$F(R)$ Gravity with an Axion-like Particle: Dynamics, Gravity Waves, Late and Early-time Phenomenology

TL;DR

This paper explores a generalized $F(R)$ gravity model coupled with an axion-like scalar, analyzing its cosmological dynamics, gravitational wave behavior, and early universe constraints, revealing potential effects on gravitational wave propagation and cosmological evolution.

Contribution

It introduces a non-minimally coupled $F(R)$ gravity model with an axion-like particle, detailing its cosmological solutions, gravitational wave modifications, and early universe constraints.

Findings

Arbitrary cosmological evolutions can be realized within the model.

Gravitational wave speed matches $F(R)$ gravity, with possible enhancement or dissipation effects.

Constraints from the radiation era limit the non-minimal coupling strength.

Abstract

In this work we investigate several theoretical and phenomenological implications of a scalar -$F(R)$ gravity containing a non-minimal coupling to the scalar curvature. This kind of model is a generalization of axion-$F(R)$ gravity models, so we shall examine several implications of the latter theory. Firstly we study in detail the Einstein frame picture of the model, and also we discuss the dynamics of the cosmological system. By appropriately using the equations of motion, we demonstrate that an arbitrary cosmological evolution can be realized. Also we study the gravitational waves of the theory, and we demonstrate that the speed of their propagation is the same as in $F(R)$ gravity, but there is the possibility of enhancement or dissipation of the gravitational waves, an effect quite similar to the propagation of gravity waves in a viscous fluid. Finally, we examine the energy momentum tensor and we investigate which quantities related to it are conserved. We also present the constraints imposed by the radiation domination era on the non-minimal coupling of the axion scalar field to the scalar curvature.

Equations

$$\mathcal{S}=\int d^{4}x\sqrt{-g}\left[\frac{1}{2{\kappa }^2}F(R)+\frac{1}{2{\kappa }^2}h(\varphi )G(R)-\frac{1}{2}{\partial }^{\mu }\varphi {\partial }_{\mu }\varphi -V(\varphi )\right],$$

$$F(R)=R+\frac{1}{36H_i^2}{R}^2.$$

$$h(\varphi )\sim \frac{1}{{\varphi }^{\delta }},$$

$$G(R)\sim R^{\gamma },$$

$$V(\varphi (t))\simeq \frac{1}{2}{m}_a^2{\varphi }_i^2(t).$$

$$\dot{\varphi }(t_i)=\zeta \ll 1,\varphi (t_i)=f_a{\theta }_a,$$

$$\ddot{\varphi }+3H\dot{\varphi }+m_a^2\varphi =0,$$

$$\varphi (t)=A(t)\cos (m_{a}t),$$

$$\frac{\dot{A}}{m_a}\sim \frac{H}{m_a}\sim ϵ\ll 1,$$

$$A\sim a^{-3/2}.$$

$$\varphi (t)=a^{-3/2}\cos (m_{a}t).$$

$$S=\int d^{4}x\sqrt{-g}\left[\frac{1}{2{\kappa }^2}\left\{F(R)+h(\varphi )G(R)\right\}-\frac{1}{2}\omega (\varphi ){\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V(\varphi )+{\mathcal{L}}_{\mathrm{matter}}\left(g_{\mu \nu },{\Phi }_i\right)\right],$$

$$S_{AB}=\int d^{4}x\sqrt{-g}\left[\frac{1}{2{\kappa }^2}\left\{B\left(R-A\right)+F(A)+h(\varphi )G(A)-\frac{1}{2}\omega (\varphi ){\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V(\varphi )\right\}+{\mathcal{L}}_{\mathrm{matter}}\left(g_{\mu \nu },{\Phi }_i\right)\right].$$

$$B=F^{\prime }(A)+h(\varphi )G^{\prime }(A).$$

$$F^{\prime }(A)+h(\varphi )G^{\prime }(A)=F^{\prime }(R)+h(\varphi )G^{\prime }(R)>0.$$

$$S_{\sigma \varphi }=\int d^{4}x\sqrt{-g}$$

$$\displaystyle+\mathcal{L}_{\mathrm{matter}}\left(g_{\mu\nu},\Phi_{i}\right)\Bigr{]}\,.$$

$$g_{\mu \nu }={\mathrm{e}}^{-\sigma }{\tilde{g}}_{\mu \nu },$$

$$S_{\mathrm{E}}=$$

$$U\left(\varphi ,\sigma \right)\equiv$$

$$\omega (\varphi )>0,$$

$$\sqrt{\frac{1}{3}{\left(\frac{\partial U}{\partial \sigma }\right)}^2+\frac{{\mathrm{e}}^{\sigma }}{\omega (\varphi )}{\left(\frac{\partial U}{\partial \varphi }\right)}^2}\ge cU,$$

$$0=-{\mathrm{e}}^{\sigma }d\sigma +\left(F^{\prime \prime }(A)+h(\varphi )G^{\prime \prime }(A)\right)dA+h^{\prime }(\varphi )G^{\prime }(A)d\varphi ,$$

