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Inflationary gravitational waves in the effective field theory of modified gravity

In the approach of the effective field theory of modified gravity, we derive the second-order action and the equation of motion for tensor perturbations on the flat isotropic cosmological background. This analysis accommodates a wide range of gravitational theories including Horndeski theories, its generalization, and the theories with spatial derivatives higher than second order (e.g., Horava-Lifshitz gravity). We obtain the inflationary power spectrum of tensor modes by taking into account corrections induced by higher-order spatial derivatives and slow-roll corrections to the de Sitter background. We also show that the leading-order spectrum in concrete modified gravitational theories can be mapped on to that in General Relativity under a disformal transformation. Our general formula will be useful to constrain inflationary models from the future precise measurement of the B-mode polarization in the cosmic microwave background.


I. INTRODUCTION
The detection of primordial gravitational waves in the Cosmic Microwave Background (CMB) offers an exciting possibility for probing the physics around the GUT scale. In particular, the inflationary paradigm [1][2][3] predicts the generation of nearly scale-invariant primordial tensor and scalar perturbations with the tensor-to-scalar ratio r less than the order of 0.1 [4,5]. Since the spectral index n s of scalar perturbations was measured by the Planck satellite in high precision (n s = 0.9603 ± 0.0073 at 68 % CL [6]), the precise bounds on r from the ongoing and upcoming CMB B-mode polarization experiments [7][8][9][10] are the next important step for approaching the origin of inflation.
Many of the single-field inflationary models proposed so far belong to a class of Horndeski theories [11]the most general Lorentz-invariant scalar-tensor theories with second-order equations of motion. In fact, the leading-order power spectra of tensor and scalar perturbations were derived for inflationary models in the framework of Horndeski theories [12,13]. These results were employed to place observational constraints on individual models (such as slow-roll inflation [3], k-inflation [14], Starobinsky inflation [1], Higgs inflation [15,16]) from the WMAP and Planck data [17,18].
There exist more general modified gravitational theories beyond the Horndeski domain. Choosing the socalled unitary gauge in which the perturbation of a scalar field φ on the flat Friedmann-Lemaître-Robertson-Walker (FLRW) background vanishes, the Horndeski Lagrangian can be expressed in terms of geometric scalar variables appearing in the 3+1 Arnowitt-Deser-Misner (ADM) decomposition of space-time [19]. In Horndeski theories the coefficients in front of such geometric scalars have particular relations, but it is possible to consider extended theories with arbitrary coefficients: Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories [20]. The Hamiltonian analysis based on linear perturbations on the flat FLRW background shows that GLPV theories have one scalar propagating degree of freedom without ghost-like Ostrogradski instabilities [20][21][22][23].
Another example outside the Horndeski domain is Hořava-Lifshitz gravity [24], in which an anisotropic scaling in time and space plays a role for the realization of a power-counting renormalizable theory. In this case there are spatial derivatives up to 6-th order, with which Lorentz invariance is explicitly broken. The building blocks of Hořava-Lifshitz gravity are the threedimensional ADM geometric scalars invariant under a foliation-preserving diffeomorphism.
The effective field theory (EFT) of cosmological perturbations is a powerful framework to deal with lowenergy degrees of freedom in a systematic and unified way [25][26][27]. This approach is not only useful to parametrize higher-order correlation functions of curvature perturbations generated during inflation [6,28,29] but also to perform a systematic study for the physics of a latetime cosmic acceleration induced by the modification of gravity [30]- [43]. In fact, recent studies [19,20,36,[40][41][42] showed that the EFT approach can encompass a wide range of theories including Horndeski/GLPV theories and Hořava-Lifshitz gravity.
The EFT approach of Ref. [19] is based upon a general Lagrangian L in unitary gauge that depends on the lapse N and several ADM geometric scalars constructed from the extrinsic curvature K µν and the three-dimensional intrinsic curvature R µν . The action expanded up to second order in scalar metric perturbations shows that the theory has one scalar degree of freedom with spatial derivatives higher than second order in general, while the time derivatives are of second order. Both Horndeski and GLPV theories satisfy conditions for the absence of such higher-order spatial derivatives [19,39,40].
Hořava-Lifshitz gravity can be incorporated in the EFT approach of Ref. [19] by taking into account additional geometric scalar quantities (associated with spatial derivatives up to 6-th order) to the Lagrangian L mentioned above [41,42]. These quantities include the 6-th derivative ∇ i R jk ∇ i R jk and the scalars constructed from the acceleration vector a i = ∇ i ln N . In the nonprojectable version of Hořava-Lifshitz gravity where the lapse N depends on both time and space [44], the contribution like a i a i in the Lagrangian can alleviate the instability problem [45,46] present in the projectable version (in which N depends on time alone).
In Ref. [43] the second-order action for scalar perturbations was derived for the generic EFT Lagrangian encompassing Horndeski/GLPV theories and Hořava-Lifshitz gravity. This result can be useful for the computation of the primordial scalar power spectrum generated during inflation and for discussing conditions under which the ghosts and instabilities are absent (see Ref. [39] for a review).
In this paper we employ such a general EFT approach for the study of gravitational waves on the flat FLRW background. Our analysis is more generic than those of Refs. [39,41,47] in that higher-order spatial derivatives appearing in Hořava-Lifshitz gravity are explicitly taken into account for the computation of the second-order action of tensor perturbations. Unlike Ref. [48], we do not consider the terms associated with the broken spatial diffeomorphism. We provide a general formula for the inflationary power spectrum of gravitational waves by taking into account slow-roll corrections to the leading-order spectrum.
This paper is organized as follows. In Sec. II we present the action of our EFT approach and briefly review how several modified gravitational theories are incorporated in our general framework. In Sec. III we derive the second-order action and the equation of motion for tensor perturbations. In Sec. IV we obtain the spectrum of gravitational waves generated during inflation and in Sec. V we apply the results to concrete modified gravitational theories. Sec. VI is devoted to conclusions.

