −
1, and the universewill expand at a larger rate than in ΛCDM. This reducesthe absolute comoving radial distance to a distant source.For relative distance measures like the SNe, the reductionin H d L ( z ) can be compensated by lowering the frac-tional contribution of matter through Ω m . The same isnot true for absolute distances, such as those measuredby the CMB, baryon oscillations, and Hubble constant,if the overall matter contribution to the expansion rateΩ m H a − remains fixed. B. Structure Formation in the Linear Regime
The same methodology of introducing an effective darkenergy component for the background expansion historyapplies as well to the linear metric perturbations thatgovern the evolution of large scale structure.To define the effective dark energy, one must first pa-rameterize the solutions to the (4 + 1)-dimensional equa-tions involving metric perturbations in the brane as wellas the bulk [17]. Three simplifications aid in this pa-rameterization [21]. The first is that at high redshift theeffect of r c and the extra dimension goes away rapidly [seeEq. (1)]. The parameterization needs only to be accuratebetween the matter dominated regime and the present. Likewise, the predictions of ΛCDM for the relation-ships between the matter and baryon density at recom-bination and morphology of the CMB acoustic peaks re-main unchanged. As a consequence, the shape of theCMB acoustic peaks still constrains the physical colddark matter and baryon density Ω c h and Ω b h as usual.With these as fundamental high redshift parameters, ina flat spatial geometry, the DGP degree of freedom r c isthen specified by either Ω m = 1 − Ω e = Ω c + Ω b or H .We shall see below that in a flat universe the competingrequirements of CMB and SNe distance measures on r c will slightly shift the values of Ω c h and Ω b h to reach acompromise at the expense of the goodness of fit.The second simplification is that well below the hori-zon, perturbations on the brane are in the quasi-staticregime where time derivatives can be neglected in com-parison with spatial gradients and propagation effectsinto the bulk are negligible. This allows the equationsto be simply closed on the brane by a modified Pois-son equation (12) that can be recast as arising from theanisotropic stress of the effective dark energy [14, 15].The final simplification is that on scales above the hori-zon, the impact of the bulk perturbations on the branebecomes scale free and depends only on time [18] throughdimensionless combinations of H , r c and K . Furthermoreon these scales, generic modifications to gravity are fullydefined by the Friedmann equation and the anisotropicstress of the effective dark energy [29, 30].As shown in Rfn. [21] and detailed in Appendix A,interpolation between these two limits leads to a sim-ple PPF parameterization of DGP on all linear scales.This parameterization has been verified to be accurateat a level substantially better than required by cosmicvariance by a direct computation of bulk perturbations[19]. With such a parameterization, efficient Einstein-Boltzmann codes such as CAMB [31] can be modified tocalculate the full range of CMB anisotropy, as is done inthis paper.The result of this calculation is that compared withΛCDM, the growth of structure in the DGP model duringthe acceleration epoch is suppressed. In the quasi-staticregime, this suppression is mostly due to the higher red-shift extent of the acceleration epoch discussed in theprevious section (see also Fig. 1) with a small compo-nent from the effective anisotropic stress or modificationto the Poisson equation [14, 15]. On scales approaching r c , the effect of the anisotropic stress becomes much moresubstantial due to the perturbations propagating into thebulk [18] resulting in an even stronger integrated Sachs-Wolfe (ISW) effect in the CMB anisotropy power at thelowest multipoles [13]. III. CONSTRAINTS FROM CURRENTOBSERVATIONS
In this section, we employ Markov Chain Monte Carlo(MCMC) techniques to explore constraints on the DGPparameter space from current observations, and comparethem with the successful ΛCDM model. The data weuse are: the Supernovae Legacy Survey (SNLS) [32], theCMB anisotropy data from the five-year WMAP [33] forboth temperature and polarization (TT + EE + TE),and the Hubble constant measurement from the HubbleSpace Telescope (HST) Key Project [34].We use the public MCMC package
CosmoMC [35],with a modified version of CAMB for DGP describedin Appendix A, to sample the posterior probability dis-tributions of model parameters. The MCMC techniqueemploys the Metropolis-Hastings algorithm [36, 37] forthe sampling, and the Gelman and Rubin R statistic[38] for the convergence test. We conservatively require R − < .
01 for the eight chains we run for each model,and this generally gave us ∼ w e and densityparameter Ω e in Eqs. (6) and (7).With only scalar perturbations in considera-tion, our basic parameter set is chosen to be { Ω b h , Ω c h , θ s , τ, n s , A s } , which in turn stand forthe density parameters of baryons and cold dark matter,angular size of the sound horizon at recombination,optical depth from reionization (assumed to be instan-taneous), spectra index of the primordial curvaturefluctuation and its amplitude at k ∗ = 0 .
