Les Houches Lectures on Physics Beyond the Standard Model of Cosmology
LLes Houches Lectures on PhysicsBeyond the Standard Model ofCosmology
Justin Khoury
Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104In these Lectures, I review various extensions of the Λ-Cold Dark Matter (ΛCDM) model,characterized by additional light degrees of freedom in the dark sector. In order to repro-duce the successful phenomenology of GR in the solar system, these fields must effectivelydecouple from matter on solar system/laboratory scales. This is achieved through screeningmechanisms, which rely on the interplay between self-interactions and coupling to matterto suppress deviations from standard gravity. The manifestation of the new degrees of free-dom depends sensitively on their environment, which in turn leads to striking experimentalsignatures. a r X i v : . [ a s t r o - ph . C O ] D ec Introduction
Cosmology can rightfully claim to have a standard model: the ΛCDM model. Withroughly 4.5% baryonic matter, 26.5% dark matter, and 69% vacuum energy, this modelprovides an exquisite fit to all known cosmological data. (For a review, see Jain andKhoury (2010).) Its empirical success lies in its parsimony — only a handful of param-eters are required to fit observations. As a result, it is highly predictive: the ΛCDMexpansion and growth histories are tightly correlated, leaving essentially no wiggleroom to account for possible discrepancies.The discomfort with ΛCDM among theorists is of course one of naturalness. Theinferred value of the cosmological constant requires an absurb conspiracy among itsvarious contributions. This is the famous Cosmological Constant (CC) problem. Morecompelling is the possibility of new physics associated with dark energy, which wouldstabilize the vacuum energy at its observed value. A parallel with the weak hierar-chy problem in particle physics seems appropriate. The recent discovery of the Higgsparticle represents the crowning achievement of the Standard Model. The StandardModel is now complete, but leaves us with an unnaturally low value for the Higgsmass. Either the Higgs mass is fine-tuned, or it is stabilized by new physics (such assupersymmetry) at the weak scale.The naturalness problem afflicting cosmology is in some sense more robust than itsparticle physics counterpart. Indeed, the weak hierarchy problem stems from radiativecorrections to the Higgs mass from hypothetical particles beyond the weak scale. In thecase of the CC problem, however, vacuum energy contributions from known particles,such as the electron, are already problematic. Furthermore, the required solution forthe CC problem is arguably more radical. While the proposed solutions to the weakhierarchy problem — supersymmetry, technicolor — are by no means trivial, they bothfit within the standard framework of local quantum field theory. As shown by Weinberg(1989), on the other hand, no dynamical solution to the CC problem is possible withinGeneral Relativity (GR).Let us begin with a lightning review of the CC problem.
A precise calculation of the vacuum energy of course requires knowledge of physics allthe way to the Planck scale. Within the effective field theory framework, however, onecan calculate robust low-energy contributions to get a sense for the required degree offine-tuning.Consider the vacuum energy of a scalar field of mass m , ignoring interactions. Eachquantum contributes an energy E k = (cid:112) (cid:126)k + m . Integrating over all quanta below hy screen? some cutoff Λ UV , the energy density is (cid:104) ρ (cid:105) = (cid:90) Λ UV d k (2 π ) (cid:113) (cid:126)k + m (cid:39) π (cid:18) Λ + m Λ − m (cid:18) Λ UV m (cid:19) + finite (cid:19) , where in the last step we have used the fact that the integral peaks close to Λ UV ,and expanded the integrand in powers of m/k . The first two terms, which diverge asa power of Λ UV , depend sensitively on the UV physics. They would be absent, forinstance, if one used instead dimensional regularization. The log term, on the otherhand, is regulator-independent and represent a robust, low-energy contribution to thecosmological constant.The UV-insensitive contribution from all particles with masses m i (cid:28) Λ UV is there-fore given by (cid:104) ρ (cid:105) = − π (cid:88) i ( − F i m i log (cid:18) Λ UV m (cid:19) , (1.1)where the sum is over all particle species, and F i is the fermion number. The electroncontribution, for instance, is ρ e ∼ m e ∼ meV , (1.2)which is of course orders of magnitude larger than the observed value of ∼ meV. Thisis the essence of the CC problem. The robust, low-energy contributions from known particles are unacceptably large. A tremendous cancellation must occur between thesecontributions and (unknown) high energy contributions to generate the observed valueof the vacuum energy.An old idea is to postulate that the vacuum energy is effectively time-dependent,and that there is some kind of relaxation mechanism driving the vacuum energy tosmall values at late times. Indeed, we need only explain why the vacuum energy issmall now, not in the very early universe. Unfortunately, as shown long time agoby Weinberg (1989), such a mechanism is impossible, under the key assumption thatgravity is described by GR. A possible loophole is therefore to go beyond GR, whichinevitably introduces new degrees of freedom (Feynman et al. , 1996; Weinberg, 1965;Deser, 1970; Khoury et al. , 2012; Khoury et al. , 2013). Although no compelling mechanism has been found to date, we can robustly infer twonecessary properties of these degrees of freedom. For concreteness, let us assume theyare scalars. Firstly, to neutralize Λ to an accuracy of order ∼ H M ∼ meV , thescalars must have a mass at most comparable to H , m φ ∼ < H . (1.3)Otherwise, if they were much more massive, they could be integrated out and thuswould be irrelevant for the low energy dynamics. Introduction
Fig. 1.1