II. THE GENERAL EFT ACTION OF MODIFIED GRAVITY
The EFT of cosmological perturbations is based upon the 3+1 decomposition of space-time described by the line element [49] (2.1) where N is the lapse function, N i is the shift vector, and h ij is the three-dimensional spatial metric. Introducing a unit vector n µ = (−N, 0, 0, 0) orthogonal to the constant t hypersurfaces Σ t , the induced metric h µν on Σ t can be expressed of the form h µν = g µν + n µ n ν .
The extrinsic curvature is defined by K µν = h λ µ n ν;λ = n ν;µ + n µ a ν , where a semicolon represents a covariant derivative and a ν ≡ n λ n ν;λ is the acceleration vector. The scalar quantities that can be constructed from the extrinsic curvature are the trace of K µν and the square of K µν , i.e., The internal geometry of Σ t is characterized by the three-dimensional Ricci tensor R µν = (3) R µν , which is dubbed the intrinsic curvature. From R µν we can construct the following scalar quantities: Since it is possible to express the Riemann tensor R µνλσ in terms of the Ricci tensor and scalar in three dimensions, we do not need to consider scalar combinations associated with R µνλσ . In Hořava-Lifshitz gravity there are other scalar quantities that generate spatial derivatives up to 6-th order: We can also take into account the terms like R j i R k j R i k and RR j i R i j , but they are irrelevant to the dynamics of linear scalar perturbations on the flat FLRW background. Hence we do not incorporate those terms in the following analysis.
In the original Hořava-Lifshitz gravity [24] the spacetime foliation is preserved by the space-independent reparametrization t → t ′ (t), so the lapse N is assumed to be a function of time t alone (which is called the projectability condition). This can be extended to a nonprojectable version in which the lapse depends on both the spatial coordinate x i (i = 1, 2, 3) and t [44]. Since the acceleration a i = ∇ i ln N does not vanish in this case, we can also consider the following scalar combinations: where ∆ ≡ ∇ i ∇ i . The action of general modified gravitational theories that depends on the above mentioned scalar quantities is given by (2.6) where g is a determinant of the metric g µν and L is a Lagrangian. As we will briefly review in the following, the action (2.6) encompasses the Horndeski/GLPV theories and Hořava-Lifshitz gravity.
First of all, Horndeski theories are described by the Lagrangian where φ ≡ (g µν φ ;ν ) ;µ , and G j (j = 2, · · · , 5) are functions in terms of a scalar field φ and its kinetic energy X = g µν ∂ µ φ∂ ν φ, R and G µν are the Ricci scalar and the Einstein tensor in four dimensions, respectively. Here and in the following, a lower index of L denotes the partial derivatives with respect to the scalar quantities represented in the index, e.g., G j,X ≡ ∂G j /∂X. In unitary gauge we have φ = φ(t) and X = −φ(t) 2 /N 2 , where a dot represents a derivative with respect to t. Hence the dependence of φ and X in the action (2.7) is interpreted as that of the lapse N and the time t. In fact, we can express the Lagrangian (2.7) of the form [19,20,40] Up to quadratic order in perturbations on the flat FLRW background we have that is the Hubble parameter. Horndeski theories have the following correspondence where F 3 and F 5 are auxiliary functions satisfying G 3 = F 3 + 2XF 3,X and G 5,X = F 5 /(2X) + F 5,X . From Eqs. (2.11)-(2.14) the following two relations hold under which the number of 6 independent functions reduces to 4. GLPV [20] generalized Horndeski theories in such a way that the coefficients A 4 , B 4 , A 4 , and B 5 are not necessarily related to each other. The general action (2.6) can incorporate both Horndeski and GLPV theories described by the Lagrangian (2.8).