002 Mpc − , i.e. , ∆ ζ = A s ( k/k ∗ ) n s − . Also, we follow Rfn. [33], andinclude A SZ , with flat prior of 0 < A SZ <
2, to accountfor the contributions to the CMB power spectra fromSunyaev-Zeldovich fluctuations. The lensing effect onthe CMB is neglected. Note that in all the three modelswe considered, i.e. self-accelerating DGP, ΛCDM andQCDM, we have the same parameter sets and priorsexcept that we restrict the DGP parameter space to H r c > .
08 so that metric fluctuations remain wellbehaved [see Eq. (16)]. We apply this prior to theQCDM model as well for a fair comparison. In practice,the excluded models are strongly disfavored by the dataand the prior is only necessary for numerical reasons.
A. Flat Models
We start with the minimal parameterization of a flatuniverse with scale free initial conditions. In this casethe three model classes ΛCDM, DGP and QCDM allhave only one parameter that describes acceleration. Inthe chain parameters this is θ s but can be equivalently TABLE I: Mean and marginalized errors for various parame-ters of the self-accelerating DGP, QCDM with the same ex-pansion history as the DGP and ΛCDM models from SNLS+ WMAP5 + HST, assuming a flat universe . The first 6 pa-rameters are directly varied when running the Markov Chains,while the others are derived parameters, as are in the follow-ing tables.parameters DGP QCDM ΛCDM100Ω b h ± ± ± c h ± ± ± θ s ± ± ± τ ± ± ± n s ± ± ± A s ] 3.02 ± ± ± H ± ± ± m . ± .
02 0.26 ± ± r c . ± .
009 .. ..TABLE II: Parameters and the likelihood values at the best-fit point of the self-accelerating DGP, QCDM with the sameexpansion history as the DGP and ΛCDM models fitting toSNLS + WMAP5 + HST, assuming a flat universe .parameters DGP QCDM ΛCDM100Ω b h c h θ s τ n s A s ] 3.01 3.06 3.18 H m r c − L defined as the derived parameters H or Ω m . The con-straints on the three model classes are given in Table Ifor the means and marginalized errors on various param-eters. Table II shows the best-fit values of the parame-ters and the corresponding likelihoods, which serve as a“goodness of fit” criterion.First, we compare the constraints on the QCDM modelwith those on ΛCDM. The differences are expected toreflect those between the background expansion historiesof the DGP and ΛCDM models. In spite of the clusteringeffects on the largest scales of a quintessence dark energy,the difference between QCDM and ΛCDM is completelyencoded in the equation of state of their dark energycomponents given the fixed sound speed of quintessence.Constraints from the SNe magnitudes come from thedimensionless luminosity distance H d L ( z ) [see Eq. (9)],once the unknown absolute magnitude is marginalized.In order to match the predictions for H d L ( z ) of a flatΛCDM model, we would expect that the QCDM modelhas a smaller Ω m to compensate the larger w e its dark
10 100 10000200040006000 10 100 100001000200030004000500060007000
WMAP5 DGP QCDM CDM K =0 ( + ) C TT / ( K ) FIG. 2: Predictions for the power spectra of the CMB tem-perature anisotropies C TT ℓ of the best-fit DGP (solid), QCDMwith the same expansion history as DGP (short-dashed), andΛCDM (dashed, coincident with QCDM at low ℓ ) models ob-tained by fitting to SNLS + WMAP5 (both temperature andpolarization) + HST, assuming a flat universe . Bands repre-sent the 68% and 95% cosmic variance regions for the DGPmodel. Points represent WMAP5 measurements; note thatnoise dominates over cosmic variance for ℓ > ∼ energy has. However, lowering Ω m also shortens moreof the distance to the last scattering surface and henceincreases the angular size of the sound horizon (see § II A).The physical scale of the sound horizon can partiallycompensate and is controlled by Ω c h and Ω b h butthese parameters are also well measured by the shapeof the peaks and can only be slightly adjusted at a costto the goodness of fit. Thus the parameter ranges inTable I for Ω c h decrease and Ω b h increase slightlywhich both have the effect of decreasing the angularsize of the horizon while Ω m remains nearly unchanged.This compromise between the energy density and dis-tance constraints results in tension between the CMBand SNe data. This tension shows up as a differ-ence between the − L values of these two models:2 ln L (ΛCDM) − L (QCDM) ≃ n s and a smaller A s . This is a consequence of the larger ISW effect in theQCDM model due to a slower growth rate. Tilting thespectrum can compensate for the excess power in thelow- ℓ modes (as shown by the near-coincidence of theshort-dashed and dashed