The tadpole diagram (on the left) involving the scalar field (dotted line) attachedto Standard Model fields (solid line) running in the loop is necessary in order to neutralizethe Standard Model vacuum energy contribution. By unitarity, the tree-level diagram (on theright) with a scalar exchanged by Standard Model fields is also allowed, implying that thescalar field mediates a 5 th force that must therefore be screened. Secondly, these scalars must couple to Standard Model fields, since Standard Modelfields contribute O (TeV ) to the vacuum energy. In other words, the tadpole diagramshown in Fig. 1.1 must be present in the theory. But by unitarity, so must the ex-change diagram. Hence φ mediates a force between Standard Model fields, whose range,given (1.3), is comparable to the present Hubble radius. If left unabated, such a long-range force would generically lead to significant deviations from General Relativity inthe solar system, in conflict with well-known constraints from tests of gravity (Will,2006).Thus we are led to conclude, on very general grounds, that these scalars must ef-fectively decouple from matter on solar system/laboratory scales in order to reproducethe successful phenomenology of GR. This can be achieved through screening mecha-nisms , which rely on the high density of the local environment (relative to the meancosmological density) to suppress deviations from standard gravity. In what followswe will describe 3 general classes of screening mechanisms and highlight some of theircharacteristic properties.Besides cosmology and the CC problem, screening mechanisms are also motivatedby the vast experimental effort aimed at testing the fundamental nature of gravity ona wide range of scales, from laboratory to solar system to extra-galactic scales. Fora review, see Will (2006). As we will see below, viable screening theories make novelpredictions for local gravitational experiments. The subtle nature of these signals haveforced experimentalists to rethink the implications of their data and have inspired thedesign of novel experimental tests. The theories of interest thus offer a rich spectrumof testable predictions for ongoing and near-future tests of gravity. Screening Mechanisms: A BriefOverview
The broad classes of screening mechanisms surveyed here can all be encompassed inthe general action L = M R E + L φ ( φ, ∂φ, ∂ φ ) + L m (cid:2) A ( φ ) g E µν , ψ (cid:3) , (2.1)where M = 1 / πG N , and all indices are contracted with the Einstein-frame metric g E µν . For simplicity, we have assumed that the scalar couples conformally and uni-versally to matter fields described by L m . In particular, matter particles universallyfollows geodesics of the Jordan-frame metric g J µν = A ( φ ) g E µν . (2.2)In most situations of interest, save of course for cosmological evolution, we will seethat the field excursions are small in Planck units, i.e. ∆ φ (cid:28) M Pl . (For chameleons,this is true even cosmologically, as shown in Wang et al. (2012).) We are thereforejustified in linearizing the coupling function A ( φ ): A ( φ ) (cid:39) gφM Pl . (2.3)The dimensionless coupling g is generally assumed to be O (1), corresponding togravitational-strength scalar force. In the Newtonian approximation, (2.2) and (2.3)imply the following relation between Jordan-frame and Einstein-frame Newtonian po-tentials (defined as usual by g = − (1 + 2Φ)):Φ J = Φ E + gφM Pl . (2.4)This will prove useful in physically interpreting the different screening conditions. With chameleons (Khoury and Weltman, 2004 b ; Khoury and Weltman, 2004 a ), one can moregenerally assume different couplings to different matter species, thereby explicitly violating the weakequivalence principle. We also ignored the possibility of disformal coupling, e.g. Koivisto et al. (2012). An important exception is the symmetron (Hinterbichler and Khoury, 2010; Olive and Pospelov,2008; Pietroni, 2005; Hinterbichler et al. , 2011 a ; Brax et al. , 2011 a ) and varying-dilaton (Brax et al. ,2011 b ) mechanisms, where a φ → − φ symmetry precludes the linear term in A ( φ ). Instead, A ( φ ) (cid:39) gφ /M in those cases. In practice, the phenomenology of the symmetron is qualitatively similarto that of the chameleon, so for simplicity we focus our discussion to the linear coupling (2.3). Screening Mechanisms: A Brief Overview
For situations relevant to tests of gravity, we can ignore the backreaction of thescalar onto the metric, and approximate φ as evolving on Minkowski space-time. More-over the matter can be treated as a non-relatisvistic source, with negligible pressure.In this approximation, the Lagrangian relevant for the scalar dynamics is L scalar = L φ ( φ, ∂φ, ∂ φ ) − gφM Pl ρ m , (2.5)where indices are now contracted with η µν . A first requirement on L φ is that it mustdescribe a single degree of freedom, i.e. , its equations of motion must be second-order in time. The most general such Lagrangian (coupled to gravity) was discovereddecades ago by Horndeski (1974). See Deffayet et al. (2011) for a recent derivation.A second requirement is that the scalar should develop non-linearities in the presenceof sufficiently massive/dense sources, in such a way that its effects are mitigated. Itturns out there are three qualitatively different ways to do so, described below. • Chameleon (Khoury and Weltman, 2004 b ; Khoury and Weltman, 2004 a ):In this example, the scalar Lagrangian consists of a standard kinetic term pluspotential: L φ = −
12 ( ∂φ ) − V ( φ ) . (2.6)Since the interactions are governed by a potential, whether or not the scalar de-velop non-linearities depends on the local value of φ . In other words, the screeningcondition in this case is schematically of the form φ < φ c . (2.7)We will see in Sec. 3 that the critical value φ c depends on the local gravitationalpotential. A useful rule of thumb to ascertain which regions of the universe arescreened or unscreened is to map out the Newtonian potential smoothed on somescale. Screened regions correspond to Φ J > Φ c .A recurring theme of all screening mechanisms is that their strong coupling scale israther low, i.e. , comparable or lower than the meV dark energy scale. In the caseof chameleons, perturbative unitarity breaks down at the meV scale (Upadhye et al. , 2012): Λ s ∼ meV . (2.8)A drawback of this screening mechanism is that L φ has no particular symmetry,hence one would not expect V ( φ ) to be protected under radiative corrections.This issue has been analyzed in detail recently in Upadhye et al. (2012), whereit was found that keeping quantum corrections under control imposes an upperlimit on the chameleon mass in the local (laboratory) environment. The upperlimit is very general and insensitive to the details of the potential. Remarkably,for gravitational-strength coupling ( g ∼ O (1)), a factor-of-two improvement overthe current experimental bounds on the range of a scalar fifth force would ruleout all viable chameleon models. creening Mechanisms: A Brief Overview On the plus side, since chameleons are described by canonical scalar fields withself-interaction potentials, they in principle admit a standard Wilsonian UV com-pletion. Unlike the other mechanisms described below, they do not suffer from su-perluminality issues. Ideally one would like to see an explicit UV-complete realiza-tion of this mechanism in string theory. As a first step in this direction, Hinterbich-ler et al. (2011 b ) presented a scenario for embedding the chameleon mechanismwithin supergravity/string theory compactifications. (See Nastase and Weltman2013 b ; Nastase and Weltman 2013 a for related work, and Brax and Davis 2012;Brax et al. et al. (2013 b ) extended the scenario and showedthat, with suitable generalization of the superpotential and Kahler potential, thevolume modulus can also drive slow-roll inflation in the early universe.The last generic property of chameleons pertains to their cosmological impact. Itwas shown recently by Wang et al. et al. ∼ Mpc. Hence, it has negligible effect on density perturbationson linear scales. Nevertheless, the chameleon mechanism remains interesting asa way to hide light scalars suggested by fundamental theories. The way to testthese theories is to study small scale phenomena, as we will see in Sec. 3.2. • Kinetic/K-mouflage (Babichev et al. , 2009; Dvali et al. , 2011):In this case, the scalar interactions are governed by a ‘kinetic’ function P ( X ),where X = − ( ∂φ ) /