A. Cosmological perturbations
The perturbed line element involving the four scalar perturbations δN , ψ, ζ, E, and tensor perturbations γ ij can be written of the form where a(t) is the scale factor, and with detĥ = 1. The tensor perturbation γ ij is traceless and divergence-free, i.e., γ ii = ∂ i γ ij = 0. The last term on the r.h.s. of Eq. (3.2) was introduced for the simplification of calculations, but it does not affect the second-order action of gravitational waves [50].
Choosing the gauge E = 0, the spatial infinitesimal transformation vector is fixed. To fix the time transformation vector in Horndeski and GLPV theories we choose the unitary gauge in which the field perturbation δφ vanishes, so the dependence on the scalar field φ and its kinetic term X does not explicitly appear in the action (2.6). In the projectable version of Hořava-Lifshitz gravity [24], the lapse N is a function of t alone and hence δN = 0. In the non-projectable Hořava-Lifshitz gravity [44], there is no such restriction for the gauge choice.
Expanding the action (2.6) up to second order in scalar perturbations for the spatial gauge choice E = 0, we can obtain the equations of motion for the background and linear scalar perturbations without fixing the temporal gauge. Varying the first-order perturbed action with respect to δN and δ √ h, respectively, the background equations are given by [19,40,43] The linear scalar perturbation equations derived by varying the second-order action in terms of δN , ψ, and ζ are presented in Ref. [43].
B. The second-order action for tensor perturbations Let us derive the second-order action of Eq. (2.6) for gravitational waves. Regarding the extrinsic curvature, tensor perturbations satisfy the relations K = 3H and δK i j = δ ikγ kj /2. Up to first order, the three-dimensional Ricci tensor reads The three-dimensional Ricci scalar from tensor perturbations is a second-order quantity, which is given by Then the quantity Z 1 is fourth-order in perturbations. On using the above relations, the second-order action for tensor modes reduces to S Note that there are no contributions to S (2) h from the scalars (2.5). The condition for avoiding the tensor ghost corresponds to L ,S > 0.
Varying the action (3.8) with respect to γ ij , we obtain the equation of motion where In the absence of spatial derivatives higher than second order, c t exactly corresponds to the propagation speed of gravitational waves. In order to avoid the small-scale Laplacian instability in this case, we require that c 2 t > 0. General Relativity corresponds to G 4 = M 2 pl /2 and G 5 = 0 in the Horndeski Lagrangian (2.7), i.e., −A 4 = B 4 = M 2 pl /2 and A 5 = B 5 = 0 in Eq. (2.8). In this case we have L ,S = M 2 pl /2, E = M 2 pl /2, c 2 t = 1, L ,Z = 0, and L ,Z2 = 0, so Eq. (3.11) reduces toγ ij +3Hγ ij −∆γ ij = 0.

IV. THE INFLATIONARY GRAVITATIONAL WAVES
In this section we derive the power spectrum of gravitational waves generated during inflation.