curves in Fig. 2).Next we compare the constraints on the DGP modelwith those on QCDM. Since the two models havethe same expansion history, the differences are entirelycaused by the differing growth rates. Due to the prop-agation of perturbations into the bulk for scales near r c and the opposite effect of dark energy clustering inQCDM, there is a substantially stronger ISW effect in TABLE III: Mean and marginalized errors for various param-eters of the self-accelerating DGP, QCDM with the same ex-pansion history as the DGP and ΛCDM models from SNLS+ WMAP5 + HST, allowing curvature .parameters DGP QCDM ΛCDM100Ω b h ± ± ± c h ± ± ± θ s ± ± ± τ ± ± ± K ± ± ± n s ± ± ± A s ] 3.02 ± ± ± H ± ± ± m ± ± ± r c ± the first few multipoles of the CMB anisotropy power[13]. Since this effect is only important on the largestangular modes in the CMB TT power spectra which isfurther limited by the large cosmic variance, the param-eter ranges for these two models do not differ signifi-cantly. Nonetheless from Fig. 2, it is clear that the best-fit DGP model over predicts the low- ℓ modes anisotropy,though as before, n s and A s adjustments try to re-duce the primordial perturbations on large scales. Thisleads to DGP being an even worse fit than QCDM with2 ln L (QCDM) − L (DGP) ≃ − L for the maximum likelihood parametersto ≃
28, where ∼
70% is driven by the background expan-sion, while ∼
30% by the dynamical effects on structuregrowth.
B. Adding in Curvature
From our analysis in the above section, the flat DGPmodel is a poor fit to the current observations mostlybecause it cannot simultaneously satisfy the geometricalrequirements of the relative luminosity distances of theSNe and the angular size of the sound horizon at recom-bination with a single parameter. Since curvature hasmore of an effect on high redshift distance measures, thetension in the distance measures can be alleviated by in-cluding Ω K in the parameter space [12, 13]. Our resultsare given in Table III and Table IV.With curvature, the − L of the maximum likeli-hood ΛCDM model almost has no improvement. Thisis consistent with the results of Rfn. [39], who foundstrong limits on curvature in ΛCDM, by fitting WMAP5data combined with SNe or HST. As expected the max-imum likelihood model in the QCDM space improves in − L by ∼
10. The QCDM model needs an open uni-verse to increase the distance to last scattering to com-pensate the smaller Ω m , consistent with the findings by TABLE IV: Parameters and likelihood values at the best-fit point of the self-accelerating DGP, QCDM with the sameexpansion history as the DGP and ΛCDM models fitting toSNLS + WMAP5 + HST, allowing curvature .parameters DGP QCDM ΛCDM100Ω b h c h θ s τ K n s A s ] 3.02 3.05 3.18 H m r c − L [12]. Even with this additional freedom, distance mea-sures remain in slight tension due to the Hubble con-stant since lowering Ω m with Ω m h well determined bythe CMB implies a higher Hubble constant [13]. The al-lowed amount of this shift is limited by the HST KeyProject constraint of H = 72 ± − Mpc − . ForQCDM the baryon acoustic oscillation constraint wouldalready disfavor such a shift [40, 41, 42, 43] but its appli-cation to DGP requires cosmological simulations of thestrong coupling regime. Note that with the distance ten-sion partially removed, the shifts in Ω c h and Ω b h arereduced.The lowering of Ω m , in addition to the large equationof state parameter of the quintessence, also causes matterdomination to terminate at an earlier redshift, and leadsto a stronger ISW effect in the QCDM model, as can beseen in Fig. 3, again with a partial compensation from n s and A s . The net difference of − L values of theQCDM model compared to ΛCDM is ≃
10, 50% smallerthan before but still a significantly poorer fit.The situation is even worse for DGP. Here the enhance-ment of the ISW effect at low Ω m is even more substan-tial. Thus the mean value of Ω K is smaller than theoptimal one for the distance constraints in QCDM. Evenadjusting the other parameters to give the maximum like-lihood model shown in Fig. 3, the poor fit is noticeableat the low multipoles. For example the probability of ob-taining a quadrupole as extreme as the observations fromthe DGP maximum likelihood model is ∼
1% comparedwith ∼
6% for the ΛCDM maximum likelihood model.The net difference in − L by including curvature asa parameter is only ∼ −
2∆ ln L ≃
23. Since the difference of −
2∆ ln L from QCDM can be attributed to the ISW effect, ∼ ∼
60% by the dynamical effects on structure growth.
