Using Cosmological Perturbation Theory to Distinguish between GR and Unimodular Gravity

Abstract

Unimodular gravity is one of the oldest geometric gravity theories and alternatives to general relativity. Essentially, it is based on the Einstein–Hilbert Lagrangian with an additional constraint on the determinant of the metric. It can be explicitly shown that unimodular gravity can be recast as general relativity in the presence of a cosmological constant. This fact has led to many discussions on the equivalence of both theories at the classical and quantum levels. Here, we present an analysis focused on the classical scalar perturbations around a cosmological background. We focus on the unusual situation in which the typical conservation laws are not adopted. The discussion is extended to the case where a non-minimal coupled scalar field is introduced. We also present a gauge-invariant analysis showing that perturbations in unimodular gravity display instabilities. Our results reinforce that the equivalence is not verified completely at a cosmological perturbative level.

1. Introduction

General relativity (GR) aims to describe gravity as a manifestation of the geometry of space–time in four dimensions. It is based on the mathematics of differential geometry. The theory is invariant by the full group of diffeomorphism transformations. It has led to important discoveries such as the existence of black holes and gravitational waves. At the same time, it permits very successful applications for the description of the universe as a whole, leading to the standard cosmological model. The latter explains, in a quite simple way, almost all observational data, even at the cost of introducing a dark sector composed of two exotic components (dark matter and dark energy), not detected directly until now, and also a primordial inflationary phase driven by a new field, the inflaton, whose nature is still a matter of debate. On the other hand, GR is plagued by the presence of classical and quantum singularities and, at the same time, it has not been quantized until now in a fully consistent way. These problems have led to an intensive search for an alternative description of the gravitational phenomena, generally in line with the geometrical approach, that can cope with the existence of the dark sector and the inflationary phase, being free of singularities and admitting a consistent quantum version.

GR was proposed in 1915 and, soon after, alternative formulations appeared. In 1919, the unimodular constrained version of GR was formulated, leading to what is now known as unimodular gravity (UG) [1,2]. In UG, the determinant of the metric is fixed as a constant, in occurrence, equal to 1. Whereas this can be viewed as a choice of the coordinate system, it has some important consequences. First of all, UG is invariant by a subclass of transformations, called transverse diffeomorphisms (TD) [3]. If reparametrizations of the metric are included, the resulting transformations are called Weyl transverse diffeomorphisms (WTD); see [4] for a detailed discussion of WTD. Moreover, the field equations are traceless, implying the absence of information about geometrical quantities, e.g., the Ricci scalar R. Matter can be coupled in a particular way to the geometrical sector, preserving the traceless nature of the equations. If the conservation of the canonical energy–momentum tensor is imposed, the UG field equations imply the GR equations in the presence of a cosmological constant $\mathsf{\Lambda }$, which appears as an integration constant. This is generally viewed as an advantage with respect to GR. Moreover, it is generally argued that UG offers some improvements over GR at a quantum level [4]. Many discussions exist, on the other hand, on the possible equivalence between RG and UG, but the restriction of the invariance of UG to the TD group seems to indicate that this equivalence is not complete (see, for example, Refs. [4,5] and references therein). This issue has been widely explored in the literature, both at the classical [6,7,8,9,10] and quantum levels [11,12,13,14,15,16,17].

Our goal in this work is to reinforce our previous claim that cosmological perturbation theory is able to provide a possible means to differentiate between GR and UG already at a classical level.

One of the possible reasons for the non-equivalence between GR and UG arises from the conservation of the canonical energy–momentum tensor. In GR, the conservation laws are direct consequences of the invariance with respect to general diffeomorphism transformations. This property is reflected in the use of the Bianchi identities. For UG, the invariance with respect to the TD (which is the UG version to be employed here) does not lead, in principle, to the same conservation laws as in GR. This is related to the fact that the UG field equations are traceless and the application of the Bianchi identities leads to a relation where the divergence of the canonical energy–momentum tensor is not necessarily zero. However, we can impose, by hand, the same conservation laws as in GR, and the GR field equations in the presence of an integration constant (identified as the cosmological constant) is obtained. Does this indicate an equivalence between GR and UG? First, we must remember that the usual conservation law constitutes a choice in UG. Moreover, in UG, even with this choice, the invariance remains dictated by the restricted TD, instead of the full diffeomorphism group.

In this text, we delve into the issue of the perturbative characteristics of UG in comparison to GR. We demonstrate that while equivalence can be achieved for the vacuum scenario, the presence of matter may alter this conclusion. Furthermore, this perturbative non-equivalence can be extended even to vacuum solutions when an extension of the unimodular structure is implemented with a non-minimally coupled scalar field.

In the next section (Section 2), we provide a brief overview of the fundamental equations governing the cosmological background in both GR and UG. Moving on to Section 3, we discuss the perturbative aspects and present the equivalence between the results obtained in both theories for the vacuum case. We also review relevant findings from the existing literature, particularly those obtained in Refs. [18,19]. Section 4 is dedicated to a gauge-invariant analysis in the absence of the usual energy–momentum conservation law, which constitutes one of the most significant outcomes of this work. For the sake of completeness, we also discuss the scenario in which matter is present. In Section 5, we examine the unimodular version of the Brans–Dicke theory, while Section 6 delves into its application to $f\left(R\right)$ theories.

Finally, in Section 7, we present our concluding remarks and summarize the key findings of this study.

