New measures to test modified gravity cosmologies

The accelerated expansion of the current Universe has been clarified by the observations of type Ia supernovae in the late 1990s [1, 2] and supported by observations of cosmic microwave background radiation (CMB) [3, 4, 5] and baryon acoustic oscillations (BAO) [6, 7, 8, 9, 10, 11, 12]. Broadly speaking, there are two approaches to explain the accelerating Universe; (i) introducing an additional energy component, called dark energy, to the total energy budget and (ii) modifying the action of gravity from the one predicted by Einstein’s theory of relativity. In dark energy models, one adopts the Einstein-Hilbert action and introduces a fluid matter with the equation-of-state parameter $w<-1/3$, where $w=-1$ corresponds to a cosmological constant. The simplest dark energy model is $\Lambda$CDM, and more generally, the $w$CDM or quintessence model has been considered [13, 14, 15, 16]. On the other hand, the Einstein-Hilbert action itself is modified in the models of modified gravity theories. The models include $F(R)$ gravity [17, 18, 19, 20, 21, 22], massive gravity [23, 24, 25], and Horndeski’s theory [26].

The differences of the models appear both in the background equations and in the linear perturbation equations. While $\Lambda$CDM is consistent with most of the observations, a tension between the values of the Hubble constant, $H_{0}$, constrained from early universe and late universe started to be recognized after the Planck mission reported the first results [4]. The descrepancy in the $H_{0}$ values between local observarions and CMB observations was first reported using Cepheid variables [27, 28, 29], and it has also been seen in the observations of strong lensing time delay and so on [30, 31, 32, 33, 34, 35]. Moreover, there is another tension in a parameter, $f\sigma_{8}$, where $f$ is the growth rate of the universe and $\sigma_{8}$ is the normalization of the density fluctuation amplitude. This parameter can be directly constrained through peculiar velocities of galaxies in galaxy redshift surveys, known as redshift-space distortions (RSD) [36, 37], and is used to distinguish modified gravity models [38, 39, 40, 41, 42]. However, the $f\sigma_{8}$ values observed so far at $z<1$ are systematically lower than the prediction of the best-fitting $\Lambda$CDM model from the Planck result [43, 44, 45, 46, 47, 48], as pointed out by, e.g., Refs. [49, 50]. Therefore, the importance of reconsidering the dynamics of the Universe in modified gravity models is increasing.

Modified gravity theories, in general, have too many degrees of freedom to be completely analyzed. Consequently, the observables of perturbation quantities (e.g. growth rate of the matter density perturbation) have been investigated only for simple models or for phenomenologically parametrized models. Moreover, the background evolution in modified gravity models is often assumed to be the same as that in the $\Lambda$CDM model. In this paper we take account of both the variations of background dynamics and those of perturbation quantities. For this purpose, we introduce new phase diagrams; the $\alpha{\rm-}f\sigma_{8}$ and $H{\rm-}f\sigma_{8}$ diagrams. Similar diagrams can be found in Refs. [51, 52, 53, 54, 55]. Here $\alpha$ describes the effect on gravitational lensing and is defined in Eq. (2.4) below in terms of the deflection potential $\Psi_{\rm defl}$, and $\alpha=0$ in the case of general relativity (e.g., [56, 57, 58, 59, 60, 61]). Thus, these diagrams will enable us to investigate how each of the two key observations of large-scale structures, RSD and gravitational lensing, can constrain a given modified gravity model.

In this paper, we consider Horndeski’s theory, which is a general scalar-tensor theory. We do not adopt phenomenological parameterizations which have been commonly used in the literature. We focus several specific examples of Horndeski’s gravity and demonstrate how deviations from the $\Lambda$CDM model appear in the $\alpha{\rm-}f\sigma_{8}$ and $H{\rm-}f\sigma_{8}$ diagrams.

The contents of the paper are as follows. We briefly overview how models of dark energy and modified gravity behave on the $\alpha{\rm-}f\sigma_{8}$ and $H{\rm-}f\sigma_{8}$ diagrams in Sec. 2. For this purpose we present the former and latter diagrams for the simple representative cases of dark energy and modified gravity, the $w$CDM and $F(R)$ gravity models, respectively. Then we describe Horndeski’s theory in Sec. 3, and considering its typical examples, we present their $\alpha{\rm-}f\sigma_{8}$ and $H{\rm-}f\sigma_{8}$ diagrams in Sec. 4. Our concluding remarks are given in Sec. 5. A more detailed description of Horndeski’s theory is presented in Appendix A. A short note on the possibility of having negative $\alpha$ is given in Appendix B.