10 100 10000200040006000 10 100 100001000200030004000500060007000
WMAP5 DGP QCDM CDMwith curvature ( + ) C TT / ( K ) FIG. 3: Predictions for the power spectra of the CMB tem-perature anisotropies C TT ℓ of the best-fit DGP (solid), QCDMwith the same expansion history as DGP (short-dashed),and ΛCDM (dashed) models obtained by fitting to SNLS +WMAP5 (both temperature and polarization) + HST, allow-ing curvature . C. Changing the Initial Power
The adjustments of A s and n s in the above examplessuggest that perhaps a more radical change in the initialpower spectrum can bring DGP back in agreement withthe data. For example, one can sharply reduce large scalepower in the temperature spectrum by cutting off the ini-tial power spectrum on large scales, below a wavenumber k min . While this is a radical modification and has no par-ticular physical motivation, it is useful to check whethersuch a loss of power could satisfy the joint temperatureand polarization constraints.CMB polarization at these scales arises from reioniza-tion which occurs at a substantially higher redshift thanDGP modifications affect. A finite polarization requiresnot only ionization but also large scale anisotropy at thisepoch. Eliminating initial power on these scales elimi-nates the polarization as well. The EE power in the lowmultipoles has now been measured at 4-5 σ level [22] lead-ing to a significant discrepancy if the large scale power isremoved in the model.Given that including this parameter does not improvethe fit, instead of adding it to the MCMC parameterspace, we illustrate its effects on the maximum likelihoodDGP model in § III B.By maximizing the likelihood for this model to theTT power alone, we find the best agreement is obtainedat k min = 8 × − Mpc − , with −
2∆ ln L TT ≃ − k min = 0. Model predictions with this k min are plotted in Fig. 4, together with the WMAP 5 yeardata.With this power truncation the over prediction prob-
10 100 10000200040006000 10 100 100001000200030004000500060007000
WMAP 5 k min =0 k min =8 x 10 -4 Mpc -1 ( + ) C TT / ( K ) FIG. 4: Predictions for the power spectra of the CMB tem-perature anisotropies C TT ℓ of the best-fit DGP model as foundin § III B without cutting off any large scale primordial per-turbations (solid) and with a cut-off scale of k min = 8 × − Mpc − (dotted) – the best-fit scale obtained when fitting tothe WMAP 5 year TT data alone, while all other parametersare fixed at their best-fit values with k min = 0. lem for the low- ℓ TT power is alleviated, but the EE po-larization power on large angular scales is significantlyreduced (see Fig. 5). When the polarization data ofTE + EE are included, we find −
2∆ ln L all ≃ k min = 8 × − Mpc − , and when k min is varied, thecombined data does not favor any positive values of k min as shown in Fig. 6. We conclude that though the DGPmodel can be made a better fit to the TT power spectrumwith a large scale cut-off, polarization measurements arenow sufficiently strong to rule out this possibility. Whilewe have only included instantaneous reionization mod-els, changing the ionization history to have an extendedhigh redshift tail can only exacerbate this problem byincreasing EE power in the ℓ ∼ −
30 regime [44].
IV. DISCUSSION
We have conducted a thorough Markov Chain MonteCarlo likelihood study of the parameter space availableto the DGP self-accelerating braneworld scenario givenCMB, SNe and Hubble constant data. To carry out thisstudy, we have introduced techniques for characterizingmodified gravity and non-canonical dark energy candi-dates with the public Einstein-Boltzmann code CAMBthat are of interest beyond the DGP calculations them-selves.We find no way to alleviate substantially the tensionbetween distance measures and the growth of horizonscale fluctuations that impact the low multipole CMBtemperature and its relationship to the polarization. In
10 100 100010 -3 -2 -1