$$\frac{\partial A(\varphi ,\sigma )}{\partial \varphi }=-\frac{h^{\prime }(\varphi )G^{\prime }(A)}{F^{\prime \prime }(A)+h(\varphi )G^{\prime \prime }(A)},\frac{\partial A(\varphi ,\sigma )}{\partial \sigma }=-\frac{{\mathrm{e}}^{\sigma }}{F^{\prime \prime }(A)+h(\varphi )G^{\prime \prime }(A)},$$

$$\frac{\partial U}{\partial \sigma }=-{\mathrm{e}}^{-\sigma }A-2{\mathrm{e}}^{-2\sigma }\left(-F(A)-h(\varphi )G(A)+V(\varphi )\right),\frac{\partial U}{\partial \varphi }={\mathrm{e}}^{-2\sigma }{V}^{\prime }(\varphi ).$$

$${\left[\frac{1}{3}{\left\{-\frac{R}{F^{\prime }(R)+h(\varphi )G^{\prime }(R)}-\frac{2\left(-F(R)-h(\varphi )G(R)+V(\varphi )\right)}{{\left(F^{\prime }(R)+h(\varphi )G^{\prime }(R)\right)}^2}\right\}}^2+\frac{\omega (\varphi ){\left(V^{\prime }(\varphi )\right)}^2}{{\left(F^{\prime }(R)+h(\varphi )G^{\prime }(R)\right)}^3}\right]}^{\frac{1}{2}}$$

$$\ge c\left[\frac{R}{F^{\prime }(R)+h(\varphi )G^{\prime }(R)}+\frac{-F(R)-h(\varphi )G(R)+V(\varphi )}{{\left(F^{\prime }(R)+h(\varphi )G^{\prime }(R)\right)}^2}\right].$$

$$F(R)=R+f_0{R}^2,h(\varphi )G(R)=h_0{\varphi }^{-\delta }{R}^{\gamma }.$$

$$B=1+2f_{0}R+\gamma h_0{\varphi }^{-\delta }{R}^{\gamma -1},$$

$$\sqrt{\frac{1}{3}{\left(R+\left(2-\gamma \right)h_0{\varphi }^{-\delta }{R}^{\gamma }-V(\varphi )\right)}^2+\omega (\varphi ){\left(V^{\prime }(\varphi )\right)}^2\left(1+2f_{0}R+\gamma h_0{\varphi }^{-\delta }{R}^{\gamma -1}\right)}$$

$$\le c\left(f_0{R}^2+\left(\gamma -1\right)h_0{\varphi }^{-\delta }{R}^{\gamma }+V(\varphi )\right),$$

$$0=$$

Full text

I Introduction

Dark matter is one of the persisting problems in modern theoretical particle physics and cosmology, and up to date, no experimental verification occurred. Observational data coming from galactic scale structures, like the bullet cluster, or from galactic rotation curves, strongly indicate that dark matter seems to control the dynamics of galaxies rotation and also seems to control the collision of galaxies. In addition, the observation of the bullet cluster, strongly indicates that dark matter is a particle, and halos of dark matter particles accompany the visible structure of a galaxy. Although, the dark matter effects can be mimicked by modified gravity itself reviews4 ; reviews5 ; reviews6 ; reviews1 ; reviews2 ; reviews3 , see also Capozziello:2006ph ; Nojiri:2016vhu , observations seem to favor the particle nature of dark matter for the moment at least. Up to present date, the dark matter searches utterly failed to identify any dark matter particle, and to our opinion this is because most dark matter searches focused on mass scales for the weakly interacting massive particle (WIMP) of the order up to GeV or even hundred GeV Oikonomou:2006mh . However, the scientific community searching for WIMPs now focuses on mass scales of eV or much more smaller mass scales. One of the most promising class of WIMPs with tiny mass is the axion, or any axion like particle predicted from low scale compactifications of string theory Marsh:2015xka ; Marsh:2017yvc ; Odintsov:2019mlf ; Cicoli:2019ulk ; Fukunaga:2019unq ; Caputo:2019joi ; maxim ; Auriol:2018ovo ; Ioannisian:2017srr . In fact several experimental and observational proposals exist already in the literature Du:2018uak ; Henning:2018ogd ; Ouellet:2018beu ; Safdi:2018oeu ; Rozner:2019gba ; Avignone:2018zpw ; Caputo:2018vmy ; Caputo:2018ljp ; Millar:2016cjp ; Majorovits:2017ppy ; Anastassopoulos:2017ftl ; TheMADMAXWorkingGroup:2016hpc , see also Lawson:2019brd , and many researchers aim to identify in the laboratory this elusive particle. To our opinion, if the axion exists, it will be found in the next 10-15 years due to extensive experimental searches now performed, mainly based on the fact that axions and photons interact in the presence of magnetic fields Balakin:2009rg ; Balakin:2012up ; Balakin:2014oya , and the axion is the last resort of particle dark matter, unless nature hides the WIMPs to a supersymmetry breaking scale, and this will be another surprise for theorists. The axion is known to provide a very good candidate for dark matter, or at least some of the dark matter existing in the Universe Marsh:2015xka , and when we refer to the axion, this should not be confused with the QCD axion, which is a pseudo-scalar field, but we refer to any axion-like particle, with a primordial $U(1)$ broken symmetry. Particularly, these are known as misalignment axion fields, which are canonical scalar fields. In Ref. Odintsov:2019evb , we investigated an effective model of $F(R)$ gravity in the presence of an misalignment axion field, and by using the axion field dynamics known from the literature Marsh:2015xka , we demonstrated that the axion via a non-minimal coupling to the scalar curvature affects the late-time era, making possible for the dark energy era to be realized. Some striking features of the model were the facts that firstly, the early-time era was controlled by the well-known $R^{2}$ gravity Starobinsky:1982ee , and the axion field after it starts oscillating, for cosmic times for which $m_{a}\geq H$, so during and after the reheating era, its energy density scales as $\rho_{a}\sim a^{-3}$. Effectively it describes a dark matter condensate field, which on average yields an effective equation of state (EoS) parameter $\langle w\rangle=0$.

