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Department of Physics Papers Department of Physics
7-27-2009
Galileon Cosmology
Nathan Chow University of Pennsylvania
Justin Khoury University of Pennsylvania, [email protected]
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Recommended Citation Chow, N., & Khoury, J. (2009). Galileon Cosmology. Retrieved from https://repository.upenn.edu/ physics_papers/71
Suggested Citation: Chow, N. and J. Khoury. (2009). "Galileon Cosmology." Physical Review D. 80, 024037.
© 2009 The American Physical Society http://dx.doi.org/10.1103/PhysRevD.80.024037
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/71 For more information, please contact [email protected]. Galileon Cosmology
Abstract We study the cosmology of a galileon scalar-tensor theory, obtained by covariantizing the decoupling Lagrangian of the Dvali-Gabadadze-Poratti (DGP) model. Despite being local in 3 + 1 dimensions, the resulting cosmological evolution is remarkably similar to that of the full 4 + 1-dimensional DGP framework, both for the expansion history and the evolution of density perturbations. As in the DGP model, the covariant galileon theory yields two branches of solutions, depending on the sign of the galileon velocity. Perturbations are stable on one branch and ghostlike on the other. An interesting effect uncovered in our analysis is a cosmological version of the Vainshtein screening mechanism: at early times, the galileon dynamics are dominated by self-interaction terms, resulting in its energy density being suppressed compared to matter or radiation; once the matter density has redshifted sufficiently, the galileon becomes an important component of the energy density and contributes to dark energy. We estimate conservatively that the resulting expansion history is consistent with the observed late-time cosmology, provided that the scale of modification satisfies rc≳ 15 Gpc.
Disciplines Physical Sciences and Mathematics | Physics
Comments Suggested Citation: Chow, N. and J. Khoury. (2009). "Galileon Cosmology." Physical Review D. 80, 024037.
© 2009 The American Physical Society http://dx.doi.org/10.1103/PhysRevD.80.024037
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/71 PHYSICAL REVIEW D 80, 024037 (2009) Galileon cosmology
Nathan Chow and Justin Khoury Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Received 22 May 2009; published 27 July 2009) We study the cosmology of a galileon scalar-tensor theory, obtained by covariantizing the decoupling Lagrangian of the Dvali-Gabadadze-Poratti (DGP) model. Despite being local in 3 þ 1 dimensions, the resulting cosmological evolution is remarkably similar to that of the full 4 þ 1-dimensional DGP framework, both for the expansion history and the evolution of density perturbations. As in the DGP model, the covariant galileon theory yields two branches of solutions, depending on the sign of the galileon velocity. Perturbations are stable on one branch and ghostlike on the other. An interesting effect uncovered in our analysis is a cosmological version of the Vainshtein screening mechanism: at early times, the galileon dynamics are dominated by self-interaction terms, resulting in its energy density being suppressed compared to matter or radiation; once the matter density has redshifted sufficiently, the galileon becomes an important component of the energy density and contributes to dark energy. We estimate conservatively that the resulting expansion history is consistent with the observed late-time cosmology, provided that the scale of modification satisfies rc * 15 Gpc.
DOI: 10.1103/PhysRevD.80.024037 PACS numbers: 04.50. h
I. INTRODUCTION mass is large in regions of high density, thereby suppress- ing any long-range interactions. Theories of fðRÞ gravity Scalar-tensor theories of gravity have experienced a [6] rely on the chameleon effect to ensure consistency with resurgence of sorts, over the last 20 years. This is due in solar system tests [7]. part to string theory, where the plethora of compactification A second mechanism is the Vainshtein screening effect moduli generically appear in the 4D effective theory with of the longitudinal graviton or brane-bending mode, usu- kinetic mixing with the graviton. Moreover, the discovery ally denoted by , in the Dvali-Gabadadze-Poratti (DGP) of accelerated expansion makes the possibility that general model [8]. As we review in Sec. II, this effect is most easily relativity is modified on the largest scales plausible. If this understood in a certain decoupling limit of the theory is the case, then the new gravitational degrees of freedom !1 [9,10]: MPl, M5 , keeping the strong coupling scale relevant on cosmological scales are likely to include a ð 2Þ1=3 scalar cousin for the graviton. MPlrc fixed. The resulting theory is local on the brane, and describes a self-interacting scalar field coupled The best-known example of a scalar-tensor theory is due þ to Brans and Dicke (BD) [1], to weak-field gravity in 3 1 dimensions: M2 Z pffiffiffiffiffiffiffi ! S ¼ Pl d4x g R BD ð@ Þ2 BD M2 2 L ¼ Pl ðE Þ þ 2 ðE Þ Z h h M h pffiffiffiffiffiffiffi 4 Pl þ 4 L ½ d x g matter g ; (1.1) 2 rc 2 1 ð@ Þ h þ h T ; (1.2) where the matter Lagrangian is independent of . MPl 2 Unfortunately, the BD parameter is so tightly constrained * 4 by solar system and pulsar observations, !BD 4 10 E ¼ h þ [2], that the cosmological effects of the BD scalar are where h h =2 ... is the linearized rendered uninterestingly small. Einstein tensor. As a vestige of 5D Lorentz transforma- A tantalizing alternative is that the apparent decoupling tions, the action is invariant under the Galilean shift of the scalar field is a local effect, owing to the large matter symmetry, @ ! @ þ c . Thus has been dubbed a 2 2 density of the solar system or pulsar environment. In other galileon field [11]. In regions of high density, MPlrc , words, the BD parameter is effectively a growing function nonlinearities in dominate and result in its decoupling. of the density. While decoupled locally, the scalar field can This is qualitatively similar to the chameleon mechanism, have interesting cosmological effects in the much sparser except that the galileon relies on derivative interactions as cosmic environment. There are only two robust mecha- opposed to a scalar potential. In this paper, we study the nisms that realize this idea. One is the chameleon mecha- cosmology of the galileon, by promoting (1.2) to a fully nism [3–5]: by adding a suitable potential Vð Þ, the scalar covariant, nonlinear theory of gravity coupled to a galileon field acquires mass which depends on the density. The field:
1550-7998=2009=80(2)=024037(13) 024037-1 Ó 2009 The American Physical Society NATHAN CHOW AND JUSTIN KHOURY PHYSICAL REVIEW D 80, 024037 (2009) Z 2 2 pffiffiffiffiffiffiffi M rc tion fluid. When the matter density has dropped suffi- S ¼ d4x g Pl e 2 =MPl R ð@ Þ2h 2 M ciently, so that Hrc 1, the galileon becomes an Pl important component of the total energy density and con- þ L ½ matter g : (1.3) tributes to dark energy. We also study the effects of the galileon on the growth of As in the DGP model, where the Galilean shift symmetry is inhomogeneities. By virtue of its nonminimal coupling to only exact in the strict decoupling limit M !1, the shift gravity, the galileon enhances the gravitational attraction Pl between particles, which translates into more efficient symmetry is now broken by MPl-suppressed operators in (1.3). Of course the above nonlinear completion is by no growth of density perturbations. The screening mechanism is also at play in the evolution of perturbations: the galileon means unique—many other Lagrangians, for instance in- ð@ Þ4=M4 enhancement is suppressed for Hrc 1, but becomes cluding a Pl term, will reduce to (1.2) in the weak- field limit. Since the cosmological predictions should be important once Hrc 1. A similar timelike Vainshtein fairly robust under such corrections, however, we take (1.3) effect was also observed in [14]. as a fiducial galileon theory and study its implications for While a full likelihood comparison to data is left for cosmology. future study, we discuss various constraints on the galileon Despite being a local theory in 3 þ 1 dimensions, the cosmology, such as from estimates of the matter density at cosmology derived from (1.3) comes remarkably close to different redshifts, the luminosity-distance relation, and reproducing that of the 4 þ 1-dimensional DGP model, at the angular-diameter distance to the last scattering surface. The resulting bound on rc is least for Hrc * 1. As we will see, the agreement holds for both the expansion history and the evolution of density perturbations. More generally, our galileon cosmology rc * 15 Gpc; (1.5) reproduces many qualitative features of DGP: (1) The Friedmann equation allows for two branches of solutions, depending on the sign of _ . In analogy, which constrains the scale of the modification to be at least the modified Friedmann equation in DGP [12] also a few times the Hubble radius today. However, since all has two branches: other cosmological parameters are kept fixed in our con- siderations, (1.5) is likely a conservative estimate. H The paper is organized as follows. In Sec. II we review H2 ¼ : (1.4) 2 how the weak-field action (1.2) arises from the decoupling 3MPl rc limit of the DGP model, and describe the origin of the self- screening mechanism near spherical sources. In Sec. III we (2) One branch of solutions has stable perturbations, discuss the nonlinear extension (1.3) and derive the cova- whereas the other is plagued with ghostlike insta- riant equations of motion. We present in Sec. IV the bilities. This again agrees with DGP [10], where the cosmology of the galileon model. In particular, we derive ‘‘ ’’ and ‘‘þ’’ branches in (1.4) are stable and an approximate analytic solution, which displays the unstable, respectively. screening mechanism, and study its stability. Our analytic (3) The effective equation of state for the galileon sat- considerations are borne out by the numerical solutions isfies w < 1 on the stable branch, and w > 1 presented in Sec. V. Turning to inhomogeneities, we study on the unstable branch. This agrees with the effec- in Sec. VI the effects of the galileon on the growth of tive equation of state inferred from the H=rc cor- density perturbations. In Sec. VII we discuss various ob- rection in (1.4)[13]. servational constraints on galileon cosmology and derive (4) Moreover, the two branches are classically discon- the bound on rc given in (1.5). We conclude in Sec. VIII nected, unless R<0. This is closely related to the with a brief summary and discuss future research avenues. 2 2 condition 3P 12MPl=rc necessary to tran- While preparing this manuscript we became aware that sition from the stable to the unstable branch of Deffayet et al. were independently studying a model with solutions in the decoupling theory [10]. some similarities to ours [15]. This paper is based on the M.Sc. thesis of N. C. at the University of Waterloo [16]. An important difference with DGP, however, is that our covariant galileon theory does not allow for self- II. DECOUPLING LIMIT OF DGP accelerated cosmology—the self-accelerated solution is spoiled by 1=MPl terms in (1.3). In the DGP model, our visible universe is confined to a One of our key results is a cosmological analogue of the 3-brane in a 4 þ 1-dimensional bulk. Despite the fact that Vainshtein screening mechanism. At early times, Hrc the extra dimension is infinite in extent, 4D gravity is 1, the dynamics of are dominated by the cubic interac- nevertheless recovered over some range of scales on the tion term, resulting in the galileon energy density being brane because of a 3 þ 1-dimensional Einstein-Hilbert suppressed by Oð1=HrcÞ compared to the matter or radia- term, intrinsic to the brane:
024037-2 GALILEON COSMOLOGY PHYSICAL REVIEW D 80, 024037 (2009) Z pffiffiffiffiffiffiffiffiffiffi M3 @ ! @ þ c ; (2.6) S ¼ d5x g 5 R DGP 5 2 5 bulk which is a vestige of the full 5D Lorentz transformations. Z pffiffiffiffiffiffiffiffiffiffi M2 þ 4 Pl þ L ½ Thus has been dubbed a galileon field [11]. d x g4 R4 matter g : (2.1) brane 2 The bulk and brane Planck masses define a crossover scale, A. Self-screening effect The approximate recovery of general relativity in the M2 ¼ Pl vicinity of astrophysical sources, through the Vainshtein rc 3 ; (2.2) 2M5 effect, can be understood at the level of (2.5)[10]. The equation of motion for the galileon, which separates the 4D and 5D regimes. At distances r 2 ð þ 2 h 2 ð Þ2Þ¼ rc on the brane, the gravitational force law scales as 1=r , @ 6MPl@ 2rc@ rc@ @ T; (2.7) whereas for r r it scales as 1=r3. c is remarkable in many respects. Even though the interac- From the point of view of a brane observer, the 5 tion term in (2.5) contains four derivatives, the equation of helicity-2 states of the massless 5D graviton combine to motion is nevertheless second-order—all higher-derivative form a massive spin-2 representation in 4D. More pre- terms cancel out when performing the variation. Moreover, cisely, the 4D graviton is a resonance—a continuum of ¼ (2.7) takes the simple form @ j T=2MPl for some massive states—whose spectral width is peaked at the scale 1 -current j , thereby allowing for a generalized ‘‘Gauss’ rc . As in massive gravity [17], the helicity-0 or longitu- dinal mode, denoted by , becomes strongly coupled at a law’’: spherically-symmetric exterior solutions for only much lower scale than M , given by [9,18] depend on the mass enclosed. 