10 100 100010 -3 -2 -1 k min =0 k min =8 x 10 -4 Mpc -1 ( + ) C EE / ( K ) FIG. 5: Predictions for the power spectra of the CMB E-modepolarization C EE ℓ of the best-fit DGP model as found in § III Bwithout cutting off any large scale primordial perturbations(solid) and with a cut-off scale of k min = 8 × − Mpc − (dot-ted) – the best-fit scale obtained when fitting to the WMAP5 year TT data alone, while all other parameters are fixedtheir best-fit values. Note here, according to Rfn. [22], thereionization feature at the lowest- ℓ modes is preferred by thedata through ∆ χ = 19 . -4 -4 -4 -4 -3 - l nL a ll k min (Mpc -1 ) FIG. 6: The total log-likelihood of TT + TE + EE as a func-tion of the cut-off scale k min of the primordial perturbations,shown as the difference from its value at k min = 0. We findthat when polarization is included, the combined data doesnot favor any positive values of k min . The local minimumshows up at the scale of k min = 8 × − Mpc − , which is theone favored by the TT data alone. Note all other parametersare fixed at their best-fit values of the DGP model as foundin § III B. particular we show that the maximum likelihood flatDGP model is a poorer fit than ΛCDM by 2∆ ln L = 28,nominally a (2∆ ln L ) / = 5 . σ result. Interestingly, asubstantial ( ∼ L = 5 with one extra parameterand the difference from ΛCDM is 2∆ ln L = 23, which isstill 4 . σ discrepant with most of the difference arisingfrom the changes to growth.Furthermore, while the excess power at large anglescan be reduced by changing the initial power spectrumto eliminate large scale power, existing WMAP5 CMBpolarization measurements already forbid this possibil-ity. Specifically, by introducing any finite cut off to theinitial power spectrum to flatten the temperature powerspectrum, the global likelihood decreases.While it is still possible that the resolution of the ghostand strong coupling issues of the theory can alter theseconsequences, it is difficult to see how they can do sowithout altering the very mechanism that makes it a can-didate for acceleration without dark energy. The failure of this model highlights the power of combining growthand distance measures in cosmology as a test of gravityon the largest scales. Acknowledgments
We thank K. Koyama and Y.S. Song for useful con-versations. This work was supported in part by the NSFgrant AST-05-07161, by the Initiatives in Science andEngineering (ISE) program at Columbia University, andby the Pol´anyi Program of the Hungarian National Of-fice for Research and Technology (NKTH). SW and WHwere supported by the KICP under NSF PHY-0114422WH was further supported by U.S. Dept. of Energy con-tract DE-FG02-90ER-40560 and the David and LucilePackard Foundation. LH was supported by the DOEgrant DE-FG02-92-ER40699, and thanks Alberto Nico-lis for useful discussions and Tai Kai Ng at the HongKong University of Science and Technology for hospital-ity. MM was supported by the DOE grant DE-AC02-98CH10886. Computational resources were provided bythe KICP-Fermilab computer cluster. [1] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett.
B 485 , 208 (2000), [arXiv:hep-th/0005016].[2] C. Deffayet, Phys. Lett.
B 502 , 199 (2001), [arXiv:hep-th/0010186].[3] M. A. Luty, M. Porrati and R. Rattazzi, JHEP , 029(2003), [arXiv:hep-th/0303116].[4] A. Nicolis and R. Rattazzi, JHEP , 059 (2004),[arXiv:hep-th/0404159].[5] K. Koyama, Phys. Rev. D 72 , 123511 (2005), [arXiv:hep-th/0503191].[6] D. Gorbunov, K. Koyama and S. Sibiryakov, Phys. Rev.
D 73 , 044016 (2006), [arXiv:hep-th/0512097].[7] C. Charmousis, R. Gregory, N. Kaloper and A. Padilla,JHEP , 066 (2006), [arXiv:hep-th/0604086].[8] C. Deffayet, G. Gabadadze and A. Iglesias, JCAP ,012 (2006), [arXiv:hep-th/0607099].[9] G. Dvali, New J. Phys. , 326 (2006), [arXiv:hep-th/0610013].[10] K. Koyama and F. P. Silva, Phys. Rev. D 75 , 084040(2007), [arXiv:hep-th/0702169].[11] M. Fairbairn and A. Goobar, Phys. Lett.
B 642 , 432(2006), [arXiv:astro-ph/0511029].[12] R. Maartens and E. Majerotto, Phys. Rev.
D 74 , 023004(2006), [arXiv:astro-ph/0603353].[13] Y.-S. Song, I. Sawicki and W. Hu, Phys. Rev.
D 75 ,064003 (2007), [arXiv:astro-ph/0606286].[14] A. Lue, R. Scoccimarro and G. D. Starkman, Phys. Rev.
D 69 , 124015 (2004), [arXiv:astro-ph/0401515].[15] K. Koyama and R. Maartens, JCAP , 016 (2006),[arXiv:astro-ph/0511634].[16] S. Wang, L. Hui, M. May and Z. Haiman, Phys. Rev.
D76 , 063503 (2007), [arXiv:0705.0165]. [17] C. Deffayet, Phys. Rev.
D 66 , 103504 (2002), [arXiv:hep-th/0205084].[18] I. Sawicki, Y.-S. Song and W. Hu, Phys. Rev.
D 75 ,064002 (2007), [arXiv:astro-ph/0606285].[19] A. Cardoso, K. Koyama, S. S. Seahra and F. P. Silva,Phys. Rev.
D 77 , 083512 (2008), [arXiv:0711.2563].[20] W. Hu, Phys. Rev.
D 77 , 103524 (2008),[arXiv:0801.2433].[21] W. Hu and I. Sawicki, Phys. Rev.
