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Observational Evidence for Cosmological-Scale Extra Dimensions Niayesh Afshordi, Ghazal Geshnizjani, and Justin Khoury Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada We present a case that current observations may already indicate new gravitational physics on cosmological scales. The excess of power seen in the Lyman-α forest and small-scale CMB experiments, the anomalously large bulk flows seen both in peculiar velocity surveys and in kinetic SZ, andthehigherISWcross-correlationallindicatethatstructuremaybemoreevolvedthanexpected from ΛCDM. We argue that these observations find a natural explanation in models with infinite- volume (or, at least, cosmological-size) extra dimensions, where the graviton is a resonance with a tiny width. The longitudinal mode of the graviton mediates an extra scalar force which speeds up structure formation at late times, thereby accounting for the above anomalies. The required graviton Compton wavelength is relatively small compared to the present Hubble radius, of order 300-600 Mpc. Moreover, with certain assumptions about the behavior of the longitudinal mode on 9 super-Hubble scales, our modified gravity framework can also alleviate the tension with the low 0 0 quadrupole and the peculiar vanishing of the CMB correlation function on large angular scales, 2 seen both in COBE and WMAP. This relies on a novel mechanism that cancels a late-time ISW contribution against the primordial Sachs-Wolfe amplitude. n a J I. INTRODUCTION 3. High resolution observations of the CMB at (cid:96) ∼ 7 3000 by the Cosmic Background Imager (CBI) in- 1 Inthetenyearssincethediscoveryofcosmicaccelera- dicate an excess power compared to the theoret- ] tionbythesupernovaeteams[1],theΛ-colddarkmatter ical primary CMB power spectrum [6]. This ex- h (ΛCDM) model has emerged as the standard paradigm cess could be explained by the thermal Sunyaev- p for cosmology. With a single parameter (Λ), the model Zel’dovich effect from galaxy clusters at z ∼ 1. - o predicts expansion and growth histories consistent with However,theamplitudeofmatterpowerspectrum, r observations. Asnear-futureexperimentswillsoonprobe necessary to explain CBI excess is about 35% (at t s the large scale structure with unprecedented accuracy, it ∼2σ level) larger than the current best-fit concor- a is worth asking at this juncture whether ten years from dance ΛCDM model [7]. [ nowtheΛCDMmodelwillbehailedastheultimatecos- 4. The Lyman-α forest in the spectra of high red- 2 mological theory, or whether it will have by then joined v shiftquasarsprovidesuswiththehighestresolution its Einstein de Sitter predecessor in the ranks of defunct 4 precision measurement of the cosmological matter cosmologies. A host of recent observations may have al- 4 power spectrum. However, Lyman-α forest con- ready revealed cracks in ΛCDM’s armor: 2 straintsonthelinearmatterpowerspectrum,based 2 1. The cross-correlation between galaxy surveys and on the SDSS quasar spectra at z ∼ 3 [8, 9] is also . 2 the cosmic microwave background (CMB), result- ∼ 35% (at ∼ 2.5σ level) larger than the ΛCDM 1 ing from the Integrated Sachs-Wolfe (ISW) effect, prediction (Fig. 1 in [10]). 8 hasrecentlybeenmeasuredatthe4σlevelusingthe 0 5. As first observed by the Cosmic Background Wilkinson Microwave Anisotropy Probe (WMAP) : Explorer (COBE) [11] and later confirmed by v dataandacombinationoflarge-scalestructuresur- i veys [2, 3]. This provides non-trivial and indepen- WMAP [12], the CMB temperature anisotropy X dentevidenceforcosmicacceleration. Remarkably, showsalackofcorrelationonlarge(>∼60◦)angular r separations. While related to the low quadrupole, a the combined amplitude is larger than the ΛCDM prediction by >∼ 2σ’s: 2.23±0.60 [2], with 1 being the real-space anomaly is more robust statistically — less than 0.03% of random realizations exhibit the ΛCDM expectation. smaller power on those scales [12, 13, 14]. The sta- 2. Independent recent analyses suggest anomalously tistical significance of this anomaly has only grown large bulk flows on large scales. Using a com- with subsequent WMAP data releases [15]. pilation of peculiar velocity surveys, Watkins et al. [4] obtained a bulk flow of 407 ± 81 km s−1 Of course, these anomalies may eventually go away, as on 50h−1 Mpc scales, in contrast with the ΛCDM systematic effects are better understood. After all, these expectationof∼190kms−1 forther.m.s. valueof represent for the most part ∼ 2σ discrepancies. But it thebulkflow,whichareinconsistentatthe2σlevel. is striking that the first four observations on this list of Meanwhile,usingthekineticSunyaev-Zeldovichef- misfitsallpointtowardsthesamephysics: thatstructure fect to estimate peculiar velocities of galaxy clus- ismoreevolvedonlargescalesthanpredictedbyΛCDM. ters, Kashlinsky et al. [5] found a coherent bulk This suggests that gravity may be stronger at late times motion of 600-1000 km s−1 out to >∼300h−1 Mpc. and on large scales. In this paper we show that this is precisely what hap- ofthestandardDGPmodel. Inspiredbytheseresults,we pens in infrared-modified gravity theories, inspired by caninfertheparametricdependenceoftherequiredPPF brane-world constructions with infinite-volume extra di- input functions and parameters for Cascading Gravity mensions. Because the extra dimensions are infinite in models. This allows us to calculate various cosmological extent (at least cosmologically