5 Let us indeed study the spherically-symmetric galileon ¼ð 2Þ1=3 ¼ 3ð Þ strong MPlrc : (2.3) profile due to a point mass: T M r . In this case, (2.7) can be integrated to give 1 ¼ 28 1 For rc H 10 cm, for instance, this gives strong 0 02ðrÞ M2 r 1000 km. The strong coupling behavior is essential to the 0ð Þþ 2 ¼ Pl Sch 6MPl r 4rc : (2.8) phenomenological viability of the model through the r r2 Vainshtein screening effect [17,18]. As we review below, Using (2.4), the solution to this algebraic equation for 0 is nonlinear interactions in are important near an astro- given by physical source and result in the decoupling of from the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi source. The characteristic scale below which is strongly 0ðrÞ 3r 4 r3 ¼ 1 þ 1 þ ? : (2.9) coupled, denoted by r?, is given by 2 3 MPl 4rc 9 r ¼ð 2 Þ1=3 r? rcrSch ; (2.4) Note that we have chosen the branch of the solution such that 0 ! 0 as r !1. The other branch, corresponding to where r is the Schwarzschild radius of the source. And 0 Sch diverging at infinity, belongs to the same branch of since r is cosmologically large (of order of the Hubble c solutions as the self-accelerated DGP cosmology, and is radius today), r is parametrically larger than r . ? Sch therefore unstable [10]. This is a general property: solu- In analogy with massive gravity [19], it is instructive to tions to (2.7) always come in a pair, with one member zoom in on the nonlinearities in by considering the continuously connected to the trivial solution, with stable decoupling limit [9,10]: M , M !1keeping the strong Pl 5 perturbations, and the other connected to the self- coupling scale strong fixed. Equivalently, around a spheri- ! accelerated cosmological solution, with unstable perturba- cal source this corresponds to sending rSch 0 keeping r? tions. It is impossible to classically move from one branch fixed. In other words, in this limit nonlinearities in the of solutions to the other without violating some energy helicity-2 (Einsteinian) modes drop out, while interactions condition [10]. In this work, we focus almost exclusively of the helicity-0 state survive. The resulting effective the- on the stable branch of solutions. ory is local on the brane and describes (weak-field) gravity þ At short distances, r r?, the galileon-mediated force plus a scalar field in 3 1 dimensions: is clearly suppressed compared to the gravitational force: 2 2 MPl 2 rc 2 jr~ j 3=2 2 3=2 L ¼ h~ ðEh~Þ 3ð@ Þ ð@ Þ h F r? r r Einstein 4 M ¼ ¼ ¼ 1: (2.10) Pl F jr~ j 2 1=2 r r grav MPl rcr Sch ? þ 1 ~ þ 1 h T T; (2.5) Thus, as advocated, the strong interactions of lead to its 2 MPl decoupling near a source, and the theory reduces to ~ ~ where E h ¼ hh =2 þ ... is the linearized Newtonian gravity. This approximate recovery of standard Einstein tensor. This Lagrangian is, up to a total derivative gravity near a source has been established in approximate term, invariant under the Galilean shift symmetry, solutions of the full DGP model [18,20,21]. The above
024037-3 NATHAN CHOW AND JUSTIN KHOURY PHYSICAL REVIEW D 80, 024037 (2009) Z 2 2 -mediated force, albeit small in the solar system, is never- pffiffiffiffiffiffiffi M rc ¼ 4 Pl 2 =MPl ð Þ2h theless constrained by lunar laser ranging observations S d x g e R @ 2 MPl [22–24]: rc * 120 Mpc. A comparable bound on rc has þ L ½ also been obtained by studying the effect on planetary matter g ; (3.1) orbits [25]. At large distances, r r?, on the other hand, the non- where h is now understood as the covariant Laplacian: linear terms in are negligible, and the resulting correc- h ¼r r . This clearly reduces to (2.13) in the weak- tion to Newtonian gravity is of order unity: field limit. The Galilean shift symmetry (2.6) is softly F broken in the action (3.1) through M -suppressed opera- ¼ 1 Pl : (2.11) tors. This also true of the full DGP model, where the Fgrav 3 Galilean symmetry arises only in the strict decoupling limit The galileon-mediated force therefore leads to an enhance- as a remnant of the full 5D Lorentz group. ment of the gravitational attraction by a factor of 4=3.In Of course the above nonlinear extension is by no means this far-field regime, the theory reduces to a scalar-tensor unique. For instance, we could consider more general theory, with the galileon acting as a Brans-Dicke scalar. functions of =MPl multiplying the Ricci scalar, or include ð Þ4 4 corrections of the form @ =MPl, all of which would !1 B. Jordan frame description drop out in the limit MPl . Be that as it may, we take Our action (2.5) is cast in Einstein frame, where the (3.1) as a fiducial covariant theory and explore its cosmo- kinetic terms are diagonal, but couples directly to matter. logical predictions. A study of more general Lagrangians is We find it more convenient to instead work in Jordan left for the future. frame, by performing the shift Remarkably, as we will see in the next section, the 4D cosmology arising from (3.1) reproduces many features of ~ 2 the full-fledged DGP model. Our Friedmann equation has h ¼ h þ : (2.12) MPl two branches of solutions, depending on the sign of the This removes the T coupling, at the price of introducing velocity of . The two branches are distinguished by kinetic mixing between h and : having stable or unstable (ghostlike) perturbations. Moreover, we uncover a cosmological analogue of the M2 Vainshtein effect: at early times, when the density of the L ¼ Pl h ðEhÞ þ M2 ðEhÞ Jordan 4 Pl universe is high, nonlinear interactions in are important, r2 1 resulting in the galileon energy density being subdominant c ð@ Þ2h þ h T : (2.13) M 2 compared to the matter or radiation fluid. Pl The covariant equation of motion for the galileon is This forms makes the Brans-Dicke nature of the theory readily obtained from (3.1): manifest, in the limit where the -interactions can be 2 M neglected, with the Brans-Dicke parameter identified as ðh Þ2 ðr r Þ2 R r r ¼ Pl Re 2 =MPl : ¼ 2 !BD 0. 