In this work, we shall further explore several theoretical and phenomenological features of the model Odintsov:2019evb , and of some extensions of this model, to include non-canonical scalar fields. So the study is focused on theoretical problems and implications of $F(R)$ gravity in the presence of a non-canonical in general scalar field, and we emphasize on the phenomenology of the theory containing the canonical scalar field, which is a subcase. Particularly, we shall consider the Einstein frame picture of the scalar-$F(R)$ gravity, and we shall derive the no-ghost conditions. Also we shall consider the general dynamics of the model and we shall demonstrate that the resulting gravitational equations of motion can be used as a reconstruction technique, which enables us to realize an arbitrary cosmological evolution. In addition, we shall study the gravitational wave spectrum, and we shall find several constraints on the scalar-$F(R)$ gravity theory. Finally, the energy conditions are considered, and by using these, for the radiation domination era, we evince how the scalar-$F(R)$ gravity theory can be constrained, and specifically how the non-minimal coupling of the axion to the curvature can be constrained.

The motivation for using extensions of $F(R)$ gravity including scalar fields is threefold. Firstly, $F(R)$ gravity has been considered and actively investigated as a model quite elegantly describing dark energy Capozziello:2003gx ; Nojiri:2003ft . In the $F(R)$ gravity, appears a scalar particle called scalaron which interacts with the standard model particles very weakly and therefore the lifetime of this particle could be rather long. Then some scenarios where the scalaron can be dark matter particle have been also considered Nojiri:2008nt ; Cembranos:2008gj ; Katsuragawa:2016yir . This scenario is very fascinating because the $F(R)$ gravity might explain both of the dark energy and dark matter. Since the scalaron is directly interacting with the standard model particles, however, the scalaron can decay into the standard particles even if the interaction is very weak and we need to consider some scenario to suppress this decay. One easy way to avoid this problem is to introduce another scalar field and we may regard the particle corresponding to the field as a dark matter. Secondly, the Starobinsky model seems to be quite robust against the Planck 2018 Akrami:2018odb constraints on inflation, and thirdly, after the Higgs particle discovery Aad:2012tfa , the scalar fields seem to be an inherent constituent of the primordial era. Thus in this work we shall further explore the theoretical and phenomenological implications of the misalignment axion-$F(R)$ gravity.

This paper is organized as follows: In section II, we briefly recall the essential features of the misalignment axion-$F(R)$ gravity model developed in Ref. Odintsov:2019evb . In section III, we study the Einstein frame picture of the model, and we derive the no-ghost criteria, while in section IV, we discuss the dynamics of the model. We also provide a general reconstruction technique that can be used to realize any arbitrary cosmological evolution. In section V we study the gravitational waves of the axion-$F(R)$ gravity model, and we consider several constraints related with the observations. In section VI, we discuss the energy conditions, and we study the dynamics of the model for several eras, and we demonstrate how the radiation domination era may impose a constraint on the non-minimal coupling of the axion to the scalar curvature. Finally, the conclusions follow at the end of the paper.

II Overview of the non-minimally Coupled Axion-$F(R)$ Gravity Model

In this section we shall briefly recall the main outcomes of the model developed in Ref. Odintsov:2019evb , in order to put the present paper in proper context, and to have a reference point for the studies that follow. The axion-like $F(R)$ gravity model of Ref. Odintsov:2019evb was based on the following gravitational action,

[TABLE]

with $\kappa^{2}=\frac{1}{8\pi G}$, and $G$ stands for Newton’s gravitational constant. The $F(R)$ gravity chosen in Ref. Odintsov:2019evb was the well-known $R^{2}$ gravity,

[TABLE]

Also the coupling $h(\phi)$ to $G(R)$, was assumed to be,

[TABLE]

choosing $\delta>0$, and also the $G(R)$ function is,

[TABLE]

with the parameter $\gamma$ belonging to the interval $0<\gamma<0.75$. The model of Ref. Odintsov:2019evb is an effective model which relies mostly on the axion field dynamics. The axion field at early times, when the axion mass $m_{a}$ satisfies $m_{a}\ll H$, is considered frozen in its vacuum expectation value $\phi_{i}$, with the potential being,

[TABLE]