large), the 4d graviton is observables, all within a 3+1-dimensional framework. no longer massless but is instead broadened into a res- One of our key observations is that in 6 or more onance — a continuum of massive states — with a tiny dimensions the background evolution becomes nearly width r−1. At first sight this may seem at odds with our indistinguishable from ΛCDM. This allows us to con- c earlier conclusion, since a massive/resonance graviton siderrelativelysmallgravitonComptonwavelengths,e.g. impliesweakergravity. Whilethisiscertainlytrueonthe r ∼300−600Mpc,withoutviolatingconstraintsonthe c largestdistances, onintermediate(albeitcosmologically- expansion history. Compared to the usual assumption relevant)scalesthelongitudinal(orhelicity-0)modeme- r ∼ H−1 in DGP, this expands considerably the dura- c 0 diates an extra scalar force which enhances gravitational tion and range of scales over which enhanced gravity is attraction by order unity. effective. This extra force is of course at the origin of the well- As mentioned earlier, the culprit for most of the inter- known van Dam-Veltman-Zakharov (vDVZ) discontinu- estingphenomenologyinthesemodelsisthelongitudinal ity[16]. AsconjecturedbyVainshtein[17],however,non- or helicity-0 mode of the massive graviton. This degree linear interactions can suppress the effects of the longi- of freedom is ultimately responsible for addressing all of tudinal mode near astrophysical sources. This screening the aforementioned anomalies: isalsoatplaycosmologically,suppressingtheextraforce atearlytimesandonsmallscales. Thusgravitybecomes • The longitudinal degree of freedom modifies the strongeronlyatlatetimesandonsufficientlylargescales. time-evolution of the lensing potential, which re- This suppression at high density or curvature is qualita- sults in a stronger ISW cross-correlation compared tively similar to the chameleon mechanism [18, 19, 20], to ΛCDM. except that the Vainshtein effect arises from derivative • Inthenon-relativisticlimit,thehelicity-0modecan interactions as opposed to a scalar potential. be thought of as mediating an extra scalar force, The theories of interest are higher-dimensional gen- which enhances gravitational attraction on inter- eralizations of the Dvali-Gabadadze-Porrati (DGP) mediate scales. This speed-up in structure forma- model [21], in which our visible universe is confined to tion at late times results in larger bulk flows, more a 3-brane. The confrontation of its cosmological predic- clustersatz ∼1,andenhancedpoweronLyman-α tions with observations has been the subject of much lit- scales. eraturesinceitsadvent[22,23,24,25,26,27,28,29,30]. Insteadofhavingoneextradimension,herethebulkcon- • As a consequence of this mode, our modified persists of 6 or more space-time dimensions. turbationequationsexhibitaneffectiveanisotropic Extending the DGP scenario to higher dimensions has stresscomponent. Withcertainassumptionsabout proven difficult historically. Early studies demonstrated theformofthiscomponentonsuper-Hubblescales, that naive extensions are plagued by ghost-like instabili- we find that the late-time ISW effect can destruc- ties,evenaroundflatspace[31,32]. Recently,however,it tively interfere with the primordial Sachs-Wolfe wasshownthatsuchinstabilitiesareabsentifour3-brane amplitude, resulting in a power deficit in the CMB lies within a succession of higher-dimensional branes, on large scales. each with their own induced gravity term, and embedded in one another in a flat bulk space-time [33, 34]. Looking forward, our modified gravity models make We refer to this framework as Cascading Gravity [35]. several key predictions that will be tested by near-future In the simplest codimension-2 case, for instance, our 3- experiments. For instance, because photons do not cou- brane is embedded in a 4-brane within a 6-dimensional ple directly to the longitudinal mode, the weak lensing bulk. A similar cascading behavior of the gravitational potentialisconsiderablylessaffectedthantheNewtonian force law was also obtained in a different codimension-2 potential. Thus a distinguishing prediction is a discrep- framework [36]. ancy in the value of σ estimated from the matter power 8 Due to the higher-dimensional nature of these con- spectrum and that inferred from either CMB or weak structions,extractingcosmologicalpredictionspresentsa lensing observations. For r ∼300−600 Mpc, the differ- c dauntingtechnicalchallenge. Evenderivingthemodified ence is a 20-30% effect, depending on the redshift of the Friedmann equation is highly non-trivial. Here we cir- observation. cumventthesedifficultiesbydevelopingaphenomenolog- The paper is organized as follows. In Sec. II, we de- ical description for the background cosmology, while us- scribe the building blocks of higher-dimensional gener- ing the parametrized post-Friedmann (PPF) [37] frame- alizations to DGP, following [38, 39]. The result is a work to encode modifications to Einstein gravity into two-paramater family of massive or resonance gravity the evolution of perturbations. This approach has been theories, with α specifying the form factor, and r the c showntoaccuratelyreproducethecosmologicalbehavior Compton wavelength for the graviton. Motivated by the 2 DGP Friedmann equation, we propose in Sec. III a gen- A. Strong coupling of the longitudinal mode eralized Friedmann equation for higher-dimensional theories. We then turn to cosmological perturbations in The propagator (1) describes a resonance, or a contin- Sec. IV, and motivate our fiducial input parameters for uum of massive states, which is peaked at the tiny scale PPF. Using these, we derive the resulting CMB power r−1. Animmediateconsequenceisthatthegravitonnow c spectrum and angular correlation function in Sec. V. In propagates 5 degrees of freedom: the 2 helicity-2 states Sec.VI,westudythepredictionsforstructureformation, ofEinsteingravity, 2helicity-1states, and1helicity-0or including the weak lensing power spectrum (Sec. VIA), longitudinal mode, usually denoted by π. the Integrated Sachs-Wolfe cross-correlation (Sec. VIB), On top of the massive spin-2 representation, higher- large scale bulk flows (Sec. VIC), the Lyman-α forest codimension extensions of DGP also have D −5 scalar (Sec. VID), and secondary CMB anisotropies from ther- fields that couple to Tµ. For simplicity, we will assume µ mal Sunayev-Zeldovich (SZ) effect (Sec. VIE). We con- thatthesescalarshavethesamepropagatorasthatofthe clude with a summary of our results in Sec. VII. massive graviton (1), although this is generally not the caseinconsistentrealizations[33,40]. Inotherwords,for ourpurposeswewillnotdistinguishbetweenthehelicity- 0 mode and the extra scalars, and will denote them col- II. PHENOMENOLOGY OF lectively as π. HIGHER-DIMENSIONAL GRAVITY Theseextrascalardegreesoffreedomcontributetothe trace part of the one-particle exchange amplitude [34]: We are interested in generalizing the phenomenology (cid:18) (cid:19) 1 1 oftheDGPmodel[21],arguablythesimplestexampleof A∼ TµνT(cid:48) − TT(cid:48) . (3) k2+r−2(1−α)k2α µν D−2 modifiedgravityintheinfrared. Inthisscenario,ourvisi- c bleuniverseisconfinedtoa3-braneina4+1-dimensional Because the resulting 1/(D −2) coefficient differs from bulk. Although the bulk has infinite volume, 4D gravity the usual 1/2 of General Relativity (GR), this linearized isrecoveredatshortdistances,duetoaninducedgravity amplitude is apparently inconsistent with solar system term on the brane. The gravitational force law therefore tests. And, remarkably, this departure from GR persists scales as the usual 1/r2 at short distances, but weakens even in the massless limit, r → ∞, a phenomenon fa- c to1/r3 atlargedistances,withthecross-overscalerc set mously known as the vDVZ discontinuity [16]. by the bulk and brane Planck masses: rc =M42/M53. As conjectured by Vainshtein in the context of mas- In momentum space, this corresponds to the graviton sive gravity [17], however, these theories can be phe- propagator 1/(k2 + r−1k). Following [38, 39], we will nomenologically viable because the linearized approxi- c consider the following power-law generalizations: mationforπ breaksdowninthevicinityofastrophysical sources [41, 42, 43, 44]. Below a macroscopic scale r , (cid:63) 1 non-linearinteractionsinπ becomeimportantandresult , (1) in its decoupling. Power counting arguments [38] give k2+r−2(1−α)k2α c (cid:16) (cid:17)1/(1+4(1−α)) r = r4(1−α)r , (4) with the DGP model described by α = 1/2. More gen- (cid:63) c Sch erally, α is a freely-specifiable parameter, modulo two wherer istheSchwarzschildradiusofthesource. And Sch constraints: Firstly, in order for the modification to be since r → ∞ as r → ∞, GR is recovered everywhere (cid:63) c relevant in the infrared, we must demand that α < 1. in this limit, as it should. Approximate explicit solu- Secondly, it is straightforward to show that this propa- tions [41, 45, 46] in DGP have confirmed the Vainshtein gator has a well-defined spectral representation, and is effect and the recovery of GR for r (cid:28)r . (cid:63) therefore ghost-free, only if α is positive definite [38]. The leading correction to GR near a source can be Thus we have the allowed range: inferredasfollows. Firstofall,thesolutionforπ(r)must beoforderr /r atr =r ,inordertomatchthelinear Sch (cid:63) (cid:63) 0≤α<1. (2) solution. Furthermore, within r we expect the solution (cid:63) to be analytic in r−2(1−α). These requirements together c fix the parametric dependence of π for r (cid:28)r [38, 39]: The above parametrization is not only phenomeno- (cid:63) l(oαgi=cal1l/y2)s,imapslwe,elbluatsiittsahlsioghmera-kceosdicmoenntasicotns gweintheraDliGzaP- π ∼ rSch (cid:18) r (cid:19)−21+2(1−α) . (5) r r tions[31,32,33,34]. SincethebulkhasD ≥6space-time (cid:63) (cid:63) dimensions in the latter class of theories, the force law Inotherwords,thecorrectiontotheNewtonianpotential scales as 1/rD−2 in the infrared, and the corresponding is of order propagator tends to a constant (D > 6) or behaves as logk (D = 6) as k → 0. Thus all higher-dimensional δΨ (cid:114) r (cid:18) r (cid:19)2(1−α) ∼ . (6) extensions of DGP correspond to α≈0 theories [34]. Ψ r r Sch c 3 An important constraint on such modification comes identical expansion history as ΛCDM cosmology. Of fromLunarLaserRangingexperiments[39,47]: δΨ/Ψ< course, higher-codimension DGP models will lead to 2.4×10−11 [48]. Substituting r = 0.886 cm for the small deviations from ΛCDM expansion history, as they Sch Earth and r = 3.84×1010 cm for the Earth-Moon dis- do not exactly correspond to α = 0. But α = 0 should tance gives offer a realistic approximation to their modified Fried- mann equation since we expect the correction to be a rc >∼4·102(19−α−α5)H0−1. (7) slowly-varying function of H/rc [56]. For small values of α, we can quantify the expansion This constraint on (α,rc) is shown in Fig. 1. Explicitly, history in terms of the usual dark energy