2rc (3.2) III. NONLINEAR COMPLETION Similarly, the Einstein equations are given by Nearly all of the interesting phenomenological features 2 =MPl 2 ¼ þ 2 ðr r hÞ 2 =MPl of the DGP model are attributable to the helicity-0 mode e MPlG T MPl g e and can be understood at the level of the decoupling theory. r2 þ c ð2r r h The Vainshtein effect, reviewed above, is one example. M The existence of a self-accelerated solution is another Pl þ r r ð Þ2 example: the equation of motion (2.7) in vacuum (T ¼ 0) g @ has a solution where M x x =r2, in agreement with 2 Pl c 2rð r Þð@ Þ Þ: (3.3) the weak-field limit of de Sitter space [10]. This motivates us to propose a 4D theory of modified Since the matter action is independent of , the matter gravity, by promoting (2.13) into a fully covariant, non- stress-energy tensor satisfies the usual conservation law: linear theory of gravity coupled to a galileon field. By r T ¼ 0. construction, this nonlinear theory will reduce to (2.13) in the limit of weak gravitational fields. Therefore its IV. COSMOLOGY predictions will agree with those of the full DGP model to leading order in 1=MPl. In this section we specialize the above equations to the Looking at (2.13), a natural nonlinear completion sug- cosmological context, by assuming homogeneity and iso- gests itself: tropy. For simplicity, we focus on the case of a spatially-flat
024037-4 GALILEON COSMOLOGY PHYSICAL REVIEW D 80, 024037 (2009) universe. Under these assumptions, (3.2) reduces to cosmological evolution. In any event, we find: 2 2 2 d 2 2 MPl 2 =M rc _ ð Þþ ð Þ¼ Pl ¼ 2 =MPl H _ 3H H _ 2 Re ; (4.1) 6H M_ Pl e ; dt 6rc M2 Pl 2 where dots represent derivatives with respect to proper rc € ¼ 2 þ 2 =MPl 2 =MPl ð þ Þ t P 2_ 2e 2MPle € 3H _ : time . Remarkably, the left-hand side is reminiscent of MPl the equation for a canonically-normalized scalar field, € þ (4.6) 3H _ , with _ identified as H _ 2. Because the effective field momentum is proportional to Note that changes sign depending on the choice of 2 _ , however, the dynamics are quite different from those of branch, i.e., whether _ is positive or negative. That the a standard scalar field. In particular, if R 0, as is the case galileon effective energy density can be negative should in a universe dominated by matter, radiation or vacuum not come as a surprise, since it is well known that non- energy, then solutions with >_ 0 and <_ 0 are classi- minimal couplings can induce violations of various energy cally disconnected. Indeed, expanding (4.1), conditions in Jordan frame [26–28]. This can also be seen 2 at the level of the effective equation of state, M 2H _ € þ ...¼ Pl Re 2 =MPl ; (4.2) 2 r2 2 6rc ð c _ 2 =MPl Þþ ð _ Þ P € 1 M2 e _ 2 M 3H ¼ ¼ Pl Pl we see that _ is driven away from zero if R>0.A w 2 2 ; rc _ 2 =MPl 3H _ ð1 2 e Þ necessary condition to transition from one branch to the MPl other is therefore R<0. This is closely related to the (4.7) condition 3P 12M2 =r2 necessary to transition Pl c from the stable to the unstable branch of solutions in the which, a priori, allows for w < 1. In fact, we will see in decoupling theory [10]. We will see that the above con- Sec. IV B that w < 1 holds at early times on the stable clusions are borne out by numerical analysis—as long as branch, when the universe is radiation- or matter- the universe is dominated by matter, radiation, or vacuum dominated. We will also see in Sec. VA that w anyway energy, _ never changes sign. contributes positively to the total effective equation of We next turn our attention to the Friedmann and state, wtot, since is negative. Raychaudhuri equations for the scale factor. For the matter, Many of these features also arise in the full DGP model. we assume as usual that the stress-energy tensor is de- Indeed, if we think of the H=rc modification in the DGP scribed by a perfect fluid, Friedmann equation [12], T ¼ð þ PÞu u þ Pg ; (4.3) H H2 ¼ ; (4.8) 3M2 r with and P denoting the energy density and pressure of Pl c the fluid. The Friedmann equation is given by the (0, 0) as an effective contribution to the matter content, then component of (3.3): clearly its energy density can have either sign, depending 2 2 on the choice of branch. By the same token, the effective rc _ 2 2 2 =MPl ¼ þ 2 =MPl 3MPlH e 6H M_ Pl e 2 : equation of state corresponding to the modification can be MPl < 1. For instance, on the normal branch, the effective (4.4) energy density is negative and weff < 1 [13]. While such phantom behavior may at first seem surprising, our analy- Similarly, the Raychaudhuri equation follows as usual sis now makes it clear that it is nothing but a natural from a combination of the ði; iÞ and (0, 0) components: consequence of the scalar-tensor nature of gravity on the 2 2 a€ 1 _ 3rc € brane. 2 2 =MPl ¼ ð þ Þ þ 2 =MPl MPle 3P 4e a 6 2 MPl 2 2 A. Self-accelerated solution? 1 rc _ þ þ 2 =MPl ð þ Þ 2MPle € 2H _ : In analogy with the full-fledged DGP model, we are 2 MPl tempted to look for a self-accelerated cosmology—a de (4.5) Sitter solution in the absence of any cosmological constant j j From the form of (4.4) and (4.5), we can read off an or matter other than itself. In the regime MPl effective energy density and pressure for the field, which where we expect agreement with DGP, the equations of we denote by and P . Of course, since is nonmini- motion at first sight do seem to allow for approximate self- mally coupled to gravity, is not conserved, hence it acceleration. Setting _ ¼ _ 0 and H ¼ H0 to be constant in should only be understood as an effective energy density, (4.1) and (4.4), one finds an approximate solution for j j informing us about the effects of the galileon on the MPl, given by