2. The prototypical example is L φ = −
12 ( ∂φ ) − L M ( ∂φ ) . (2.9)Whether or not φ becomes non-linear near a source clearly depends on its localgradient, i.e. , the screening condition in this case takes the form | ∂φ | > L − M Pl . (2.10)In light of (2.4), the screening condition can equivalently be cast as a conditionon the local gravitational acceleration, |∇ Φ J | > | (cid:126)a c | . A useful rule of thumb toascertain which regions of the universe are screened or unscreened is to mapout the gravitational acceleration smoothed on some scale. Cosmologically, thedepartures from standard gravity are most pronounced on scales ∼ > Mpc, that is,on linear scales.Choosing L ∼ H , the strong coupling scale is conveniently of the order of thedark energy scale: Λ s ∼ (cid:112) L − M Pl ∼ meV . (2.11)The scalar field enjoys a shift symmetry φ → φ + c . (2.12) Screening Mechanisms: A Brief Overview
This is mildly broken by the coupling to matter, but given the low value of thecutoff, loop corrections to the mass are small, i.e. , δm φ ∼ Λ /M Pl ∼ H , which ispotentially interesting for late-time cosmology.Screening only works for the particular sign of the ( ∂φ ) term given in (2.9).Indeed, assuming spherical symmetry, the equation of motion outside a pointsource of mass M reduces to, upon using Gauss’ law,d φ d r (cid:18) L M (cid:18) d φ d r (cid:19) (cid:19) = gM πr M Pl . (2.13)Screening relies on the non-linear term dominating close to the source. Very farfrom the source, on the other hand, the equation is to a good approximation linear.Only for this particular choice of sign can the linear and non-linear regimes matchcontinously. Unfortunately, this choice of sign implies that radial perturbationson top of this background propagate superluminally. Correspondingly, the 2 → et al. , 2006). • Vainshtein (Vainshtein, 1972; Arkani-Hamed et al. , 2003; Deffayet et al. , 2002):In this case, the scalar interactions involve second-derivatives of φ . The prototyp-ical Lagrangian is the so-called cubic Galileon (Deffayet et al. , 2002; Luty et al. ,2003): L φ = −
12 ( ∂φ ) − L M Pl (cid:50) φ ( ∂φ ) . (2.14)Whether or not φ develops non-linearities near a massive source, that is, whetheror not the cubic term is large compared to the kinetic term, depends on themagnitude of (cid:50) φ . The screening condition is | ∂ φ | > L − M Pl . (2.15)In light of (2.4), this is equivalent to a condition on the local curvature, |∇ Φ J | > | R c | . A useful rule of thumb to ascertain which regions of the universe are screenedor unscreened is to map out the curvature smoothed on some scale. Like the kineticmechanism, the departures from standard gravity are most significant today onscales ∼ > Mpc.Choosing L ∼ H , the strong coupling scale isΛ s ∼ (cid:112) L − M Pl ∼ (1000 km) − . (2.16)As we will see in Sec. 4, however, the strong coupling scale is renormalized tomuch higher values in the vicinity of massive objects, such as in the solar system.The scalar field enjoys a galilean-like internal symmetry φ → φ + c + b µ x µ , (2.17)where b µ is an arbitrary constant 4-vector. Indeed, under this transformation bothterms in (2.14) shift by a total derivative. This is mildly broken by the coupling creening Mechanisms: A Brief Overview Table 2.1
Classification of screening mechanisms and their properties.
Mechanism Screening Condition Symmetry Cutoff Superluminality ScalesChameleon φ < φ c None Λ s ∼ mm − No ∼ < MpcKinetic | ∂φ | > Λ δφ = c Λ s ∼ mm − Yes ∼ > MpcVainshtein | ∂ φ | > Λ δφ = c + b µ x µ Λ s ∼ (1000 km) − Yes ∼ > Mpcto matter, but the low cutoff value implies a tiny quantum correction to the mass δm φ ∼ Λ /M Pl (cid:28) H .This theory also suffers from superluminal propagation around certain back-grounds. In particular, radial perturbations around a spherically-symmetric sourcepropagate strictly superluminally, as we will see explicitly in Sec. 4. Nevertheless,it has been argued that galileons are protected against the formation of closedtime-like curves (CTC) by a Chronology Protection (Burrage et al. , 2012; Evs-lin, 2012), analogously to what happens in GR (Hawking, 1992). Specifically, ifone tries to create a closed causal time-like curve from healthy initial conditions,the galileon will become strongly coupled ( i.e. , the effective description will breakdown) before the CTC can form.One may wonder whether there are other possibilities, such as screening conditionsinvolving higher-derivatives of the field, e.g. | ∂ φ | > M ? The answer is no, at least inthe context of Lorentz-invariant theories. The Lagrangian in this case would includeterms with n ≥ φ , ∂φ and ∂ φ , are the only possibilities. Of course onecould always consider hybrid versions of these mechanisms. Chameleons
The chameleon mechanism operates whenever a scalar field couples to matter in such away that its effective mass depends on the local matter density (Khoury and Weltman,2004 b ; Khoury and Weltman, 2004 a ; Gubser and Khoury, 2004; Brax et al. , 2004; Motaand Shaw, 2006; Mota and Shaw, 2007). The scalar-mediated force between matterparticles can be of gravitational strength, but its range is a decreasing function ofambient matter density, thereby avoiding detection in regions of high density. Deepin space, where the mass density is low, the scalar is light and mediates a fifth forceof gravitational strength, but near the Earth, where experiments are performed, andwhere the local density is high, it acquires a large mass, making its effects short rangeand hence unobservable.This is achieved with a canonical scalar field with suitable self-interaction potential V ( φ ). The theory (in the weak-field limit and for non-relativistic matter) is givenby (2.5) and (2.6): L cham = −