D 76 , 104043 (2007),[arXiv:0708.1190].[22] WMAP, M. R. Nolta et al. , arXiv:0803.0593.[23] S. Bashinsky, arXiv:0707.0692.[24] M. Kunz and D. Sapone, Phys. Rev. Lett. , 121301(2007), [arXiv:astro-ph/0612452].[25] R. Caldwell, A. Cooray and A. Melchiorri, Phys. Rev. D76 , 023507 (2007), [arXiv:astro-ph/0703375].[26] B. Jain and P. Zhang, arXiv:0709.2375.[27] E. Bertschinger and P. Zukin, arXiv:0801.2431.[28] C. Deffayet, S. J. Landau, J. Raux, M. Zaldarriaga andP. Astier, Phys. Rev.
D 66 , 024019 (2002), [arXiv:astro-ph/0201164].[29] W. Hu and D. J. Eisenstein, Phys. Rev.
D 59 , 083509(1999), [arXiv:astro-ph/9809368].[30] E. Bertschinger, Astrophys. J. , 797 (2006),[arXiv:astro-ph/0604485].[31] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. , 473 (2000), [arXiv:astro-ph/9911177].[32] The SNLS, P. Astier et al. , Astron. Astrophys. , 31(2006), [arXiv:astro-ph/0510447].[33] WMAP, J. Dunkley et al. , arXiv:0803.0586, see alsohttp://lambda.gsfc.nasa.gov.[34] HST, W. L. Freedman et al. , Astrophys. J. , 47 (2001), [arXiv:astro-ph/0012376].[35] A. Lewis and S. Bridle, Phys. Rev.
D 66 , 103511(2002), [arXiv:astro-ph/0205436], the code and de-scription of its features are available on the Web site:http://cosmologist.info/cosmomc.[36] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth,A. H. Teller and E. Teller, The Journal of ChemicalPhysics , 1087 (1953).[37] W. K. Hastings, Biometrika , 97 (1970).[38] A. Gelman and D. B. Rubin, Statistical Science , 457(1992).[39] WMAP, E. Komatsu et al. , arXiv:0803.0547.[40] D. J. Eisenstein et al. , Astrophys. J. , 560 (2005),[arXiv:arXiv:astro-ph/0501171]. [41] G. H¨utsi, Astron. Astrophys. , 375 (2006),[arXiv:arXiv:astro-ph/0604129].[42] W. J. Percival et al. , Mon. Not. Roy. Astron. Soc. ,1053 (2007), [arXiv:arXiv:0705.3323].[43] E. Gaztanaga, A. Cabre and L. Hui, ArXiv e-prints(2008), [arXiv:0807.3551].[44] M. J. Mortonson and W. Hu, arXiv:0804.2631.[45] T. Giannantonio, Y.-S. Song and K. Koyama,arXiv:0803.2238.[46] M. Zaldarriaga, U. Seljak and E. Bertschinger, Astro-phys. J. , 491 (1998), [arXiv:astro-ph/9704265].[47] J. Zhang, L. Hui and A. Stebbins, Astrophys. J. ,806 (2005), [arXiv:arXiv:astro-ph/0312348]. APPENDIX A
In Appendix A, we give details of the modification to CAMB employed in the main text to calculate the modifiedgrowth of perturbations in the DGP model under the PPF prescription. For perturbations of both the metric andmatter in the various gauges, we use the same notations as in Rfn. [20].
