Stable massless scalar polarization of $f(R)$ gravity

The action of $f(R)$ gravity is as follows:

where $\kappa=8\pi$, $R$ is the Ricci scalar, $f(R)$ is an arbitrary function of $R$, and $S_{m}$ is the matter action. In order to study gravitational wave in the absence of matter, we let ${S_{m}}=0$ for the vacuum field. In the metric formalism, the vacuum field equation of eq. (1) is obtained by varying its action:

where $f^{\prime}(R)={{df(R)}\mathord{\left/{\vphantom{{df(R)}{dR}}}\right.\kern-1.2pt}{dR}}$ and $\Box={g^{\mu\nu}}{\nabla_{\mu}}{\nabla_{\nu}}$. Taking the trace of eq. (2), we have:

For $f(R)$ gravity, its de Sitter stage corresponds to a vacuum solution with a positive constant background curvature $R_{d}$ [29]. By perturbing eq. (3) with the de Sitter background curvature $R_{d}$, the wave equation for the scalar field can be yielded:

Here the de Sitter stage is viewed as homogeneous and static, and $m$ is defined as:

According to [35], $R_{d}$ should meet the following condition:

At the scale size of gravitational wave detectors, the background metric ${\tilde{g}}_{\mu\nu}$ can be nearly approximated to the Minkowski metric [26, 29]. Adding a perturbation $h_{\mu\nu}$ to the background metric, one has:

where ${{\tilde{g}}_{\mu\nu}}\approx{\eta_{\mu\nu}}$ and $\eta_{\mu\nu}$ is the Minkowski metric. From here on, the subsequent discussion is based on the particular model $f(R)=R+\alpha{R^{2}}$ ($\alpha>0$) for the simplicity of process [21]. A new tensor is introduced:

Following [21], the transverse traceless gauge conditions are allowed to use:

Combining eqs. (7), (8) and (9), the Ricci tensor under the perturbation of the metric becomes:

For the particular model $f(R)=R+\alpha{R^{2}}$, under the first-order perturbation of the metric, eq. (2) becomes:

Comparing eq. (10) with eq. (11), one gets:

where ${m^{2}}={1\mathord{\left/{\vphantom{1{\left({6\alpha}\right)}}}\right.\kern-1.2pt}{\left({6\alpha}\right)}}$ is obtained from eq. (5) for $f(R)=R+\alpha{R^{2}}$. According to eq. (3) and $f(R)=R+\alpha{R^{2}}$, one has $\left({\Box-{m^{2}}}\right)R=0$ and eq. (12) can be reduced to:

which is the wave equation for the tensor field. Eq. (13) is consistent with the wave equation of General Relativity, and it means that their tensor polarizations are the same (including the plus and the cross modes) [36]. Moreover, general $f(R)$ model will also lead to the same tensor polarizations, see [29] for more details.

The scalar polarization in $f(R)$ gravity is an extra polarization that General Relativity does not have, and let us start with the wave equation eq. (4) for the scalar field. For the plane wave traveling along the $z$ direction, the solution to eq. (4) is:

where $\rm{c.c.}$ is the complex conjugation, $\phi_{1}$ is the amplitude, ${\eta_{\mu\nu}}{p^{\mu}}{p^{\nu}}=-{m^{2}}$ and ${p^{\mu}}=\left({\Omega,\ 0,\ 0,\ \sqrt{{\Omega^{2}}-{m^{2}}}}\right)$ [21, 36]. Keep ${{\bar{h}}_{\mu\nu}}=0$ to remove the tensor part of eq. (8) and $\delta R$ can be rewritten as the function of $vt-z$:

where $v={{\sqrt{{\Omega^{2}}-{m^{2}}}}\mathord{\left/{\vphantom{{\sqrt{{\Omega^{2}}-{m^{2}}}}\Omega}}\right.\kern-1.2pt}\Omega}$. The following linear approximation is applied:

Bring eq. (15) into eq. (16) to obtain:

and hence the geodesic deviation ${{{d^{2}}{x^{i}}}\mathord{\left/{\vphantom{{{d^{2}}{x^{i}}}{d{t^{2}}=-}}}\right.\kern-1.2pt}{d{t^{2}}=-}}{R_{itjt}}{x^{j}}$ can be turned into:

where ${m^{2}}={1\mathord{\left/{\vphantom{1{\left({6\alpha}\right)}}}\right.\kern-1.2pt}{\left({6\alpha}\right)}}$ for the particular model $f(R)=R+\alpha{R^{2}}$, eqs. (18) and (19) together stand for the breathing mode, and eq. (20) represents the longitudinal mode. It is shown that the scalar polarization of $f(R)$ is a mix of the longitudinal and the breathing modes [21]. And the mixed mode will be reduced to a pure breathing mode when the scalar mass $m$ of $f(R)$ (defined in eq. (5)) vanishes. Although the above discussions are nearly based on the particular model $f(R)=R+\alpha{R^{2}}$, some similar results have been given in general $f(R)$ gravity [26] and even in other extended gravity theories which includes $f(R)$ gravity [25, 26].

In order to explore the possibility of massless scalar polarization in $f(R)$ gravity, we are going to discuss several specific viable $f(R)$ models including Hu-Sawicki [37], Starobinsky [38] and Gogoi-Dev [34]:

where ${\mathop{\rm arccot}\nolimits}\left(x\right)$ is the inverse function of ${\mathop{\rm cot}\nolimits}\left(x\right)$, $\alpha_{i}$ and $\beta_{i}$ ($i=1,\ 2,\ 3$) are positive dimensionless parameters, while ${R_{c}}>0$ is a parameter in unit of curvature and roughly corresponds to the order of present Ricci scalar [6]. These above $f(R)$ models are proposed for constructing viable dark energy models, and they all satisfy two assumptions: $f(0)=0$ and $f(R)\to R-2\Lambda$ as $R\gg{R_{c}}$ [38]. Here $f(R)=R-2\Lambda$ is the traditional $\Lambda$CDM model, $\Lambda$ is the effective cosmological constant, and $\Lambda$ is considered to be unrelated to the quantum vacuum energy in above models [38]. Moreover, it should be noted that only 3-parameter $f(R)$ models are involved. This is because that the subsequent constraints in section III will contain three independent equations and the models with 2 parameters or less should be naturally excluded.

In the following part, we will first show a series of substitutions and calculation results for the simplification of the subsequent processes. Let us make the following definitions:

where $i=1,\ 2,\ 3$. It is worth noting that because the scalar mass in eq. (5) is defined by $R_{d}$, the following stability conditions almost based on $x_{d}$ instead of $x$. According to eq. (25) and eq. (26), ones have:

where ${f_{i}}^{\prime}(R)={{d{f_{i}}(R)}\mathord{\left/{\vphantom{{d{f_{i}}(R)}{dR,}}}\right.\kern-1.2pt}{dR}}$, ${f_{i}}^{\prime}({R_{c}}x)={{d{f_{i}}({R_{c}}x)}\mathord{\left/{\vphantom{{d{f_{i}}({R_{c}}x)}{dx}}}\right.\kern-1.2pt}{dx}}$, ${g_{i}}^{\prime}(x)={{d{g_{i}}(x)}\mathord{\left/{\vphantom{{d{g_{i}}(x)}{dx}}}\right.\kern-1.2pt}{dx}}$ and the same goes for higher derivatives. An agreement is made in our follow-up parts: the object of derivation depends on what the variable is in the parentheses behind the function. Besides, when the variable is changed to a specific value with a subscript ‘$d$’, it means taking the derivative of the variable at its specific value such as ${\left.{{g_{i}}^{\prime}({x_{d}})={g_{i}}^{\prime}(x)}\right|_{x={x_{d}}}}$ (although $x_{d}$ is also a variable). Bring eqs. (21), (22) and (23) into eq. (26) to individually get:

And eqs. (5) and (6) can be updated by eqs. (24), (26), (27) and (28) to:

For $R\geq{R_{0}}>0$, the necessary stability conditions for $f(R)$ need to be met:

where $R_{0}$ stands for the Ricci scalar in the infinite future [38]. The first condition eq. (36) guarantees that the graviton is not a ghost [39]. If eq. (36) is violated, the homogeneity and isotropy of the Universe would be lost [40, 41]. The second condition eq. (37) is provided to avoid the Dolgov-Kawasaki instability [39, 42]. Besides, eq. (37) was also considered to prevent the scalaron from being a tachyon and a ghost [38]. Employing eqs. (27) and (28), we can turn eqs. (36) and (37) into:

where $0<{x_{0}}\leq x$ and ${x_{0}}\equiv{{{R_{0}}}\mathord{\left/{\vphantom{{{R_{0}}}{{R_{c}}}}}\right.\kern-1.2pt}{{R_{c}}}}$. For the de Sitter curvature, we have ${x_{d}}\in\left[{\left.{{x_{0}},\ +\infty}\right)}\right.$, and hence the above inequations lead to:

What needs to be distinguished is: eqs. (40) and (41) will be applied first to check the stability at the curvature point, and if they are met, eqs. (38) and (39) will be applied later to check the stability in the curvature range.

In addition to the basic settings ${\alpha_{i}},{\beta_{i}}>0$ ($i=1,\ 2,\ 3$), the parameter constraints of ${{f_{i}}(R)}$ from cosmology are provided by:

and

Eq. (42) is given by considering the violations of the weak and strong equivalence principles, whose constraint range is stricter than the solar system constraints ${\beta_{1,2}}>0.5$ [43]. Eq. (43) is given by using the gravitational wave event GW170817 [34, 44]. The parameter $\alpha_{i}$ can be expressed by $x_{d}$ and ${\beta_{i}}$ according to eq. (35). Substitute eqs. (31), (32) and (33) into eq. (35) to gain:

It is important to point out that ${\alpha_{i}}\left({{x_{d}}}\right)$ are just the values determined by $x_{d}$ rather than the functions of $x$, and $x_{d}$ in ${\alpha_{i}}\left({{x_{d}}}\right)$ cannot be treated as a variable in the process of taking the all-order derivatives of ${g_{i}}(x)$ at $x_{d}$ (namely for: ${g_{i}}^{\prime}({x_{d}})$, ${g_{i}}^{\prime\prime}({x_{d}})$, ${g_{i}}^{\prime\prime\prime}({x_{d}})$, ${g_{i}}^{\prime\prime\prime\prime}({x_{d}})\cdots$). In addition, it should be noted that eq. (35) is satisfied naturally when eqs. (44), (45) and (46) are used. In other words, using them will remove the need of considering eq. (35).

Following [29, 45], by regarding eq. (3) as a Klein-Gordon equation in vacuum for an effective scalar field, we are allowed to define the scalar field $\Phi$ and the effective potential $V(\Phi)$ as follows:

In order to keep a settled perturbation of space-time, the background scalar $\Phi_{d}$ is strongly required to stay at a local minimum of the effective potential $V(\Phi)$ [29, 46]. The most direct and simple constraints to satisfy this requirement are:

where ${\Phi_{d}}\equiv f^{\prime}({R_{d}})$ and eq. (49) is actually equivalent to eq. (6). Due to eq. (5), one has ${V^{\prime\prime}}({\Phi_{d}})={m^{2}}$, so eq. (50) means ${m^{2}}>0$, agreeing with most suggestions for restricting the scalar mass squared of $f(R)$ [21, 28, 29]. As a result, the massless scalar polarization of $f(R)$ gravity seems to be excluded.