12 ( ∂φ ) − V ( φ ) − gφM Pl ρ m . (3.1)The dimensionless coupling parameter g is assumed to be O (1), corresponding togravitational strength coupling. The equation of motion for φ that derives from thisLagrangian is ∇ φ = V ,φ + gM Pl ρ m . (3.2)The immediate thing to notice is that, because of its coupling to matter, the scalarfield is governed by a density-dependent effective potential V eff ( φ ) = V ( φ ) + gφM Pl ρ m . (3.3)For suitably chosen V ( φ ), the effective potential can develop a minimum at somefinite field value φ min in the presence of background matter density. This is illustratedin Fig. 3.1 for a monotonically-decreasing V ( φ ) and monotonically-increasing A ( φ ).As shown in Fig. 3.2, the mass of small fluctuations around the minimum, m φ = V ,φφ ( φ min ) , (3.4) increases , while φ min decreases, with increasing density.A prototypical chameleon potential satisfying these properties is the inverse power-law form, V ( φ ) = Λ n φ n . (3.5) hameleons V eff ( φ ) φ V ( φ ) A ( φ ) ρ Fig. 3.1
Schematic of the effective potential felt by a chameleon field (solid line). Theeffective potential is a sum of the bare potential of runaway form, V ( φ ) (dashed line), and adensity-dependent piece, from coupling to matter (dotted line). (Potentials with positive powers of the field, V ( φ ) ∼ φ s with s an integer ≥
2, arealso good candidates for chameleon theories (Gubser and Khoury, 2004).) The effectivepotential in this case is given by V eff ( φ ) = Λ n φ n + gφM Pl ρ m . (3.6)For g >
0, this displays a minimum at φ min = (cid:18) n Λ n M Pl gρ m (cid:19) n +1 . (3.7)The mass of small fluctuations around the minimum is m φ = n ( n + 1)Λ n φ n +2min ∼ ρ n +2 n +1 m . (3.8)Thus φ min and m φ are respectively decreasing and increasing functions of the back-ground density ρ m , as desired.The tightest constraint on the model comes from laboratory tests of the inversesquare law, which set an upper limit of ≈ µ m on the fifth-force range assuminggravitational strength coupling (Adelberger et al. , 2007). This imposes the followingbound at laboratory density ρ lab ∼ − g / cm : m − φ ( ρ lab ) ∼ < µ m . (3.9)Plugging in numbers for the inverse power-law potential, this translates to an upperbound on the scale M (Khoury and Weltman, 2004 b ; Khoury and Weltman, 2004 a ) Chameleons
Fig. 3.2
Effective potential for a ) low ambient matter density and b ) high ambient density.As the density increases, the minimum of the effective potential, φ min , shifts to smaller values,while the mass of small fluctuations, m φ , increases. Λ ∼ < meV , (3.10)which, remarkably, coincides with the dark energy scale. (There is some mild depen-dence on n , which we ignore for simplicity.) This also ensures consistency with allknown constraints on deviations from GR, including post-Newtonian tests in the so-lar system and binary pulsar observations (Khoury and Weltman, 2004 b ; Khoury andWeltman, 2004 a ). Equation (3.9) is at best a rough guess, however, since tests of theinverse square law are performed in vacuum chambers. A careful modeling of the E¨ot-Wash set-up was done in Khoury and Weltman 2004 b ; Khoury and Weltman 2004 a ,including a calculation of the chameleon profile inside the vacuum chamber. The endresult is that Ref. (3.10) is a fairly accurate bound.Using (3.8), the bound (3.9) translates to the following range for cosmic density ρ cosmos ∼ − g / cm : m − ( ρ cosmos ) ∼ < Mpc . (3.11)(This bound is achieved assuming for n (cid:28)
1. For n (cid:29)
1, on the other hand, theupper bound becomes 10 − pc.) The Compton wavelength of the chameleon cosmo-logically is at least a factor of 10 shorter than the Hubble radius H − , which wouldbe a desirable value for quintessence-like behavior. In particular, chameleon effectsare Yukawa-suppressed, and GR is recovered, above the Mpc scale. This confirms theclaim made in Sec. 2 that chameleons have negligible impact on linear-scale densityperturbations today. Although derived above assuming an inverse power-law poten-tial, the Mpc barrier is actually more general. As shown in Wang et al. (2012), thisholds under very general conditions whenever the screening condition is determined bythe local φ value, as is the case for chameleon, symmetron and varying-dilaton scalarfields.Nevertheless the range of the chameleon-mediated force spans a remarkable factorof 10 from the laboratory to the cosmos! The scalar field effectively uses the expo- pherically symmetric source and thin-shell effect Fig. 3.3
Sketch of the set-up for thin-shell calculation. nentially large density of the local environment (compared to cosmological density) tohide itself from experiments, which is why it aptly deserves the name “chameleon”.At a typical place in the solar system, the matter density in baryonic gas and darkmatter is approximately ρ solar − system ∼ − g / cm . The corresponding Comptonwavelength for the chameleon is m − φ ( ρ solar − system ) ∼ < AU , (3.12)which is orders of magnitude larger than the size of the solar system! In other words,the chameleon mediates a long-range force in the solar system, which should have beenseen in post-Newtonian tests of GR. Fortunately, we will see that the chameleon hasanother way of hiding itself, by suppressing its coupling to massive objects, such asthe Sun or the Earth. This is the so-called thin-shell effect, which we describe next. In order to understand in detail how the chameleon force is suppressed in the presenceof high ambient density, we want to solve the field profile in the presence of a massivecompact object, following Khoury and Weltman (2004 a ). We consider a sphericallysymmetric object with radius R , homogeneous density ρ in and total mass M . Further,we imagine that this object is immersed in a homogeneous medium with density ρ out .In the case of the Sun or the Earth, this is meant to model the density of the ambientbaryonic gas and dark matter.The scalar equation of motion (3.2) for a static and spherically-symmetric back-ground reduces to d φ d r + 2 r d φ d r = V ,φ + g ρM Pl , (3.13) Chameleons where the density profile is given by ρ m ( r ) = (cid:26) ρ in ; r < Rρ out ; r > R , (3.14)with corresponding minima of the effective potential denoted by φ min and φ out , respec-tively. The corresponding mass m φ around these minima will be similarly denoted.The situation is sketched in Fig. 3.3. As boundary conditions, the field profile mustbe regular at the origin, d φ/ d r = 0 at r = 0, and minimize the effective potential farfrom the source, φ → φ out as r → ∞ .It is instructive to derive the general solution through simple analytical argu-ments (Khoury and Weltman, 2004 b ; Khoury and Weltman, 2004 a ). For a sufficientlylarge body, in a sense that will be made precise below, the field approaches the mini-mum of its effective potential deep in the interior: φ (cid:39) φ in ; r < R . (3.15)Outside the object, but still within an ambient Compton wavelength away ( r < m − ),the field profile is approximately 1 /r : φ (cid:39) Cr + φ out ; R < r < m − , (3.16)where we have imposed the boundary condition φ → φ out as r → ∞ . The constant C is fixed by matching (3.15) and (3.16) at the surface of the object, i.e. , φ ( r = R ) = φ in .As a result, the exterior solution is φ (cid:39) − Rr ( φ out − φ in ) + φ out . (3.17)The above solution admits a nice analogy with electrostatics (Jones-Smith andFerrer, 2012; Pourhasan et al. , 2011). Indeed, since ∇ φ (cid:39) R near the surface. The surface “charge density” gρ in ∆ R/M Pl supports the discontinuity in field gradients:d φ d r (cid:12)(cid:12)(cid:12)(cid:12) r = R + = gρ in M Pl ∆ R . (3.18)Substituting (3.17), we can solve for the shell thickness:∆ RR = φ out − φ in gM Pl Φ , (3.19)where Φ ≡ M πM R is the surface gravitational potential. For consistency, we shouldhave ∆ R (cid:28) R . An object is therefore said to be screened if φ out − φ in gM Pl Φ (cid:28) . (3.20) xperimental/Observational Tests The exterior field profile in this case can be written as φ ( r > R ) (cid:39) − g πM Pl RR M e − m out ( r − R ) r (screened) , (3.21)where we have restored the Yukawa exponential. This profile is identical to that ofa massive scalar of mass m out , except that the coupling is reduced by the thin-shellfactor ∆ R/R (cid:28) φ out − φ in gM Pl Φ ∼ >
1. For fixed densitycontrast, this corresponds to a weak source ( i.e. , one with small Φ). In this regime,the coupling is not suppressed, and the object is said to be unscreened : φ ( r > R ) (cid:39) − g πM Pl M e − m out ( r − R ) r (unscreened) . (3.22)This is the usual Yukawa profile for a massive scalar.This chameleon screening effect can also be understood qualitatively as follows.If the object is sufficiently massive such that deep inside the object the chameleonminimizes the effective potential for the interior density, then the mass of chameleonfluctuations is relatively large inside the object. As a result, the contribution from thecore to the exterior profile is Yukawa-suppressed. Only the contribution from within athin shell beneath the surface contributes significantly to the exterior profile. In otherwords, since the chameleon effectively couples only to the shell, whereas gravity ofcourse couples to the entire bulk of the object, the chameleon force on an exterior testmass is suppressed compared to the gravitational force. The idea that the manifestation of a fifth force is sensitive to the environment hasspurred a lot of activity. Astrophysically, chameleon scalars affect the internal dy-namics (Hui et al. , 2009; Jain and VanderPlas, 2011) and stellar evolution (Changand Hui, 2011; Davis et al. , 2012; Jain et al. , 2012) of dwarf galaxies residing in voidor mildly overdense regions. In the laboratory, chameleons have motivated multipleexperimental efforts aimed at searching for their signatures: • The E¨ot-Wash experiment searches for deviations from the inverse-square-law atdistances > ∼ µ m. Based on detailed theoretical predictions (Upadhye et al. , 2006),the E¨ot-Wash group analyzed their data to constrain part of the chameleon parameterspace (Adelberger et al. , 2007). • If chameleons interact with the electromagnetic field via e β γ φ F µν F µν , then photonstraveling in a magnetic field will undergo photon/chameleon oscillations, similar toaxions. The CHameleon Afterglow SEarch (CHASE) experiment (Chou et al. , 2009;Steffen et al. , 2010) has looked for an afterglow from trapped chameleons convert-ing into photons. Similarly, the Axion Dark Matter eXperiment (ADMX) resonantmicrowave cavity was used to search for chameleons (Rybka et al. , 2010). Photon-chameleon mixing can occur deep inside the Sun (Brax and Zioutas, 2010) and affectthe spectrum of distant astrophysical objects (Burrage et al. , 2009). Chameleons
Through the nice analogy between chameleon screening and electrostatics (Jones-Smith and Ferrer, 2012; Pourhasan et al. , 2011) mentioned above, a novel experimentaltest of chameleons has been proposed recently that would exploit an enhancementof the scalar field near the tip of pointy objects — a“lightning rod” effect (Jones-Smith and Ferrer, 2012). See Brax et al. (2010) for a discussion of collider signatures,and Upadhye (2013) for signatures of symmetron in the laboratory.The most striking signature of chameleons can be found by testing gravity in space.Because the screening condition depends on the ambient density, small bodies that arescreened in the laboratory may be unscreened in space. This leads to striking predic-tions for future satellite tests of gravity, such as the planned MicroSCOPE mission and STE-QUEST . In particular, chameleons can result in violations of the (weak)Equivalence Principle in orbit with η ≡ ∆ a/a (cid:29) − , in blatant conflict with lab-oratory constraints. Similarly, the total force — gravitational + chameleon-mediated— between unscreened particles can be O (1) larger than in standard gravity, whichwould appear as O (1) deviations from the value of G N measured on Earth. http://microscope.onera.fr/ Galileons
Galileons are scalar field theories with a number of remarkable properties. The simplestgalileon theory (Deffayet et al. , 2002; Luty et al. , 2003; Nicolis and Rattazzi, 2004) wasoriginally discovered as a particular decoupling limit of the Dvali-Gabadadze-Porratimodel (Dvali et al. , 2000; Dvali and Gabadadze, 2001). This was generalized by Nico-lis et al. (2009) to enumerate all possible galileon terms. Galileons also appear inthe decoupling limit of the de Rham-Gabadadze-Tolley (dRGT) massive gravity theo-ries (de Rham et al. , 2011; de Rham et al. , 2011; Hassan and Rosen, 2012; Hinterbich-ler, 2012) and its extensions (D’Amico et al. , 2013; Gabadadze et al. , 2012), as wellas cascading gravity (de Rham et al. , 2008 a ; de Rham et al. , 2008 b ; de Rham et al. ,2009; de Rham et al. , 2010; Agarwal et al. , 2010; Agarwal et al. , 2011).Galileons have been covariantized (Deffayet et al. , 2009 b ; Deffayet et al. , 2009 a ),used to drive cosmic acceleration (Chow and Khoury, 2009; Silva and Koyama, 2009;De Felice et al. , 2011; Deffayet et al. , 2010 b ), applied to inflation (Burrage et al. ,2011 b ; Creminelli et al. , 2011; Mizuno and Koyama, 2010; De Felice and Tsujikawa,2011; Kobayashi et al. , 2011; Renaux-Petel, 2011; Gao and Steer, 2011; Renaux-Petel et al. , 2011), and employed as a self-tuning mechanism (Charmousis et al. , 2012;Copeland et al. , 2012). They have inspired novel scenarios of the early universe, specif-ically Galilean Genesis and related scenarios (Creminelli et al. , 2010; Hinterbichler andKhoury, 2012; Perreault Levasseur et al. , 2011; Liu et al. , 2011; Qiu et al. , 2011; Wangand Brandenberger, 2012; Liu and Piao, 