Throughout this paper, we adopt Natural units, $\hbar=c=k_{B}=1$, and the gravitational constant $8\pi G$ is denoted by ${\kappa}^{2}\equiv 8\pi/{M_{Pl}}^{2}$ with the Planck mass of $M_{Pl}=G^{-1/2}=1.2\times 10^{19}$GeV.

In models in which dark energy is given by a matter field, commonly its energy momentum tensor is decoupled from the other matter components. In this case, the main effect of dark energy is to modify the background evolution of the Universe, and hence its effect to the growth of linear perturbations is indirect. On the other hand, in modified gravity theories of dark energy, its effective energy momentum tensor is likely to be coupled to the matter components. As a result, not only the background evolution of the Universe but also the linear perturbation equations may significantly deviate from those in the $\Lambda$CDM model.

In the following, we assume the metric,

and adopt notations in Fourier space as

where $k$ is the wave number, $D(t,k)$ is the growing mode of the matter density perturbation, and $\hat{\delta}(k)$ describes the scale dependence of $\delta(t,k)$ at initial time $t=t_{i}$ under the normalization of $D$ as $D(t_{i},k)=1$.

In Eq. (2.2), we introduced a quantity $\alpha$, which vanishes in Einstein gravity (i.e., $\Psi+\Phi=0$). The relation of $\alpha$ with the well-known gravitational slip parameter, $\eta\equiv-\Phi/\Psi$ [39], is $\eta=1/(1+\alpha)$. Thus $\alpha$ characterizes how much a given gravity model deviates from General Relativity (GR). Because the strength of gravitational lensing is determined by the deflection potential [62],

we see that the gravitational lensing effect is reduced if $\alpha>0$ or $\alpha<-2$ and enhanced if $-2<\alpha<0$. We note that there is a subtlety in this interpretation. It will be shown in section 4 that, for a class of modified gravity models considered in this paper, the lensing effect is unaffected if we compare it for the same mass distribution.

To see this, let us first introduce $G_{\rm eff}$ which relates the Newton potential to the matter density perturbation in modified gravity:

Let us also introduce $G_{\rm light}$ that would relate the deflection potential $\Psi_{\rm defl}$ to $\delta\rho$,

We have $G_{\rm eff}=G$ and $G_{\rm light}=2G$ in GR. On the other hand, for modified gravity, the $\alpha$-dependence of $G_{\rm eff}$ is found in section 4 as

From Eqs. (2.4) and (2.5), we find

Hence we obtain

Inserting Eq. (2.7) into the above $G_{\rm eff}$ gives $G_{\rm light}=2G$, that is, the resulting $\Psi_{\rm defl}$ will be the same as that in GR, independent of $\alpha$.

However, as the mass distribution is measured by its gravitational effect that includes the modification in the effective gravitational constant, what can be compared is the lensing effect for the same $\Psi$. This means our original interpretation is correct from an observational point of view. More precisely speaking, it is the gravitationally inferred mass distribution that is overestimated if $\alpha>0$ or $\alpha<-2$ and otherwise if $-2<\alpha<0$, while the gravitational lensing effect remains the same.

To quantify the growth of linear perturbations, one usually considers the combination, $f\sigma_{8}$, inferred from galaxy surveys, where $f=d\ln D/d\ln a$ and $\sigma_{8}$ is the normalization parameter of the density perturbation spectrum. In the following, to test modified gravity models, we evaluate the ratio of $f\sigma_{8}$ in a given model to that in the $\Lambda$CDM model, $f\sigma_{8}/f\sigma_{8,\Lambda{\rm CDM}}$. Note that the value of $f$ should approach unity in any viable model of gravity in the high-redshift limit. We also assume that the value of $\sigma_{8}$ coincides with that of the $\Lambda$CDM model in the high redshift limit. Thus, this ratio becomes unity at high redshifts. Since $f\sigma_{8}$ depends not only on the modifications of the linear perturbation equations but also on the background evolution of the Universe, below we will study correlations between $f\sigma_{8}$ and $\alpha$ and between $f\sigma_{8}$ and the Hubble expansion rate $H$ in various models of gravity. The evolution equation of the quasi-static mode of the matter density contrast $\delta=\delta\rho_{m}/\rho_{m}$ is expressed as [63, 61]

The parameter $f\sigma_{8}$ will be evaluated by using Eq. (2.10) with appropriate initial conditions.