B. Equation of motion for a canonical field
Then the kinetic term in the action (3.8) can be expressed as S K = dτ d 3 x λ=+,× |v ′ λ | 2 /2, where a prime represents a derivative with respect to τ . From Eq.
where the function K(k, τ ) is defined as and In the context of low-energy effective field theories, we will discuss the case where K(k, τ ) ≃ c 2 t k 2 , such that the linear form of the dispersion relation, ω = c t k, is not modified by the nonlinear terms in Eq. (4.6) well below the cut-off of the theories. Otherwise, we would need to know the UV-completion of the theories, or our treatment would break down. In fact, the non-linear terms are suppressed for the physical wavenumber k phys = k/a much below the cut-off value k max phys ≡ M pl /|c 1 | 1/2 or M pl /|c 2 | 1/4 . In Hořava-Lifshitz gravity [24] and in the trans-Planckian physics studied in Refs. [51][52][53][54][55], k max phys is close to M pl , i.e., |c 1 | ∼ |c 2 | ∼ O(1). In the EFT approach of inflation advocated by Weinberg [27], the cutoff scale is slightly smaller than M pl , say √ ǫM pl , where ǫ = −Ḣ/H 2 is the slow-roll parameter typically of the order of 0.01.
In the following we shall focus on the situation in which the cut-off scale k max phys is much larger than the Hubble parameter H during inflation. In this case the Hubble radius crossing occurs in the linear regime of the dispersion relation (i.e., K ≃ c 2 t k 2 ), so that the second and third terms in the parenthesis of Eq. (4.6) are regarded as small corrections to the first term. In other words, the parameter defined by is much smaller than 1, where H k is the Hubble parameter at c t k = aH. According to the previous discussion, we will solve Eq. (4.5) iteratively, and write its solution in the form where the leading-order perturbation v In order to solve Eq. (4.10), we take into account the slow-roll inflationary corrections to the leading-order solution on the de Sitter background [56]. We then substitute the leading-order solution into Eq. (4.11) to obtain an iterative solution of v (1) λ .

C. Solutions to the tensor equations of motion
In the following we assume that the parameters defined by are much smaller than unity during inflation. Then the quantity z ′′ /z, up to next to leading-order corrections, can be estimated as Introducing a dimensionless variable y ≡ c t k aH , (4.14) its time derivative obeys y ′ = −aHy(1 − ǫ − s). Then, Eq. (4.10) can be expressed as Here and in the following, we drop contributions of the slow-roll corrections of the order of ǫ 2 . In other words, we deal with the first-order slow-roll parameters as constants. The solution to Eq. (4.15), after approximating by neglecting non-linear terms in the slow-roll parameters, is given by where α k and β k are integration constants, H ν (x) and H (2) ν (x) are Hankel functions of the first and second kinds respectively, and The Bunch-Davies vacuum corresponds to the choice β k = 0. On using the property H (4.18) The coefficient α k is determined by the Wronskian condition v = i, such that (up to second order in the slow-roll parameters) where c tk is the value of c t at c t k = aH (i.e., at y = 1). For the derivation of Eq. (4.19) we used the property that any time-dependent function f (τ ) on the quasi de Sitter background can be expanded around y = 1 (denoted by the subscript k), as f (τ ) = f (τ k ) − (ḟ /H k ) ln(τ /τ k ) [57]. For µ much smaller than 1 the quantity y is also expanded as y µ ≃ 1 + µ ln(τ /τ k ), so the variation of c t , H, and L ,S can be quantified as Expanding the function 2 ν Γ(ν) around ν = 3/2, it follows that The speed of propagation for this mode, for large k's, coincides, by construction, with c t , such that this choice is consistent with the assumption that the corrections do not modify the standard propagation of gravitational waves. Integrating Eq. (4.11) after substitution of the leading-order solution of v (0) λ , the resulting particular solution is given by The correction v (4.25) Since we are not interested in the next-order solution to Eq. (4.25), we can replace H, c t , and L ,S for H k , c tk , and L ,Sk , respectively.

D. The spectrum of inflationary gravitational waves
The tensor power spectrum is known by substituting where C = γ − 2 + ln 2 = −0.729637... and σ is defined by Eq. (4.8). The leading-order power spectrum is given by P lead h (k) = H 2 k /(π 2 L ,Sk c 3 tk ). The last two terms in the square bracket of Eq. (4.26), which correspond to the corrections induced by spatial derivatives higher than second order, are suppressed by the factor σ ≈ H 2 k /(k max phys ) 2 . Provided that σ/c 2 tk ≪ ǫ, these terms are smaller than the slow-roll corrections.
We introduce the tensor spectral index n t , as On using the property d ln k/dt| ctk=aH = H(1 − ǫ − s) and defining the following slow-roll parameters η ≡ǫ Hǫ , η S ≡ǫ S Hǫ S , δ s ≡ṡ Hs , (4.28) it follows that which should be evaluated at c t k = aH. The leadingorder spectral index is given by n lead t = −2ǫ − ǫ S − 3s.