The following initial conditions at early times are assumed,

[TABLE]

with $t_{i}$ being the cosmic time corresponding to the inflationary era, and $f_{a}$ being the axion decay constant. In view of the initial conditions, the axion is considered frozen, thus it contributes a minor overdamped cosmological constant during inflation, therefore the evolution during the inflationary era is governed solely by the $R^{2}$ gravity. As the Universe expands and cools, when $m_{a}\geq H$, the axion field starts to oscillate. For all cosmic times, the axion field equation of motion is,

[TABLE]

so for $m_{a}\geq H$, by assuming a slow-varying oscillatory behavior for the axion, of the form,

[TABLE]

with $A(t)$ quantifying the slow-varying behavior, since it satisfies,

[TABLE]

by solving the equation of motion (7) with the ansatz (8), we get the solution,

[TABLE]

Hence, for cosmic times after the inflationary era, during the reheating and until late times, the scalar field behaves as,

[TABLE]

Thus, at early times, the axion field has an effective equation of state (EoS) parameter $w_{a}=-1$, while for cosmic times $m_{a}\gg H$, it has an average EoS of the form $w_{a}=0$, which describes dark matter. Finally, as it was shown in Ref. Odintsov:2019evb , the late-time dynamics of the model is governed solely by the term $\sim h(\phi)G(R)$, with $h(\phi)\sim\phi^{-\delta}$ and $G(R)\sim R^{\gamma}$ and $0<\gamma<0.75$.

In the following sections we shall discuss several issues related with the phenomenological aspects of the axion-$F(R)$ gravity model. Particularly, we shall consider the Einstein frame counterpart theory, and we discuss the constraints imposed by the non-ghost condition, and several other theoretical issues related to the Einstein frame picture. Also we shall present the reconstruction techniques for the general scalar-$F(R)$ gravity, which enables us to realize any cosmological solution, given the Hubble rate. This is particularly useful for cosmological eras in between the inflationary and dark energy era. Also we shall discuss the gravitational waves solutions corresponding to early times, and finally, we shall examine if the energy conditions are satisfied, and how the radiation domination era may constrain the coupling of the scalar field $h(\phi)$ to the $G(R)$ gravity term.

III The Einstein Frame Picture: Constraints and Limitations

Let us firstly consider the Einstein frame picture of the scalar-$F(R)$ gravity theory, and we discuss in some detail some theoretical implications and constraints of the model.

Let us consider a more general gravitational action in comparison to the one appearing in Eq. (1), of the form,

[TABLE]

where $F(R)$ and $G(R)$ are functions of the scalar curvature $R$ and $h(\phi)$, $\omega(\phi)$, and $V(\phi)$ are functions of the scalar field $\phi$. Also, $\mathcal{L}_{\mathrm{matter}}\left(g_{\mu\nu},\Phi_{i}\right)$ is the Lagrangian density of the matter fluids present -if any- and $\Phi_{i}$’s express the matter fields. We now rewrite the action (12) by introducing the auxiliary fields $A$ and $B$ as follows,

[TABLE]

Then by varying the action with respect to the auxiliary scalar $A$, we obtain,

[TABLE]

Then the condition for the absence of the anti-gravity or the condition that the graviton is not ghost is given by,

[TABLE]

Obviously, this condition is satisfied by the gravitational action (1). Upon redefining the scalar field $B$ by using a new scalar field $\sigma$ as $B=\mathrm{e}^{\sigma}$, we assume that Eq. (14) can be solved with respect to $A$ as $A=A(\phi,\sigma)$. Then the action (13) can be rewritten as follows,

[TABLE]

By the scale transformation of the metric,

[TABLE]

the action (III) can be rewritten in the Einstein frame, and it is equal to,

[TABLE]

In addition to Eq. (15), the condition that ensures the absence of ghost degrees of freedom is,

[TABLE]

which is clearly satisfied by the gravitational action (1). Also, let us quote here a marginal remark, related to the swampland criteria, which is the following, the constraint given by the so-called swampland conjecture could have the following form,

[TABLE]

with a constant $c$. Since $B=\mathrm{e}^{\sigma}$, Eq. (14) yields,

[TABLE]

and we obtain,

[TABLE]

which in conjunction with Eqs. (14) and (III), indicates that,

[TABLE]

Then since $A=R$ in the original Jordan frame action, and $B=\mathrm{e}^{\sigma}$, by using Eq. (14), we can rewrite the condition (20), as follows,

[TABLE]

As an example, we consider the following model,

[TABLE]

Here $f_{0}$, $h_{0}$, $\delta$, and $\gamma$ are constants and we may assume $\delta>0$ and $0<\gamma<3/4$. Then Eq. (14) has the following form

[TABLE]

and the condition (III) gives

[TABLE]

which constrains the value of $h_{0}$ and the non-minimal coupling $h(\phi)$ in general.

IV General Dynamics in the Jordan Frame: Inflation and Intermediate Eras

In this section we shall discuss the dynamical evolution of the scalar-$F(R)$ gravity gravitational system, which is governed by the equations of motion. Also we shall show how the gravitational equations can be used as a reconstruction technique, and in effect an arbitrary cosmological evolution may be realized by the theory at hand.