Before presenting detailed studies on modified gravity models, let us first consider the simplest dark energy model, i.e. the $w$CDM model. Note that $\alpha=0$ because there is no deviation from GR in this model. The $H{\rm-}f\sigma_{8}$ diagram is depicted in Fig. 2. The initial conditions are set at $z=10$ where they coincides with those of the $\Lambda$CDM model. The figure shows that the deviation in $f\sigma_{8}$ is almost proportional to that in the Hubble rate at each redshift, with negative proportionality coefficients. Thus $f\sigma_{8}$ in $w$CDM models becomes smaller than that in the $\Lambda$CDM model for models with a larger Hubble rate $H$ which corresponds to $w<-1$. We note that the linear perturbation equations in the $w$CDM model are unchanged from those of the $\Lambda$CDM model. Therefore, the smaller value of $f\sigma_{8}$ due to a larger $H$ may be regarded as a purely background effect. We should note that if we apply a different boundary condition, e.g., $H_{wCDM}=H_{\Lambda CDM}$ and $\Omega_{m}=0.31$ at $z=0$, the quantitative results will be different though the qualitative tendency will remain the same.

As we mentioned in the above, the behavior of the matter density perturbation is not independent from the background evolution of the Universe in general. In some theories, e.g. $F(R)$ gravity theories, even if the difference between the model and GR at the background level is negligibly small, there can be $O(1)$ difference at perturbation level due to the scale dependence of the matter density perturbation that enhances the deviation from GR on small scales. As a characteristic example for such a case, let us consider $F(R)$ gravity with

where $R_{0}$ and $\Lambda$ are the current values of the Ricci scalar and the cosmological constant, respectively. For the parameter $F_{R0}$ in the range $10^{-6}<|F_{R0}|<10^{-4}\ll\Lambda/R_{0}$, this model is known to reproduce the background evolution of the $\Lambda$CDM model almost exactly [64]. In Fig. 2, we plot the predicted values of $\alpha$ and $f\sigma$ for several redshifts. In general, the function $\delta(t,k)$ is not separable, in the other words, the $k$-dependence of $D(t,k)$ in (2.3) cannot be ignored in $F(R)$ gravity theories. Here, for definiteness, we plot $f\sigma_{8}$, i.e. $f\sigma$ at $k=(8h^{-1}$Mpc$)^{-1}$. It shows that the density perturbation in this model always evolves faster than that in the case of the $\Lambda$CDM model, and $\alpha$ is always positive.

In the previous section, we studied two examples; $w$CDM model and $F(R)$ gravity theory. In the $w$CDM case, the linear perturbation equations are the same as those in the $\Lambda$CDM model, while the background evolution of the Universe is different. On the other hand, in the $F(R)$ gravity case, the background evolution of the Universe is the same as the $\Lambda$CDM model, while the linear perturbation equations are different. In more general modified gravity theories, both of them will be different from the $\Lambda$CDM model. Here we consider Horndeski’s theory. It is a general scalar-tensor theory which includes $F(R)$ gravity as a special case [63, 65, 66].

The action in Horndeski’s theory is given by [26, 67, 68]

where

Here, $K$, $G_{3}$, $G_{4}$, and $G_{5}$ are generic functions of $\phi$ and $X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$, and the subscript $X$ means a derivative with respect to $X$. The total action is the sum of $S_{H}$ and the matter action $S_{\mathrm{matter}}$ which contains baryons and cold dark matter. More details are referred to Appendix A

The recent observations of gravitational wave event GW170817 [69] and its electromagnetic counterparts [70, 71, 72] showed that the propagation speed of gravitational waves should satisfy

in the relatively recent Universe. This bound implies the sound speed of the tensor mode, given by

should be almost unity, where and below the subscript $\phi$ means a derivative with respect to $\phi$. If the terms $XG_{5\phi}$, $XG_{4X}$, $\cdots$ are relevant for the evolution of the Universe, then one expects a substantial deviation of $c_{T}^{2}$ from unity. Therefore, it is reasonable to assume that the terms proportional to $G_{4X}$, $G_{5\phi}$, and $G_{5X}$ are not relevant for the current accelerated expansion of the Universe. Hence we set $G_{4X}=G_{5}=0$ in the following.