024037-5 NATHAN CHOW AND JUSTIN KHOURY PHYSICAL REVIEW D 80, 024037 (2009) sffiffiffi 1. Cosmological screening 2 M t 2 3=2 1 Pl ; H ; (4.9) 0 3 r 0 3 r The above constant- _ solution is only valid provided c c that the galileon energy density is a negligible contribution 2 =MPl where the ‘‘ ’’’s indicate the assumption j j MPl.Up to the total energy. With our approximation e 1, to a trivial redefinition of rc, this agrees with the self- the first of (4.6) reduces to accelerated DGP solution that follows from the ‘‘þ’’ DGP ¼ 2 _ r2 _ 2 branch of (4.8): H0 1=rc. ¼ 1 c j j 3H2M2 2 Unfortunately, the approximation MPl breaks Pl H MPl MPl down within a time t r , which is of the order of a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c Hubble time. In other words, H evolves significantly over ¼ 4 1 3w 1 1 ð Þ ð Þ : (4.13) a Hubble time, and hence cannot be approximated as Hrc 3 1 w 3 1 w Hrc constant. Self-accelerated cosmology is spoiled in our This elucidates the timelike screening effect advocated galileon theory by =MPl corrections. earlier: at early times, when Hrc 1 (or, equivalently, t r ), nonlinear galileon interactions are important, and B. Early-time solution and cosmological screening c as a result its gravitational backreaction is negligible. In In this section we derive approximate analytic solutions particular, this ensures that nucleosynthesis and recombi- for when the universe is dominated by other components nation proceed as in standard cosmology, with negligible than , such as matter, radiation or dark energy. We will corrections coming from . We note in passing that the see that the dynamics of the galileon exhibit a timelike parametric dependence of =3H2M2 is consistent with Pl analogue of the Vainshtein effect: at early times, t rc, the modified Friedmann equation in the full DGP model: nonlinearities in are important, resulting in the galileon looking back at (4.8), the relative contribution of the H=r c energy density being negligible. Once t rc, however, modification term to the total expansion rate is indeed exits the strongly-coupled regime, and becomes a sig- suppressed by 1=Hrc. nificant contribution to the total energy density. The self-screening effect breaks down after a time of To simplify the analysis, suppose that the universe is order rc, at which point becomes a significant contri- dominated by a single matter component with constant bution to the expansion rate. As we will see in Sec. VII, equation of state w. Moreover, we assume that the variation constraints on the late-time expansion history will enforce in throughout this phase is small in Planck units: a lower bound on rc. Incidentally, t rc also signals the j j M ðt ¼ Þ¼ j j Pl. Since we can always set 0 0 by moment when the approximation MPl breaks down, 2 =MPl trivial rescaling of MPl, it follows that e 1. The since consistency of these approximations will be checked a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi posteriori. 1 3w t ¼ : (4.14) With these assumptions, the Friedmann Eq. (4.4) re- M 3ð1 wÞ r 2 2 Pl c duces to its standard form, 3H MPl , with the usual solution To summarize, the constant- _ solution in (4.12) is valid at early times, t r , when the galileon is strongly coupled. ð Þ ð2=ð3ð1þwÞÞÞ c a t t : (4.10) In this regime, the galileon energy density can be consis- tently neglected, and the galileon excursion in field space is Substituting this into (4.1), small in Planck units. 2 3 M2 An illustrative way to understand this cosmological _ € þ ð1 wÞ _ 2 Pl ð1 3wÞ; (4.11) screening is to consider the nonlinear galileon interactions 2 2 H 2rc as an effective BD parameter. By comparing the BD and galileon actions, given by (1.1) and (3.1) respectively, and we see that the galileon equation of motion allows for a ¼ 2 =MPl solution with _ ¼ constant: making the identification e , we can define an eff sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi effective dynamical !BD _ 1 1 3w sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : (4.12) 2 rc 1 3ð1 3wÞ MPl r 3ð1 wÞ eff ð Þ¼ 2 =MPl h c !BD t e Hrc; 2MPl 2 1 w Therefore a real solution exists if either w 1=3 or w>1, (4.15) but, as we will see shortly, only for w 1=3 is the solution a dynamical attractor. Moreover, we will see in Sec. IV B 3 where in the second line we have made the approximation that among the two branches of solutions in (4.12), only e2 =MPl 1, and substituted the constant- _ solution (4.12). eff <_ 0 has stable fluctuations. The other branch, with >_ Following the discussion of the previous paragraph, !BD is 0, is plagued with ghostlike instabilities. large at early times, when Hrc 1, and standard cosmol-
024037-6 GALILEON COSMOLOGY PHYSICAL REVIEW D 80, 024037 (2009)
2 Z ogy is recovered. When Hrc 1, however, we enter the r pffiffiffiffiffiffiffi ¼ c 4 ð Þ2h scalar-tensor gravity regime with !eff 1. Seff d x g @ : (4.21) BD MPl For completeness, we also derive the galileon equation of state during the strong coupling phase. Substituting Once again, as in the analysis of Sec. IV B 2, we neglect (4.12) into (4.7), we obtain perturbations in the metric, since the cosmological back- ground is driven by some other source of stress-energy. 3 1 Expanding (4.21) to quadratic order in ’, then after some w ¼ ð1 wÞþO : (4.16) 2 Hrc integration by parts we obtain 2 Z Interestingly, in this regime w is completely fixed by the ð Þ 2r pffiffiffiffiffiffiffi S 2 ¼ c d4x gðh g r r Þ@ ’@ ’: background equation of state. In particular, this says that eff M Pl w 1 for w 1=3. As discussed earlier, however, this (4.22) does not signal the presence of ghost instabilities, but instead is a natural consequence of the nonminimal cou- Note that this holds for arbitrary background ðx;~ tÞ.To pling to gravity. make contact with the decoupling results of [10], it is ¼ 2r r useful to define K rc =M Pl, which measures 2. Dynamical attractor the extrinsic curvature of the brane in the decoupling limit of the full DGP model. In terms of K ,(4.22) takes the We next prove that our constant- _ solution is a dynami- form cal attractor for physically-relevant values of w. Perturb the Z pffiffiffiffiffiffiffi galileon as ð2Þ ¼ 4 ð Þ Seff 2 d x g K g K @ ’@ ’: (4.23) ðtÞ¼ ðtÞþ’ðtÞ; (4.17) In the weak-field limit, where we expect consistency with with _ given by (4.12). Since the backreaction of on the the decoupling limit, this indeed agrees with Eq. (22) of expansion has already been shown negligible in this [10] in the strong coupling regime: K 1. Remarkably, strongly-coupled regime, HðtÞ and RðtÞ are oblivious to the form of the moduli space metric, K g K ,is ’ and hence can be left unperturbed. Expanding (4.1)to preserved in our 4D covariant theory. 2 =M linear order in ’, while remembering that e Pl 1 in Specializing to the constant- _ solution, (4.22) reduces to our approximation, we get 2 Z ð Þ 6r pffiffiffiffiffiffiffi 2 2 ¼ c 4 _ 2 þ ðr~ Þ2 d Seff d x gH ’_ ’ : (4.24) ðH’_ Þþ3HðH’_ Þ 0: (4.18) MPl 3 dt Thus the kinetic term of fluctuations is proportional to _ — 3 It follows that H’_ 1=a ,or the kinetic term is positive if < _ 0, corresponding to _ 1 stable perturbations, and is negative if > 0, correspond- ’_ t ðð1 wÞ=ð1þwÞÞ: (4.19) ing to ghostlike perturbations. Hence, just like in the full Ha3 DGP model, one branch of solutions is stable, whereas the Thus perturbations redshift away compared to _ ¼ other is unstable. Moreover, looking back at (4.13), the constant for 1
