Towards a fully consistent parameterization of modified gravity
Tessa Baker, Pedro G. Ferreira, Constantinos Skordis, Joe Zuntz
aa r X i v : . [ a s t r o - ph . C O ] J a n Towards a fully consistent parameterization of modified gravity
Tessa Baker ∗ Astrophysics, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK
Pedro G. Ferreira † Astrophysics and Oxford Martin School, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK
Constantinos Skordis ‡ School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD,UK
Joe Zuntz § Astrophysics and Oxford Martin School, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UKDept. of Physics & Astronomy, University College London, WC1E 6BT, UK
There is a distinct possibility that current and future cosmological data can be used to constrainEinstein’s theory of gravity on the very largest scales. To be able to do this in a model-independentway, it makes sense to work with a general parameterization of modified gravity. Such an approachwould be analogous to the Parameterized Post-Newtonian (PPN) approach which is used on the scaleof the Solar System. A few such parameterizations have been proposed and preliminary constraintshave been obtained. We show that the majority of such parameterizations are only exactly applicablein the quasistatic regime. On larger scales they fail to encapsulate the full behaviour of typical modelscurrently under consideration. We suggest that it may be possible to capture the additions to the‘Parameterized Post-Friedmann’ (PPF) formalism by treating them akin to fluid perturbations.
I. INTRODUCTION
It is possible that we live in a Universe in which morethan 96% of the energy and matter density is in the formof an exotic dark substance. The conventional view isthat roughly a quarter of this obscure substance is in theform of dark matter and the remainder is in the formof dark energy. Theories abound that propose explana-tions for dark matter and dark energy and there is anactive programme of research attempting to understandand measure them.It may also be possible that our understanding of grav-ity is lacking, and that Einstein’s theory of General Rela-tivity (and more specifically, the Einstein field equations)are not entirely applicable on cosmological scales. Thepast decade has seen unprecedented growth, from a hand-ful to a veritable menagerie of possible modifications togravity that may be perceived as a fictitious dark sector[1].The proliferation of theories of modified gravitycouldn’t have come at a better time. Observational cos-mology has entered what some have called an era of ‘pre-cision cosmology’. Hubristic as such a point of viewmight be, it is certainly true that cosmology is beinginundated by data, from measurements of the CosmicMicrowave Background (CMB) [2, 3], galaxy surveys [4],weak lensing surveys [5] and measurements of distance ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] and luminosity at high redshift with supernovae Ia [6].With such data in hand it is possible to test cosmolog-ical models and constrain their parameters with someprecision. With the forthcoming experiments currentlyon the drawing board [7–9], great things are expected.In particular, there is a hope that it may be possible todistinguish between the two paradigms: the dark sectorversus modified gravity.The situation in cosmology is reminiscent of that inGeneral Relativity in the late 1960’s and early 1970’s.Then, Einstein’s theory was undergoing a golden age withdiscoveries in radio and X-ray astronomy, as well as pre-cision measurements in the Solar System and beyond,making it increasingly relevant. As a result, a plethoraof alternative theories of gravity were proposed whichcould all in principle be tested (and ruled out) by ob-servations [10–12]. Out of this situation a phenomeno-logical model of modified gravity emerged, the Param-eterized Post-Newtonian (PPN) approximation [13–16],which could be used as a bridge between theory and ob-servations. In other words, from observations it is possi-ble to find the constraints on the parameters in the PPNapproximation. The constraints are model-independent.From any given theory it is then possible to calculate thecorresponding PPN parameters and find if they conformto observations. The PPN approximation is sufficientlygeneral that it can encompass almost all modified theo-ries of gravity that were then proposed.Clearly something like the PPN approach is desirablein cosmology. Given the rapid increase in the number ofmodified theories of gravity, it would make sense to con-struct a parameterization that could serve as a bridge be-tween theory and observation. Observers could expresstheir constraints in terms of a set of convenient parame-ters; theorists could then make predictions for these pa-rameters and check if their theories are observationallyviable. Instead of performing many constraint analyseson individual theories, one could run just a single con-straint analysis on the parameterized framework. Witha dictionary of translations between theories and the pa-rameterization in hand, these general constraints couldbe immediately applied to any particular theory.Another key advantage of a parameterized approach isthat it allows one to explore regions of theory space forwhich the underlying action is not known. For example,in § VI B we will see how the Lagrangian f ( R ) is related toour framework ‘parameters’ (which are really functions,not single numbers – see below). Cosmological data willexclude certain regions of parameter space. If a new formof f ( R ) is proposed in the future, it should be a quickoperation to see whether it falls into the excluded region- even though that particular Lagrangian was not knownat the time that the constraint analysis was performed.In this paper we will discuss the requirements of howto parametrize modifications to gravity on cosmologicalscales. It develops the principles first put forward in [17]and explores how they may be applied more generally.The layout of this paper is as follows. In § II we discussthe idea behind the PPN approach and show that it can’tbe imported wholesale into cosmology. We then brieflylook at attempts at parametrizing gravity and point outtheir limitations. In § III we discuss a possible formal-ism in detail – we co-opt the name Parameterized Post-Friedmann approach from [18] – and argue that it maybe sufficiently general to encompass a broad class of the-ories. In § IV we construct the hierarchy of equationsthat should be satisfied in the case in which there are noextra fields contributing to the modifications to gravity.In § V we discuss the more general case with extra fieldsand how this affects the relations between the differentcoefficients. In § VI we focus on four modified theories ofgravity and analyse how they fit into the formalism thatwe are proposing. Finally, in § VII we summarize thestate of play of the parameterization we are proposing.
II. THE CONVENTIONAL APPROACH
The plan is to construct a parameterization that mightmimic the PPN approach on cosmological scales. Itis therefore useful to look very briefly at the PPN ap-proach, which proceeds as a perturbative expansion in v/c (though we set c=1 in what follows). Consider themodified, linearized Schwarzschild solution: ds = − (1 − dt + (1 − dr + r d Ω Ψ = G Mr Φ = γ PPN
Ψ (1)where G is Newton’s constant, M is the central mass, Ωis the two-dimensional volume element of a sphere, (Ψ, Φ) are gravitational potentials and γ PPN is a PPN param-eter, equivalent to one of the older Eddington-Robertson-Schiff parameters.There are a few properties which are of note in this ex-pression. First of all, the parameterization is constructedaround a solution of the Einstein field equations, theSchwarzschild solution (with γ PPN = 1) – the field equa-tions do not come into play. Second (and this isn’t obvi-ous from the expressions above), the parameter γ PPN onlydepends on parameters in the theory and not on inte-gration constants or ‘environmental’ parameters such asthe central mass. This means that, given a theory, itis possible to predict γ PPN solely in terms of fundamen-tal parameters of the theory (i.e. the parameters in theaction). Finally, we see that the mismatch between thegravitational potentials can be expressed as [14]Φ − Ψ = ζ PPN
Φ (2)with ζ PPN = ( γ PPN − /γ PPN often called the gravitationalslip . General Relativity is recovered when ζ PPN = 0.The idea of applying an equation of the form of eqn.(2)to cosmology emerged from the work of Bertschinger in[19]. Bertschinger showed that on large scales it was pos-sible to calculate the evolution of Ψ and Φ using only in-formation about the background evolution and assuminga closure relationship between the two potentials. Thesimplest assumption is a closure relation of the form ofeqn.(2), but in no way was it implied that this would bea realistic relationship that would be valid in the generalspace of theories of modified gravity.Nevertheless, over the last few years the simplifiedequation for gravitational slip has been adopted as a gen-eral parameterization which should be valid in cosmology[20–22]. It has been shown to be valid in a few cases, inthe quasistatic regime (i.e. on small scales), and explicitexpressions have been found for ζ in terms of fundamentalparameters of those theories (some examples are collectedin [1]). Such a parameterization has been extended to in-clude another parameter, a modified Newton’s constant G eff , which may differ from G . The method is then touse eqn.(2) and a modified Poisson equation, − k Φ = 4 πG eff a X i ρ i ∆ i (3)(where ρ i is the energy density of fluid i and ∆ i is the co-moving energy density) to modify the evolution equationsfor cosmological perturbations. A modified Einstein-Boltzman solver is then used to calculate cosmologicalobservables. The two parameters ( G eff , ζ ) have beenadopted more generally and have been used to find pre-liminary constraints on modified gravity theories by anumber of groups [21–24].Clearly such an approach to parametrizing modifiedgravity has some significant differences with the PPNapproach. For a start, modifications are applied to thefield equations and not to specific solutions of Einstein’sfield equations. This is understandable – the solutionsof interest in cosmology are not only inhomogeneousbut time-varying, unlike the incredible simplicity of theSchwarzschild solution that arises due to Birkhoff’s the-orem. Also, unlike in PPN, the parameters at play – ζ and G eff – will not only depend on fundamental param-eters of the theory but also on the time evolution of thecosmological background.Ideally, any time-dependence in the parameterizationwill be simply related to background cosmological quanti-ties (like the scale factor, energy densities or any auxiliaryfields that are part of the modifications) and not depen-dent upon the time evolution of Φ, Ψ or any other per-turbation variables. For such a requirement to be possi-ble it is essential that any parameterization is sufficientlygeneral to encompass a broad range of theories. As wewill show in this paper, parameterizations using eqns.(2)and (3) are simply not general enough to capture the fullrange of behaviour of modified theories of gravity. It hasbeen argued that such a parameterization can be used asa diagnostic; that is, for example, a non-zero measure-ment of ζ might indicate modifications of gravity [25].This may be true, but such a measurement cannot thenbe used to go further and constrain specific theories. Itwould be more useful to build a fully consistent parame-terization which can be used as a diagnostic and can belinked to theoretical proposals. The purpose of this paperis to take the first steps towards such a parameterization. III. THE FORMALISM