However, eqs. (49) and (50) are overly strict conditions for local minimum. For some specific $f(R)$ models, a local minimum is also able to be reached in the case of $m=0$ [30]. Let us give more general constraints for building a local minimum in the effective potential:

where $K=1,\ 2,\ 3\cdots$ and the superscript $\left(N\right)$ stands for the $N$th derivative of $V(\Phi)$ at $\Phi={\Phi_{d}}$. When $K=1$, eqs. (51) and (52) can return to the original constraints eqs. (49) and (50). As $K$ increases, there are more and more constraints for $f(R)$ to satisfy, and it means that more and more free parameters needs to be included in $f(R)$. In order to make it easier for $f(R)$ to survive in above constraints, the constraints of effective potential in the case of $K=2$ will be discussed:

where the subscript $i$ is added to represent for different effective potential coming from ${f_{i}}(R)$. The above constraints gives three independent equations, so $f(R)$ models need at least 3 parameters to satisfy them. That is why 3-parameter $f(R)$ models are shown in subsection II.2 for subsequent testing. Take eqs. (25), (26) and (27) into eqs. (47) and (48) to obtain:

For $x=x_{d}$, the above equations lead to:

By bringing eqs. (57), (58), (59) and (60) into the constraints of effective potential eqs. (53) and (54) and considering eqs. (34), (40) and (41), we have:

where eq. (64) is obtained under the premise ${g_{i}}^{\prime\prime\prime}({x_{d}})=0$. Eq. (61) indicates that the de Sitter points are stationary points and it is equal to eq. (35); eq. (62) is used to build up the massless case of $f(R)$; eq. (63) is applied to ensure that stationary points are extreme points instead of inflection points; eq. (64) is given to make sure that extreme points are local minimum points instead of local maximum points. In addition, the above constraints can be converted to the forms described by $f(R)$ if eqs. (24), (26), (27), (28), (29) and (30) are used.

In figure 1, a possible intersection of ${m_{1}}^{2}=0$ and ${g_{1}}^{\prime\prime\prime}({x_{d}})=0$ sits in the light red area, but it is definitely outside the light orange area. It implies that the constraints cannot be met concurrently, and hence the Hu-Sawicki model is inapplicable for a stable massless scalar polarization. In figure 2, it can be seen that the light orange area wraps ${m_{2}}^{2}=0$ and ${g_{2}}^{\prime\prime\prime}({x_{d}})=0$ while the light red area wraps ${g_{2}}^{\prime\prime\prime}({x_{d}})=0$. Because there is no intersection between the two lines (even for ${\beta_{2}}\gg 1$), the Starobinsky model is also inapplicable for a stable massless scalar polarization.

In figure 3, there is an intersection between ${m_{3}}^{2}=0$ and ${g_{3}}^{\prime\prime\prime}({x_{d}})=0$, and its coordinate is around (${\beta_{3}}=1.0441$, ${x_{d}}=0.9077$). By using eq. (46), one obtains ${\alpha_{3}}=0.3511$, agreeing with the required range $0<{\alpha_{3}}<0.7$. Different from figure 1 and figure 2, the intersection occurs in both the light red and the light orange areas at the same time, which means that all constraint conditions can be met and an active local minimum point can be reached. Therefore, the Gogoi-Dev model is feasible for a stable massless scalar polarization.

The results indicate that the models of Hu-Sawicki and Starobinsky cannot be used to build up a stable massless scalar polarization, while Gogoi-Dev model can do that. Because these involved models have the same freedom degree, the reason for the difference should lie in the model structure rather than the number of free parameters. Obvious evidence can be seen in the active areas constructed by a series of inequalities. The active areas in figures 1, 2 and 3 visibly own different shapes, and these shapes are drawn by the derivatives and parameter bounds of $f(R)$. In the next subsection, we are going to examine the feasibility of Gogoi-Dev model in massless case in more detail.