By varying the action (12) with respect to the metric, we obtain the following equations of motion,

[TABLE]

where $T_{\mathrm{matter}\,\mu\nu}$ is the energy-momentum tensor of the matter fluids present. On the other hand, the variation of the action (12) with respect to the scalar field $\phi$ gives,

[TABLE]

In a spatially flat FRW universe,

[TABLE]

and by assuming that the scalar field $\phi$ depends only on the cosmological time $t$, the $(t,t)$ and $(i,j)$ components of (IV) have the following form,

[TABLE]

where, the Hubble rate $H$ is defined by $H=\dot{a}/a$ and the scalar curvature $R$ is given by $R=12H^{2}+6\dot{H}$. Furthermore $\rho$ and $p$ are the energy density and the pressure of the matter fluids, respectively. By using the ambiguity for the redefinition of the scalar field $\phi$, we may identify $\phi=t$. Then Eqs. (IV) and (IV) can be rewritten as,

[TABLE]

Then given an arbitrary cosmological evolution $H(t)$, by using the equation of state and the continuity equation, we may find the explicit $t$ dependencies of $\rho$ and $p$. Then the right hand side of both Eqs. (33) and (IV) is expressed as a function of the cosmic time $t$. Since $\phi=t$, by replacing $t$ with $\phi$ in the right hand side of both Eqs. (33) and (IV), we obtain the explicit forms of $\omega(\phi)$ and $V(\phi)$ as functions of the scalar field $\phi$. This indicates that an arbitrary expansion history of the Universe described by $H=H(t)$ can be realized by choosing $\omega(\phi)$ and $V(\phi)$ to satisfy Eqs. (33) and (IV) for any form of $F(R)$, $G(R)$, and $h(\phi)$. As an example, we consider the model in which the scale factor is,

[TABLE]

where $a_{0}$, $H_{I}$, $H_{T}$, and $H_{L}$ are dimensionful constants. Then, Eq. (35) yields,

[TABLE]

In effect, cosmic times for which $t\ll\frac{1}{H_{T}}$, $H\sim H_{I}$, correspond to the inflationary era in the early Universe and cosmic times for which $t\gg\frac{1}{H_{T}}$, $H\sim H_{L}$, correspond to the accelerated expansion of the late Universe. We also find that the scalar curvature $R$ is given by,

[TABLE]

When the matter has a constant EoS parameter $w\equiv\frac{p}{\rho}$, $\rho$ is given by $\rho=\rho_{0}a^{-3\left(1+w\right)}$ with a constant $\rho_{0}$. Then Eqs. (33) and (IV) indicate that the Universe whose evolution in the expansion is given by (35) is realized by choosing,

[TABLE]

Here the functions $F(R)$, $G(R)$, and $h(\phi)$ can be arbitrary but we may choose them as in (25).

V Gravitational Waves of the Axion-$F(R)$ Gravity Theory

We now consider the gravitational wave based on the action (III) in the Einstein frame by considering the perturbation of the background ${\tilde{g}}_{\mu\nu}={\tilde{g}^{(0)}}_{\mu\nu}$ as ${\tilde{g}}_{\mu\nu}={\tilde{g}^{(0)}}_{\mu\nu}+{\tilde{h}}_{\mu\nu}$ in the Einstein equation,

[TABLE]

Here the matter energy momentum tensor ${\tilde{T}}_{\mathrm{matter}}^{\mu\nu}$ in the Einstein frame is,

[TABLE]

If the matter fluids couple minimally with gravity, that is, if the matter Lagrangian $\mathcal{L}_{\mathrm{matter}}\left(\mathrm{e}^{-\sigma}{\tilde{g}}_{\mu\nu},\Phi_{i}\right)=\mathcal{L}_{\mathrm{matter}}\left(g_{\mu\nu},\Phi_{i}\right)$ does not include any derivative of the metric $g_{\mu\nu}$, the matter energy momentum tensor ${\tilde{T}}_{\mathrm{matter}}^{\mu\nu}$ in the Einstein frame is related with the matter energy momentum tensor $T_{\mathrm{matter}}^{\mu\nu}$ in the original Jordan frame as ${\tilde{T}}_{\mathrm{matter}}^{\mu\nu}=\mathrm{e}^{-3\sigma}T_{\mathrm{matter}}^{\mu\nu}$, that is, ${\tilde{T}}_{\mathrm{matter}\,\mu\nu}=\mathrm{e}^{-\sigma}T_{\mathrm{matter}\,\mu\nu}$. When we consider the gravitational wave, we often use the transverse and traceless gauge conditions. Since we are considering the scale transformation (17), if $h_{\mu\nu}$, which is defined by the fluctuation from the background metric $g_{\mu\nu}=g^{(0)}_{\mu\nu}$ as $g_{\mu\nu}=g^{(0)}_{\mu\nu}+h_{\mu\nu}$ in the original frame in the action (12), satisfies the transverse and traceless gauge conditions,

[TABLE]