parameters forstandardDGP(α=1/2)thisgivesrc >∼120Mpc. For w,wa,...: α=0, on the other hand, the bound is far weaker: rc >∼ 1 pc. Future lunar precession experiments will improve 3 (cid:18) H2α (cid:19) ρ ≡ Λ− , these bound by an order of magnitude [48]. Somewhat DE,eff 8πG r2(1−α) c weaker constraints on DGP have also been obtained by 1dlnρ studying the effect on planetary orbits [49], and it will w ≡ −1− DE,eff 3 dlna be very interesting to extend these studies to general α 2αΩ +O(α2) theories. = −1− m , (Hr )2(1−Ω ) c m dw 6αΩ2 +O(α2) III. BACKGROUND COSMOLOGY wa ≡ dlna = (Hr m)2(1−Ω ). (10) c m The standard DGP model has a modified Friedmann Note that since α is positive, the modified gravity cor- equation of the form [50] rectionleadstoaneffectivedarkenergycomponentwith w <−1. This is indeed the case in the DGP model [57]. 8πG H H2 = ρ∓ , (8) While such phantom behavior may at first seem surpris- 3 rc ing,itisinfactanaturalconsequenceofthescalar-tensor nature of the effective theory in DGP. Indeed, at the 4D where we have assumed spatial flatness. Much atten- effective level the Friedmann equation is cast in Jordan tion has been paid to the “plus” (or self-accelerating) frame,whichcanleadtoaneffectivew <−1[58,59,60]. branch of this equation, because of its asymptotically The current (mainly) geometrical constraint w = de Sitter solution even in the absence of vacuum en- −0.972 ± 0.060 (1σ error), from a combination of ergy. However, various arguments have by now estab- WMAP5, baryonic acoustic oscillations, and super- lished that this branch suffers from ghost-like instabili- novae [70], implies 1+w > −0.067 at 95% confidence ties [43, 44, 51, 52, 53, 54]. We shall henceforth focus level if we impose the prior that w ≤−1, which leads to: on the “minus” or healthy branch. Since the 1/Hr cor- c rections slows down the expansion rate in this case, we must include a cosmological constant to account for cos- r >3.2α1/2H−1(cid:2)1+O(α2)(cid:3) . (11) mic acceleration. c 0 A simple phenomenological extension of this equation In other words, unless α (cid:46) 0.1, r should be larger than suggests itself: H2 = 8πGρ/3−H2−γ/rγ. (The “plus” c c today’s Hubble radius. branch version of this equation was introduced in [55] Figure1comparesthesolarsystemconstraintonr — to study generalized self-accelerated solutions.) We can c see (7) — with the cosmological constraint, if we inter- fix γ in terms of α as follows. First, from (8) we see pret (11) as a constraint on the distance to the last scat- that γ =1 for standard DGP, corresponding to α=1/2. tering surface (assuming a spatially flat universe). We Furthermore, note from (1) that α = 1 can be absorbed see that for α(cid:46)0.5, the main constraint on the value of into a redefinition of G; if this is also the case for the r comes from the background cosmology. However, as Friedmann equation, then γ = 0 for α = 1. Assuming c we noted above, in the α → 0 limit (corresponding to 2 forsimplicityalineardependenceofγ onα,weconclude or more extra dimensions), the background cosmology is that γ =2(1−α), or indistinguishable from ΛCDM, and thus r remains un- c 8πG H2α constrained. We will see in the next section that cosmo- H2 = ρ+Λ− , (9) logical perturbations will be sensitive to r even in this 3 rc2(1−α) limit. c where Λ is a cosmological term. In the higher- dimensional framework, Λ should find a natural origin, for instance as a remnant vacuum energy component IV. COSMOLOGICAL PERTURBATIONS from the degravitation mechanism [39]. Remarkably, this parametrization implies that α = 0 Since the study of cosmological perturbations in ex- (corresponding to 2 or more extra dimensions) leads to plicit higher-codimension extensions of DGP has yet to 4 (cid:36) = −2g/(1 + g). A key difference is that these parameters were all assumed to depend on a only, whereas our g(a,k) will also be a function of scale. For future reference, it is convenient to introduce the lensing potential Φ−Ψ Φ ≡ . (14) − 2 Gravitational lensing observations, as well as the Inte- grated Sachs-Wolfe (ISW) effect, are both sensitive to Φ . Galaxy peculiar velocity measurements, on the − other hand, are determined by the Newtonian potential Ψ=(g−1)Φ . See [61] for a pedagogical description of − how various observables relate to different metric combi- nations. In the next few subsections, we will specify g(a,k) for our modified gravity theories, based on known results in standard DGP. An important lesson from studies of DGP is that g(a,k) has qualitatively different behaviors on sub- or super-horizon scales. Hence we will explore these two regimes separately. FIG.1: 95%lowerlimitsonr ,thescaleoftransitiontohigher c A. Sub-horizon evolution dimensional gravity, from background cosmology (see (11)) andlunarlaserranging(see(7)),asafunctionofα. Standard DGPcorrespondstoα=1/2,whereasitshigher-dimensional Focusing on sub-Hubble scales and non-relativistic extensions all have α≈0. sources in standard DGP, Lue et al. [22] obtained 1 1 g | =− · , (15) be worked out, for our analysis we must resort to a phe- DGP k(cid:29)aH 3 (cid:2)1+2Hrc(cid:0)1+ 3HH(cid:48)(cid:1)(cid:3) nomenologicalparametrizationofthesemodifications. In where (cid:48) ≡ d/dlna. Note the plus sign within the brack- this paper we take the parametrized post-Friedmann ets, which is appropriate for the normal branch [29]. (PPF) approach introduced recently [37]. This formal- Apart from this sign, this result was also obtained