Even after this simplification, the theory still remains quite complicated. For example, the effective Newton constant that can be defined on sufficiently small scales $k^{2}\gg a^{2}H^{2}$ takes the form [61],

where we have introduced an effective coefficient of the kinetic term $C_{\rm kin}$

As for $\alpha$ which describes the modification of the lensing effect, it is expressed using Eq. (3.11) as [61]

The above equation shows that $\alpha$ is non-vanishing only if $G_{4}$ is non-trivial, i.e. is $\phi$-dependent. It also shows that $\alpha<0$ is realized only if $G_{3X}\neq 0$ provided that we have $K_{X}>0$ and $G_{4}>0$ which are satisfied for most healthy theories of gravity (see Fig. 3). Thus $\alpha=0$ if $G_{4\phi}=0$, $\alpha>0$ if $G_{4\phi}\neq 0$ and $G_{3}=0$, while $\alpha$ can be positive or negative if $G_{3}\neq 0$ and $G_{4\phi}\neq 0$.

In passing we note that $f(R)$ gravity theories correspond to the case [63, 65, 66];

Because $G_{3}=0$, $\alpha$ is always positive in $f(R)$ gravity with $K_{X}>0$. We should note that $O(a^{2}H^{2}/k^{2})$ terms in Eqs. (3.8) and (3.12) are to be taken into account in F(R) gravity models [63]. The case $K_{X}\leq 0$ with $G_{3}=0$ or $K_{X}-2G_{3\phi}\leq 0$ may be also acceptable if instabilities are absent. Such a case is discussed in Appendix B, but we will not consider it in the main text for simplicity.

Let us now consider a few specific examples of Horndeski’s gravity that show small but observationally interesting deviations from the $\Lambda$CDM model. To compare with the $\Lambda$CDM model, we will assume the same amount of matter as that in the $\Lambda$CDM model, i.e. $\Omega_{m,0}h^{2}$ is fixed, and fix the ratio $f\sigma_{8,\Lambda CDM}/f\sigma_{8}=1$ at sufficiently high redshifts, $z\geq 10$. In what follows, to evaluate the effects of functions $K(\phi,X)$, $G_{3}(\phi,X)$, and $G_{4}(\phi)$, we assume their forms as

where $K_{2}$, $V_{0}$, $V_{1}$, $m^{2}$, $g$ and $\lambda$ are constant parameters.

In this above class of models, the expressions for $\alpha$ and $G_{\rm eff}$ are greatly simplified as

where $C_{\rm kin}$ defined in Eq. (3.11) is also simplified as

As $G_{4\phi}$ should be small enough to guarantee the proximity to Newton gravity in the large scale structure formation, the above two equations imply that $\alpha$ is small and positive if $C_{\rm kin}=O(1)$ and $>0$, and $G_{\rm eff}\approx G$.

Now we are in a position to evaluate the effect of $\alpha$ on gravitational lensing for the same mass density distribution. Since the resulting gravitational potential is proportional to the effective gravitational constant $G_{\rm eff}$, the resulting gravitational lensing effect is proportional to the deflection potential given by

where $\Psi_{GR}$ is the gravitational potential of the mass distribution if gravity were GR. Using Eqs. (4.4) and (4.5), we find

which gives

where $\Psi_{{\rm defl},GR}$ is the deflection potential in the case of GR. Thus we see that the gravitational lensing effect is independent of $\alpha$ for the same mass density distribution. In particular, it remains essentially the same as that in GR if $\lambda\phi/M_{pl}\ll 1$, as we mentioned in section 2.

We have investigated a class of dark energy models based on Horndeski’s theory of modified gravity which exhibit small but interesting deviations from the $\Lambda$CDM model not only in the background evolution but also in the linear perturbation level. The models include the $w$CDM, $F(R)$ gravity, and four kinds of Horndeski gravity models. To classify the properties of these models, we have introduced two diagrams; $H{\rm-}f\sigma_{8}$ and $\alpha{\rm-}f\sigma_{8}$ diagrams. We have found that these two diagrams provide a useful tool to distinguish the differences in the observational predictions of different models from each other.