When considering modified gravity theories it can behelpful to cleanly separate the non-standard parts fromthe familiar terms that arise in General Relativity (hence-forth GR). We can always write the modifications as aan additional tensor appearing in the Einstein field equa-tions, i.e. G µν = 8 πG a T µν + a U µν (4)The diagonal components of the tensor U µν are equiva-lent to an effective dark fluid with energy density X andisotropic pressure Y (where the constants have been ab-sorbed). The zeroth-order Einstein equations are then: E F ≡ H + 3 K = 8 πG a X i ρ i + a X (5) E R ≡ − (2 ˙ H + H + K ) = 8 πG a X i P i + a Y (6)where H = H/a is the conformal Hubble parameter and K is the curvature. We will use E F and E R as definedabove throughout this paper. For future use we define E = E F + E R . The summations in the above expressionsare taken over all conventional fluids and dark matter,and dots denote derivatives with respect to conformaltime. In this paper we will largely adhere to the defini-tions and conventions used in [17]. We also note that the Bianchi identity ∇ µ G µν = 0 implies the relation:˙ E F + H ( E F + 3 E R ) = 0 (7)Assuming that the conservation law ∇ µ T µν = 0 holds sep-arately for ordinary matter and the effective dark fluid, X and Y must be related by the equation ˙ X +3 H ( X + Y ) =0. Continuing in this vein, our goal is to write the linearlyperturbed Einstein equations as: δG µν = 8 πG a δT µν + a δU µν (8)In general the tensor δU µν will contain both metric per-turbations and extra degrees of freedom (hereafter d.o.f)introduced by a theory of modified gravity. We can sep-arate δU µν into three parts: i) a part containing onlymetric perturbations, ii) a part containing perturbationsto the extra d.o.f., iii) a part mixing the extra d.o.f. andperturbations to the ordinary matter components: δU µν = δU metric µν ( ˆΦ , ˆΓ ... ) + δU dof µν ( χ, ˙ χ, ¨ χ... ) + δU mix µν ( δρ... )(9)The argument variables in this expression will be intro-duced shortly. We have written the Einstein field equa-tions such that T µν contains only standard, uncoupledmatter terms and hence obeys the usual (perturbed) con-servation equations, δ ( ∇ µ T µν ) = 0. As a result U µν mustobey its own independent conservation equations, so thatat linear order we have δ ( ∇ µ U µν ) = 0. We will use the fol-lowing notation to denote components of δU µν from hereonwards: U ∆ = − a δU , ~ ∇ i U Θ = − a δU i (10) U P = a δU ii , D ij U Σ = a ( U ij − δU kk δ ij )where D ij = ~ ∇ i ~ ∇ j − / q ij ~ ∇ projects out the longitu-dinal, traceless part of δU µν and q ij is a maximally sym-metric metric of constant curvature K . The definitionof U Σ . In the case of unmodified background equationsperturbed conservation equation for U µν gives us the fol-lowing two constraint equations at the linearized level[17]:˙ U ∆ + H U ∆ − ~ ∇ U Θ + 12 a ( X + Y )( ˙ β + 2 ~ ∇ ǫ )+ H U P = 0 (11)˙ U Θ + 2 H U Θ − U P − ~ ∇ U Σ + a ( X + Y )Ξ = 0 (12)where the metric fluctuations β, ǫ and Ξ are defined ineqn.(13).In this paper we will initially present general forms forthe construction of metric-only δU µν that satisfy equa-tions (11) and (12), then impose the restriction thatthe field equations can contain at most second-orderderivatives. Gravitational theories containing derivativesgreater than second-order are generally disfavoured asthey typically result in instabilities or the presence ofghost solutions [1, 26, 27]. However, we note that somespecial cases of higher-order theories are acceptable e.g. f ( R ) theories [28] (see § VI B). Hence we start with thegeneral case in order to indicate how our results may beextended to higher-derivative theories [29].The requirement of second-order field equations meansthat U ∆ and U Θ can only contain first-order derivativeswith respect to conformal time, as can be seen fromeqns.(11) and (12). The specific implications this hasdepends on which of the tensors in eqn.(9) are present.In § IV we will explore the structure of theories with onlymetric perturbations, whilst theories with extra degreesof freedom will be presented in § V. In Appendix B wedisplay formulae for generating constraint equations inan arbitrary-order theory of gravity with no additionaldegrees of freedom.We write the perturbed line element (for scalar pertur-bations only) as: ds = − a (1 − dt − a ( ~ ∇ i ǫ ) dt dx + a (cid:20)(cid:18) β (cid:19) q ij + D ij ν (cid:21) dx i dx j (13)It will prove useful to define the gauge-variant combina-tion: V = ˙ ν + 2 ǫ (14)We also define the gauge-invariant potentials:ˆΦ = −
16 ( β − ~ ∇ ν ) + 12 H V (15)ˆΨ = − Ξ −
12 ˙ V − H V (16)ˆΦ and ˆΨ are equivalent to the Bardeen potentials − Ψ H and Φ A respectively. Note that ˆΨ contains a second-order time-derivative. In the first four sections of thispaper we will frequently use a linear combination of thesevariables that remains first-order in time derivatives ofperturbations, due to a cancellation between the ˙ V terms:ˆΓ = 1 k (cid:16) ˙ˆΦ + H ˆΨ (cid:17) (17)From § V onwards we specialise to the conformal Newto-nian gauge, and hence revert to the familiar potentialsΦ and Ψ. We introduce a shorthand notation for thecomponents of δG µν in exact analogy to that introducedfor δU µν , i.e. E ∆ = − a δG etc. Hereafter the left-handsides of the perturbed Einstein equations will be denotedby: E ∆ = 2( ~ ∇ + 3 K ) ˆΦ − H k ˆΓ − H EVE Θ = 2 k ˆΓ + 12 EVE P = 6 k d ˆΓ dτ + 12 H k ˆΓ − ~ ∇ + 3 K )( ˆΦ − ˆΨ) − E ˆΨ + 32 (cid:16) ˙ E R − H E R (cid:17) VE Σ = ˆΦ − ˆΨ (18) In terms of these variables the perturbed Einstein equa-tions are [17]: E ∆ = 8 πGa X i ρ i δ i + U ∆ (19) E Θ = 8 πGa X i ( ρ i + P i ) θ i + U Θ (20) E P = 24 πGa X i ρ i Π i + U P (21) E Σ = 8 πGa X i ( ρ i + P i )Σ i + U Σ (22)For simplicity we will hereafter consider only the case ofa universe with zero spatial curvature, K = 0. IV. THE GENERAL PARAMETERIZATION -NO EXTRA FIELDSA. General case - unmodified background
Let us begin with the simplest case by applying tworestrictions: i) We consider the case of modifications togravity that appear only at the perturbative level, thatis, they maintain the background equations of GR fora Friedmann-Robertson-Walker metric; ii) there are nonew d.o.f. present in the theory, so δU µν contains onlymetric perturbations. We will relax restriction i) in § IV Dand restriction ii) in § V. We will see shortly that thetreatment presented in this subsection is also applicableto ΛCDM, because the X + Y terms in eqns.(11) and(12) vanish for a cosmological constant. The agreementbetween an exact ΛCDM background and current datameans that theories obeying restrictions i) and ii) areof particular interest, even though they correspond to alimited region of theory space.The requirement of gauge form-invariance placesstrong restrictions on the forms that δU µν can take [17].We will postpone a detailed discussion of these restric-tions until § IV D, where they will be a useful tool in guid-ing us to allowed combinations of metric perturbations.In this subsection it suffices to point out that the stan-dard Einstein field equations of GR are of course alreadygauge form-invariant; so any additive modification like δU µν must be independently gauge form-invariant in or-der to preserve the invariance of the whole expression.This property is a direct consequence of the fact that wehave not yet modified the background equations. In thiscase the only objects that can be present in the tensor δU µν are the gauge-invariant metric potentials ˆΦ and ˆΓ.So, we can construct the tensor δU µν from series ofall the possible derivatives of ˆΦ and ˆΓ. This structureshould be general enough to encompass any metric the-ories, where the action is constructed purely from curva-ture invariants, e.g. f ( R ) gravity, Gauss-Bonnet gravity[30] and Lovelock gravity [31]. If we wish to parameterizeonly second-order theories then we will need to truncatethese series at N = 2, as discussed in section III. Thecomponents of U µν are given by: U ∆ = N − X n =0 k − n (cid:16) A n ˆΦ ( n ) + F n ˆΓ ( n ) (cid:17) (23) U Θ = N − X n =0 k − n (cid:16) B n ˆΦ ( n ) + I n ˆΓ ( n ) (cid:17) (24) U P = N − X n =0 k − n (cid:16) C n ˆΦ ( n ) + J n ˆΓ ( n ) (cid:17) (25) U Σ = N − X n =0 k − n (cid:16) D n ˆΦ ( n ) + K n ˆΓ ( n ) (cid:17) (26)The coefficients A n - K n are functions of the scale factor a , wavenumber k and background quantities such as ˙ ρ i –for the sake of clarity we will suppress these dependenciesthroughout. The factors of k ensure that the coefficientfunctions are dimensionless.Let us take a moment to explain the upper limits onthe summations in eqns.(23)-(26). ˆΦ and ˆΓ are first-orderin time derivatives (see eqns.(15) and (17)). U ∆ is differ-entiated in eqn.(11), so truncating the series in eqn.(23)at ˆΦ ( N − gives field equations containing time deriva-tives of order N. U Θ is treated analogously to U ∆ . As U P and U Σ are not differentiated in the components of theBianchi identity, the series in eqns.(25) and (26) are al-lowed to extend one order higher than those in eqns.(23)and (24).We substitute our forms for U ∆ , U Θ , U P and U Σ intothe components of the Bianchi identity (11) and (12). ˆΦand ˆΓ are non-dynamical fields and so will not evolvein the absence of source terms. Yet when we performthe substitution, the Bianchi identity appears to give ustwo evolution equations for ˆΦ and ˆΓ. The only way thiscan be avoided is if the coefficients of each term ˆΦ ( n ) and ˆΓ ( n ) vanish individually, which provides us with con-straint equations on the functions A n - K n (this procedurewill be clarified with an example shortly). Each compo-nent of the Bianchi identity results in N constraint equa-tions from each of the ˆΦ ( n ) terms and ˆΓ ( n ) terms, andeqns.(23)-(26) contain 8N-4 coefficient functions in total.Hence we have 4N − B. Second-order case - unmodified background
In Appendix B we give formulae for generating theconstraint equations of an arbitrary-order theory withunmodified background equations. We will now explic-itly present the second-order case, which corresponds tosetting N=2 in eqns.(23)-(26). In a general case this willgive us four free functions. However, if the backgroundequations are unaltered then we must set F = I = 0 be-cause ˆΓ contains a second-order conformal time deriva-tive of the scale factor. One might consider cancelling this ¨ a term by adding a term proportional to EV , butthis would break the gauge-invariance of the perturbedEinstein equations. We will see later that modification ofthe background equations allows us to add an EV termwithout violating gauge-invariance, which in turns meansthat ˆΓ can be present in U ∆ and U Θ .Using eqns.(11) and (12) we find that setting F = I =0 forces J = K = 0 also. Then, for the second-ordercase, the remaining terms in δU µν are: U ∆ = A k ˆΦ U Θ = B k ˆΦ U P = C k ˆΦ + C k ˙ˆΦ + J k ˆΓ U Σ = D ˆΦ + D k ˙ˆΦ + K ˆΓ (27)The constraint equations are given in Table I, indicat-ing the terms and Bianchi identity from which they arise( B ⇒ eqn.(11), B ⇒ eqn.(12)). These expressions canbe generated using the formulae in Appendix B. We cansee immediately that the ˆΓ terms in U P and U Σ vanish,leaving δU µν expressed entirely in terms of ˆΦ and ˙ˆΦ. Wehave two free functions remaining, which we will chooseto be D and D . Eliminating C from the two ˙ˆΦ con-straints gives (where H k = H /k ): H k D = −
12 ( A + 3 H k B ) (28)The combination on the right-hand side appears when weform the (Fourier-space) Poisson equation from eqns.(19)and (20), where it acts to modify the value of Newton’sgravitational constant: − k ˆΦ = 4 π G − ˜ g a X i ρ i ∆ i (29)where ∆ i = δ i + 3 H (1 + w i ) θ i is a gauge-invariant matterperturbation and˜ g = −
12 ( A + 3 H k B ) (30)The sum in eqn.(29) is over all known fluids and darkmatter, and G denotes the canonical value of Newton’sconstant. From here on we will replace D /k in eqn.(27)by ˜ g/ H to remind us of the connection between the mod-ifications to the slip relation and the Poisson equation.We will also replace D by ζ to distinguish it from theother coefficient functions, which can all be expressed interms of ˜ g and ζ using the constraint equations. We con-tinue to suppress the arguments of ˜ g and ζ .The effective gravitational constant appearing in thePoisson equation is G eff = G / (1 − ˜ g ). The tracelessspace-space component of the Einstein equations be-comes:ˆΦ − ˆΨ = 8 πG X i ( ρ i + P i )Σ i + ζ ˆΦ + ˜ g H ˙ˆΦ (31) Origin Constraint equation1 [B1] ˆΦ ˙ A + H A + kB + H C = 02 [B1] ˙ˆΦ A + H k C = 03 [B1] ˆΓ J = 04 [B2] ˆΦ ˙ B + 2 H B − kC + kD = 05 [B2] ˙ˆΦ B − C + D = 06 [B2] ˆΓ 2 K − J = 0TABLE I: Table of the constraint equations for the second-order metric theory specified in § IV B. These can be generatedusing the formulae in Appendix B.