in ism is reviewed in Appendix A. cosmologicalperturbationtheoryusingaquasi-staticap- In Newtonian gauge, scalar metric perturbations are proximation [24]. specified by the gauge-invariant potentials Ψ and Φ: The dependence on Hr in (15) can be attributed to c the dynamics of π. Indeed, for the universe as a whole, ds2 =−(1+2Ψ)dt2+a2(1+2Φ)d(cid:126)x2. (12) both r and r can be approximated by H−1. Setting Sch α = 1/2 in this case, (6) gives the leading correction to In the absence of anisotropic stress, the momentum con- standard cosmology, δΨ/Ψ| ∼ 1/Hr , in agreement straint in GR fixes Φ = −Ψ. However, this is no longer DGP c with (15) in the strong coupling regime Hr (cid:29) 1. Re- true in DGP because of the helicity-0 mode. At the 4D c peating the argument for general α, (6) gives effective level, this can be understood tracing back to thescalar-tensornatureofthetheory,asitiswell-known δΨ 1 ∼ . (16) that Φ(cid:54)=Ψ in Jordan frame. This deviation from GR is Ψ (Hr )2(1−α) c parametrized in PPF through Meanwhile, in the weak coupling regime, from the ex- Φ+Ψ change amplitude (3) we can read off δΨ/Ψ = −(D − g(a,k)≡ . (13) Φ−Ψ 4)/(D − 2), consistent with the numerical coefficient of (15) for D = 5. Taking into account this numerical This function can be specified independently from the coefficient, this suggests generalizing (15) to expansion history. (cid:18) (cid:19) D−4 1 At this point we should mention closely related g| =− . parametrizations of modified gravity that have appeared k(cid:29)aH D−2 1+2(Hr )2(1−α)(cid:0)1+ H(cid:48)(cid:1) c 3H in the literature. Jain and Zhang [61] parametrized the (17) deviation from Φ = −Ψ through η = 1 + g/(1 − g). Asimilarparametrizationofg forgeneralα theorieswas This convention was also adopted by Bertschinger and also studied in [64], albeit for self-accelerated cosmolo- Zukin [62], except that the parameter was denoted by gies. Our parametrization is also related to the slip pa- γ. Daniel et al. [63], on the other hand, introduced rameter(cid:36) introducedin[63]. Theprecisetranslation,as 5 mentioned earlier, is (cid:36) = −2g/(1+g). In the matter- resulting from the modified growth history can cancel dominated era, these authors assumed (cid:36) ∼1/H2, which the primordial Sachs-Wolfe contribution, resulting in a coincides with the scaling of g in the case α=0. power deficit on large angular scales. This novel mechanism can therefore naturally account for the observed low quadrupole as well as the lack of angular correlations B. Super-horizon evolution and decay of above ∼60◦ [11, 12, 13, 14, 15]. See Fig. 3. Newtonian potential The actual value of A in explicit higher-codimension modificationsofgravityremainsatpresentunknownand, In standard DGP, the evolution of super-horizon fluc- in any case, is likely to be model-dependent. Thus we tuations was studied in [25] using a scaling ansatz for view the agreement with large-angular CMB anomalies bulk perturbations. On the self-accelerating branch, the for A=1 as a strong observational guide towards build- solution is well-fitted by [37] ing successful models of infrared modified gravity. The onus is then on model-builders to find explicit brane- (cid:18) (cid:19) 9 0.51 world constructions where A=1 is realized. For present g | = 1+ . (18) DGP k(cid:28)aH 8Hr −1 Hr −1.08 purposes, we shall assume this can be achieved and ex- c c plore the consequences for other observables. Let us The above parametric dependence and the form of our therefore take sub-Hubble parametrization (17) suggest the following ansatz for general α, 1 g| = . (20) k(cid:28)aH 1+2(Hr )2(1−α)(cid:0)1+ H(cid:48)(cid:1) A c 3H g| = . (19) k(cid:28)aH 1+2(Hrc)2(1−α)(cid:0)1+ 3HH(cid:48)(cid:1) Note that the transition g → 1 has a natural physical interpretation: from the definition of g in (13), this cor- The normalization constant A will be fixed shortly. respondstoadecayoftheNewtonianpotentialΨrelative Using a simple interpolation between the self- to the lensing potential Φ . − acceleratedanalogueoftheshort-wavelengthg given DGP At this point the astute reader might worry about the in (15) and its long-wavelength counterpart (18), Wang signofg| . Onsub-Hubblescales,apositiveg would k(cid:28)aH et al. [30] recently showed that the self-accelerated DGP beinterpretedasπmediatingarepulsiveforceandthere- modelisdisfavoredatthe∼5σlevelcomparedtoΛCDM. fore representing a ghost. Fortunately, such reasoning TheiranalysisincludesCMBtemperatureanisotropy,su- doesnotapplyonsuper-horizonscales,anditisstraight- pernovae data, and constraints on the Hubble constant. forwardtodevisetoyscalar-tensortheoriesin3+1dimen- Roughly70%ofthetensioncomesfromthemodification sionsthatleadtoapositivegonsuper-Hubblescales. For in the Friedmann equation, while the remaining 30% is instance, as shown in Appendix B, all Brans-Dicke the- due to the difference in growth history, resulting in too ories satisfy this property during the matter-dominated large an ISW contribution to the low-(cid:96) multipoles of the era. CMB. Similarly, we find that an unacceptably large ISW component is a generic outcome for general α theories, unless r is much larger than the present Hubble ra- C. Summary of fiducial model c dius. But there is one important and very interesting exception: by choosing A ≈ 1 and making a few mi- Thefull g(a,k)must interpolate betweenthesub-and nor assumptions on the remaining PPF parameters to super-Hubble behavior given in (17) and (20), respec- be discussed shortly, the Integrated Sachs-Wolfe effect tively. Forthispurpose,wechoosetheinterpolationform 1 (cid:34) 1 (cid:18)D−4(cid:19) (cid:0)c k (cid:1)b (cid:35) g(a,k)= − gaH , (21) 1+2(Hrc)2(1−α)(cid:0)1+ 3HH(cid:48)(cid:1) 1+(cid:0)cgakH(cid:1)a D−2 1+(cid:0)cgakH(cid:1)b wherea,bandc areconstants. Whileourresultsarenot g overly sensitive to the precise values for these constant a=b=0.5; c =0.5. (22) parameters, in practice we have found that smoother in- g terpolationsofferabetterfittothelowmultipolesofthe Although the interpolation is smooth, one can think of CMB power spectrum. Hence, as fiducial values we take c as setting the scale below which g becomes negative. g Note that for the above choice of c , this scale is compa- g 6 rabletothehorizon. Largervaluesofa,bandc diminish gk,a g 1.0 the effectiveness of the cancelation mechanism. AnotherkeyinputparameterinPPFiscΓ, whichper- 0.8(cid:72) (cid:76) tains to the evolution of the curvature perturbation ζ 0.6 (see (A2) for its definition). Independent of the theory of gravity under consideration, energy-momentum con- 0.4 servationaloneimposesthatζ isconservedintheinfinite- 0.2 wavelength limit [65]. Using this fact, Bertschinger [66] derived the following consistency relation 0.0 ζ =const.= H (cid:2)(g−1)Φ −g(cid:48)Φ −(g+1)Φ(cid:48) (cid:3) (cid:45)0.2 H(cid:48) − − − +(g+1)Φ . (23) (cid:45)0.4 k − 10(cid:45)4 0.01 1 100 104aH Ingeneral,thisconditionmustholdwelloutsidethehori- zon. Aswewillseeshortly,however,hereweimposethat FIG. 2: Dependence of g(k,a) (see (21)) in our model on ζ isconservedonallscalesinourfiducialmodifiedgravity k/aH in the matter-dominated era for r = 0.3,1,3, and 10 c model. Hubble radius. The number of space-time dimensions, D, is While ζ is actually conserved on all scales in ΛCDM assumedtobe6(i.e. 2extradimensions). GeneralRelativity cosmology, this is not necessarily the case for modified is recovered when rc (cid:29)H−1 as, g→0. gravity theories. In the PPF approach, one specifies a parameter c which, as described in Appendix A, de- Γ The remaining parameters to vary are of course α, D termines an effective horizon c /aH above which ζ is Γ and r . Since we are interested in higher dimensional conserved. Our mechanism for canceling the large-scale c generalizations of DGP, we shall fix CMB power requires that conservation of ζ persist on a sufficientlywiderangeofsub-Hubblemodes,correspond- α=0, (25) ingtosmallvaluesofc . Inpractice,wehavefoundthat Γ cΓ <∼ 10−2 is desirable. For simplicity, however, as as discussed below (2). Since the expansion history re- fiducial value we shall set duces to ΛCDM in this case, we are free to consider val- uesofr thatarerelativelysmallcomparedtotheHubble c c =0, (24) Γ radius. We will find that our modified gravity theories thereby enforcing ζ conservation on all scales. The evo- can explain the anomalies listed in the Introduction and lution of super-horizon modes is therefore uniquely de- match other observations with rc ∼ 300−600 Mpc. Fi- termined through (23) by specifying g and the expan- nally, the total number of space-time dimensions D sets sionhistory. Notethattheassumptionofconstantζ was the strength of the extra scalar force at short distances, also made by Bertschinger and Zukin [62] in what they as seen in (17), with larger D corresponding to stronger dubbed scale-independent modified gravity models. (A gravity. To explain the CBI excess and the anomalous key difference, however, is that their g was also indepen- Lyman-α power, it will suffice to consider two extra di- dentofscale.) InAppendixA,weshalldescribethePPF mensions, or formalism in more generality, allowing for non-zero c . Γ D =6. (26) In light of the relatively large number of input functions and parameters inherent to the PPF framework, This completes the description of our fiducial model. let us synthesize the essential qualitative and quantita- tive features of our fiducial parametrization: V. CMB TEMPERATURE ANISOTROPY AND • The difference in metric potentials g(a,k) is cho- LARGE-SCALE ANOMALIES sen to interpolate between the sub-horizon behavior, dictated by well-understood features of mas- Letusstepbackandexplainwhythechoiceforg| sive/resonance gravity outlined in Sec. IIA, and k(cid:28)aH in (20) can explain the low quadrupole and the lack of themoremodel-dependentsuper-horizonbehavior. Forthelatter,thelarge-scaleCMBanisotropycon- correlation on >∼ 60◦ scales. On large angles, the CMB anisotropyiswellapproximatedbythesumoftheSachs- strains us to choose A = 1, so that g| makes k(cid:28)aH Wolfe and the Integrated Sachs-Wolfe (ISW) effects [67]: atransitionfrom0atearlytimesto1atlatetimes — see Fig. 2. Physically, this corresponds to a decay of the Newtonian potential Ψ relative to Φ. δTCMB = 1Φ +2(cid:90) dt∂Φ− (cid:39) 1Φ +2∆Φ . (27) Smoother interpolations are preferable. T 3 − ∂t 3 − − CMB • The curvature perturbation ζ must be nearly con- The integral in the ISW effect is taken along the light served,whichrequiressmallvaluesforc . Forsim- cone,whilethepartialderivativeisatfixedcomovingspa- Γ plicity, we set c =0 in our fiducial model. tial position. Therefore, the last approximation in (27) Γ 7 is only valid when the spatial variations of Φ are much − isnmtearlelesrtitnhgantoitnsottiemtehvaatroiantitohne:se|∇(cid:126)scΦa−le|s(cid:28), ∆|ΦΦ˙−−|.caIntibsethoebn- 53000000 