3. General stability analysis V. NUMERICAL ANALYSIS We can easily extend the above stability analysis to We can solve the equation of motion (4.1) and the general perturbations: Raychaudhuri Eq. (4.5) numerically to obtain exact cos- ðx;~ tÞ¼ ðtÞþ’ðx;~ tÞ: (4.20) mological solutions. We focus exclusively on the stable branch, by setting initial conditions with <_ 0. Since We work at the level of the action, since here it is trans- < 0 on this branch, we must include a dark energy parent whether perturbations have a right-sign kinetic term component to obtain late-time acceleration, which we or are ghostlike—a diagnosis that is trickier to make with take to be a cosmological constant for simplicity. In linearized equations of motion. other words, in addition to we include radiation, matter Since is strongly coupled at early times, its perturba- and a cosmological term. And since contributes to the tions are dominated by the cubic interaction term in (3.1): effective dark energy component, for each value for rc we
024037-7 NATHAN CHOW AND JUSTIN KHOURY PHYSICAL REVIEW D 80, 024037 (2009) adjust so that the fractional contribution in matter today converges to the analytic prediction. The exact solution ð0Þ is kept fixed to the fiducial value m ¼ 0:26. starts to deviate from the analytic prediction around the present time, however, when the Hubble radius H 1 be- A. Results comes comparable to rc—this is consistent with the dis- cussion of Sec. IV B 1. Note that the galileon energy Figs. 1 and 2 show the evolution of the energy density density is a more significant contribution to the dark energy for the various components as a function of redshift, for today for r ¼ 10 Gpc than for 20 Gpc. ¼ c rc 10 and 20 Gpc, respectively. (Since these are log-log Figure 3 shows the fractional contributions to the total plots and < 0, for the galileon component we instead energy density, defined for the various components as plot j j.) These figures confirm the analytic results de- rived in Sec. IV B. The dotted and dashed lines in Figs. 1 e2 =MPl ¼ i : (5.2) and 2, respectively, trace the galileon energy density as i 2 2 3H MPl predicted by the constant- _ solution of Sec. IV B. Note that, to plot the analytic prediction, we have substituted in Note that the galileon contribution is negative, since <_ (4.12) the total equation of state wtot defined by 0. This figure shows once again that the galileon is sub- dominant at early times, until Hrc 1 when becomes a _ H ¼ 3 ð þ Þ significant contribution to the expansion rate. 1 wtot : (5.1) H2 2 Plotted in Fig. 4 is the effective equation of state for the whole evolution, wtot, defined in (5.1). Here we plot the From the figures, we see that for most of the cosmologi- CDM behavior (solid curve) compared to our galileon cal evolution, the actual galileon energy density agrees well with the constant- _ prediction. Moreover, the constant- _ solution is manifestly an attractor: as seen in Fig. 2, for instance, the galileon energy density quickly
FIG. 3. The fractional contributions to the total energy density from matter ( m), radiation ( r), cosmological constant ( , dashed line) and the galileon field ( , bold dashed line) for r ¼ 20 Gpc, as a function of redshift. FIG. 1. Result of numerically integrating the cosmological c evolution equations with rc ¼ 10 Gpc. The solid curves denote the matter, radiation, cosmological constant and galileon energy density. The dotted line is the galileon energy density as pre- dicted from the analytic solution.
FIG. 4. The effective equation of state for the expansion his- tory, defined in terms of the Hubble parameter by H=H_ 2 ¼ 3ð1 þ wtotÞ=2, for CDM (solid), and the galileon model with rc ¼ 10 Gpc (dotted), 15 Gpc (dash-dotted) and 20 Gpc FIG. 2. Same as Fig. 1, this time for rc ¼ 20 Gpc. (dashed).
024037-8 GALILEON COSMOLOGY PHYSICAL REVIEW D 80, 024037 (2009) model with rc ¼ 10 Gpc (dotted curve), 15 Gpc (dash- dotted) and 20 Gpc (dashed). The early-time behavior is as expected from standard cosmology—the equation of state goes through successive stages of wtot 1=3 and wtot 0, corresponding to radiation- and matter- dominated eras, respectively. At late times, the energy density is dominated by an effective dark energy compo- nent composed of and , with negative equation of state. Compared to CDM, the galileon contribution pushes the dark energy equation of state to values larger than 1. And the smaller rc is, the further wtot is from 1 at late times. Conversely, wtot approaches 1 today as rc !1. While it may seem surprising at first sight that FIG. 6. Comparison of the galileon energy density (j j) with wtot > 1 at late times, since we argued in Sec. IV that w < 1 on the branch of interest, one must keep in mind the effective energy density coming from the modification to the j j _ Friedmann equation in the DGP model ( DGP ). We have used that < 0—the galileon contribution to H, being propor- ¼ tional to ð1 þ w Þ , is therefore negative. rc 15 Gpc in each case and kept the matter, radiation and cosmological constant densities fixed. The densities and To say a few words about the asymptotic behavior, since DGP agree well for the entire expansion history until today, is negative and decreasing, it eventually catches up with and begin to diverge in the future when MPl. . At this point the Hubble parameter vanishes, and the universe starts to contract. Galileon models thus generi- cally predict a late-time contracting phase. This could effective energy density component: naturally match onto an ekpyrotic [29–33], New DGP H ; (5.3) Ekpyrotic [34–38] or cyclic [39,40] contracting phase, 3M2 r for instance. See [41] for a review of these models. Pl c Figure 5 shows the effective BD parameter !eff , intro- where we have chosen the normal branch. Figure 6 com- BD j j duced in (4.15), for the whole evolution, with rc ¼ 10 Gpc pares the evolution of DGP in DGP cosmology with the (dotted curve), 15 Gpc (dash-dotted) and 20 Gpc (dashed). galileon energy density j j in our galileon theory, in each eff 8 ¼ At early times the !BD 10 is large, and standard cos- case with rc 15 Gpc. For simplicity, we fix the matter, eff radiation and cosmological constant contributions. We see mology is recovered. At late times, !BD 1, and the theory reduces to scalar-tensor gravity. As expected, the that DGP and agree remarkably well for the entire scalar-tensor regime is achieved at earlier times for smaller expansion history until z ¼ 0, and begin to diverge in the j j values of rc. future (z<0) when MPl.