The anisotropic stress perturbation Σ i is automaticallygauge-invariant, but negligible for standard fluids at latetimes. The above expression echoes its PPN equivalent,eqn.(2); but note that, as discussed in § II, ζ is a func-tion of background quantities (which potentially intro-duce time- and scale-dependence), whereas ζ PPN dependedonly upon fundamental parameters of a gravitational the-ory.Other authors have made numerous different choicesfor the two free functions of a second-order theory; auseful summary of some of these is provided by [22] . Acommon choice is to introduce a function Q = G eff /G ,related to our ˜ g by Q = (1 − ˜ g ) − [21] (though dif-ferent notation is in no short supply) . The relation-ship between the two potentials is often parameterizedas ˆΦ = η slip ( a, k ) ˆΨ in the spirit of the PPN parameter γ P P N [13]. It might be felt that by introducing yet an-other parameterization of PPF we are adding to this dis-array. However, in the next subsection we will argue thata two-function slip relation such as eqn.(31) is needed toavoid implicitly introducing higher-order derivatives intoa purely metric theory.Writing the relationship between the two gauge-invariant potentials as ˆΦ = η slip ( a, k ) ˆΨ implies that thespatial off-diagonal component of the Einstein field equa-tions is:ˆΦ − ˆΨ = (cid:18) − η slip (cid:19) ˆΦ (32)Comparing the above equation with eqn.(31) implies: η − = 1 − ζ − ˜ g H d ln ˆΦ dτ (33)Now η slip has an environmental dependence, which isproblematic. We would require detailed knowledge of theenvironment in which we wish to test a theory a priori ,and the PPF functions would need to be recalculated fornumerous different situations. Unless ˙ˆΦ = 0, the param-eterizations in eqns.(31) and (32) do not have a simpleequivalence.A degeneracy arises between ˜ g and ζ when comparingto data from weak gravitational lensing, which probes thecombination Φ + Ψ in the conformal Newtonian gauge. In parameterizations equivalent to ( Q, η slip ) the degener-acy is Q (1 + 1 /η slip ), so for lensing applications it makessense to define new parameters along and perpendicularto the degeneracy direction [32–34]. In the (˜ g, ζ ) parame-terization a degeneracy remains. The dominant contribu-tions to lensing signals come from quasistatic scales, onwhich time derivatives of perturbations can be neglected(see later for a fuller discussion). The degeneracy is then: − k (Φ + Ψ) = 4 πG a X i ρ i δ i (2 − ζ )1 − ˜ g (34)It seems that neither of the two parameterizations pre-sented so far are optimal for weak lensing constraints. C. Why neglecting ˜ g in the slip relation implies ahigher-derivative theory We have seen in the previous section that in a metric-based second-order theory of modified gravity the mostgeneral form of the gravitational slip should be expressedin terms of the gauge-invariant potential ˆΦ and its firstderivative with respect to conformal time. Two free func-tions ζ and ˜ g were used as the coefficients of these termsrespectively, where ˜ g resulted in a modification to New-ton’s gravitational constant in the Poisson equation. Us-ing a single function to relate ˆΦ and ˆΨ is equivalent tosetting ˜ g = 0 (see eqn.(31)), which is inconsistent withwith allowing a second free function to modify Newton’sconstant. Making the choice ˜ g = 0 uses up one degree offreedom, leaving us only a single free function with whichto describe the system.The above reasoning is set within the confines of asecond-order theory. We will now show that using asingle function to relate ˆΦ and ˆΨ whilst maintaining G eff = G is equivalent to invoking a higher-derivativetheory of gravity. To do this, let us consider the formthat the tensor δU µν would take in a third-order theory.Its components would be: U ∆ = A k ˆΦ + A k ˙ˆΦ + F k ˆΓ U Θ = B k ˆΦ + B ˙ˆΦ + I k ˆΓ U P = C k ˆΦ + C k ˙ˆΦ + C ¨ˆΦ + J k ˆΓ + J k ˙ˆΓ U ∆ = D ˆΦ + D k ˆ˙Φ + D k ¨ˆΦ + K ˆΓ + K k ˙ˆΓ (35)The constraint equations for this system are given inTable II. We will continue to define the combination thatmodifies G as ˜ g = − . A +3 H k B ), but note that thisis no longer equal to H k D as it was in the second-ordercase. Consider the case where we set D = D = K = K = 0, that is, we use a single function to relate ˆΦand ˆΨ. Through linear combinations of the constraints Origin Constraint equation1 [B1] ˆΦ ˙ A + H A + kB + H C = 02 [B1] ˙ˆΦ ˙ A + H A + kA + kB + H C = 03 [B1] ¨ˆΦ kA + H C = 04 [B1] ˆΓ ˙ F + H F + kI + H J = 05 [B1] ˙ˆΓ kF + H J = 06 [B2] ˆΦ ˙ B + 2 H B − kC + kD = 07 [B2] ˙ˆΦ ˙ B + kB + 2 H B − kC + kD = 08 [B2] ¨ˆΦ B − C + D = 09 [B2] ˆΓ ˙ I + 2 H I − kJ + kK = 010 [B2] ˙ˆΓ I − J + K = 0TABLE II: Table of the constraint equations for the third-order metric theory specified in § IV C. These can be generatedusing the formulae in Appendix B. in Table II we derive the expressions: A + 3 H k B = 0 (36) F + 3 H k I = 0 (37)˜ g = 32 B H k + 13 − ˙ H k k ! = 12 k B (cid:18) E k (cid:19) (38)The first two of these expressions are the combinationsthat appear when we form the Poisson equation. Theyindicate that the potential additive modifications pro-portional to ˙ˆΦ and ˆΓ disappear; the format of eqn.(29) isretained. Eqn.(38) shows that we can only have a modi-fication to the effective gravitational constant if B = 0,and so from eqn.(36) A = 0 also. Using the third equa-tion in Table II, C = 0 in this case. Hence we are forcedto include ˙ˆΦ terms in U ∆ and U Θ , and a ¨ˆΦ term in U P .Since ˆΦ contains a first-order time derivative already (seeeqn.(15), the ˙ U ∆ in eqn.(11) will result in field equationscontaining third-order time derivatives - a higher-ordergravitational theory.This result is a direct consequence of choosing D = 0in U Σ . Removing this constraint changes eqn.(38) to:˜ g = 12 k B (cid:18) E k (cid:19) + H k D (39)which permits B = 0 , ˜ g = 0, as we had in § IV B.The above findings make sense within the context ofthe Lovelock-Grigore theorem [35, 36], which states thatunder the assumptions of four-dimensional Riemanniangeometry and no additional fields, the Einstein-Hilbertaction (plus a cosmological constant) is the only possibleaction that leads to local second-order field equations.In eqn.(27) the presence of D /k in U Σ means that thisparameterization implies a non-local theory. This is notin itself problematic – nonlocal theories can arise whena degree of freedom has been integrated out, or elimi-nated from the action using an integral solution of thecorresponding equation of motion. If D = D = K = 0 in eqns.(35) there are no nonlocal terms present in thegravitational field equations, so we should not be sur-prised that the Lovelock-Grigore theorem prevents usfrom obtaining a second-order theory. In § VI we willmeet theories which evade the Lovelock-Grigore theoremin a number of different ways: by introducing new d.o.f.(e.g. scalar-tensor theory), higher-order field equations (cid:0) f ( R ) gravity (cid:1) , or through nonlocality and extra dimen-sions (DGP). D. Cases with ‘XY’ backgrounds
The previous examples have all assumed that thebackground field equations are those of a Friedmann-Robertson-Walker metric plus standard cosmological flu-ids. We now relax this assumption and consider theorieswhich modify the Einstein field equations at both thebackground and perturbative levels. It is well-known thatany modification to background-level field equations is in-distinguishable from the effects of a dark fluid [37]; hencewe can write any background equations as the standardFRW ones with an additional energy density and pres-sure, see eqns.(5) and (6). We will refer to such theoriesas having ‘XY backgrounds’.Any extension to GR must preserve the property of dif-feomorphism invariance. Invariance under passive diffeo-morphisms corresponds to the familiar principle of gen-eral covariance. Applying a passive diffeomorphism willgenerally result in field equations which look differentto those in the old co-ordinate system. In contrast, in-variance under active diffeomorphisms requires that theactual form of field equations remains unchanged by agauge transformation. In Table III we list the gaugetransformations for relevant variables. In practical terms,gauge form-invariance means that the extra terms thatappear under a gauge transformation must cancel eachother (using identities from the zero-order field equationsif need be). This places tight restrictions on our form for δU µν .To see how this happens in ordinary GR, consider thelinearly perturbed ‘00’ component of the Einstein equa-tions (that is, eqn.(19) with U ∆ set to zero). Whenwe apply a gauge transformation the left-hand side ac-quires a term − H Eξ/a . This is cancelled by the trans-formation of δ on the right-hand side, provided that E = E F + E R = 8 πGa P i ρ i (1 + w i ), i.e. provided thatthe zeroth-order equations are satisfied. This is whywe were only able to use gauge-invariant potentials in δU µν in § IV A and § IV B: if we don’t alter the zeroth-order equations, adding anything else breaks gauge form-invariance.Now that we wish to consider XY backgroundsthis procedure no longer works, because E =8 πG a P i ρ i (1 + w i ). We must add a new term to U ∆ that will produce a part like a H a ( X + Y ) ξ under a gaugetransformation. Only then will the gauge-variant partscancel by virtue of the zeroth-order equation. Metric variables Fluid variablesΞ → Ξ − ˙ ξa δ → δ − a (1 + w ) H ξǫ → ǫ + a [ ξ + H ψ − ˙ ψ ] θ → θ + a ξβ → β + a [6 H ξ − k ψ ] Π → Π + a [ ˙ w − w H (1 + w )] ξν → ν + a ψ Σ → Σ V → V + a ξ Components of δG µν E ∆ → E ∆ − a H ( E F + E R ) ξ E Θ → E Θ + a ( E F + E R ) ξE P → E P + ξa ( ˙ E R − H E R ) E Σ → E Σ TABLE III: Behaviour of metric and fluid variables underinfinitesimal diffeomorphisms generated by the vector field ξ µ = a ( − ξ, ~ ∇ i ψ ). Note that the shear Σ is gauge-invariant. As a toy example, consider a simple theory whichmodifies the zeroth-order equations solely by introduc-ing time-dependence to Newton’s gravitational constant.Following the notation of previous sections, we can writethe sum of the modified Friedmann and Raychaudhuriequations as: E = 8 π G − ˜ g b ( a ) a X i ρ i (1 + w i ) (40)Rewriting this in the form of ordinary GR (and hereaftersuppressing the argument of ˜ g b ): E = 8 πG a X i ρ i (1 + w i ) + ˜ g b E (41)from which we can identify a ( X + Y ) = ˜ g b E (see eqns.(5)and (6)). A possible form for U ∆ is then: U ∆ = − g b H (cid:20) k ˆΓ + 14 EV (cid:21) + A k ˆΦ (42)Unlike the second-order example of § IV B, U ∆ now con-tains a ˆΓ term. The offending second-order derivativeof the scale factor present in ˆΓ is cancelled by the termproportional to EV . We did not have the freedom toadd such a term in the case of unmodified backgroundequations, because V is gauge-variant.Using eqn.(42) in eqn.(19) and the transformationsgiven in Table III, it can be verified that the gauge-variant parts cancel by satisfying eqn.(40). Note thatthe need to have gauge-invariant second-order equationshas totally fixed the first term in eqn.(42); all freedom re-sides in the gauge-invariant part of U ∆ via the function A .As a quick sanity check, one can verify that the fieldequations remain second order in the conformal Newto-nian gauge. In this gauge V = ˙ V = 0, Ξ = − Ψ and β = − /k ( ˙ˆΦ + H Ψ)). U ∆ is then explicitly first-order in derivatives, resulting in second-order equations. U Θ is treated analogously to U ∆ , and has the form: U Θ = 2˜ g b (cid:20) k ˆΓ + 14 EV (cid:21) + B k ˆΦ (43)Combining eqns.(42) and (43), we find that the Poissonequation has the same form as it did in the case with un-modified background equations, eqn.(29). We define thecombination ˜ g = − / A + 3 H k B ) as we did in § IV B.We will assume that the modifications to Newton’s grav-itational constant appearing in the zeroth-order and per-turbed equations are the same, i.e. ˜ g b = ˜ g , noting thatwe have not formally proved this to be the case.Once we have deduced the form of U ∆ and U Θ , U P and U Σ can be found using the Bianchi identities (cid:0) eqns.(11)and (12) (cid:1) . Eliminating the free function A in favour of˜ g and B , these are: H U P = 3 ˆΦ (cid:20) k H ˙ B + kB L + 23 k (cid:0) ˙˜ g + H ˜ g (cid:1)(cid:21) +3 ˙ˆΦ (cid:20) k H B + ˜ g (cid:18) E k (cid:19)(cid:21) +6 (cid:0) ˙˜ g H + ˜ gL (cid:1) (cid:20) k ˆΓ + 14 EV (cid:21) +6˜ g H ddτ (cid:20) k ˆΓ + 14 EV (cid:21)