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) sttahuiapnteerdt-hhaeonrcaizuloyrntviacstaculalryleesaps—eartfsuuerenbca(tt2iio3on)n.oζIfngrep(ama,raktii)nc,usflracoorm,nsiftthagnetgfoaoecnst 2KΠΜ 21050000(cid:72)(cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) from 0 to a finite (constant) value g during the matter 2l 1000(cid:144) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) f (cid:99) (cid:230) (cid:230) era, from (23) we see that: 1 (cid:76) (cid:230) (cid:230) (cid:43) (cid:123) (cid:18) 5 (cid:19) (cid:18) g (cid:19) (cid:123) (cid:72) (cid:76)CDM Φ → Φ ⇒∆Φ =− f Φ . (28) − 5+g − − 5+g − f f (cid:230) Forexample, forthetransitionfrom0tog =1wehave: f 5 10 50 100 500 1000 (cid:123) (cid:20) (cid:21) δT 1 5(1−g ) CMB (cid:39) Φ +2∆Φ = f Φ =0! (29) TCMB 3 − − 3(5+gf) − 5000 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) In other words, if g → 1 at late times in the modified 3000 (cid:230)(cid:230) (cid:230) gcscaraanlvceiestlyaottnhtlehaoerrgyte,imathneegolIefSstW—heactnordarrneSssaiptciohonsn-dWi—nog,lftweohesiffucehpcetcrso-hueoxldraiczptoolny- 22KΠΜl 121005000000(cid:144)(cid:72)(cid:76) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) tentially explain the vanishing of CMB power on scales (cid:99) (cid:230) (cid:230) (cid:38) 60◦. Note that g → 1 is equivalent to a vanish- 1(cid:43) (cid:76) (cid:230) (cid:230) (cid:123) ingNewtonianpotentialrelativetothelensingpotential, (cid:123) (cid:72) rc(cid:61)600Mpc Ψ (cid:28) Φ , at late times. A partial cancelation between − the primordial Sachs-Wolfe and a late-time ISW contri- (cid:230) bution arising from g was also noticed in [62, 63]. 5 10 50 100 500 1000 This phenomenological observation, in addition to the (cid:123) fact that Brans-Dicke theories have g > 0 on super- horizonscales—seeAppendixB—,justifiestheinfrared limFiitgoufreou3rdaenmsaotnzstforartges(ah,okw) gtihviesnciannc(2el1a)t.ion can work 5000 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) 3000 (cid:230) (cid:230) smaptoeldcotiwfiraed(cid:96)w’sbeyrfoeWrg.deniFffeaernragetnettdoviunaslciunluegsdeaofPvPrecrF.sipoTnarhaoemfCeCtMAerBMs [Bp69o][w.6e8r] 2KΠΜ 21050000(cid:72)(cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) The apparent lack of power on large angles was first (cid:99)2l 1000(cid:144) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) quantified by Spergel et al. 2003 in the first data release 1(cid:43) (cid:76) (cid:230) (cid:230) (cid:123) of WMAP [71], through the quantity: (cid:123) (cid:72) rc(cid:61)300Mpc (cid:90) 1/2 S ≡ [C(θ)]2dcosθ, (30) (cid:230) 1/2 −1 5 10 50 100 500 1000 whereC(θ)istheangularcorrelationfunctionoftheCMB (cid:123) temperatureanisotropiesshowninFig.4. Mostrecently, Copiet al.[13, 15]arguedthattheprobabilityofS to 1/2 FIG. 3: CMB angular power spectra for best-fit ΛCDM be lower than it is observed outside the Galactic plane is (blue, solid curve), r = 600 Mpc (green, dashed curve) and c only0.025%intheconcordanceΛCDMmodel. However, r = 300 Mpc (red, short-dashed curve). The vertical bars c different estimators for the correlation function can lead show the cosmic variance spread centered on the respective to larger values of S . Figure 5 compares this proba- theoreticalpredictions. Inthemiddleandbottompanels,the 1/2 bility for our modified gravity models with ΛCDM, and previous curves are shown for comparison. The data points shows that it is increased by about a factor of 2-3 in are from the WMAP-5 data release [70]. the relevant range. While this is not enough to explain the low values of S in direct measurements of corre- 1/2 lation function [e.g., the two smaller values in Fig. 5], anisotropy,theχ2 marginallyincreasesby1.21(0.32)for it can bring the prediction within 90% probability range r = 300 (600) Mpc, with other cosmological parameters c for maximum likelihood estimates of the WMAP5 power fixed. The origin of this increase can be seen in Fig. 6, spectrum [the larger value in Fig. 5]. where the suppression of power at small (cid:96)’s decreases Despite the improvement of the consistency be- the TE temperature-polarization cross-power spectrum tween models and data at large angles for temperature below the observed value. However, given that the low- 8 (cid:76)CDM 1.5 rc(cid:61)600Mpc rc(cid:61)300Mpc 2K 1.0 (cid:76) Μ (cid:72) 2Π (cid:230) E (cid:144) T Cl 0.5 1(cid:43) (cid:76) (cid:230) l (cid:72) (cid:230) (cid:230)(cid:230)(cid:230) 0.0 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230) 5 10 50 100 500 1000 l 10 (cid:76)CDM 2KΜ (cid:76) rrcc(cid:61)(cid:61)630000MMppcc Π 1(cid:72) 2 E (cid:144) E Cl FIG. 4: Square of the correlation function of CMB tempera- 1(cid:43) 0.1(cid:76) ture anisotropy, as a function of the cosine of the separation ll (cid:72) angle in the sky. The curves show ΛCDM (solid), r = 300 c Mpc (dotted) and rc = 600 Mpc (short-dashed). The long- 0.01 dashedcurveistheLegendretransformoftheWMAP5max- imum likelihood power spectrum [72]. The observed correla- 5 10 50 100 500 1000 tionissystematicallybelowtheΛCDMpredictionforθ(cid:38)60◦, l or cosθ<0.5. FIG. 6: Temperature-Polarization (TE) and polarization (EE) power spectra for ΛCDM (blue, solid curve), r = 300 c Mpc (red, short-dashed curve) and r = 600 Mpc (green, c dashed curve). The points are from the WMAP5 data release [72]. est (cid:96)-bin in the TE spectrum decreased by a factor of 3 from the 1st year to the 5-year WMAP data release due to an improved foreground cleaning, it is still con- ceivable that the systematics in this bin are underesti- mated. Therefore, we predict