B. Comparison with DGP cosmology VI. GROWTH OF DENSITY PERTURBATIONS How closely does our 4D galileon theory come to re- By virtue of its nonminimal coupling to gravity, the producing the cosmology of the 5D DGP model? To make galileon enhances the gravitational attraction between par- the comparison it is convenient to think of the H=rc ticles. Translated to the cosmological context, we expect correction term in the DGP Friedmann Eq. (4.8)asan more rapid growth of density perturbations. Furthermore, from the discussion of Sec. II, the galileon enhancement should be suppressed at early times, due to self-screening, but should become important once the matter density has dropped sufficiently. Since the matter action is independent of the galileon in Jordan frame, the evolution of matter density perturba- tions, m m= m, on sub-Hubble scales is governed by the standard expression r~ 2 € þ 2H _ ¼ ; (6.1) m m a2 where is the Newtonian potential in Jordan frame. The effects of are all encoded in its contribution to the Poisson equation. eff ¼ FIG. 5. The effective BD parameter, !BD, with rc 10 Gpc Consider expanding the galileon around its cosmologi- (dotted), 15 Gpc (dash-dotted) and 20 Gpc (dashed). cal profile, ðx;~ tÞ¼ ðtÞþ’ðx;~ tÞ. In the Newtonian ap-
024037-9 NATHAN CHOW AND JUSTIN KHOURY PHYSICAL REVIEW D 80, 024037 (2009) 0.07 proximation, we are justified in neglecting time derivatives ~ 0.06 DGP of ’ relative to spatial gradients: j’_ j jr’j. We can thus π expand (3.2) to linear order in ’ as follows, keeping in 0.05 2 =MPl 0.04 mind that e 1 for most of the expansion history, β r~ 2 1/3 0.03 2 ’ 2 4r ð € þ 2H _ Þ ¼ M R: (6.2) 0.02 c a2 Pl 0.01 Meanwhile, from the trace of (3.3), 0 2 1 0 2 10 10 10 r~ ’ 1+z M2 R T þ 6M : (6.3) Pl Pl a2 FIG. 7. Comparison of the perturbation enhancement factor Note that we have neglected the -dependent terms in the 1=3 in our galileon model (dotted) with the corresponding second line of (3.3), since the backreaction of the galileon factor 1=3 DGP in DGP cosmology (dashed). As in Fig. 6,we is negligible for all times, except in the recent past. Since have used rc ¼ 15 Gpc in each case and kept the matter, ¼ T m m for nonrelativistic sources, combining (6.2) radiation and cosmological constant densities fixed. and (6.3) gives r~ 2 4 G our galileon theory to make nearly identical predictions for ’ ¼ 2 a m; (6.4) structure formation as the full DGP model. We leave for 3 MPl future work a detailed study of the evolution of perturba- where tions in the covariant galileon theory. 2r2 1 c ð € þ 2H _ Þ: (6.5) VII. OBSERVATIONAL CONSTRAINTS 3M Pl A rigorous comparison of the galileon cosmological Moreover, using (6.4), it is straightforward to obtain the scenario with observations requires a full likelihood analy- modified evolution equation for density perturbations: sis, which is beyond the scope of this paper. For the purpose of this section, we restrict ourselves to a compari- € þ _ ¼ þ 1 m 2H m 4 G m m 1 : (6.6) son with a few observables and derive a lower bound on r . 3 c Since rc is the only parameter that we allow to vary, our This Poisson equation for ’ exhibits the expected features: estimate is likely conservative. 2j þ j at early times, when rc € 2H _ =MPl 1, the coupling !1 to m is much suppressed since . This is the cos- A. Mass estimates mological self-screening mechanism, which decouples the Because of the nonminimal coupling of the galileon, the galileon from matter inhomogeneities. At late times, on the fractional matter density, m, depends on , as seen from other hand, becomes of order unity, and the galileon (5.2): couples to matter with strength comparable to that of gravity. The evolution Eq. (6.6) takes on a standard form e2 =MPl ¼ m : (7.1) in the late-time regime, except for the fact that the gravi- m 2 2 3H MPl tational attraction is enhanced by the galileon factor. ð0Þ For comparison, the linearized perturbation equation Therefore, for fixed matter density today, m , the matter derived for the full DGP model, derived in [42] for density in the past will differ from the standard gravity spherical top-hat perturbations, takes on an identical prediction by form to (6.6), with in this case given by m ¼ 2 0=MPl 2 2 e ; (7.2) 1 þ 2r H þ r H std * DGP ¼ c c : (6.7) m z 1 1 þ rcH where 0 is the present value of the galileon. And since (Note that the choice of sign is consistent with the normal 0 < 0 on the stable branch, m is larger at early times DGP ð0Þ branch of DGP.) Clearly the asymptotic behavior of than predicted by standard gravity, again keeping m for Hrc 1 and Hrc 1 agrees with that of (6.5). fixed. Hence 0 is constrained by estimates of the matter Figure 7 compares 1=3 in our model with 1=3 DGP in density at various redshifts, such as from cluster counts, DGP cosmology, by solving numerically for the respective Lyman- forest and weak lensing observations. (Similar cosmological backgrounds. As in the discussion of considerations apply to coupled dark matter-dark energy Sec. VB, we take rc ¼ 15 Gpc in each case and fix the models [43].) energy density in matter, radiation and cosmological term. Although the constraints from these observables should Given the close agreement shown in the figure, we expect be revisited in the presence of a galileon, it has been argued
024037-10 GALILEON COSMOLOGY PHYSICAL REVIEW D 80, 024037 (2009) 0.8 4 Λ CDM r = 20 Gpc 0.7 3.5 c r = 15 Gpc c 3 r = 10 Gpc 0.6 c 2.5 0.5
(z) 2 L | d 0 0.4 π | 1.5 0.3 1
0.2 0.5