32 ˜ gEV (cid:20) k − ˙ H (cid:21) − E ˜ g H ˙ V (44) U Σ = ˆΦ2 H k (cid:20) k (cid:0) ˙˜ g + H ˜ g (cid:1) − B (cid:18) E k (cid:19)(cid:21) − ˜ g H (cid:16) k ˆΓ − ˙ˆΦ (cid:17) where L = H + ˙ H − k . (45)As expected, we find that U Σ contains only gauge-invariant perturbation variables. This must be the casesince all other terms in eqn.(22) are gauge-invariant, sothere is nothing to cancel against.If we express the last term in U Σ in terms of ˆΨ wefind that we can write the relationship between the twopotentials as ˆΦ = η slip ( a, k ) ˆΨ for this toy example. Wehave already chosen our two PPF functions to be ˜ g and B , so η slip is simply a particular combination of these: η slip = H (1 − ˜ g ) H (1 − ˜ g ) − ˙˜ g + B k (cid:0) E + k (cid:1) (46)The key result of this toy example is that (for purelymetric theories) δU µν can have a more complex formwhen the background equations are not standard FRWplus standard cosmological components (baryons, CDM,etc.) The hierarchy of constraint equations becomes morecomplex due to the non-zero X and Y terms in eqns.(11)and (12). We can use the principles of energy conserva-tion, gauge-invariance and second-order field equations asa shortcut to the correct forms; the same results would beobtained by solving the hierarchy of constraint equationsdirectly. V. THE GENERAL PARAMETERIZATION-EXTRA FIELDS
The formalism we have developed so far is only ap-plicable to purely metric theories. Yet, as discussed in § II, the majority of modified gravity theories introducenew degrees of freedom, often as additional scalar, vec-tor or tensor fields. It is not immediately obvious thatthe behaviour of these theories can be be encapsulatedby a either a (
Q, η slip ) or (˜ g, ζ ) parameterization. Thereis a risk that we might develop a model-independent for-malism that does not map onto most of our well-studiedtheories.In § VI we will study several example cases, chosen tobe representative of common classes of modified gravitymodels, and ask whether they can be expressed in thetwo-function format of quasistatic PPF. Attempting tomap disparate theories onto a single framework is onlyplausible if those theories share some common features.Hence, before turning to specific examples, we wish toconsider what general statements can be made about thestructure of the the field equations in theories with extradegrees of freedom.Let the scalar perturbations to an extra degree of free-dom be denoted by χ , e.g. if the new degree of freedom isa scalar field φ then χ = δφ . (We have succumbed to thecommon but unwise choice of terminology by referringto ‘scalar perturbations’, even though the new degree offreedom may itself be a vector or tensor field. ‘Spin-0 per-turbations’ would be a better choice of terminology [38]).If we are to obtain second-order field equations then weknow that U ∆ and U Θ can only contain the perturbations χ and ˙ χ .More generally we can introduce multiple new d.o.f.and denote their scalar perturbations by the vector ~χ ,with components χ ( i ) . The perturbed field equations ina general gauge are awkward and rarely used; hence wewill specialise to the conformal Newtonian gauge for theremainder of this paper. The relevant expressions forscalar-tensor theory ( § VI A) are presented in a generalgauge in Appendix A.In the conformal Newtonian gauge the 00-element ofthe tensor δU µν can be represented as: U ∆ = k ~α T ~χ + k~α T ˙ ~χ + A k Φ + A k ˙Φ+ F k Ψ + F k ˙Ψ + M k δρ + M k δ ˙ ρ (47)where δρ is the total energy density fluctuation of stan-dard cosmological fluids (similarly for θ, Π and Σ to beused shortly). ~α and ~α denote vectors of functions with the same dimensionality as ~χ . ~α i , A i , F i and M i arefunctions of background quantities such as ρ and a ; thesedependencies have been suppressed for clarity. Note thateqn.(47) has the form indicated schematically in eqn.(9).The term M δ ˙ ρ represents a modification which dependson the rate of change of the density fluctuations of or-dinary matter. Whilst formally this term is permittedto be present in U ∆ , we are unaware of any theory ofmodified gravity that results in a perturbed 00-equationwith a term like this. Theories employing a chameleonmechanism introduce modifications to GR that dependupon the environmental matter density, but not upon its rate of change . We will therefore choose M = 0 in whatfollows. The δρ term in eqn.(47) can be eliminated usingthe relation:8 πG a δρ = E ∆ − U ∆ (48) E ∆ can be expressed in terms of Φ , ˙Φ and Ψ in the con-formal Newtonian gauge, see eqns.(18). Rearranging andredefining the coefficient functions, we obtain: U ∆ = k ~α T ~χ + k~α T ˙ ~χ + A k Φ + A k ˙Φ+ F k Ψ + F k ˙Ψ (49)This procedure has enabled us to eliminate energydensity fluctuations of ordinary matter from U ∆ . If thenew d.o.f. are not coupled to ordinary matter then δρ does not appear in U ∆ anyway. It is worth remindingthe reader that in this section we are working in theconformal Newtonian gauge, so Φ and Ψ should notbe confused with their gauge-invariant counterparts ˆΦand ˆΨ, which already contain first- and second-orderderivatives respectively. If we were to consider eqn.(49)in a general gauge we would find that ˙ˆΦ and ˆΨ onlyappear in the combination ˆΓ. The ˙Ψ in eqn.(49) wouldbecome − ˙Ξ (cid:0) recall that in the conformal Newtoniangauge ˆΦ ≡ Φ ≡ − χ and ˆΨ ≡ Ψ ≡ − Ξ, using eqns.(15)and (16) (cid:1) .We apply a similar treatment to the remaining compo-nents of δU µν . Recall that U P and U Σ are permitted tocontain second-order terms such as ¨Φ and ¨ χ . Terms suchas ˙ θ, ˙Π , ¨Π , ˙Σ and ¨Σ are discarded to maintain containconsistency with our treatment of U ∆ ; we stress againthat this is done only on an intuitive basis. The result-ing expressions for δU µν (together with eqn.(49)) are asfollows: U Θ = k~β T ~χ + ~β T ˙ ~χ + B k Φ + B ˙Φ + I k Ψ + I ˙Ψ (50) U P = k ~γ T ~χ + k~γ T ˙ ~χ + ~γ T ¨ ~χ + C k Φ + C k ˙Φ+ C ¨Φ + J k Ψ + J k ˙Ψ + J ¨Ψ (51) U Σ = ~ε T ~χ + 1 k ~ε T ˙ ~χ + 1 k ~ε T ¨ ~χ + D Φ + D k ˙Φ+ D k ¨Φ + K Ψ + K k ˙Ψ + K k ¨Ψ (52)where β i , . . . ε i and B i , . . . K i denote functions of back-ground quantities.0The Bianchi identities then give us two constraintequations coupling terms in χ ( i ) , Φ , Ψ and their deriva-tives. In contrast to the previous section we can no longerset the coefficients of the each term to zero individually.In the case without extra fields this was possible becauseall our variables were non-dynamical, so obtaining evo-lution equations for them would be unphysical. But nowthat extra fields appear in δU µν , the Bianchi identitiesyield equations describing how the metric variables re-spond to the set of perturbations ~χ . Therefore we nolonger have a hierarchy of constraint equations for thecoeffcients α , . . . K that allow us to reduce them downto two functions. This is not problematic in itself. Tomap a specific theory onto the parameterization we cansimply pull the necessary coefficients out of the perturbedfield equations. We will see shortly ( § VI) that in manycases that these are relatively simple functions.To constrain a general parameterized theory usingMarkov Chain Monte Carlo (MCMC) analysis we insteadchoose some sensible ansatz for the functions α , . . . K .For example, a Taylor series up to cubic order in Ω Λ wasused in [39, 40]; the MCMC then constrains the coeffi-cients of the terms in the Taylor series. Rigidly fixingthe format of the parameterization in this way meansthat we simply have to constrain real numbers. Thissimplicity is a key advantage of explicitly parameterizingfor the new fields as in eqns.(49-52). The alternative ap-proach – absorbing the new fields into an evolving G eff and slip parameter where possible – will give G eff and ζ very complicated forms that are difficult to parameter-ize (for example, see eqns.(76) and (96)). The trade-offis that our method requires considerably more than twocoefficient functions. We expect that some of these willbe well-constrained by the data, others less so.In the case of just one or two new d.o.f., the systemconsisting of the Einstein equations, the two conservationequations for ordinary matter and two Bianchi identitiesfor the U -tensor can be solved. In order to avoid a con-tradiction, the Bianchi identities for the U -tensor mustbe equivalent to the equations of motion for the extradegrees of freedom (obtained by varying the action withrespect to the extra fields or similar). Futhermore, whenonly a single d.o.f. is present the solutions of the twocomponents of the Bianchi identity must be consistentwith each other.When more than two new d.o.f. are present the Bianchiidentities do not provide sufficient information to solvethe system, and we must supply additional relations be-tween the new d.o.f, metric variables and matter vari-ables. With our goal of an abstract, unified frameworkin mind, we will introduce a general structure to tacklesuch cases. We make the conjecture that one can use the‘generalized dark matter’ (GDM) formalism developed byHu [41] in order to obtain the necessary closure relations.GDM is a phenomenological model in which specificationof three parameters - an equation of state, a rest-framesound speed and a viscous sound speed - suffice to recon-struct the full perturbed stress-energy tensor of a fluid. Cold dark matter, radiation, massive neutrinos, WIMPs,scalar fields and a cosmological constant can all be re-covered as limiting cases of GDM. In our case the d.o.f.parameterized as GDM may be genuine fluid components(e.g. scalar or vector fields), or effective fluids (eg. thescalaron of f ( R ) gravity, the Weyl fluid of DGP gravity- see § VI). For example, in eqn.(49) we identify the extrad.o.f. with an energy density perturbation: ~α T ˆ ~χ + ~α T ˙ˆ ~χ = 8 πG a ρ E ∆ E (53)where the ‘hat’ symbol indicates that we have foldedin the necessary metric perturbations to make gauge-invariant versions of χ and ˙ χ . Similarly we can con-struct gauge-invariant ˆ ~χ, ˙ˆ ~χ and ¨ˆ ~χ from the terms ineqns.(50)-(52), and identify these with velocity, pressureand anisotropic stress perturbations respectively. A sub-script E will be used to indicate these effective perturba-tions.The GDM formalism then provides a way of reducingthese four fluid perturbations to just two, which can thenbe related to the metric potentials via the perturbed con-servation equations. These are [41]:˙ δ E = − (1 + w E ) (cid:16) k θ E − (cid:17) + ˙ w E δ E w E − H w E Γ E +[ . . . ] Φ + [ . . . ] ˙Φ + [ . . . ] ¨Φ + [ . . . ] Ψ + [ . . . ] ˙Ψ + [ . . . ] ¨Ψ (54)˙ θ E = −H (cid:0) − c (cid:1) θ E + c δ E + w E Γ E (1 + w E ) −
23 Σ E +[ . . . ] Φ + [ . . . ] ˙Φ + [ . . . ] ¨Φ + [ . . . ] Ψ + [ . . . ] ˙Ψ + [ . . . ] ¨Ψ (55)where the square brackets denote combinations of thefunctions B i , . . . K i . We should remember that these arereally second-order equations, due to the ˙ˆ ~χ in eqn.(53).In eqn.(55) the pressure perturbation of the effectivefluid has been decomposed into an adiabatic and a non-adiabatic part:Π E = c δ E + w E Γ E (56)We have adopted common notation by using Γ E to rep-resent the dimensionless non-adiabatic pressure pertur-bation; this should not be confused with our metric po-tential ˆΓ. The adiabatic sound speed is fully determinedby the equation of state parameter w E : c = w E − H ˙ w E w E (57)The non-adiabatic pressure perturbation is specified byintroducing a parameter c , interpreted as the soundspeed of the fluid in its rest frame: w E Γ E = (cid:0) c − c (cid:1) (cid:0) δ E + 3 H (cid:0) w E (cid:1) θ E (cid:1) (58)A third and final parameter is needed to relate theanisotropic stress, Σ E , to the velocity perturbations.This is the viscosity parameter c :(1 + w E ) (cid:16) ˙Σ E + 3 H Σ E (cid:17) − ˙ w E w E Σ E = 4 c k θ E (59)1By combining eqns.(57)-(59) with eqns.(54) and (55) wecan obtain two equations relating any two of the fluid per-turbations to the two metric potentials (although we notethat the presence of the ˙Σ E term in eqn.(59) might makethis step non-trivial). These expressions will contain thethree GDM parameters { w E , c , c } . An equivalentthree-parameter framework was studied in the contextof dark energy in [42], and constraints from current andfuture data sets were investigated in [43].We have already mentioned the degeneracy bewteenmodifications to gravity and fluid components in thezeroth-order field equations [37]. An explicit exampleof this is presented in [44] for the case of Eddington-Born-Infeld gravity. We stress that the effective fluidsrepresenting the extra d.o.f. at the background and per-turbed levels need not have the same properties. Indeed,it is most likely that they will be different. If they arenot, we have no way of deciding whether dark energy isreally a modification to gravity or a dark fluid, even usingobservations that reflect the rate of growth of structure[45].As described above, for theories with two or less extrad.o.f. the GDM prescription is not strictly needed; thesystem of equations is already closed. But if we wish toconstrain modified gravity in a model-independent waywe cannot make assumptions about the number of newd.o.f. introduced. The GDM approach allows us to ob-tain model-independent closure relations at the expenseof introducing three new parameters, which would needto be constrained in a MCMC analysis. VI. EXAMPLES