1. PPF Description of Modified Gravity in the Linear Regime
Structure formation in the linear regime for a certain class of modified gravity theories can be equivalently describedas an effective dark energy component under ordinary gravity. The requirements of this class are that members remainmetric theories in a statistically homogeneous and isotropic universe where energy-momentum is conserved. This classincludes the self-accelerating branch of the DGP model.Following Rfns. [20, 21], scalar perturbations for the DGP self-accelerating scenario can be parameterized by threefree functions, g ( η, k ), f ζ ( η ), and f G ( η ), and one free parameter c Γ . We shall see in § g ( η, k ) is defined asΦ + ≡ g ( η, k )Φ − − πGH k H P T Π T , (10)where Φ + ≡ (Φ + Ψ) /
2, Φ − ≡ (Φ − Ψ) /
2, with Φ = δg ij / g ij , Ψ = δg / g as the space-space and time-timemetric perturbations in the Newtonian gauge, k H = ( k/aH ), Π stands for the anisotropic stress, and the subscript“ T ” denotes sum over all the true components. When anisotropic stresses of the true components are negligible, g just parameterizes deviations from GR in the metric ratio of Φ + to Φ − .There are two key features of the PPF parameterization and these define the two additional functions f ζ ( η ) and f G ( η ). The first is that the curvature perturbation in the total matter comoving gauge ζ is conserved up to order k H in the super-horizon (SH) regime in the absence of non-adiabatic fluctuations and background curvaturelim k H ≪ ˙ ζaH = − ∆ P T − c K P T Π T ρ T + P T − Kk k H V T + 13 c K f ζ ( η ) k H V T , (11)where f ζ ( η ) parameterizes the relationship between the metric and the matter. The second is that in the quasi-static(QS) regime, one recovers a modified Poisson equation with a potentially time dependent effective Newton constantlim k H ≫ Φ − = 4 πGc K k H H ∆ T ρ T + c K P T Π T f G ( η ) . (12)Note that even if f G = 0, the Poisson equation for Ψ may also be modified by a non-zero g ( η, k ). In the above twoequations, ∆, V and ∆ P ( = P ∆) are density, velocity, pressure perturbations in the total matter comoving gauge,and c K = 1 − K/k . To bridge the two regimes, an intermediate quantity Γ is introduced,Φ − + Γ = 4 πGc K k H H (∆ T ρ T + c K P T Π T ) , (13)0and an interpolating equation is adopted to make sure it dynamically recovers the behavior specified by Eqs. (11) and(12) (1 + c k H ) " ˙Γ aH + Γ + c k H (Γ − f G Φ − ) = S . (14)Here the free parameter c Γ gives the transition scale between the two regimes in terms of the Hubble scale, and thesource term S is given by, S = ˙ g/ ( aH ) − gg + 1 Φ − + 4 πG ( g + 1) k H H ( g " ˙( P T Π T ) aH + P T Π T − [( g + f ζ + gf ζ )( ρ T + P T ) − ( ρ e + P e )] k H V T ) . (15)By interpolating between two exact behaviors specified by functions of time alone f ζ ( η ) and f G ( η ), the PPF param-eterization is both simple and general. Moreover, the one function of time and scale g ( η, k ) also interpolates betweentwo well defined functions of time alone for many models, including DGP. The same is not true of parameterizationsthat involve the effective anisotropic stress alone or the metric functions directly.
2. PPF Parameterization for DGP
The PPF parameterization for the self-accelerating DGP is given in Rfn. [21], which we summarize as the follows.On super-horizon scales, the iterative scaling solution developed in Rfn. [18] is well described by g SH ( η ) = 98 Hr c − (cid:18) . Hr c − . (cid:19) , (16)and f ζ ( η ) = 0 . g SH ( η ), while in the QS regime, the solution is parameterized by [15] g QS ( η ) = − " − Hr c H aH ! − , (17)and f G ( η ) = 0. On an arbitrary scale in the linear regime, g is then interpolated by g ( η, k ) = g SH + g QS ( c g k H ) n g c g k H ) n g , (18)with c g = 0 .
14 and n g = 3. This fitting formula has been shown to give an accurate prediction for the evolution of Φ − according to the dynamical scaling solution [21]. The transition scale for Γ is set to be c Γ = 1, i.e. at the horizon scale.We note here that the solutions developed in Rfns. [15, 18] are for flat universes, so strictly speaking, the above PPFparameterization only works for DGP with Ω K →
0. However for the small curvatures that are allowed by the data,we would expect its effect on structure formation to be small and arise from terms such as H → H + K/a (see, e.g. , Rfn. [45]). Given the cosmic variance of the low- ℓ multipoles, these corrections should have negligible impact onthe results.
3. “Dark Energy” Representation of PPF
By comparing the equations that the PPF quantities satisfy with their counterparts in a dark energy system undergeneral relativity, we obtain the following relations for the perturbations of PPF’s corresponding effective dark energy.These relations act as the closure conditions for the stress energy conservation equations of the effective dark energy.The first closure condition is a relationship between the PPF Γ variable and the components of the stress energytensor of the effective dark energy ρ e ∆ e + 3( ρ e + P e ) V e − V T k H + c K P e Π e = − k c K πGa Γ . (19)The second closure condition is a relationship for the anisotropic stress P e Π e = − k H H πG g Φ − . (20)1Stress energy conservation then defines the velocity perturbation V e − V T k H = − H πG ( ρ e + P e ) g + 1 F " S − Γ − ˙Γ aH + f ζ πG ( ρ T + P T ) H V T k H , (21)with F = 1 + 12 πGa k c K ( g + 1)( ρ T + P T ) . (22)For details of all the derivations, see Rfn. [20]. The PPF equation for Γ then replaces the continuity and Navier-Stokesequations for the dark energy. Note that the effective dark energy pressure perturbation, which obeys complicateddynamics to enforce the large and small scale behavior, is not used.