V. APPLICATION TO CONCRETE THEORIES
We estimate the inflationary tensor power spectrum and its spectral index in concrete modified gravitational theories by using the general results derived in Sec. IV.
The contributions from the terms A 2 (N, t) and Since the leading-order spectrum is P lead h (k) = 2H 2 k /(π 2 M 2 pl ), the energy scale of inflation is directly known from the measurement of primordial gravitational waves. More concretely, we have H k /M pl ≃ π rP s (k)/2, where P s (k) ≃ 2.2 × 10 −9 is the observed scalar power spectrum [6] and r = P lead h (k)/P s (k) is the tensor-to-scalar ratio. On using the observational bound r 0.2 [6], we have that H k /M pl 4 × 10 −5 . Hence, for |g 3 |, |g 5 | 1, the corrections induced by spatial derivatives higher than second order are suppressed compared to the slow-roll corrections (typically of the order of 0.01).
The tensor power spectrum and its spectral index, up to next to leading-order terms, read where η F ≡ǫ F /(Hǫ F ).
Compared to Eq. (5.2), the leading-order power spectrum P lead h (k) = 2H 2 k /(π 2 M 2 pl F ) of Eq. (5.6) is divided by the term F . This term is associated with the conformal factor Ω 2 under the transformationĝ µν = Ω 2 (φ, X)g µν . In the following we study the case in which the conformal factor depends on φ alone, i.e., on t in unitary gauge. This assumption is justified provided that the X dependence in Ω 2 works only as slow-roll corrections to the leading-order φ-dependent term. Under the conformal transformationĝ µν = Ω 2 (t)g µν , the coefficients A 4 and B 4 in Eq. (2.8) transform, respectively, as [22] [22]. This means that, for the choice Ω 2 = F , the leading-order tensor spectrum in the transformed (Einstein) frame can be derived by setting L ,S = M 2 pl /2 and c t = 1 in Eq. (4.26), i.e.,P lead h (k) = 2Ĥ 2 k /(π 2 M 2 pl ). Since the Hubble parameters in two frames are related to each other asĤ = [H +Ḟ /(2N F )]/ √ F , the spectrum P lead h (k) is equivalent to P lead h (k) = 2H 2 k /(π 2 M 2 pl F ) at leading order in slow-roll. Provided that the null energy condition is not violated in the Einstein frame the Hubble parameterĤ decreases, in which case the tensor power spectrum is red-tilted.
The above properties can be notably seen in the Higgs inflationary scenario with the scalar-field potential V (φ) = (λ/4)(φ 2 − v 2 ) 2 and the function F = 1 + ζφ 2 /M 2 pl , where ζ is a non-minimal coupling [15] (see also Refs. [59]). In order to realize the self-coupling λ of the order of 0.1, the non-minimal coupling is constrained to be ζ = O(10 4 ) from the CMB normalization. For ζ ≫ 1 the quantity F is related to the number of efoldings N e from the end of inflation, as F ≃ 4N e /3 [60], which is much larger than 1 on scales relevant to the CMB anisotropies. The action in the Einstein frame is characterized by a canonically normalized field with the potentialV = V (φ)/F 2 [61], in which case the tensor spectrum P lead h (k) = 2Ĥ 2 k /(π 2 M 2 pl ) is red-tilted due to the decrease ofĤ.