2013; Hinterbichler et al. , 2012 a ; Creminelli et al. , 2013 a ; Creminelli et al. , 2013 b ; Hinterbichler et al. , 2012 b ; Hinterbichler et al. ,2013 a ).Galileons have been extended to p -forms (Deffayet et al. , 2010 a ), multiple fields (Padilla et al. , 2010; Padilla et al. , 2011 b ; Padilla et al. , 2011 a ; Hinterbichler et al. , 2010), andto more general backgrounds (Goon et al. , 2011 c ; Burrage et al. , 2011 a ; Goon et al. ,2011 a ; Goon et al. , 2011 b ). See Khoury et al. a ; Khoury et al. b ; Koehn et al. et al. b ; Koehn et al. a for recent embeddings of galileons andgeneral higher-derivative scalar theories in supersymmetry and supergravity.At the heart of the phenomenological viability of these theories is the Vainshteinscreening mechanism (Vainshtein, 1972; Arkani-Hamed et al. , 2003; Deffayet et al. ,2002). This relies on derivative couplings of a scalar field becoming large in the vicinityof massive sources. These non-linearities crank up the kinetic term of perturbations,thereby weakening their interactions with matter. Historically, these scalar field theories were actually discovered much earlier (Horndeski, 1974;Fairlie and Govaerts, 1993; Fairlie and Govaerts, 1992; Fairlie et al. , 1992). Galileons
The defining properties of galileons are: i ) their equations of motion are second-order; ii ) they enjoy the “galilean” shift symmetry π → π + c + b µ x µ . (4.1)The simplest, non-trivial theory with these properties is the cubic galileon theory givenin Eq. (2.14): L gal = − ∂π ) − (cid:50) π ( ∂π ) − gM Pl πρ m . (4.2)(To conform to the literature, we denote the galileon scalar by π , with appropriatelyrescaled its kinetic term.) To check property ii ), it is clear that, apart from the couplingto matter, the scalar terms are strictly invariant under the ordinary shift symmetry δπ = c . They also shift by a total derivative under δπ = b µ x µ , as can be checkedexplicitly: δ ( ∂π ) = 2 b µ ∂ µ π = 2 ∂ µ ( b µ π ) ; δ (cid:0) ( ∂π ) (cid:50) π (cid:1) = 2 b µ ∂ µ π (cid:50) π = − b µ ∂ µ ∂ ν π∂ ν π = − ∂ µ (cid:0) b µ ( ∂π ) (cid:1) , (4.3)where we have integrated by parts a bunch of times. Hence this is a symmetry of theaction.To check property i ), we vary (4.2) to obtain the equation of motion:3 (cid:50) π + 1Λ (cid:18) ( (cid:50) π ) − ( ∂ µ ∂ ν π ) (cid:19) = g M Pl ρ m . (4.4)This equation is of course non-linear, but nevertheless second-order — the sameamount of initial data as for an ordinary scalar field must be supplied to obtain aunique solution.Since the cubic interaction term is non-renormalizable, we should treat (4.2) as aneffective field theory with cutoff Λ s and allow all possible operators consistent withthe symmetries L = − ∂π ) − (cid:50) π ( ∂π ) + ∞ (cid:88) n =3 c n Λ n − ( (cid:50) π ) n − gM Pl πρ m , (4.5)where the c n ’s are all of O (1). Even if the c n ’s are set to zero classically, they will begenerated by quantum corrections. Now, the Vainshtein mechanism (Vainshtein, 1972;Deffayet et al. , 2002) relies on the (cid:50) π ( ∂π ) term becoming large compared to thekinetic term ( ∂π ) near massive objects, that is, (cid:50) π (cid:29) Λ . In this regime, one wouldexpect that all higher-order operators ( (cid:50) π ) n become important as well, signaling abreakdown of the effective field theory.Contrary to this naive expectation, there is in fact a regime where the galileon termcan dominate while all other operators are negligible. This is because the action (4.5) We ignore higher-order galileon terms for simplicitly. This is a technically natural choice sincethese terms do not get generated by quantum corrections (Hinterbichler et al. , 2010). olution around spherically-symmetric source actually involves two expansion parameters (Luty et al. , 2003; Nicolis and Rattazzi,2004): a classical expansion parameter, α cl ≡ (cid:50) π Λ , (4.6)measuring the strength of classical non-linearities; and a quantum expansion parame-ter, α q ≡ ∂ Λ , (4.7)measuring the relevance of quantum corrections. We will see that there are situationswhere classical non-linearities are important ( α cl (cid:29)
1) while quantum corrections re-main under control ( α q (cid:28) (cid:50) π ) n operators have two derivativesper field, and hence are suppressed by powers of α q relative to the galileon term.Another remarkable fact is that the cubic galileon term does not get renormal-ized (Luty et al. , 2003; Hinterbichler et al. , 2010). Indeed, all terms generated atone-loop are of the schematic formΓ − loop ∼ (cid:88) m (cid:20) Λ + Λ ∂ + log (cid:18) ∂ Λ (cid:19) ∂ (cid:21) (cid:18) ∂∂π Λ (cid:19) m , (4.8)and hence involve 2 derivatives per π . The galileon term, having 3 π ’s and 4 ∂ ’s, istherefore not renormalized. Let us illustrate this in the context of Vainshtein screening for a static point sourceof mass M , so that T µµ = − M δ (3) ( (cid:126)x ). For static, spherically-symmetric ansatz, (4.4)reduces to (Nicolis and Rattazzi, 2004) (cid:126) ∇ · (cid:18) (cid:126)E + ˆ r E r (cid:19) = gMM Pl δ (3) ( (cid:126)x ) , (4.10)where (cid:126)E ≡ (cid:126) ∇ π . (4.11)Integrating over a sphere centered at the origin, this implies6 E + 4Λ E r = gM πr M Pl . (4.12)Remarkably, this equation is algebraic in E , and can be readily solved: This traces back to the special structure of the cubic vertex. Through integration by parts, onecan always arrange for an external leg π ext to be hit by 2 derivatives, e.g. , ∂ µ π∂ µ π ext (cid:50) π = − ∂ ν ∂ µ π∂ ν π∂ µ π ext − ∂ µ π∂ ν π∂ µ ∂ ν π ext = 12 ( ∂π ) (cid:50) π ext − ∂ µ π∂ ν ∂ µ ∂ ν π ext . (4.9) Galileons E ± = Λ r (cid:18) ± (cid:113) r + 4 r r − r (cid:19) (4.13)where we have introduced the Vainshtein radius r V ≡ s (cid:18) gM πM Pl (cid:19) / = (cid:0) g r Sch L (cid:1) / , (4.14)where r Sch = M πM is the Schwarzschild radius of the source, and where, as in (2.16),we have expressed the strong coupling scale as Λ s ≡ ( L − M Pl ) / .This nontrivial profile is crucial to the Vainshtein effect. Below we consider tworegimes. For this purpose, we will focus on the “+” branch, with E → r → ∞ . (The“ − ” branch, which asymptotically matches to the self-accelerated solution (Deffayet,2001), has unstable ( i.e. , ghost-like) perturbations (Luty et al. , 2003; Charmousis et al. ,2006; Gregory et al. , 2007).) • r (cid:29) r V : Far from the source, the solution approximates a 1 /r profile, E ( r (cid:29) r V ) = 3Λ r (cid:32)(cid:114) r r − (cid:33) (cid:39) g · M πM Pl r . (4.15)Correspondingly, the galileon force, F π = g |∇ π | , is g / F π F gravity (cid:12)(cid:12)(cid:12)(cid:12) r (cid:29) r V (cid:39) g . (4.16)(For DGP, for which g = 1, this reproduces the famous 1 / : α cl ∼ (cid:16) r V r (cid:17) (cid:28) α q ∼ r Λ s ) (cid:28) , (4.17)hence classical non-linearities and quantum corrections are both unimportant. • r (cid:28) r V : Close to the source, the solution reduces to E (cid:39) Λ r / √ r ∼ √ r . (4.18)The galileon force is therefore suppressed compared to gravity at distances muchless than the Vainshtein radius: F φ F gravity (cid:12)(cid:12)(cid:12)(cid:12) r (cid:28) r V ∼ (cid:18) rr V (cid:19) / (cid:28) . (4.19)In this regime, the classical non-linearity parameter is of course large, We assume M (cid:29) M Pl , so that Λ − (cid:28) r V . olution around spherically-symmetric source Fig. 4.1