The constraint conditions eqs. (61), (62), (63) and (64) are aimed at giving a local minimum of the effective potential $V(\Phi)$. In order to show this clearly, some figures will be drawn for Gogoi-Dev model to reflect how ${V_{3}}(\Phi)$ change with $\Phi$. To start with, based on the intersection coordinate given in subsection IV.1, the model parameters $\alpha_{3}$ and $\beta_{3}$ are determined. After that, we apply eqs. (55), (56) and (57) to convert $\Phi$, ${V_{3}}^{\prime}(\Phi)$ and ${V_{3}}(\Phi)$ to the functions described by the parameter $x$, so that the figures of parametric functions about ${V_{3}}^{\prime}(\Phi)$ and ${V_{3}}(\Phi)$ versus $\Phi$ can be drawn with the change of $x$. In figure 4, there is a zero value for ${V_{3}}^{\prime}(\Phi)$ at ${\Phi_{d}}=0.4579$ , and ${V_{3}}^{\prime}(\Phi)<0$ on the left-hand side of $\Phi_{d}$ while ${V_{3}}^{\prime}(\Phi)>0$ on the right-hand side of $\Phi_{d}$. Thus, ${V_{3}}(\Phi)$ has a local minimum at $\Phi_{d}$, agreeing with the result of the picture of ${{{V_{3}}(\Phi)}\mathord{\left/{\vphantom{{{V_{3}}(\Phi)}{{R_{c}}}}}\right.\kern-1.2pt}{{R_{c}}}}$ . Here ${{{V_{3}}(\Phi)}\mathord{\left/{\vphantom{{{V_{3}}(\Phi)}{{R_{c}}}}}\right.\kern-1.2pt}{{R_{c}}}}$ is drawn by integrating ${{{V_{3}}^{\prime}(\Phi)}\mathord{\left/{\vphantom{{{V_{3}}^{\prime}(\Phi)}{{R_{c}}}}}\right.\kern-1.2pt}{{R_{c}}}}$ from $x=0.8$ to $t$ ($0.8\leq t\leq 1.0$) for avoiding the effect of extreme points.

Eqs. (40) and (41), the stability conditions for $f(R)$ at $x_{d}$, have been satisfied for Gogoi-Dev model. Therefore, eqs. (38) and (39), the stability conditions for $f(R)$ in $\left[{{x_{0}},\ +\infty}\right)$, should be checked in the following part. Because $0<{x_{0}}\leq{x_{d}}$, both ${g_{i}}^{\prime}(x)>0$ and ${g_{i}}^{\prime\prime}(x)>0$ need to be met in $\left[{{x_{0}},\ {x_{d}}}\right]$ and $\left[{{x_{d}},\ +\infty}\right)$. The range of $\left[{{x_{0}},\ {x_{d}}}\right]$ will not be strictly defined, since its range depends on the different values of the model parameter $R_{c}$. In other words, if there is a continuous range tightly attached to the left-hand side of $x_{d}$, and ${g_{i}}^{\prime}(x)>0$ and ${g_{i}}^{\prime\prime}(x)>0$ are met in this continuous range, we believe that these stability conditions could be satisfied in $\left[{{x_{0}},\ {x_{d}}}\right]$. In figure 5, we can see that ${g_{3}}^{\prime\prime}(x)>0$ in $\left[{0,\ +\infty}\right)$, so the stability condition eq. (39) is true for Gogoi-Dev model. On the other hand, there are ${g_{3}}^{\prime}(x)>0$ in $\left[{h,\ {x_{d}}}\right]$ ($0<h<{x_{d}}$) and $\left[{{x_{d}},\ +\infty}\right)$. Because $\left[{h,\ {x_{d}}}\right]$ can be regarded as $\left[{{x_{0}},\ {x_{d}}}\right]$ with the adjustable parameter $R_{c}$, the stability condition eq. (38) is also true for Gogoi-Dev model. To sum up, the stability in the curvature range is not violated for Gogoi-Dev model.