Thus far our discussion of the PPF parameterizationhas been purely formal. Studying some specific casesof gravitational theories should help to consolidate theideas outlined in this paper. The four examples pre-sented here are chosen to represent some common classesof theories of modified gravity; a comprehensive review ofmany other theories is presented in [1]. A simple scalar-tensor theory and Einstein-Aether theory represent the-ories that introduce additional fields to GR (scalar andvector fields respectively). Theories with additional ten-sor fields such as Eddington-Born-Infeld gravity [44, 46]and bimetric theories [47] also belong to this broad cate-gory. f ( R ) gravity is studied as an example of a higher-derivative theory; Hoˇrava-Liftschitz gravity [48–50] andGalileon theories [51, 52] also fall into this class. Our fi-nal example is Dvali-Gabadazze-Porrati gravity (DGP)[53, 54], which we study as a representative higher-dimensional theory. Whilst DGP itself is now disfavouredby observations, it incorporates features common to otherbraneworld theories [55, 56] such as Randall-Sundrummodels [57, 58] and cascading gravity [59].For each theory we will consider the extent to whichthe gravitational field equations can be modelled by thePPF parameterization. Is it possible to describe such a rich variety of theories using only two functions? Theusual arena for PPF is the quasistatic limit, in whichtime derivatives of perturbations can be neglected rela-tive to spatial derivatives. This is the dominant regimefor measures of the rate of structure growth, such as weaklensing and peculiar velocity surveys. However, we arealso interested in using constraints from the IntegratedSachs-Wolfe effect, which requires consideration of scalesabove the quasistatic regime. There has been much workrecently highlighting the importance of a correct rela-tivistic treatment of large scales, for both the theoreticaland observed matter power spectrum [60–63]. We willconsider whether the PPF parameterization can be ex-tended to this regime. A. Scalar-tensor theory
Scalar-tensor theories, first considered by Jordan,Brans and Dicke [64], modifiy GR by introducing a scalarfield which couples to the Ricci scalar in the gravita-tional action. The concept is closely linked to that ofquintessence, in which a scalar field is used a dark en-ergy fluid-type component but without the explicit non-minimal coupling to the Ricci scalar. Particle physicsis in no short supply of candidate scalar fields, and thereduction of higher-dimensional theories to effective four-dimensional field theories also gives rise to candidatescalars (moduli). However, finding a field with exactlythe right properties to account for dark energy has proveddifficult. There is an obvious aesthetic appeal in connect-ing the scalar field with the inflaton; however, they neednot a priori be the same field, and introducing such aconnection creates further obstacles to constructing vi-able models.In general, three functions are required to specify ascalar-tensor theory: the coupling to the Ricci scalar, F ( φ ), a potential for the scalar field, U ( φ ), and a func-tion multiplying the kinetic term of the scalar, Z ( φ ).However, it is possible to reduce F ( φ ) and Z ( φ ) tojust one function through a field redefinition, result-ing in F ( φ ) = φ and Z ( φ ) = ω ( φ ) /φ [65]. The choice ω ( φ ) = constant is termed a Brans-Dicke theory, whichrecovers GR plus a cosmological constant in the limit ω → ∞ , U ( φ ) → Λ. Measurements from the Cassinispacecraft constrain ω & ,
000 (2 σ limits) in the solarsystem [66].A generic property of scalar-tensor theories is that theyresult in a time-dependent gravitational ‘constant’. Thisis precisely one of the features of the PPF formalism(through ˜ g in our parameterization), which gives hopethat scalar-tensor gravity might be fully accommodatedby PPF. For simplicity we will focus on a scalar-tensortheory with variable coupling ω ( φ ) and no potential,working in the conformal Newtonian gauge. The action2in the Jordan frame is: S = 116 π Z d x √− g (cid:20) φR − ω ( φ ) φ ( ∇ φ ) (cid:21) + S m [ ψ ( i ) , g µν ](60)where S m [ ψ ( i ) , g µν ] is the matter action and ψ ( i ) the mat-ter fields. Varying this action with respect to the metricyields the gravitational field equations [67]: G µν = 8 πG φ T µν + ω ( φ ) φ (cid:18) ∇ µ φ ∇ ν φ − g µν ( ∇ φ ) (cid:19) + 1 φ ( ∇ µ ∇ ν φ − g µν (cid:3) φ ) (61)where (cid:3) = g µν ∇ µ ∇ ν . G µν is the usual Einstein tensorof GR and the scalar field has been rescaled so that itis dimensionless, φ → φ/G . In a smooth, unperturbedFRW universe this gives us the background equations(c.f. eqns.(5) and (6)): E F = 8 πG φ a X i ρ i + 12 ω ( φ ) ˙ φ φ − H ˙ φφ (62) E R = 8 πG φ a X P i + 12 ω ( φ ) ˙ φ φ + ¨ φφ + H ˙ φφ (63)Variation of the action with respect to φ gives the equa-tion of motion for the scalar field [67]: (cid:3) φ = 12 ω ( φ ) + 3 (cid:18) πG a T µµ − d ω ( φ )d φ ( ∇ φ ) (cid:19) (64)Rewriting eqn.(61) in the form of eqn.(4) indicates thatthe form of U µν must be: U µν = G µν (1 − φ ) + ω ( φ ) φ (cid:18) ∇ µ φ ∇ ν φ − g µν ( ∇ φ ) (cid:19) + ∇ µ ∇ ν φ − g µν (cid:3) φ (65)Linearly perturbing this expression (and raising an in- dex) will give us U ∆ , U Θ , U P and U Σ . We obtain: U ∆ = E ∆ (1 − φ ) + ˙Φ [3 ˙ φ ] + Ψ " H ˙ φ − ω ( φ ) ˙ φ φ + δφ "
12 d ω ( φ )d φ ˙ φ φ − ω ( φ ) ˙ φ φ − k − H + ˙ δφ " − H + ω ( φ ) ˙ φφ (66) U Θ = E Θ (1 − φ ) + δφ " ω ( φ ) ˙ φφ − H + ˙ δφ + Ψ[ − ˙ φ ] (67) U P = E P (1 − φ ) + ˙Φ h − φ i + Ψ " − φ − H ˙ φ − ω ( φ ) ˙ φ φ + ˙Ψ [ − φ ]+ δφ " − ω ( φ ) ˙ φ φ + 32 d ω ( φ )d φ ˙ φ φ + 3 H + 6 ˙ H + 2 k + ˙ δφ " H + 3 ω ( φ ) ˙ φφ + 3 ¨ δφ (68) U Σ = E Σ (1 − φ ) + δφ (69)Specialising to the conformal Newtonian gauge (in which V =0) means that we lose the time derivatives in eqns.(15)and (16). Hence the appearance of Ψ in U ∆ and U Θ above gives us no cause for concern; the Bianchi identi-ties will remain second-order equations. If we had kept toa general gauge additional terms in V, ǫ and ν would bepresent in the above expressions, ensuring that ˆΨ only ap-peared within the combination ˙ˆΦ + H ˆΨ and that any ¨ a/a terms cancel. The (lengthy) corresponding expressionsfor a general gauge are displayed in Appendix A. Therewe also demonstrate that the general-gauge expressionsstill obey the constraints of yielding second-order, gauge-invariant perturbed field equations.We can see that eqn.(66) has the form indicated ineqn.(49); in this particular case we can pick out the co-efficient functions: A = − − φ ) A = 3 ˙ φk − H k (1 − φ ) F = − H k (1 − φ ) + 6 H k k ˙ φ − ω ( φ ) k ˙ φ φF = 0 α = 12 k d ω ( φ )d φ ˙ φ φ − k ω ( φ ) ˙ φ φ − − H k α = − H k + ω ( φ ) k ˙ φφ (70)Similarly one can read off the coefficients β i , . . . ε i and B i , . . . K i for scalar-tensor theory by matching eqns.(50)-(52) with eqns.(67)-(69).3Using the expressions for U ∆ and U Θ in eqns.(8) wecan form the modified (Fourier-space) Poisson equation.Some of the terms in δφ and δ ˙ φ can be combined to formthe gauge-invariant density perturbation of a scalar field: ρ φ ∆ φ = δρ φ + 3 H ( ρ φ + P φ ) θ φ = ˙ φ (cid:16) δ ˙ φ − ˙ φ Ψ + 3 H δφ (cid:17) (71)The Poisson equation for scalar-tensor gravity is then: − k Φ = 8 πG a φ X i ρ i ∆ i + ρ φ ω ( φ ) φ ∆ φ ! (72)+3 ˙ φφ h ˙Φ + H Ψ i − δφφ " k + 6 H + 12 ˙ φ φ (cid:18) ω ( φ ) φ − d ω ( φ )d φ (cid:19) ∆ i is a gauge-invariant density perturbation to matter(CDM, baryons, radiation), and we have redefined thescalar field so as to pull a factor of 8 πG a out of thesecond term. We have chosen to write the Poisson equa-tion in this form because it delineates the extra termsthat arise in a scalar-tensor theory compared to uncou-pled quintessence. We reach a quintessence-like limit bysetting ω ( φ ) = φ and removing the φ - R coupling in theaction. As a result, the last two lines of eqn.(72) do notarise and the prefactor is just 8 πG a (because we don’trescale φ → φ/G ). It is the last two lines that offer thedistinction between quintessence and scalar-tensor grav-ity. The slip relation is:Φ − Ψ = δφφ (73)It is useful to consider some simplifying limits of theabove expressions. The ‘smooth’ limit, in which the per-turbations of the scalar field are negligible relative to thematter perturbations, gives us: − k Φ = 8 π G φ a X i ρ i ∆ i + 3 ˙ φφ h ˙Φ + H Ψ i (74)The second term on the right-hand side cannot be ab-sorbed into the first term without giving G eff an unde-sired environmental dependence.However, if we take the quasistatic limit of eqn.(72) –that is, we neglect time derivatives of perturbation vari-ables and take k ≫ H – we find that the modifications tothe Poisson equation can indeed be repackaged as a mod-ified gravitational constant. In this limit it is possible towrite the relation between the two metric potentials asΦ = η slip Ψ; e.g. for the choice ω ( φ ) = φ the form of η slip is particularly simple [68, 69]: η slip = φ + 1 φ (75) Combining eqns.(72) and (73) with the slip relation, weobtain: − k Φ = 8 π G φ a X i ρ i ∆ i + ω ( φ ) φ ρ φ ∆ φ ! × (76) " η slip − (cid:18) − η slip (cid:19) ˙ φ k φ (cid:20) ω ( φ ) φ − d ω ( φ )d φ (cid:21) + 3 H k η slip ˙ φφ − where the expression in square brackets gives the time-and scale-dependence of Newton’s gravitational constant.As mentioned earlier, it is important that we also con-sider the (super)horizon-scale limit of these theories forcorrect treatment of their predicted effects on the matterpower spectrum and large-angle CMB power spectrum.Taking the limit k ≪ H allows us to neglect the k δφ/φ term of eqn.(72), but there are no other obvious simpli-fications unless δφ is negligible on these scales. It seemsthat in order to cope with the superhorizon limit we needa parameterization that allows for additive modificationsto the Poisson equation, as well as a modified G eff . Ofcourse, there is no barrier to using the standard PPFformat to model part of the modifications, but we needto remember that the correspondence between parame-terization and theory would no longer be exact on largescales. B. f ( R ) gravity Another commonly-studied theory is f ( R ) gravity, forwhich the action is a general function of the Ricci scalar, f ( R ); see [70] for a detailed review. Two approachesto f ( R ) gravity are possible. In the metric formulationthe affine connection Γ ρµν is defined in terms of the met-ric components in the usual way, and gravitational fieldequations are obtained by varying the metric with respectto g µν only. In the Palatini formulation of f ( R ) gravitythe connection and the metric are treated as independentvariables and the action is varied with respect them indi-vidually. One finds that in the metric formulation there isa propagating scalar degree of freedom f R = ∂f ( R ) /∂R .Since the Ricci scalar is constructed from second deriva-tives of the metric, the kinetic term of the scalar degreeof freedom contains fourth-order derivatives and hencemetric f ( R ) gravity corresponds to a higher-order the-ory.The presence of a scalar degree of freedom within f ( R )theories can be made explicit by applying a conformaltransformation that maps f ( R ) gravity onto a scalar-tensor theory. Metric and Palatini f ( R ) gravity maponto Brans-Dicke theories with ω = 0 , − respectively[70]. The scalar field arising under the conformal trans-formation is sometimes referred to as the ‘scalaron’. Inthe conformally-transformed frame (the Einstein frame)4the scalaron acquires a coupling to the matter fields ψ ( i ) ,leading to non-standard conservation equations. Hencethe non-transformed frame (the Jordan frame) is re-garded as the physical frame in which observations aremade.The action in the Jordan frame is; S = 116 πG Z d x √− g f ( R ) + S m [ ψ ( i ) , g µν ] (77)Variation with respect to the metric leads to the fieldequations: f R R µν − f ( R ) g µν − ∇ u ∇ v f R + g µν (cid:3) f R = 8 πa G T µν (78)This can be rewritten in the form of eqn.(4) with thefollowing expression for U µν : U µν = R µν (1 − f R ) − g µν ( R − f ( R ))+ ∇ u ∇ v f R − g µν (cid:3) f R (79)We will let χ denote the perturbation to the extra de-gree of freedom, ie. χ = δ ( f R ) = f RR δR . Perturbing the above expression U ∆ = E ∆ (1 − f R ) + ˙Φ [3 ˙ f R ] + Ψ h H ˙ f R i + χ h H − k i + ˙ χ [ − H ] (80) U Θ = E Θ (1 − f R ) + χ [ −H ] + ˙ χ + Ψ[ − ˙ f R ] (81) U P = E P (1 − f R ) + ˙Φ h − f R i + Ψ h − f R − H ˙ f R i (82)+ ˙Ψ [ − f R ] + χ h − H − H + 2 k i + ˙ χ [3 H ] + 3 ¨ χU Σ = E Σ (1 − f R ) + χ (83)The similarity between the sets of eqns.(66)-(69) and(80)-(83) is immediately apparent, suggesting the identi-fication of f R as the scalaron. However, in f ( R ) gravitythe ‘extra’ d.o.f, χ , can be expressed in terms of metricpotentials. ˙ χ and ¨ χ are given by the expressions:˙ χ = f R ˙ R δR + f RR ˙( δR ) (84)¨ χ = δR ( f R ˙ R + f R ¨ R ) + 2 f R ˙ R ˙( δR ) + f RR ¨( δR ) (85)where f R indicates the fourth derivative of f with re-spect to R etc., and a δR = − k Φ − H ˙Φ − − (cid:16) H + 6 H − k (cid:17) Ψ − H ˙Ψ (86) a ( ˙ δR ) = − a H δR − ˙Φ (cid:16) k + 18 ˙ H (cid:17) − H ¨Φ − (3) − (cid:16) ¨ H + 2 H ˙ H (cid:17) − ˙Ψ (cid:16)