Figure 8 shows the summary of our results. The right panel shows the $\alpha-f\sigma_{8}$ diagram, which exhibits deviations in the behavior of linear perturbations from GR, at $z=0.5$ for various models with various parameters. The canonical scalar model with linear potential (denoted by V₁, discussed in section 4.1) does not have a direct coupling to gravity, hence $\alpha=0$. The non-minimally coupled scalar model with flat potential (denoted by G4, discussed in section 4.2) gives non-vanishing $\alpha$ but the values are too small ($\sim O(0.01)$) to be seen in the figure, whereas the non-minimally coupled scalar model with quadratic potential (denoted by G4m², discussed in section 4.3) . As for the non-minimally coupled scalar model with non-canonical kinetic term (denoted by G4Kx, discussed in section 4.4), $\alpha=1$, that is the dynamics of the linear perturbation is quite different from GR.

In all of the models studied in this paper, we have found $\alpha>0$. As discussed in Appendix B, there exist theoretically acceptable models with negative $\alpha$, but they all satisfy $\alpha<-2$. In fact, the condition $\alpha<-2$ is necessary to guarantee the positivity of $G_{\rm eff}$ as can be seen from Eq. (4.8). Thus either $\alpha>0$ or $\alpha<-2$, our result implies that gravity becomes effectively stronger than GR in all the models. In other words, if we are to measure the mass density distribution, we would overestimate it if we use the gravitational dynamics, while we would obtain a correct estimate if we use gravitational lensing observations. Whether this has any substantial implications to observational data analysis is an issue to be studied.

The left panel shows the $H{\rm-}f\sigma_{8}$ diagram at $z=0.5$ for various models with various parameters. It shows the dependence of the growth rate of linear perturbations on the difference in the background evolution of the Universe. We see that in all models except for the non-minimally coupled scalar model with quadratic potential or non-canonical kinetic term, there is a tendency that larger $H$ gives smaller $f\sigma_{8}$ in comparison with $\Lambda$CDM. In the case of the quadratic potential model (G4m²), because of the oscillatory feature in $H$, the fact that $f\sigma_{8}$ are smaller for smaller $H$ depends on the redshift, as well as on the choice of the model parameters. So it is difficult to discuss a general tendency in this model. In the case of the non-canonical kinetic term model (G4Kx), the dependence of $f\sigma_{8}$ on $H$ seems very small. This is explained by the fact that this model can mimic the $\Lambda$CDM model very well as long as the background evolution of the Universe is concerned.

The cosmic shear power spectrum in weak lensing surveys and the redshift-space power spectrum in galaxy surveys directly constrain $\alpha$ and $f\sigma_{8}$, respectively. The latter observable can further probe $H$ using BAO as a standard ruler. The blue bars denote observational $1\sigma$ error bars obtained by Alam $et\leavevmode\nobreak\ al.$ [77]. Since it is a $1\sigma$ constraint, no reliable conclusion can be drawn, but it seems that when compared to the $\Lambda$CDM model, those models that yield slightly larger $H$ with slightly smaller $f\sigma_{8}$ are preferred. It is, however, important to note that most of the current observational analyses have been performed assuming $\Lambda$CDM and GR, a so-called consistency test. In order to test a modified gravity model, one needs to use a theoretical template of the power spectrum with the given gravity model [78, 79, 47]. Ref. [80] has derived an analytic formula of the matter power spectrum in real space under Horndeski’s theory. Further theoretical efforts are required in order to constrain general modified gravity models on the $H-f\sigma_{8}-\alpha$ diagrams proposed in this paper.

So far, observational constraints are not so severe yet to exclude any of these models. But eventually we will be able to exclude most of the models as observational accuracies improve. The $H{\rm-}f\sigma$ and $\alpha{\rm-}f\sigma_{8}$ diagrams we introduced, or their variants, may play an important role at such a stage.

We would like to thank T. Namikawa for discussions in the early stage of this work. T. O. acknowledges support from the Ministry of Science and Technology of Taiwan under Grants No. MOST 106-2119-M-001-031-MY3 and the Career Development Award, Academia Sinina (AS-CDA-108-M02) for the period of 2019 to 2023. The work of M. S. is supported in part by JSPS KAKENHI No. 20H04727.