However, the scale transformed fluctuation ${\tilde{h}}_{\mu\nu}=\mathrm{e}^{\sigma}h_{\mu\nu}$ does not always satisfies the first condition in (42), although the second condition is trivially satisfied, ${\tilde{g}}^{(0)\,\mu\nu}{\tilde{h}}_{\mu\nu}=\mathrm{e}^{-\sigma}g^{(0)\,\mu\nu}\mathrm{e}^{\sigma}h_{\mu\nu}=g^{(0)\,\mu\nu}h_{\mu\nu}=0$. For the first condition in (42), under the scale transformation, we find,

[TABLE]

Then if we assume that a homogeneous and isometric background metric, and therefore $\sigma$ only depends on the cosmological time $t$ and also $g^{(0)}_{ti}=0$, then if we consider the perturbation with $h_{t\mu}=0$ since we are considering the massless spin 2 mode, we find,

[TABLE]

Therefore the gauge conditions in (42) for the graviton are not changed by the scale transformation,

[TABLE]

Then under the condition (45), the equation for the gravitational wave can be written as follows,

[TABLE]

We are now interested in the massless spin two mode, which satisfies,

[TABLE]

In the spatially flat FRW universe in the Einstein frame,

[TABLE]

where $d{\tilde{t}}\equiv\mathrm{e}^{\frac{\sigma}{2}}dt$ and $\tilde{a}\left(\tilde{t}\right)\equiv\mathrm{e}^{\frac{\sigma}{2}}a(t)$, due to the isometry in the spacial part, we may assume,

[TABLE]

We may further assume that the matter energy-momentum tensor (41) in the Einstein frame has the following form as in the perfect fluid,

[TABLE]

Here $\left(\tilde{U}^{\mu}\right)$ is the four velocity of the matter fluid and we now assume $\tilde{U}^{0}=1$ and $\tilde{U}^{i}=0$. In Eq. (50), ${\tilde{\gamma}}_{\mu\nu}$ is the projection tensor to the spatial directions perpendicular to $\tilde{U}^{\mu}$,

[TABLE]

We now also assume that the matter fluid minimally couples with the metric ${\tilde{g}}_{\mu\nu}$, that is, the coupling between the matter fluids and the metric does not include the derivative of the metric. Then, under the perturbation ${\tilde{g}}_{\mu\nu}={\tilde{g}}^{(0)}_{\mu\nu}+{\tilde{h}}_{\mu\nu}$, we find,

[TABLE]

On the other hand, the variation of ${\tilde{U}}_{\mu}$ is given by using the condition ${\tilde{U}}^{\mu}{\tilde{U}}_{\mu}=-1$, that is,

[TABLE]

and hence we have,

[TABLE]

or equivalently,

[TABLE]

Note that the arbitrary vector $l_{\mu}$ satisfies ${\tilde{U}}^{\mu}l_{\mu}=0$, but we choose $l_{\mu}=0$ by assuming the isometry. Due to the isometry, we may assume $\rho^{ij}$ and $p^{ij}$ are proportional to $\delta^{ij}$, Then the $(t,t)$ and $(t,i)$ components of equation for gravitational wave (V) are trivially satisfied and the $(i,j)$ component is given by,

[TABLE]

Here we have used $\left(\frac{d\phi}{d\tilde{t}}\right)^{2}=\left(\frac{dt}{d\tilde{t}}\right)^{2}=\mathrm{e}^{-\sigma}$. We may rewrite Eq. (V) in the original Jordan frame. Since, ${\tilde{h}}_{\mu\nu}=\mathrm{e}^{\sigma}h_{\mu\nu}$ and $\tilde{a}=\mathrm{e}^{\frac{\sigma}{2}}a$, we find,

[TABLE]

Then, due to the fact that $\tilde{p}=\mathrm{e}^{-2\sigma}p$ from (50) (${\tilde{T}}_{\mathrm{matter}\,\mu\nu}=\mathrm{e}^{-\sigma}T_{\mathrm{matter}\,\mu\nu}$ and ${\tilde{\gamma}}_{\mu\nu}=\mathrm{e}^{\sigma}\gamma_{\mu\nu}$), Eq. (V) is rewritten as follows,

[TABLE]

We should note that $\mathrm{e}^{\sigma}$ is given in Eq. (14) and $U\left(\phi,\sigma\right)$ is given in Eq. (III). Then in terms of the Jordan frame, we find,

[TABLE]

By choosing $\phi=t$ and using (33) and (IV), we obtain,

[TABLE]

and we can further rewrite Eq. (V) as follows,

[TABLE]

As an example, we consider the quasi-de Sitter spacetime case,

[TABLE]

Here $H_{0}$, $H_{1}$, and $t_{0}$ are constants and we assume the second term in (62) is much smaller than the first one, and the third term is also much smaller than the second term, $H_{0}\gg\left|H_{1}\left(t-t_{0}\right)\right|\gg\left|H_{2}\left(t-t_{0}\right)^{2}\right|$ by assuming $H_{0}$ is positive. We may also assume that $H_{0}^{2}\sim H_{1}$. Then we find,

[TABLE]

By further assuming $\left|t-t_{0}\right|\ll\left|t_{0}\right|$, we find,

[TABLE]