18 ˙ H + 12 H − k (cid:17) − H ¨Ψ (87) a ( ¨ δR ) = − δR a (cid:16) H + 4 H (cid:17) − a H ( ˙ δR ) −
18 ¨ H ˙Φ − ¨Φ (cid:16)
36 ˙ H + 4 k (cid:17) − H Φ (3) − (4) − (cid:16) H (3) + 2 ˙ H + 2 H ¨ H (cid:17) − ˙Ψ (cid:16)
30 ¨ H + 48 ˙ HH (cid:17) − ¨Ψ (cid:16)
24 ˙ H + 12 H − k (cid:17) − H Ψ (3) (88)The coefficients corresponding to the general expressionsin § V can be computed. For example, matching ontoeqn.(49): A = − − f R ) − f RR a (cid:16) H − k + 6 H (cid:17) +12 H f R a ˙ RA = − H k (1 − f R ) + 3 ˙ f R k + 6 f RR H k a (cid:0) k − H (cid:1) +54 f R ka H ˙ RA = 6 f RR a (cid:16) k + 3 H − H (cid:17) + 18 f R a H ˙ RA = 18 k f RR a H F = − H k (1 − f R ) + 6 H k k ˙ f R + 6 f RR a " − H H k − H k ˙ H − H k + 6 H k ¨ H k + 4 H + 3 ˙ H − k +6 f R a H ˙ R " H k + 6 ˙ H k k − F = 18 H k a (cid:16) f RR ˙ H + f R H ˙ R (cid:17) F = 18 H f RR a α ( i )0 = α ( i )1 = α ( i )2 = α ( i )3 = 0 (89)The slip relation and modified Poisson equation for f ( R )5gravity become:Φ − Ψ = χf R (90) − k Φ = 8 π G f R a X i ρ i ∆ i + 3 ˙ f R f R h ˙Φ + H Ψ i − χf R (cid:20) E + k (cid:21) (91)Again we see that these expressions are largely similarto the scalar-tensor case with ω ( φ ) = 0, f R replacing φ and χ ≡ δφ . However, we note that the similaritiesare aesthetic only; the equivalence between scalar-tensortheory and f ( R ) gravity is only formally realised undera conformal transformation as described above. δφ rep-resents a perturbation to a new field that is genuinelyadditional to GR, whereas χ is really a placeholder forthe combination of metric perturbations in eqn.(86).In the ‘smooth’ limit χ → − k Φ = 8 π G f R a X i ρ i ∆ i + 3 ˙ f R f R h ˙Φ + H Ψ i (92)with the same second background-dependent term as thescalar-tensor case.For measures of late-time structure growth we are pre-dominantly interested in the quasistatic regime. Using χ = f RR δR and eqn.(86) we can replace χ in the Pois-son equation and slip relation by sequence of the metricpotentials and their derivatives, just as we laid out ineqns.(23)-(26). For example, in the quasistatic limit theslip relation becomes:Φ − Ψ = f RR f R a (cid:16) − H + k )Ψ − k Φ (cid:17) (93)This matches the result of Pogosian & Silvestri [71] ifwe further neglect the ˙ H term. They expressed the slipparameter of f ( R ) gravity in the quasistatic limit as: η slip = 3 + 2 Q Q (94)where Q is defined as the squared ratio of the Comptonwavelength of the scalaron to the physical wavelength ofa mode: Q = 3 k a f RR f R ≈ (cid:18) λ C λ (cid:19) (95)In this limit the Poisson equation can be rewritten as: − k Φ = 8 π G f R a X i ρ i ∆ i × (96) " η slip f R f R H k ! − Ek (cid:18) − η slip (cid:19) − The term in square brackets controls the time- and scale-dependence of the effective Newton’s constant.It is no surprise to find that f ( R ) gravity behaves sim-ilar to scalar-tensor theory in the superhorizon limit. Wecan neglect k in eqn.(91) and eliminate χ using eqn.(90),but the resulting expression will still have terms in Ψ and˙Φ on the right-hand side that can not be written as a G eff . The conclusion is analogous to that of § VI A: themapping between a ( G eff , η slip ) parameterization is ex-act in the quasistatic limit, but only approximate in thelarge-scale regime. C. Einstein-Aether theory
The emergence of string theory as a candidate theoryof quantum gravity leads to the possibility that space-time coordinates are non-commutative [72]. Under thesecircumstances Lorentz symmetry may be violated. Hencemuch effort has been invested in exploring ways in whichLorentz violation can be implemented without marringthe key successful features of GR, such as general covari-ance. A minimal way to do this is to introduce a vectorfield into the action, which defines a preferred referenceframe at every point in spacetime. In Minkowski space-time one can simply introduce a constant vector fieldinto the background, but in a curved spacetime this isnot possible; the Lorentz-violating vector field must bepromoted to a dynamical field derived from a generallycovariant action. If invariance under three-dimensionalspatial rotations is preserved then the vector field mustbe time-like. Einstein-Aether theories [73] introduce sucha vector field (‘the aether’) of unit length, which is con-strained not to vanish so that Lorentz violation is main-tained even in a vacuum. The unit, time-like nature ofthe vector field is enforced through means of a Lagrangemultiplier in the action.TeVeS [74] is a well-known example of another theorythat contains Lorentz-violating vector fields. When for-mulated in the Einstein frame, TeVeS introduces both aunit, timelike vector field and a scalar field, and employsa free function to ensure that it reduces to Modified New-tonian Dynamics (MOND) [75] in the non-relativisitclimit. Upon transformation to the Jordan frame - inwhich particles follow geodesics of the metric - the unitlength of the vector field is not preserved. It was shownby Zlosnik [76] that the scalar field present in the Ein-stein frame is absorbed by the vector field in the Jordanframe, and dynamically determinines the modulus of thevector. Hence TeVeS is equivalent to an Einstein-Aethertheory in which the aether field has variable length. Ourstudy of the compatibility of Einstein-Aether theory withthe PPF framework therefore has implications for TeVeSalso.The most general action for Einstein-Aether theoriesis: S = 12 κ Z d x √− g [ R + L EA ( g µν , A µ )] + S m [ ψ ( i ) , g µν ]6(97)where the aether Lagrangian is L EA ( g µν , A µ ) = M F ( K ) + λ ( A α A α + 1) (98) M has the dimensions of mass. K is a scalar formed fromthe kinetic term of the vector field: K = M − K αβγσ ∇ α A γ ∇ β A σ K αβγσ = c g αβ g γσ + c δ αγ δ βσ + c δ ασ δ βγ (99)where c i are dimensionless constants. Barbero &Villase˜nor [77] have identified special choices of c i forwhich Einstein-Aether theory becomes equivalent to GRunder a field redefinition. More generally, the c i mustobey certain restrictions if the linearized field equationsare to be hyperbolic – hence admitting a well-posed ini-tial value problem – and exclude superluminal propaga-tion of aether perturbations and gravitational waves [78].In general there is potentially a fourth term in eqn.(99),but in the case of purely spin-0 perturbations this can beabsorbed by suitable redefinitions of c and c . We willassume there is no direct coupling between matter andthe aether; they interact only gravitationally.The gravitational field equations obtained by varyingthe Einstein-Aether action with respect to the metric canbe written in the form of eqn.(4), where the modificationtensor U µν is effectively the stress-energy tensor of theaether: U µν = ∇ σ (cid:16) F K (cid:16) J σ ( µ A ν ) ) − J σ ( µ A ν ) − J ( µν ) A σ (cid:17)(cid:17) −F K Y µν + 12 g µν F + λA µ A ν (100)where round brackets around subscripts denotes sym-metrization with weight 1/2. We use the notation F K = d F /d K , and the following definitions: J σµ = K σβµγ ∇ β A γ (101) Y µν = c ( ∇ α A µ ∇ α A ν − ∇ µ A α ∇ ν A α ) (102)Varying the action with respect to the aether field givesthe equation of motion: ∇ µ ( F K J µν ) = 2 λA ν (103)Finally, varying the action with respect to the Lagrangemultiplier λ gives the constraint A µ A µ = −
1. The re-quirement of a spatially isotropic background fixes A µ =(1 , , , λ = − A µ ∇ ν (cid:0) F K J νµ (cid:1) (104)which can be substituted back into eqn.(100) to eliminate λ . Defining α = c + 3 c + c , the resulting zeroth-order field equations are:[1 − F K α ] H + 16 F M a = 8 πG a X i ρ i (105) − [1 − α F K ] H − H (cid:20) − α F K (cid:21) +[ ˙ F K −
12 ] F M = 8 πG a X i P i (106) ∇ µ A µ = 3 H (107)and the kinetic scalar K simplifies to K = 3 α H M (108)The first two equations above can be written in the formof eqns.(5) and (6) with the identifications a X = 3 F K α H − a F M (109) a Y = − F K α H + 12 a F M − α ( F K H ) ˙ (110)For simplicity we will treat only the linear Einstein-Aether theory, in which F ( K ) = K . In this particularinstance eqns.(105) and (106) can alternatively be rewrit-ten (making use of eqn.(108) such that they differ fromthe equivalent expressions in GR only through a modifi-cation to Newton’s gravitational constant: G → G − α (111)The situation in Einstein-Aether theory is similar to thetoy model we considered in § IV D, with the instance˜ g = α/
2. We therefore expect the perturbations of U µν to look similar to eqns.(42)-(45). However, we have nowintroduced an extra field into the theory that was notpresent in the toy model, and this will give rise to newterms. Recall that when we applied a gauge transfor-mation to eqn.(42) any terms produced by gauge-variantquantities cancelled due to the background equation (40).From this we can deduce that any new terms introducedby the vector field must be explicitly gauge-invariant.Now we wish to consider linearly perturbed Einstein-Aether theory; we write the perturbations to the vectorfield as A µ = (cid:18) Z, a ∂ i Q (cid:19) (112)Taking the linear perturbation of the constraint A µ A µ = − Z = Ξ = − Ψ, the last equality being true in the confor-mal Newtonian gauge only. The perturbed equation ofmotion for the vector field is: c (cid:16) ¨ Q + k Q + 2 H ˙ Q + 2 H Q + ˙Ψ + ˙Φ + 2 H Ψ (cid:17) +(3 c + c ) (cid:18) k Q + 2 H Q − ¨ aa + ˙Φ + H Ψ (cid:19) = 0 (113)7One can then find the linear perturbations of U µν . Thisis a lengthy but straightforward exercise [78, 79]. Usingeqn.(113) to remove second-order derivatives from U Θ ,the results are: U ∆ = ( c − α ) k H Q + c k (cid:16) ˙ Q + Ψ (cid:17) − α H h ˙Φ + H Ψ i (114) U Θ = α h ˙Φ + H Ψ i + ( c + c + c ) k Q (115) U P = αk (cid:16) ˙ Q + 2 H Q (cid:17) +3 α (cid:16) ¨Φ + 2 H ˙Φ + (2 ˙ H + H )Ψ + H ˙Ψ (cid:17) (116) U Σ = − ( c + c ) (cid:16) ˙ Q + 2 H Q (cid:17) (117)It is straightforward to read off the coefficients introducedin eqn.(49): A = − α H k F = c − α H k α = H ( c − α ) α = c kA = F = 0 (118)and similarly for the remaining coefficients β i , . . . ε i and B i , . . . K i of eqns.(50)-(52). Note from eqn.(113) that thed.o.f. χ ( ≡ Q ) is not dimensionless, so neither are α and α .The Poisson equation in Einstein-Aether theory is: − k Φ = 8 πG a X i ρ i ∆ i + k (cid:16) H Q (3 c + 2 c ) + c ( ˙ Q + Ψ) (cid:17) (119)Consideration of some special cases of the c i should helpus gain some understanding of this expression. Firstlywe note that the slip between the metric potentials issourced by spatial perturbations to the vector field, soin the smooth limit we recover Φ = Ψ. In this limit theonly modification to the Poisson equation is through aconstant rescaling of Newton’s constant: − k Φ = 8 π G c a X i ρ i ∆ i (120)With Q = 0 all explicit traces of extra fields disappearfrom the components of U µν (though we ought to re-member that the perturbation to the time component ofthe vector field still remains, ‘disguised’ as the metricperturbation Ψ). We note immediately that the con-stants renormalizing Newton’s gravitational constant inthe background and linear-order equations are not gener-ally the same, being α and − c respectively. If we makethe choice α = − c , this special case of Einstein-Aethertheory can be compared to the toy model considered in § IV D. In that example the background gravitational fieldequations were modified but no extra fields were present,and we assumed that the same quantity renormalized G at both unperturbed and linearly perturbed order. In-deed we find that with the choices ˙˜ g = B = 0, α = − c and V = ˙ V = 0 (to recover the conformal Newtoniangauge) the eqns.(42)-(43) reproduce eqns.(114)-(117) inthe limit Q = 0.Another case of interest is the choice c + c = 0 , c = 0.This causes the first part of the stress-energy tensor ofthe aether (line 1 in eqn.(100)) to adopt a form akinto the Maxwell tensor of electromagnetism [80]. Un-der these conditions the gravitational slip again vanishes.Note that this differs from scalar-tensor and f ( R ) gravity,where the slip could only be zero if the extra fields (treat-ing the scalaron as an extra field) were unperturbed. ThePoisson equation now becomes: − k Φ = 8 π G c a X i ρ i ∆ i − k c c (cid:16) ˙ Q + H Q (cid:17) (121)In the quasistatic limit the ˙ Q term can be neglected.However, the ‘electromagnetic condition’ has preventedus from obtaining a relation between Q , Φ and Ψ. Soin this special case we are unable to package the modifi-cations to the Poisson equation as a modified Newton’sconstant, even in the quasistatic regime. This is an inter-esting result, because thus far we have always found thisto be possible. Indeed, for choices other than c = − c wecan write the slip relation in the usual format Φ = η slip Ψ,with slip parameter: η slip = 1 − H k c + c c + c + c (cid:16) − α (cid:17) (122)The Poisson equation is then: − k Φ = 8 πG a X i ρ i ∆ i × (123) (cid:20) c η slip − c + 2 c c + c ) (cid:18) − η slip (cid:19)(cid:21) − As before, the term in square brackets acts like an evolv-ing G eff .Whilst Einstein-Aether gravity has a similar qua-sistatic form to scalar-tensor gravity and f ( R ) gravity,its large-scale behaviour is distinctly different. In thelimit k → Q ( ˙Φ + H Ψ), such as occurred in the scalar-tensor and f ( R ) cases. Terms like this originate fromcouplings between new degrees of freedom and the cur-vature scalar R in the action. A direct coupling of thistype isn’t present in the Einstein-Aether case – instead,the coupling between the aether and the metric is en-forced through a Lagrange multiplier. However, it is nottrue that Einstein-Aether theory reduces to GR in thesuper-horizon regime, because the slip relation still has anon-GR form (eqn.(117)).8 D. DGP gravity