0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 z 0 2 4 6 8 10 12 14 16 18 20 FIG. 9. Luminosity distance (d ) as a function of redshift for rc (Gpc) L CDM (solid), and our galileon model with rc ¼ 10 Gpc (dot- ¼ ¼ FIG. 8. Value of the galileon field at the present time as a ted), rc 15 Gpc (dash-dotted) and rc 20 Gpc (dashed). In ð0Þ ¼ function of rc. For each value of rc, the cosmological term is each case, we have fixed m 0:26. ð0Þ adjusted to keep m ¼ 0:26 fixed. The dashed line represents : M the threshold value of 0 18 Pl from mass estimates—see the z<1:5. Therefore, from the percentage differences shown bound in (7.3). in the figure, we see that luminosity-distance observations tighten the bound on rc to ð Þ that the allowed range of 0 is generally insensitive to the m * specifics of dark energy [44]. A general analysis combin- rc 15 Mpc: (7.5) ing SNIa Gold data set [45], Wilkinson Anisotropy Microwave Probe (WMAP) power spectra [46], and Two- This constitutes our main constraint on the scale of the Degree Field (2dF) galaxy survey [47] obtained 0:23 & modification. ð Þ Another distance constraint comes from the angular- 0 & 0:33 [44]. (See also [48,49].) In the absence of a m diameter distance to the last scattering surface, full likelihood analysis, we can obtain a conservative 2 0=MPl & 0 bound on 0 by requiring that e 0:33=0:23,or Z z ð Þ¼ 1 rec dz dA zrec þ ð 0Þ ; (7.6) j 0j & 0:18MPl: (7.3) 1 zrec 0 H z j j Figure 8 shows 0 for various values of rc, as obtained which determines the position of the cosmic microwave j j numerically. We can read off that the above bound on 0 background (CMB) acoustic peaks. For rc * 15 Gpc, the * ð Þ is satisfied for rc 10 Gpc, or roughly 3 times larger than difference in dA zrec compared to CDM is less than the radius of the observable universe. We will see shortly ð0Þ 10%, again keeping m fixed, which is within current that luminosity-distance observations tighten the bound to CMB uncertainties. Note that a similar constraint (r * * c rc 15 Gpc. 1 3 3:5H0 ) was obtained recently by confronting the B. Cosmological distances Next we turn our attention to cosmological distance tests, in particular, the luminosity-distance relation, 0 Z z ð Þ¼ð þ Þ dz dL z 1 z 0 ; (7.4) 0 Hðz Þ as constrained by Type Ia supernovae (SNIa). Figure 9 ð0Þ shows the luminosity-distance relation with m ¼ 0:26 for the CDM model, and for our galileon model with rc ¼ 10, 15 and 20 Gpc. In the galileon examples, was ð0Þ adjusted in order to keep m fixed. Figure 10 shows the percentage difference between the various galileon examples and the CDM fiducial model. The uncertainties in present SNIa data constrain the lumi- FIG. 10. Percentage difference between the galileon examples nosity distance to no better than 7% over the range 0 < and CDM.
024037-11 NATHAN CHOW AND JUSTIN KHOURY PHYSICAL REVIEW D 80, 024037 (2009) DGP normal-branch cosmology against CMB, SNIa and (i) Following up on the discussion of the last paragraph, Hubble constant observations [50]. a thorough comparison with observations requires a full likelihood analysis, allowing various cosmologi- ð0Þ VII. CONCLUSIONS cal parameters (such as m , h, etc.) to vary. (ii) A study of the implications for structure formation In this paper we have studied the cosmology of a gali- requires a more rigorous treatment of inhomogene- leon field theory, obtained by covariantizing the ities in the presence of the galileon. To probe the -Lagrangian of the DGP model. Despite being a local nonlinear regime, it should be straightforward to theory in 3 þ 1 dimensions, the resulting cosmological generalize the N-body simulations of [51,52]to evolution is remarkably similar to that of the full 4 þ include the galileon field theory studied here. * 1-dimensional DGP framework, at least for Hrc >1 (iii) In this work we have focused exclusively on the j j & (or MPl). The similarity holds for both the expansion cubic interaction term that arises in the decoupling history (Fig. 6) and the evolution of density perturbations limit of DGP. Recently, [11] derived the most gen- (Fig. 7). eral galileon field theory (without gravity) which is In particular, as in the DGP model our covariant galileon invariant under the Galilean shift symmetry theory yields two branches of solutions, depending on the @ ! @ þ c , and whose equations of motion sign of _ . Perturbations are stable on one branch and are second order. It was shown in [53] that the ghostlike on the other. The effective equation of state for equations remain of second order in the presence the galileon is phantomlike (w < 1) on the stable of gravity, provided is suitably coupled to grav- branch, and standard (w > 1) on the unstable branch. ity. It would be very interesting to extend our A key difference, however, is that the galileon field theory analysis to include these higher-order interactions does not generate a self-accelerated solution—as shown in terms. Sec. IVA, the would-be self-accelerated solution with H (iv) The extension of our 4D theory to include multiple 1=rc is spoiled by 1=Hrc corrections in our theory. galileon fields should offer a reliable proxy for the An interesting effect uncovered in our analysis is a cosmology of higher-dimensional DGP models, cosmological version of the self-screening (or such as Cascading Gravity [54–56]. In this con- Vainshtein) mechanism. At early times, Hrc 1, the struction, our 3-brane lies within a succession of evolution of is dominated by the self-interactions terms. higher-dimensional DGP branes, embedded in one In turn, this results in being a negligible component, with another within a flat bulk space-time. The corre- Oð Þ its energy density suppressed by a factor 1=Hrc com- sponding 4D covariant theory should therefore in- pared to matter and radiation. This cosmological self- clude 2 interacting galileon fields, each with its own screening is crucial in the recovery of standard cosmology rc scale. at early times. Once the expansion rate drops to Hr 1, however, the c ACKNOWLEDGMENTS galileon becomes an important player in the Friedmann equation. A preliminary analysis of observational con- We thank Pier-Stefano Corasaniti, Michel Gingras, straints in Sec. VII shows that the modifications to the James Taylor, Brian McNamara, Rob Myers, Alberto expansion history are consistent with the observed late- Nicolis, Roman Scoccimarro, Andrew Tolley, Mark time cosmology provided rc * 15 Gpc. We should empha- Trodden, and especially Mark Wyman for helpful discus- size that this bound is most certainly conservative and will sions, as well as Wayne Hu and Kazuya Koyama for initial be relaxed by allowing other cosmological parameters to discussions. This work was supported in part at the vary. Perimeter Institute by the Government of Canada through Our analysis offers a host of interesting avenues to NSERC and by the Province of Ontario through the explore: Ministry of Research & Innovation.
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