a significantly lower TE cross-power spectrum at (cid:96) < 10, which should be clearly distinguishedfromΛCDMbythePlancksatellite,dueto itsbetterpolarizationsensitivityandforegroundcleaning capabilities [73]. Meanwhile,asshowninthebottompanelofFig.6,the EE spectrum is not appreciably affected by our modification of gravity. FIG. 5: The cumulative probability for S1/2, defined in (30), Weshouldalso pointoutthat itis possible toimprove measuring the CMB correlation on scales >60◦. The curves the full χ2 (relative to ΛCDM) without extra marginal- are for ΛCDM (solid), r = 300 Mpc (dotted) and r = 600 c c ization, simply by decreasing b in (21). However, this Mpc(dashed). Theverticallinesshowtheobservedvaluesof will spoil our prediction of extra growth for small-scale S from different estimators (see [15] for details). 1/2 structures, which we shall discuss next. 9 ln a should, (31) reduces to the standard evolution equation in the GR limit r →∞. Supp. Growth TODAY At late times (Hc <∼ rc−1), when the transition is com- plete, g(k,a) only varies slowly as the mode wavelength Hrc = 1 relative to the horizon shrinks with time — see Fig. 2. aH/cg GR In this regime we can ignore Φ(cid:48) and g(cid:48) in the ζ equa- − aH tion (23), which leads to (28) in the matter era. With this in mind, we can view (31) as a forced damped har- monic oscillator, where the force is enhanced by a factor of 5(1−g)/(5+g) compared to ΛCDM. Therefore, prior ln k to the domination of Λ the inhomogeneous solution becomes FIG. 7: Different growth regimes as function of scale factor and comoving wavenumber k. Colored regions correspond 5 5(1−g) to modes inside the horizon today. The solid line labeled Φ ≈ ΦCDM; ∆ ≈ ∆CDM. (32) − 5+g − m 5+g m by aH denotes the Hubble horizon; the dotted line traces the characteristic scale set by c . At early times, H > r−1, growth proceeds as in GR (Gregen region); for H < r−1 acnd On very small scales, where g →−(D−4)/(D−2), this c gives onscalesmuchsmallerthanthec -scale,growthisenhanced g thanks to the helicity-0 mode (Blue region); for H < r−1 c 5(D−2) 5(D−3) and on large scales, the decay of the Newtonian potential Ψ Φ → ΦCDM ; ∆ → ∆CDM. suppresses growth (Red region). − 2(2D−3) − m 2D−3 m (33) In particular, for our fiducial case of D =6, VI. STRUCTURE FORMATION WITH 10 5 MASSIVE GRAVITY Φ → ΦCDM ; ∆ → ∆CDM. (34) − 9 − m 3 m The growth of perturbations in our model can be un- Meanwhile, on large scales we have g → 1, and thus derstood analytically by making a few simplifying ap- ∆ → 0 while Φ decreases by a factor of 5/6, as we m − proximations. The essential physics is illustrated in discussed in the previous section. Fig. 7. The solid line labeled by aH denotes the Hubble Figure 8 demonstrates this behavior for Φ and ∆ − m horizon. Meanwhile,thedottedlinedenotesthescaleset with D = 6 over the range k = 10−4−10 Mpc−1 cover- by cg where g makes a transition from positive values on ing both super-horizon and sub-horizon modes. (What larger scales to negative values on smaller scales. is plotted in the right panel can be thought of as the At early times, H (cid:29) r−1, we have a GR regime on transfer function for an effective potential related to ∆ c m all scales, as g ≈ 0. For modes much smaller than the through the standard Poisson equation.) As expected, cg-scale, once H drops below rc−1 the scalar force medi- prior to the domination of Λ, large scale (super-horizon) ated by the longitudinal mode kicks in, enhancing the modes decay, while small-scale sub-horizon modes grow growth of perturbations. This is the blue-shaded region with time. inFig.(7). Thusthereisexcessofpoweronsmallscales, Once Λ dominates, however, all modes start to decay. compared to ΛCDM. Indeed, since H(cid:48) →0 in this regime, from (23) we find: For modes with wavelength above the c -scale, the g (cid:18) (cid:19) fcuonrrcetsiopnongdimngakteosaadtercaanysiotifotnhefroNmew0totnoia1naptoHten∼tiarlc−Ψ1,. Φ(cid:48)−+ 11+−gg Φ− =0 =⇒ Φ− ∝a−11−+gg . (35) This is the red-shaded region in Fig. 7. This suppression of growth results in a power deficit in the spectrum of Thus, while Φ decays as a−1 at late times in ΛCDM − density perturbations compared to ΛCDM. cosmology, here it decays even faster, as a−D+3, since Before numerically solving for the evolution of pertur- g = −(D−4)/(D−2) on small scales in our model. In bations, we can gain analytic insights by studying dif- particular, for D = 6, we have Φ− ∼ a−3 at late times. ferent regimes. Although our perturbed metric is cast This behavior can be seen in Fig. 8, where Φ− initially in Newtonian gauge, we find it more convenient to track grows by up to ∼ 10% as the graviton Compton wave- comoving gauge density perturbation, ∆ . Its evolution length enters the horizon, in agreement with (34), but m equation follows as usual by combining energy and mo- then decays faster than in the ΛCDM model after the mentum conservation [see (A1)] onset of cosmic acceleration. The competition between these two effects is why, as we will see below, gravita- (cid:18) H(cid:48)(cid:19) k2 tional weak lensing is not substantially affected by the ∆(cid:48)(cid:48) + 2+ ∆(cid:48) =(1−g) Φ , (31) m H m a2H2 − graviton Compton wavelength in our model at low red- shifts. wherewehaveusedthefactthatthecurvatureperturbar- Inthefollowingsections,wewilldiscussthepredictions ionisassumedtobeconservedonallscales,ζ(cid:48) =0. Asit ofourmodelfordifferentmeasuresofstructureformation 10