Dvali-Gabadadze-Porrati (DGP) gravity [53] has re-ceived much attention over the past decade. The modelconsiders our (3+1)-dimensional spacetime to be a hy-persurface (brane) embedded in a five-dimensional bulk,with matter fields confined to the brane but gravity freeto propagate into the extra dimension. There are twobranches to the theory, arising from a choice of sign ac-companying a square root in the Friedmann equation[55]: H − ǫHr c = 8 πG X i ρ i (124)where ǫ = ±
1. Note that we have returned to usingphysical time rather than conformal time here in order tomake the late-time behaviour more explicit. The param-eter r c in this expression defines a crossover scale belowwhich gravity is effectively four-dimensional, but abovewhich five-dimensional effects become important. It isdetermined by the ratio of the four- and five-dimensionalgravitational coupling constants: r c = κ κ (125)In a CDM-dominated universe we have H → ǫ/r c atasymptotically late times. The choice ǫ = 1 correspondsto a universe that accelerates without the need for a cos-mological constant, although the crossover scale muststill be fine-tuned to fit current supernovae data, with r c ≈ H − [55]. Alas, the self-accelerating branch of DGPhas been effectively ruled out as a viable theory due to aghost-like instability [81–83], and its failure to fit multipleobservational data sets simultaneously – a feat achievedby ΛCDM [84]. Nevertheless, we propose to describethe relations between DGP gravity and the PPF formal-ism as a typical model for cosmological perturbations inhigher-dimensional theories.We will adopt the Gaussian Normal longitudinal(GNL) gauge, in which the brane remains unperturbedat the hypersurface y = 0 and we recover the familiar4D conformal Newtonian gauge on the brane only . Theaction for a simple DGP model is: S = 12 κ Z d X p − (5) g h (5) R − i + Z d x p − (4) g (cid:20) κ
24 (4) R + L B (cid:21) (126)where (5) g and (5) R are the five-dimensional metric andcurvature scalar of the bulk; X A and x µ denote five-dimensional and brane coordinates respectively. L B isthe Lagrangian of the (brane-confined) matter fields andpossible brane tension λ , which is related to the 5D andinduced 4D cosmological constants [85].It will be useful to write the modifications to the Fried-mann and Raychaudhuri equations in the form of an ‘XY’ background, as specified in eqns.(5) and (6) [1]: X = 3 ǫr c HY = − ǫ (cid:0) dHdt + 3 H (cid:1) r c H (127)where t denotes physical time. Effective 4D field equa-tions are obtained by projecting onto the brane [86]: (4) G µν + Λ g µν = κ Π µν − E µν (128)Here g µν is the induced metric on the brane. The tensorΠ µν (not to be confused with the scalar pressure pertur-bation ρ Π) is given by:Π µν = −
14 ˜ T µα ˜ T αν + 112 ˜ T ˜ T µν + 18 g µν ˜ T αβ ˜ T αβ − g µν ˜ T (129)where˜ T µν = T µν − κ G µν (130) E µν is the projection of the 5D Weyl tensor onto thebrane: E µν = (5) C ABCD n A n C g Bµ g Dν (131)By supplementing the Bianchi identities with the Codazziequation and Israel junction conditions [87] one finds thatthe matter energy-momentum tensor is separately con-served from Π µν and E µν [86]: ∇ µ T µν = 0 (132) ∇ µ E µν = κ ∇ µ Π µν (133)where ∇ µ is the covariant derivative associated with theinduced 4D metric on the brane.Comparison of eqns.(4) and (128) enables us tostraightforwardly write down the tensor U µν for DGPgravity: U µν = − Λ g µν + κ Π µν − E µν − κ T µν (134)Now the key question is: are we able to perturb thisexpression for U µν to obtain U ∆ , U Θ , U P and U Σ in thesame way that we have done for scalar-tensor gravity, f ( R ) theories and Einstein-Aether theory ( § VI A- § VI C)?The answer is a qualified yes and no. We are able to writedown expressions for these quantities and determine thecoefficients laid out in § V. However, the perturbed com-ponents of U µν contain four new d.o.f. arising from theperturbations to the projected Weyl tensor. Hence, un-like the previous three examples presented in this section,the system is not closed using just the Bianchi identities.We cannot obtain the closure relations needed to solve forthese new d.o.f. using the purely 4D formalism developedin this paper.9Let us elaborate. We will label quantities related tothe Weyl tensor by the letter E, as we shall see it playsthe role of the effective fluid discussed in § V. Since theWeyl tensor is traceless by construction (and we are as-suming isotropy amongst the three spatial dimensions ofthe brane) we deduce that the Weyl ‘fluid’ must have aradiation-like equation of state, w E = 1 /
3. It is commonpractice to neglect the small contribution of the Weylfluid to the cosmological background, and define its com-ponents at the perturbed level: − E = µ E − E i = ~ ∇ i Θ E E ij = 13 µ E δ ij + D ij σ E (135)The perturbations of eqn.(134) are then: U ∆ = 16 κ a ˜ ρ ˜ δ − a µ E − κ a ρδ (136) U Θ = 16 κ a ˜ ρ (˜ ρ + ˜ P )˜ θ − a Θ − κ a ( ρ + P ) θU P = 12 κ a [(˜ ρ + ˜ P )˜ ρ ˜ δ + ˜ ρ ˜Π] − a µ E − κ a ρ Π U Σ = − κ a (˜ ρ + 3 ˜ P )(˜ ρ + ˜ P ) ˜Σ − a σ E − κ a ( ρ + P )ΣThe quantities marked by tildes are components of ˜ T µν defined in eqn.(130), and the unsubscripted quantitiesrefer to ordinary matter components, as in previous sec-tions. By using eqns.(4) and (130) we can rewrite theabove expressions as: U ∆ = 32 r c X (cid:0) E ∆ + a µ E (cid:1) U Θ = 32 r c X (cid:0) E Θ + a Θ E (cid:1) U P = 32 r c X E P − a µ E − Xr c Xr c − w E ) (cid:0) E ∆ − U ∆ + a µ E (cid:1) ! U Σ = − r c ( X + 3 Y ) (cid:0) E Σ + a σ E (cid:1) (137)Well below the crossover scale we recover GR, as U i → r c → ∞ . Using the definitions for the E i , we can thenput the above expressions into the form of eqns.(49)-(52).For the U ∆ component the non-zero coefficients are: A = − r c X A = − H k r c XF = − H k r c X α (1)0 = 3 a r c X (138)where we have dropped the factor of k in the first termof eqn.(49) in order to keep α (1)0 dimensionless. Unlike the previous examples, we now have a vector of threeextra d.o.f.: ~χ = { µ E , Θ E , σ E } .The two perturbed components of eqn.(133) are [1]:˙ µ E + 4 H µ E − ∇ Θ E = 0 (139)˙Θ E + 4 H Θ E − µ E + (1 + w E ) (cid:18) µ E + 3 H a Θ E (cid:19) + ∇ w E (cid:18) σ E + 2(1 + w E ) a [(2 + 3 w E )Φ − Ψ)] (cid:19) = 0(140)Now the difficulty is apparent – we have two equations forthe three variables µ E , Θ E and σ E . We could eliminate µ E or Θ E from the above equations, but the anisotropicstress of the Weyl fluid remains a free function. If we stayentirely within the bounds of a 4D formalism it mustbe treated as an additional source in the 4D effectiveEinstein equations. Some authors have obtained a closedsystem on the brane by setting σ E to zero, e.g. [88], butthis will not be the case is general.It is not surprising that σ E appears as a free function inthe perturbed effective Einstein equations. Gravitationalwaves propagating in the bulk spacetime contribute to σ E when they impinge upon the brane, and these cannotbe fully described by brane-bound perturbations. How-ever, by tackling the full system of perturbations in thebulk it is possible to express the impact of the gravita-tional waves on the brane in terms of other brane-boundquantities. Mukohyama [89] has shown that all 5D met-ric, matter and Weyl-fluid perturbations can be relatedto a master variable Ω which obeys a partial differentialequation in the bulk. This is the master equation [54]: ∂∂t (cid:18) na ∂ Ω ∂t (cid:19) − ∂ i ∂ i a (cid:18) n Ω a (cid:19) − (cid:18) n Ω ′ a (cid:19) ′ = 0 (141)where primes denote derivatives with respect to the bulkco-ordinate y , and we have assumed a Minkowski bulk.The perturbations of the Weyl fluid are given in terms ofΩ by [54]: µ E = − k a Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b (142)Θ E = k a (cid:18) ∂ Ω ∂t − H Ω (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b (143) σ E = − a (cid:20) ∂ Ω ∂t − H ∂ Ω ∂t + k a Ω − H dHdt Ω ′ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b (144)where a subscript b denotes evaluation on the brane andk is the 3-momentum on the homogeneous background.Eqns.(141) and (144) are the expressions we need to closethe system of perturbations. We have introduced an ex-tra variable, Ω, but this has been compensated for by theaddition of two new equations. A solution of the masterequation is then sufficient to fully determine the systemof perturbations. The fact that all perturbations can be0related to a single scalar is a result of the high degreeof symmetry present in this system (a maximally sym-metric brane in a maximally symmetric bulk); Ω is notgenerally believed to have a physical interpretation.But we are not quite out of the woods yet. Thoughthe system of equations is now closed, it is not closedon the brane alone. Eqns.(141) and (144) depend onderivatives of the master variable normal to the brane,and hence in general knowledge of the bulk perturba-tions will be required to solve the system. Koyama &Maartens [90] neatly sidestepped this issue by consider-ing perturbations in the small-scale, quasistatic regimeof a Minkowski bulk. This enabled them to make theapproximation | H Ω ′ | ≪ k Ω /a so that the troublesometerms can be neglected, as well as time derivatives. Un-der these assumptions one finds that the energy densityand anisotropic stress perturbations of the Weyl fluid aresimply related by µ E = 2 k σ E . In the quasistatic regimethe master equation then has an analytic solution [90]:Ω = C (1 + Hy ) − kaH (145)where C is a constant, and regularity of the bulk per-turbations has been used to eliminate a second possiblesolution. Solving for the Weyl fluid and metric pertur-bations, the modified Poisson equation and slip relationare: − k Φ = 8 πG a X i ρ i ∆ i (cid:18) − β (cid:19) (146) k (Φ − Ψ) = 8 π G β a X i ρ i ∆ i (147)where β = 1 − r c H (cid:18) H dHdt (cid:19) It is easy to see that our PPF function ˜ g should be iden-tified with (1 − β ) − . Combining eqns.(146) and (147)to eliminate matter terms, we obtain:Φ − Ψ = 21 − β Φ (148)From this we can deduce the PPF function ζ = 2 / (1 − β ). Alternatively, in the ( Q, η slip ) formatwe obtain: Q = (cid:16) − β (cid:17) , η slip = (3 β − β + 1) − .For scales larger than the quasistatic regime the deriva-tives normal to the brane cannot be ignored. Sawicki,Song and Hu were able to evolve large-scale modes nu-merically by implementing a scaling ansatz [91]:Ω = A ( p ) a p G ( x ) (149)where x is the distance from the brane in units of thecausal horizon and p is a constant. The authors tooka trial value for p and solved the system of equationsiteratively to obtain subsequent corrections. They recov-ered the quasistatic solution of Koyama and Maartensand were able to calculate the behaviour of horizon-scale modes. However, the labour involved in obtaining thesesolutions is decidedly non-trivial. If we wish to constraingeneral classes of gravitational theories simultaneouslywe need a method that does not require detailed numer-ical evolution for each theory individually. This is themotivation behind the GDM-based approach we put for-ward in § V. In the case of DGP gravity the new d.o.f.are already written in the form of fluid perturbations,and we suggest that other theories may be amenable toa similar treatment.Returning briefly to the quasistatic limit of eqns.(146)and (147), we note that the master-variable route to ob-taining a modified Poisson equation and a slip relationwas decidedly different to that taken in § VI A-VI C. Soit is interesting to find that we have reached the sameconclusion in all four cases: the usual two-function PPFparameterization works well in the quasistatic limit, butfor scales larger than this it no longer captures the fullbehaviour of the theories studied here.We should not be particularly surprised by this conclu-sion. Given the different physical mechanisms employedby the four theories studied here, it is quite a remarkablefeat that they can be mapped onto a common frame-work at all, in any limit – let alone a framework simpleenough to express the departures from GR using onlytwo functions. It is not unexpected to find that the cor-respondence between parameterization and theory doesnot hold perfectly for all scales.