C. GLPV theories
Let us proceed to the GLPV theories in unitary gauge, i.e., the Lagrangian (2.8). In this case the functions L ,S and E are given by L ,S = −A 4 (1 + ǫ 1 ) and E = B 4 (1 + ǫ 2 ) respectively, where Provided that ǫ 1 and ǫ 2 are regarded as slow-roll corrections to the leading-order terms of L ,S and E, we have . The difference from Horndeski theories is that A 4 and B 4 are not related with each other, so c 2 t generally differs from 1. Then the leadingorder tensor spectrum is given by where c 2 tk,lead = −B 4 /A 4 . We perform the disformal transformation given bỹ g µν = g µν + Γ(φ, X)∂ µ φ∂ ν φ, where Γ(φ, X) is a function in terms of φ and X [62,63]. In Ref. [22] it was shown that the structure of the GLPV action is preserved under this transformation. The coefficients A 4 and B 4 in the Lagrangian (2.8) are transformed as In the new frame the tensor propagation speed square is given byc 2 t,lead = −B 4 /Ã 4 = c 2 t,lead /(1 + ΓX). If we choose the function then it follows thatc 2 t,lead = 1. In this case, the coefficients in Eq. (2.8) are transformed asÃ 2 = A 2 /c t,lead , t,lead A 5 , andB 5 = B 5 . Sincec 2 t,lead = 1 in the new frame, the leading-order spectrum becomes P lead h (k) = H 2 k /(π 2 |Ã 4 |). If we make the conformal transformation g µν = Ω 2 (t)g µν further with Ω 2 = 2|Ã 4 |/M 2 pl , the resulting leading-order spectrum reduces to P lead h (k) = 2Ĥ 2 k /(π 2 M 2 pl ). The above discussion shows that the combination of the disformal and conformal transformations,ĝ µν = Ω 2 (φ)g µν + Γ(φ, X)∂ µ φ∂ ν φ, can lead to a metric frame in which the leading-order tensor power spectrum is of the standard form that depends on the Hubble param-eterĤ k alone. This conclusion is consistent with the recent results of Ref. [47] in which the authors took the EFT approach without having the direct connection to particular modified gravitational theories.

VI. CONCLUSIONS
We have studied gravitational waves on the flat FLRW background for the general action (2.6) that encompasses most of the modified gravitational theories proposed in the literature-including Horndeski theories, GLPV theories, and Hořava-Lifshitz gravity. The equation of motion (3.11) for tensor perturbations, which follows from the second-order action (3.8), involves the spatial derivatives higher than second order for the theories where the Lagrangian L depends on Z or Z 2 .
We derived the inflationary power spectrum of gravitational waves under the condition that the cut-off scale k max phys associated with the non-linear terms of Eq. (4.6) is much larger than the Hubble parameter H k at c t k = aH during inflation. On using the small parameter σ of the order of H 2 k /(k max phys ) 2 , the solution to Eq. (4.5) is obtained iteratively on the de Sitter background. Taking into account the slow-roll corrections to the leading-order solution as well, the resulting tensor power spectrum is given by Eq. (4.26) with the spectral index (4.29).
The corrections from the higher-order spatial derivatives to the leading-order power spectrum are suppressed by the factor σ/c 2 tk . This conclusion is consistent with the effect of modified trans-Planckian dispersion relations on the inflationary power spectrum [52][53][54][55]. For k max phys close to M pl and for c tk not very much smaller than 1, the corrections induced by the spatial derivatives higher than second order are smaller than the slow-roll corrections arising from the deviation from the de Sitter background.
We applied our general formula of the inflationary tensor power spectrum to a number of concrete modified gravitational theories. For the Lagrangian (5.1), which encompasses kinetic braiding models and Hořava-Lifshitz gravity, the leading-order spectrum is directly related to H k , as P lead h (k) = 2H 2 k /(π 2 M 2 pl ).
In Horndeski theories, where the tensor propagation speed is 1 at leading-order in slow-roll, P lead h (k) involves a dimensionless factor F = 2G 4 /M 2 pl in the denominator. Under the conformal transformationĝ µν = F g µν , the spectrum in the Einstein frame simply reduces to P lead h (k) = 2Ĥ 2 k /(π 2 M 2 pl ). In GLPV theories the leading-order tensor spectrum (5.11) involves the terms A 4 and c 2 tk,lead = −B 4 /A 4 . We showed that, under the disformal transformation g µν = Ω 2 (φ)g µν + Γ(φ, X)∂ µ φ∂ ν φ, it is possible to find a frame in whichĉ 2 tk,lead = 1 andÂ 4 = −M 2 pl /2 up to slow-roll corrections. Thus the prediction of inflationary gravitational waves is robust in that there exists the metric frame in which the leading-order spectrum is simply proportional toĤ 2 k in a vast class of modified gravitational theories.