Then Eq. (62) can be approximated as follows,

[TABLE]

Here we have neglected $\ddot{\sigma}$ and we put $a=1$ because we now considering the propagation of the gravitational wave for a time scale much shorter than the scale of the expansion of the Universe. For simplicity, we neglect the contribution from the matter fluids by putting $\rho=p=0$. Then we may separate the variables by assuming $h_{ij}=\mathrm{e}^{i\bm{k}\cdot\bm{x}}\hat{h}_{ij}(t)$. Then from Eq. (65 we get,

[TABLE]

where $k^{2}\equiv\bm{k}\cdot\bm{k}$. By assuming $\hat{h}_{ij}(t)\propto\mathrm{e}^{-i\omega t}$, we find,

[TABLE]

For large wavenumbers, $k^{2}\gg\left|\frac{9}{16}\sigma_{1}^{2}+6H_{1}+20H_{0}^{2}+\frac{17}{2}\sigma_{1}^{2}20\sigma_{1}H_{0}\right|$, we find $\omega\sim k$, and therefore there is no change in the propagation speed of the gravitational wave as in the standard $F(R)$ gravity case Nojiri:2017hai . If $\sigma_{1}>0$, however, the gravitational wave is enhanced and if $\sigma_{1}<0$, dissipation of the gravity wave occurs. The situation is very similar to the propagation in a viscous fluid Brevik:2019yma . The enhancement or the dissipation of the gravity wave occurs due to the term $\frac{3}{2}\sigma_{1}\partial_{t}h_{ij}$ in Eq. (67), which includes the first derivative of $h_{ij}$. In Eq. (V) in the Einstein frame, such a first derivative term does not appear. Then the enhancement or the dissipation of the gravity wave occurs as an effect originating from the scale transformation, ${\tilde{h}}_{\mu\nu}=\mathrm{e}^{\sigma}h_{\mu\nu}$. The effect of the enhancement or the dissipation from the scale transformation has been also found in the standard $F(R)$ gravity Capozziello:2017vdi . For the recently observed gravitational waves, Abbott:2016blz ; Abbott:2016nmj ; Abbott:2017vtc ; Abbott:2017oio ; Abbott:2017gyy ; TheLIGOScientific:2017qsa , the distances between the sources and the earth are about a few hundreds Mpc. Because no dissipation or enhancement of the gravitational waves has been observed we find the following constraint in the present Universe,

[TABLE]

VI Energy Momentum Tensor, Energy Conditions and Constraints of the Axion-like Particle Minimal Coupling $h(\phi)$

We now consider the conserved quantities like energy-momentum tensors as in Capozziello:2018wul , and also we shall constrain the parameter $\delta$ appearing in the non-minimal coupling of $h(\phi)$ to the scalar curvature term $G(R)$. In (IV), the energy-momentum tensor $T_{\mathrm{matter}\,\mu\nu}$ of the matter fluids is, of course, conserved, $\nabla^{\mu}T_{\mathrm{matter}\,\mu\nu}=0$. As in the standard $F(R)$ gravity, the Bianchi identity indicates that the following quantity is conserved,

[TABLE]

Therefore by using (IV), we may define the conserved energy momentum tensor for the scalar field $\phi$ as follows,

[TABLE]

Furthermore Eq. (IV) can be rewritten as follows,

[TABLE]

Then by using the Bianchi identity, we find the following conserved tensor,

[TABLE]

We should note that any linear combination of $T_{\mathrm{matter}\,\mu\nu}$, $T_{F(R)\,\mu\nu}$, $T_{\phi\,\mu\nu}$, and ${\hat{T}}_{\mu\nu}$, that is, ${\bar{T}}_{\mu\nu}\equiv c_{1}T_{\mathrm{matter}\,\mu\nu}+c_{2}T_{F(R)\,\mu\nu}+c_{3}T_{\phi\,\mu\nu}+c_{4}\tilde{T}_{\mu\nu}$ with constants $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$, is conserved. On the other hand, in the Einstein frame, the energy-momentum tensor ${\tilde{T}}_{\mathrm{matter}\,\mu\nu}$ is not conserved but Eq. (40) indicates that the following quantity is conserved,

[TABLE]

In the standard $F(R)$ gravity, when $F(R)$ behaves as $F(R)\propto R^{m}$, if we include the contributions from the matter with a constant EoS parameter $w$, for a flat FRW universe (30), the solution is given by,

[TABLE]

Then the effective EoS parameter is,

[TABLE]

Then even if $w>-1$, when $m<0$, all the energy conditions are effectively not satisfied for the energy momentum tensor correspond to (VI). However, in the axion-$F(R)$ gravity model, all the curvature related terms, namely $F(R)$ and $G(R)$, contain positive powers of the curvature, hence the energy conditions are satisfied. Let us elaborate on this in a more detailed manner. When the curvature related terms dominate the evolution, namely at early and late-times, the axion has two different behaviors. Particularly, for early times, the axion is frozen, and its average EoS parameter is $w=-1$, recalling that the axion is the sole matter content of the model (1), and it is a dynamical field. Thus at early times we have from Eq. (75) $w_{\mathrm{eff}}=-1$, since $w=-1$. On the other hand, after the axion starts to oscillate, the axion always scales as $\phi(t)\sim a^{-3/2}$, as it can be seen from Eq. (11). It therefore gives an average EoS $\langle w\rangle=0$, for all cosmic times $m_{a}\geq H$.