VII. DISCUSSION
The wide array of modified gravity theories nowpresent in the literature renders the individual testing oftheories time-consuming and impractical. There is alsothe possibility that none of the theories currently underconsideration are correct, or that General Relativity re-mains valid for all length scales and environments. TheParameterized Post-Friedmann formalism provides a use-ful framework for constraining deviations from GeneralRelativity without recourse to a specific underlying the-ory of modified gravity.In this paper we have shown how the constraints ofenergy conservation, gauge invariance and second-orderfield equations can be used to reduce the components of apurely metric theory to two free functions. This processhas highlighted some consistency conditions that arisewhen adopting a parameterized form for modificationsto GR. We have put forward a general structure for theperturbed Einstein equations that should be applicableto any single-metric theory for which the gravitationalaction is built from only curvature invariants. However,when additional degrees of freedom are introduced into agravitational theory (such as extra fields), our ability tomake precise statements about the form of the modifica-tions to the Einstein equations is reduced.Though a rich zoo of underlying physical mechanismshas been put forward, ultimately we are concerned with1their effects on observables. The observables themselves(galaxies, CMB photons) are predominantly controlledby the evolution of matter perturbations. There are onlya limited number of ways that a theory of modified grav-ity can affect the matter perturbations: for example, bychanging the strength of gravitational coupling or the re-lationship between the metric potentials ˆΦ and ˆΨ. Butthese effects are, to a certain extent, degenerate with thepresence of a second fluid [45]. A reduced gravitationalcoupling could be mimicked by pressure support fromanother fluid. A non-zero gravitational slip could be in-troduced by a fluid with non-negligible viscosity. Hencewe suggest that it may be possible to treat the extra d.o.f.of a theory by an effective fluid; Generalized Dark Mat-ter (GDM) provides one method of doing this in the caseof spin-0 perturbations. An effective fluid-type approachof this kind is already present in DGP.Such treatment is not strictly necessary when two orfewer new degrees of freedom are present. But if we wishto keep our parameterization as general as possible, thenwe need a method to deal with more than two degrees offreedom. The GDM approach provides closure relationsfor such cases. The GDM parameters { w E , c , c vis } can-not be assigned the direct interpretation they possess forreal fluids. But if constraints on these parameters favourbizarre-seeming values that would be unphysical for a realfluid than this could indicate their origin to be modifiedgravitational laws and not dark energy.To date, the mapping of theories onto the PPF frame-work has only been computed explicitly for a small num-ber of cases. In this paper we have added Einstein-Aether to this collection. In the quasistatic limit thetwo-function parameterization has always been found ap-plicable, but it is possible that there are some classes oftheories which cannot be reduced to such a simple for-mat. In addition, we have found that on horizon scalesthe parameterization of the modified Poisson and sliprelations no longer matches onto the underlying expres-sions exactly. These large scales are important for accu-rate calculation of how the ISW effect and matter powerspectrum are affected by modified gravity theories. Soshould we give up on the goal of a unified parameteriza-tion?Not at all. Consider the analogous problem in darkenergy; the equation of state is commonly reduced tojust two numbers via the CPL parameterization [92, 93]: w ( a ) = w + w a (1 − a ) (150)Whilst it is unlikely that a physically-motivated darkenergy model will map neatly onto this expression,eqn.(150) provides a useful way to obtain constraints onthe expected behaviour of dark energy. Of course theapproach is not ideal – if the equation of state were tobehave in a radically different way to our expectationsthen it would not be adequately described by { w , w a } .But the form of eqn.(150) has physical motivation, and gives us a way to tackle large classes of models withoutspecialising to a particular theory.The PPF formalism should be viewed in a similar way.Eqns.(29) and (31) may not match up exactly to all the-ories on all scales, but they do provide a phenomeno-logical way to search for the approximate signatures weexpect modified gravity to leave. Similarly, our proposalof mapping additional degrees of freedom onto a GDMframework may not hold exactly for all possible theories.But whilst we have not yet reached an ideal parameteri-zation of modified gravity, we believe that the approachoutlined in this paper is likely to have a wider range ofapplicability than most of the forms currently in com-mon use. If, when combined with the next generation ofexperiments, we fail to make progress on breaking the de-generacy between dark energy and modified gravity, thenwe will need to rethink our tools. For the present, use ofthe parameterizations described here is a justifiable sim-plification, provided that we are not lulled into thinkingthat their correspondence to underlying theories is exactin all situations.If we are to use such a phenomenological methodologywe need to choose a parameterization that closely mod-els our expectations. Small differences in the structureof the modifications to Einstein’s equations can lead tosignificant effects, independent of the ansatz being usedfor the PPF functions. We will demonstrate these differ-ences and how they affect the constraints obtained in afuture work [40]. Acknowledgements
We are grateful for useful discussions with R. Bean,E. Bertschinger, L. Pogosian, A. Silvestri, D. Wands, C.Will and T. Zlosnik. TB is supported by the STFC.PGF acknowledges support from the STFC, the BeecroftInstitute for Particle Astrophysics and Cosmology, andthe Oxford Martin School. CS is supported by a RoyalSociety University Research Fellowship. JZ is supportedby the Dennis Sciama fellowship and a James Martinfellowship.
Appendix A: Gauge-Invariant Equations forScalar-Tensor Theory
In this Appendix we display explicitly the gauge-invariant form of the modifications to the perturbed Ein-stein equations in the case of scalar-tensor gravity. Atthe level of the homogeneous, background universe themodifications are contained in the diagonal componentsof a tensor U µν , see eqn.(4). The perturbed componentsof this tensor appearing in eqns.(19)-(22) are denotedby U ∆ , U Θ , U P and U Σ as given in eqns.(10). In § VI Athese components were derived in the conformal Newto-nian gauge for simplicity. Using the notation introducedin § III, the expressions in a general gauge are:2 U ∆ = E ∆ (1 − φ ) + 3 ˙ φ ˆΓ + (cid:18) δφ − ˙ φ V (cid:19) "
12 d ω ( φ )d φ ˙ φ φ − ω ( φ ) ˙ φ φ − k − H + (cid:18) δ ˙ φ + ˙ φ Ξ + V H ˙ φ − ¨ φ ) (cid:19) " ω ( φ ) ˙ φφ − H − V H " ω ( φ ) ˙ φ φ + ¨ φ − H ˙ φ (A1) U Θ = E Θ (1 − φ ) + (cid:18) δφ − ˙ φ V (cid:19) " ω ( φ ) ˙ φφ − H + (cid:18) ˙ δφ + ˙ φ Ξ + V H ˙ φ − ¨ φ ) (cid:19) + V " ω ( φ ) ˙ φ φ + ¨ φ − H φ (A2) U P = E P (1 − φ ) − φ ˆΓ + 3 (cid:18) δφ − ˙ φ V (cid:19) " − ω ( φ ) ˙ φ φ + 12 d ω ( φ )d φ ˙ φ φ + H + 2 ˙ H + 23 k +3 (cid:18) ˙ δφ + ˙ φ Ξ + V H ˙ φ − ¨ φ ) (cid:19) " H + ω ( φ ) ˙ φφ + 3 δ ¨ φ + 2Ξ ( ¨ φ − H ˙ φ ) + ˙ φ ( ˙Ξ + H Ξ) + V H ¨ φ − E ˙ φ − φ (3) !! + 32 V φ " ∂∂τ ω ( φ ) ˙ φ φ + ¨ φφ + H ˙ φφ ! − H ω ( φ ) ˙ φ φ + ¨ φφ + H ˙ φφ ! (A3) U Σ = E Σ (1 − φ ) + (cid:18) δφ − ˙ φ V (cid:19) (A4)In these expressions the scalar field perturbations δφ, δ ˙ φ and δ ¨ φ appear only in gauge-invariant combinations withthe variables V and Ξ. ˆΓ is also gauge-invariant. Thelast terms in U ∆ , U Θ and U P are not gauge-invariant,and neither are E ∆ , E Θ and E P . But we find that theadditional terms produced by these parts under a gaugetransformation cancel by virtue of the background equa-tions (62) and (63) - see Table III for the transformationproperties. Note that all terms in the spatial, tracelessEinstein equation are individually gauge-invariant.So the perturbed Einstein equations with the addi-tions above remain fully gauge-invariant, as they mustdo. However, on first inspection it looks like we haveviolated the constraint of having second-order field equa-tions. Recall that this constraint restricts U ∆ and U Θ to contain at first-order time derivatives at most, due toeqns.(11) and (12). It is easy to see that the ¨ φ termspresent in U Θ cancel, but in U ∆ this is not explicitlyobvious. In addition, U ∆ appears to contain a second-order time derivative of the scale factor coming from the˙ˆΦ term (see eqn.(15)). To show this is not a problem wewill need to use the zero-order Bianchi identity, eqn.(7).In the case of scalar-tensor theory this becomes: ω ( φ ) ¨ φφ −
12 ˙ φ φ + 2 H ˙ φφ ! + 12 d ω d φ ˙ φ φ − (cid:16) ˙ H + H (cid:17) = 0(A5)where we have made use of eqn.(62). Using this to sub-stitute for the ω ¨ φ term in eqn.(A1) and simplifying we obtain the alternative form: U ∆ = E ∆ (1 − φ ) −
12 ˙ φ ˙ J − H ˙ φ Ξ − δφ (cid:0) k + 3 H (cid:1) + 12 k ˙ φ V + δφ "
12 d ω d φ ˙ φ φ − ω ( φ ) ˙ φ φ + (cid:16) δ ˙ φ + ˙ φ Ξ (cid:17) " ω ( φ ) ˙ φφ − H (A6)We see that the terms containing second derivatives of φ and the scale factor have both cancelled. The aboveexpression is now demonstrably first-order in time deriva-tives, but its ability to yield gauge-invariant Einsteinequations is no longer obvious. Appendix B: Constraint Equations
In the case of unmodified background equations, con-straints are obtained by applying the Bianchi identityto a U -tensor of the general format shown in eqns.(23)-(26). It is possible to deduce the general structure ofthese constraint equations and use this as a shortcut.The constraint equations arising from the coefficient ofΦ ( n ) can be generated from the following formulae:[ B
1] ˙ A n + kA n − + H A n + kB n + H C n = 0 (B1)[ B
2] ˙ B n + kB n − + 2 H B n − kC n + 23 kD n = 0 (B2)3where B B ( n ) are analogous:[ B
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