Different regimes for a ) Galileons and b ) GR. In both case, the far region ( r (cid:29) r V for Galileons; r (cid:29) r Sch for GR) corresponds to the weak-field regime, where the description isboth weakly-coupled and classical. In the intermediate region (Λ − (cid:28) r (cid:28) r V for Galileons; M − (cid:28) r (cid:28) r Sch for GR), the description is still classical but strongly non-linear. Below thestrong coupling scale ( r (cid:28) Λ − for Galileons; r (cid:28) M − for GR), the description becomesfully quantum. α cl ∼ (cid:16) r V r (cid:17) / (cid:29) , (4.20)while the quantum parameter is the same as before: α q ∼ r Λ s ) . (4.21)At distances r (cid:29) Λ − , quantum corrections are under control and the classicalsolution can be trusted. Sufficiently close to the source, r (cid:28) Λ − , quantum cor-rections become important and the effective field theory breaks down. (In fact,this conclusion is too conservative — we will review shortly that perturbationsacquire a large kinetic term ∼ ( r V /r ) / , which upon canonical normalizationtranslates to a higher effective cutoff.)Thus, as advocated, there is a window, namely Λ − (cid:28) r (cid:28) r V , where classical non-linearities are important while quantum effects are under control. The various regimesare illustrated in Fig. 4.1a).This is closely analogous to what happens in GR (Hinterbichler et al. , 2010). TheEinstein-Hilbert action, expanded in powers of the canonically-normalized metric per-turbation g µν = η µν + h µν M Pl , is schematically of the form L GR = M √− gR = h∂∂h + (cid:88) n ≥ h n − ∂∂hM n − , (4.22) Galileons where we have suppressed indices for simplicity. In other words, the action consistsof a kinetic term h∂∂h , and an infinite number of interaction terms with exactly twoderivatives and arbitrary powers of h/M Pl . Like the galileon cubic term, the relativecoefficients of these terms are not renormalized, thanks to diffeomorphism invariance.The measure of classical non-linearity is α grav . cl = hM Pl . (4.23)Quantum effects generate higher-curvature terms, which expanded in h are of the form L higher − curv . = √− gR , √− gR µν R µν . . . = (cid:88) n ≥ , m ≥ ∂ m h n M m + n − . (4.24)These are suppressed relative to classical operators by powers of the factor α grav . q = ∂ M . (4.25)A point source induces the spherically-symmetric profile h ∼ r Sch r , for which α grav . cl ∼ r Sch r ; α grav . q ∼ M r . (4.26)Therefore, for r (cid:29) r Sch (such as in the solar system), classical non-linearities areunimportant, whereas for r (cid:28) r Sch (such as inside and near the horizon of a blackhole) they dominate. Meanwhile, quantum effects are negligible for r (cid:29) M Pl butbecome important near and below the Planck length. The black hole horizon is theanalogue of the Vainshtein radius: this is where classical non-linearities are large andproduce important effects, while quantum effects are under control. This is illustratedin Fig. 4.1b). Further light can be shed on the Vainshtein mechanism by considering perturbationsaround the spherically-symmetric background. Expanding (4.2) in perturbations ϕ = φ − ¯ φ gives (Adams et al. , 2006) L ϕ = (cid:20) (cid:18) E (cid:48) + 2 Er (cid:19)(cid:21) (cid:0) ˙ ϕ − ( ∂ Ω ϕ ) (cid:1) − (cid:20) Er (cid:21) ( ∂ r ϕ ) − (cid:50) ϕ ( ∂ϕ ) + gM Pl ϕT µµ ∼ (cid:16) r V r (cid:17) / (cid:18) ˙ ϕ − ( ∂ Ω ϕ ) −
43 ( ∂ r ϕ ) (cid:19) − (cid:50) ϕ ( ∂ϕ ) + gM Pl ϕT µµ . (4.27)where in the last step we have assumed r (cid:28) r V and substituted the expression (4.18)for E deep inside the Vainshtein radius. Here ∂ Ω denotes the usual angular derivatives. bservational Test The key point to notice is that the kinetic term is multiplied by an enhancementfactor of (cid:0) r V r (cid:1) / (cid:29)
1, telling us that perturbations acquire a large inertia near amassive source. Said differently, in terms of the canonically-normalized variable ϕ c ∼ (cid:0) r V r (cid:1) / ϕ , the effective coupling to matter is reduced to g eff ∼ g (cid:18) rr V (cid:19) / (cid:28) g , (4.28)indicating that galileon perturbations decouple from matter. Moreover, the strongcoupling scale Λ s is renormalized to a higher scaleΛ effs ∼ Λ s (cid:16) r V r (cid:17) / (cid:29) Λ s , (4.29)indicating that perturbations have weaker self-interactions.Another noticeable feature of (4.27) is that the speed of propagation in the radialdirection is superluminal : c radials = (cid:114) . (4.30)This is a generic feature of galileons — since galileon interactions are derivative inter-actions, a galileon background (even an arbitrarily weak one) will deform the light-cone for perturbations in such a way that there is always a direction along whichthe speed of propagation is superluminal (Nicolis et al. , 2010). When galileons arecoupled to Lorentz-invariant matter, this allows for the formation of closed time-likecurves (CTCs). Nevertheless, it has been conjectured (Burrage et al. , 2012; Evslin,2012) that galileons may be protected from the formation of CTCs by a ChronologyProtection, analogously to GR (Hawking, 1992). In other words, if one tries to create aCTC from healthy initial conditions, the galileon effective field theory will break downbefore the formation of the CTC (Burrage et al. , 2012). Moreover, it was pointed outrecently (Creminelli et al. , 2013 c ; de Rham et al. , 2013) that galileon theories admit-ting superluminality can sometimes be mapped through field redefinitions to healthygalileon theories, indicating that the apparent superluminalty is unphysical. This issueclearly deserves further scrutiny. The strongest bound on the cubic galileon comes from Lunary Laser Ranging (LLR)observations. (For a review, see Nordtvedt (2003).) Integrated over the last 30 years,LLR monitoring constrain the Moon’s orbit to ∼ < cm accuracy.Deep inside the Vainshtein radius, the galileon-mediated force given by (4.19),albeit weak, gives a small correction to the Newtonian potential: δ ΦΦ (cid:39) g (cid:18) rr V (cid:19) / . (4.31)The current constraint from LLR observations is (Murphy et al. , 2012) Galileons δ ΦΦ ∼ < . × − . (4.32)Recalling the definition of the Vainshtein radius r V ≡ (cid:0) r Sch L (cid:1) / , and substituting r Sch = 0 .