In principle, the radiation domination era solely can be used to constrain the parameter $\delta$ appearing in the non-minimal coupling $h(\phi)$. Let us elaborate on this issue, because this point can be tricky, since during the early-time era, the total energy density of the cosmological system is determined by the $R^{2}$ gravity, due to the fact that the axion is frozen in its vacuum expectation value, and during the matter domination era, the axion potential $V\sim\phi^{2}$ dominates the evolution, yielding $a(t)\sim t^{2/3}$. Thus the second part of Eq. (74) which determines the parameter $h_{0}$ as a function of the parameter $m$ which is the power of the $F(R)$ gravity, no longer holds true. The $F(R)$ gravity-curvature related terms do not control the matter domination era. However, at late times, the term $h(\phi)G(R)$ controls the evolution, and it actually yields a asymptotically de Sitter evolution, asymptotic referring to large cosmic times Odintsov:2019evb . Thus the evolution is not of power-law type at late-times. During the matter domination era, the potential dominates the evolution, thus since $\phi^{2}\sim a^{-3}$ the solution $V\sim\phi^{2}\sim a^{-3}\sim t^{-2}$, when equated to the total energy density evolution it yields $-3h_{0}=-2$, hence $h_{0}=2/3$ which is the expected behavior during the matter domination. Hence the only era for which both the potential and the term $h(\phi)G(R)$ might be equally dominant is the radiation domination era, and the combined action of the two yields a power law scale factor of the form $a\sim t^{1/2}$. If one of the two was dominant, we would either have a matter domination era, or an asymptotic de Sitter solution as was evinced in Ref. Odintsov:2019evb and we previously demonstrated. This is a speculation though and strong numerical analysis is needed, but let us assume that this is the case. In such a case, $h_{0}=1/2$ during the radiation domination, and hence $\rho\sim t^{-3/2}$ so by equating this to $h(\phi)G(R)\sim\phi^{-\delta}R^{\gamma}\propto\phi^{-\delta}t^{-2\gamma}$ and solving with respect to $\gamma$, we get,

[TABLE]

Since $0<\gamma<0.75$, Eq. (76) imposes a constraint on the values of the parameter $\delta$, which is the following,

[TABLE]

which is obtained by the assumption that during the radiation domination era, both the potential and the non-minimal coupling term are dominant, however this a crude estimate and rigid numerical analysis is needed to effectively quantify the dynamics of this era, so we refrain from going into further details. Let us now consider the energy conditions, to efficiently examine if these hold true for the model (1). In the standard Einstein gravity, the energy conditions are given for $\rho$ and $p$ in the FRW universe,

[TABLE]

The above conditions are rewritten in terms of the EoS parameter $w$ if we assume $\rho\geq 0$.

[TABLE]

Even for the models in this paper, we may require,

[TABLE]

It is obvious that the matter content of the model (1) satisfies the energy conditions quantified in terms of $w$, since for early times $w=-1$, while for all later times, $w=0$. With regard to the total EoS parameter $w_{\mathrm{eff}}$, the values it takes during all eras are as follows,

[TABLE]

so all the energy conditions quantified in terms of $w_{\mathrm{eff}}$ are satisfied.

VII Conclusions

In this paper we investigated several theoretical and phenomenological issues related to axion-$F(R)$ gravity in the presence of a non-minimal coupling of the axion scalar to the scalar curvature. Particularly, we investigated the ghost-free conditions and the Einstein frame implications of the model in general. Also we demonstrated how a general cosmological solution may be realized by the scalar-$F(R)$ gravity, by using the equations of motion in the Jordan frame as the main constituents of a reconstruction method. Also, we examined the gravitational waves of the scalar-$F(R)$ gravity theory, and we demonstrated that the gravitational wave modes have the same propagation speed as in $F(R)$ gravity, but in some cases enhancement or dissipation may occur, an effect similar to that of a propagation of a gravity wave in a viscous fluid. Finally, we performed a general study of the conserved quantities and of the energy momentum tensor, and we investigated whether the energy conditions are satisfied. Also we discussed the phenomenological constraints that are imposed on the non-minimal coupling of the axion to the scalar curvature during the radiation domination era. A future perspective of this work is to extend the gravitational wave study in terms of string theory originating Chern-Simons gravity, as in Ref. Odintsov:2019mlf . The existence of polarization asymmetry in the gravitational wave spectrum, may indicate that a potential candidate theory that can harbor such effects is an axion-$F(R)$ gravity with axion Chern-Simons terms.

Acknowledgments

This work is supported by MINECO (Spain), FIS2016-76363-P, and by project 2017 SGR247 (AGAUR, Catalonia) (S.D.O). This work is also supported by MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas “Cosmic Acceleration” No. 15H05890 (S.N.) and the JSPS Grant-in-Aid for Scientific Research (C) No. 18K03615 (S.N.).