4. Modifying CAMB to Include PPF
Given the dark energy representation of PPF, we only need to modify the parts in CAMB where dark energyperturbations appear explicitly or are needed, in addition to its equation of state which will be specified by thedesired background expansion. These include the Einstein equations and the source term for the CMB temperatureanisotropy. Since CAMB adopts the synchronous gauge, in this section, we will express everything in the same gauge.The two Einstein equations used in CAMB are,˙ h L k − c K k H η T = 4 πGk H H ( ρ T δ sT + ρ e δ se ) , (23) k ˙ η T − K ( ˙ h L + 6 ˙ η T )2 k = 4 πGa [( ρ T + P T ) v sT + ( ρ e + P e ) v se ] , (24)where h L , η T are metric perturbations, δ , v are density and velocity perturbations, and superscript “s” labels thesynchronous gauge. Here, we only need to provide δ se and v se . Given the gauge transformation relation for velocity V = v s + kα , (25)with α ≡ ( ˙ h L + 6 ˙ η T ) / k , the following expression for v se is easily obtained from Eq. (21)( ρ e + P e ) v se = ( ρ e + P e ) v sT − k H H πG ( g + 1) F " S − Γ − ˙Γ aH + f ζ πG ( ρ T + P T )( v sT + kα ) k H H . (26)In order to calculate v se , we need to evaluate α , which we find, with the help of the first closure condition and the twoEinstein equations, to be given by kα = k H η T + 4 πGc K k H H (cid:20) ρ T δ sT + 3( ρ T + P T ) v sT k H (cid:21) − πGk H H P e Π e − k H Γ . (27)Here, the anisotropic stress is gauge-independent, and is given by Eq. (20), where Φ − is given by gauge-transformingthe density perturbation in Eq. (13) according to ρ ∆ = ρδ s − ˙ ρ v sT k . (28)In addition to α , we need also to specify S and ˙Γ in order to get v se . Given Φ − , S can be calculated by gauge-transforming V T in Eq. (15), and ˙Γ then follows from Eq. (14). Provided v se and P e Π e , δ se can be obtained bygauge-transforming the first closure condition ρ e δ se = − c K P e Π e − ρ e + P e ) v se k H − c K k H H πG Γ . (29)The source term for the CMB temperature anisotropy is given by [46] S T ( η, k ) = G ∆ T + 2 ˙ α + ˙ v sb k + Σ4¯ b + 3 ¨Σ4 k ¯ b ! + e − κ ( ˙ η T + ¨ α ) + ˙ G v sb k + α + 3 ˙Σ2 k ¯ b ! + 3 ¨ G Σ4 k ¯ b , (30)2
10 100 10001000
PPF param-splittingbest-fit flat DGP ( + ) C TT / ( K ) FIG. 7: A comparison of the PPF prediction (upper solid line) with the prediction by parameter-splitting (lower dashed line)for the DGP model. where the visibility function
G ≡ − ˙ κ exp( − κ ), with κ ( η ) the optical depth from today to η for Thomson scattering,∆ T ℓ is the multipole of the temperature anisotropy, v sb is the baryon velocity, Σ ≡ ∆ T − ∆ P , with ∆ P thequadrupole of the polarization anisotropy (for more explicit definitions for ∆ T ℓ and ∆ P ℓ , see Rfn. [46]), and ¯ b = c K .Here we need to specify ˙ α and ¨ α . ˙ α is given by the Einstein equation˙ α + 2 ˙ aa α − η T = − πGk H H ( P T Π T + P e Π e ) , (31)Differentiating this equation with respect to η also gives us ¨ α , which will need the derivative of the effective darkenergy’s anisotropic stress. From energy-momentum conservation and the first closure condition, we obtain˙( P e Π e ) = ˙ h L ρ e + P e ) c K − k H H πG ˙Γ + aHc K ( δρ se + ( ρ e + P e ) v se " k H + k H − k ˙ aa + ˙ HH ! , (32)which completes our modification of CAMB. APPENDIX B
In Appendix B, we briefly contrast the PPF prediction for the CMB temperature power spectrum with an attempt toapproximate the DGP prediction via the parameter-splitting technique [16, 47]. The technique works by splitting thedark energy equation of state w into two separate parameters, with one, w geometry , determining geometric distancesand the other, w growth , determining the growth of structure. We choose w geometry = − . w growth = − . H ( z ) (for z = 0 to 10) and fitting to the quasi-static growth factor(for z = 0 to 2) according to the best-fit flat DGP model (Table II). The result is shown in Fig. 7. One can seethat parameter-splitting falls short of the PPF prediction on large angular scales. This illustrates the importanceof correctly modeling the perturbation growth on horizon scales. The parameter-split of ww