886 cm for the Earth, and r = 3 . × cm for the Earth-Moon distance,the LLR constraint translates to a bound on L (Dvali et al. , 2003; Dvali et al. , 2007;Afshordi et al. , 2009): L ∼ > H − g / (cid:39) g − / Mpc , (4.33)where H − (cid:39) cm (cid:39) et al. , 2012) is expected toimprove this bound by a factor of 10. For g ∼ L to values largerthan H − . Somewhat weaker constraints on L have also been obtained (in the contextof the DGP model) by studying the effect on planetary orbits (Battat et al. , 2008).An important property of galileons is that black holes carry no galileon hair (Huiand Nicolis, 2013; Kaloper et al. , 2011), while stars of course couple to the galileon.This leads to an interesting observational signatures: in the presence of an exter-nal linear gradient, an astrophysical black hole should be offset (by an observableamount) from the center of its host galaxy (Hui and Nicolis, 2012). Cosmologically,the galileon-mediated force becomes important at late times and on large scales. Thisaffects various linear-scale observables (Song, 2008; Afshordi et al. , 2009; Lombriser et al. , 2009), such as enhanced large scale bulk flows (Wyman and Khoury, 2010;Khoury and Wyman, 2009), infall velocities (Zu et al. , 2013) and weak-lensing sig-nals (Wyman, 2011). See Khoury and Wyman 2009; Schmidt 2009; Chan and Scocci-marro 2009; Wyman et al. Beyond the cubic galileon (4.2), there are exactly 5 independent terms in 4 dimen-sions which shift by a total derivative under (4.1), and whose equations of motion aresecond order. (More generally, there are D + 1 galileon terms in D dimensions.) In 4dimensions, they take the form L = π ; L = − ( ∂π ) ; L = − ( ∂π ) (cid:50) π ; L = − ( ∂π ) (cid:18) ( (cid:50) π ) − ( ∂ µ ∂ ν π ) (cid:19) ; L = − ( ∂π ) (cid:18) ( (cid:50) π ) + 2( ∂ µ ∂ ν π ) − (cid:50) π ( ∂ µ ∂ ν π ) (cid:19) . (4.34)In other words, to the cubic action (4.2), which comprises L , L and L , one can alsoadd L and L . None of these terms are renormalized at any order in perturbation This is a consequence of the fact that these are Wess-Zumino terms for spontaneously brokenspace-time symmetries (Goon et al. , 2012). eneral galileons theory (Hinterbichler et al. , 2010). These more general galileons arise, for instance, inthe decoupling limit of dRGT massive gravity (de Rham et al. , 2011; de Rham et al. ,2011; Hinterbichler, 2012).Remarkably, the galileons form an Euler hierarchy (Fairlie and Govaerts, 1993;Fairlie and Govaerts, 1992; Fairlie et al. , 1992; Fairlie, 2011). An Euler hierarchy is asequence of Lagrangians { L , L , . . . } such that: • L n +1 = L ( E L n ), for all n ≥
2, where
E L n denotes the Euler-Lagrange equationfor L n . • L n depends on ∂π , ∂ π , but not on higher derivatives of the field. • The procedure terminates. In D dimensions, one gets exactly D terms in thesequence: { L , . . . , L D +1 } . In particular, L D +2 is a total derivative.To see how (4.34) can be built as an Euler hierarchy, we start with L = − ( ∂π ) and obtain E L = (cid:50) π = ⇒ L = L ( E L ) = − ( ∂π ) (cid:50) π , (4.35)which indeed agrees with L . Similarly, E L = ( (cid:50) π ) − ( ∂ µ ∂ ν π ) = ⇒ L = L ( E L ) = − ( ∂π ) (cid:18) ( (cid:50) π ) − ( ∂ µ ∂ ν π ) (cid:19) , (4.36)which correctly reproduces L , and so on. Exercise : Derive the Euler hierarchy starting from L = − (cid:112) ∂π ) to obtain the DBI galileons . See Eqs. (38)-(41) of de Rham and Tolley (2010) for the answer.The galileon Lagrangians (4.34) can be expressed succinctly as L n +1 = nη µ ν µ ν ··· µ n ν n ∂ µ π∂ ν π∂ µ ∂ ν π · · · ∂ µ n ∂ ν n π . (4.37)The tensor η is defined by η µ ν ··· µ n ν n ≡ n ! (cid:88) p ( − p η µ p ( ν ) η µ p ( ν ) · · · η µ n p ( ν n ) (4.38)where the sum is over all permutations of the ν indices, with ( − p denoting the signof the permutation. The η tensor is • anti-symmetric in the µ indices; • anti-symmetric in the ν indices; • symmetric under ( µ i , ν i ) ↔ ( µ j , ν j ).In this form it is clear why the equations of motion are second order. Indeed, thedangerous higher-derivative terms are of the form E L n +1 ⊃ η µ ν ··· µ n ν n ∂ ν π∂ µ ∂ ν π · · · ∂ µ ∂ µ n ∂ ν n π , (4.39)but this vanishes because η µ ν ··· µ n ν n ∂ µ ∂ µ n π = 0 by the anti-symmetry property of η in the µ indices. Hence the equation of motion involves exactly two derivatives per π , and is therefore manifestly invariant under the galilean symmetry (4.1). Summary
This is truly an exciting time for cosmology. We have a standard cosmological model,the ΛCDM model, which thus far successfully accounts for all observational data.It is remarkably predictive: with a single number Λ parametrizing dark energy, theexpansion and growth histories are uniquely predicted. In the next few years, thestandard model will be confronted by a host of increasingly powerful probes of thelarge scale structure, such as the Dark Energy Survey , BigBOSS , the Large SynopticSurvey Telescope and EUCLID . These precision experiments will test the ΛCDMmodel to unprecedented accuracy and may well reveal the existence of “beyond-the-standard-model” physics, in the form of new light degrees of freedom in the darksector.In these Lectures, we reviewed various extensions of the ΛCDM model character-ized by additional scalar fields. In order to reproduce the successful phenomenology ofGR in the solar system, these scalars must effectively decouple from matter on solarsystem/laboratory scales. This can be achieved through screening mechanisms , whichrely on the high density of the local environment (relative to the mean cosmologi-cal density) to suppress deviations from standard gravity. The manifestation of thenew degrees of freedom (generally scalar fields) therefore depends sensitively on theirenvironment, which in turn leads to striking experimental signatures.We presented 3 screening mechanisms, characterized by whether non-linearitiesare triggered by the local field value φ (chameleon), the first derivative of the field ∂φ (k-Mouflage), or its second derivative ∂ φ (Vainshtein). Observationally, the resultingmodifications are most pronounced, respectively, in regions of low Newtonian potentialΦ (chameleon), low acceleration a = |∇ Φ | (k-Mouflage), and low curvature R = |∇ Φ | (Vainshtein). Chameleon effects are important on non-linear scales today ( ∼ < Mpc),while k-Mouflage/Vainshtein effects are important on linear scales today ( ∼ > Mpc).A tantaliziing feature of these mechanisms is that they also make testable predic-tions for local tests of gravity. The most striking signature of chameleons can be foundby testing gravity in space. Because the screening condition depends on the ambientdensity, small bodies that are screened in the laboratory may be unscreened in space.This leads to striking predictions for future satellite tests of gravity, such as the planned http://cosmology.lbl.gov/BOSS/. ummary MicroSCOPE mission and STE-QUEST . In particular, chameleons can result in vio-lations of the (weak) Equivalence Principle in orbit with η ≡ ∆ a/a (cid:29) − , in blatantconflict with laboratory constraints. Similarly, the total force between unscreened par-ticles can be O (1) larger than in standard gravity, which would be interpreted as O (1)deviations from the value of G N measured on Earth.The Vainshtein mechanism lead to small, but testable, deviations from 1 /r gravityin the solar system. The most sensitive probe is Lunar Laser Ranging, which monitorsthe orbit of the Moon to incredible accuracy. The ongoing APOLLO experiment has thesensitivity to place interesting constraints on the simplest galileon theories. Specifically,it can probe galileons whose characteristic scale L is of order of the present Hubbleradius. http://microscope.onera.fr/ cknowledgements I wish to warmly thank the organizers of the 2013 Post-Planck Cosmology SummerSchool for inviting me to lecture at Les Houches, and all the students at the Schoolfor your enthusiasm and many stimulating discussions. This work was supported inpart by NSF CAREER Award PHY-1145525, NASA ATP grant NNX11AI95G, andthe Alfred P. Sloan Foundation. eferences
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