2. Background Cosmological Structure

2.1. Friedmann Equations

The GR equations in the presence of a cosmological constant and of a matter sector, as deduced from the Einstein–Hilbert Lagrangian, are

$$\begin{array}{c}\hfill R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=8\pi G{T}_{\mu \nu }+g_{\mu \nu }\mathsf{\Lambda }.\end{array}$$

(1)

In the above equation, $R_{\mu \nu }$ is the Ricci tensor. Its contraction with the metric tensor provides the Ricci scalar $R=R_{\mu \nu }{g}^{\mu \nu }$, and $\mathsf{\Lambda }$ denotes the cosmological constant. The energy–momentum tensor is denoted by $T_{\mu \nu }$. We will use here the metric signature $(+---)$.

The application of the Bianchi identities leads to the energy–momentum tensor $T^{\mu \nu }$ conservation:

$$\begin{array}{c}\hfill {T^{\mu \nu }}_{;\mu }=0.\end{array}$$

(2)

The conservation laws related to the energy–momentum tensor can be alternatively deduced from the invariance of the Einstein–Hilbert Lagrangian by diffeomorphic transformation [20].

The UG equations can also be deduced from the Einstein–Hilbert Lagrangian but through the introduction of a constraint on the determinant of the metric. For details, see [18,19]. With this procedure, the UG field equations read

$$\begin{array}{c}\hfill R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R=8\pi G\{T_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }T\}.\end{array}$$

(3)

Equation (3) is valid even if we introduce a cosmological constant in the action, i.e., after solving the constraints represented by the Lagrangian multiplier, the cosmological term disappears from the final equations. Then, the application of the Bianchi identities leads to the non-standard equation

$$\begin{array}{c}\hfill \frac{R^{;\nu }}{4}=8\pi G\{{T^{\mu \nu }}_{;\mu }-\frac{T^{;\nu }}{4}\}.\end{array}$$

(4)

Indeed, Equation (4) deserves some comments. The Newtonian approximation (4) leads to the usual Poisson equation, as in the appropriate limit of GR. However, in contrast to what happens in GR, the TD, on which UG is based, does not lead to the conservation Equation (2). Instead, it predicts the non-vanishing divergence of the energy–momentum tensor given by

$$\begin{array}{c}\hfill {T^{\mu \nu }}_{;\mu }={\mathsf{\Theta }}^{;\nu },\end{array}$$

(5)

$\mathsf{\Theta }$ is an (until now) unknown scalar function. This equation characterizes the general class of non-conservative theories of gravity. Such theories have been extensively studied in the literature. Ref. [21] reviews the numerous scenarios involving the non-conservation of the energy–momentum tensor. It is also possible to impose an extra condition given by (2). If this extra condition is imposed, (4) can be integrated, leading to (1). However, in doing so, $\mathsf{\Lambda }$ becomes a simple integration constant. This is the basis of the usual remark that UG may alleviate the cosmological constant problem, since now it is not associated with the vacuum energy or a geometric zero-order Lovelock invariant.

In fact, if the conservation of the energy–momentum tensor is imposed, after removing the right-hand side of (5), Equation (4) implies

$$\begin{array}{c}\hfill R^{;\nu }=-8\pi G{T}^{;\nu }\Rightarrow R+8\pi GT=\mathrm{constant}.\end{array}$$

(6)

Identifying the constant on the right-hand side of the above result with $-4\mathsf{\Lambda }$, the GR equations with a cosmological term are recovered.

Of course, we can also keep (4), since it results from the application of the Bianchi identities via the identification

$$\begin{array}{c}\hfill \mathsf{\Theta }=\frac{R}{4}+2\pi GT.\end{array}$$

(7)

However, again, this is a choice and many others are possible since $\mathsf{\Theta }$ is not determined from the unimodular construction.

Let us, for the moment, make the choice (7) in order to verify the consequences in a specific cosmological context. In any case, the choice $\mathsf{\Theta }=0$ implies GR with a cosmological constant, with the remaining issue of the interpretation of this constant.

Let us focus on the homogeneous, isotropic and expanding cosmological background considering the flat Friedmann–Lemaitre–Robertson–Walker (FLRW) metric given by

$$\begin{array}{c}\hfill d{s}^2=d{t}^2-a^2\left(t\right)(d{x}^2+d{y}^2+d{z}^2).\end{array}$$

(8)

In our approach to UG, the unimodular condition is $\sqrt{-g}=\xi$, where $\xi$ is an arbitrary reference (external, if one prefers) tensorial density, allowing us to choose freely the coordinate system, in contrast to the original choice $\sqrt{-g}=1$, which fixes the coordinate system.

Applying the metric (8) to the GR equations, we obtain the following (Friedmann) equations of motion:

$$\begin{array}{ccc}\hfill H^2& =& \frac{8\pi G}{3}\rho +\frac{\mathsf{\Lambda }}{3},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \dot{\rho }+3H(\rho +p)& =& 0,\hfill \end{array}$$

(10)

with $H=\dot{a}/a$, the Hubble function. We must specify an equation of state connecting $\rho$ and p. In doing so, we obtain two equations for two variables, $\rho$ and a.

On the other hand, the same cosmological metric applied to Equations (3) and (4) implies

$$\begin{array}{ccc}\hfill \dot{H}& =& -4\pi G(\rho +p),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \ddot{H}+4H\dot{H}& =& -4\pi G[\dot{\rho }+\dot{p}+4H(\rho +p)].\hfill \end{array}$$

(12)

The main point here is that by inserting (11) into (12), we obtain an identity. Hence, both equations have the same content: the system is underdetermined, having more variables than equations. This is a consequence of the fact that the UG Equation (3) is traceless, leading to a lack of information on the Ricci scalar, in contrast to the GR equations. Moreover, Equation (11)—and also (12)—is sensitive only to the combination $\rho +p$, the enthalpy density of the system. In Ref. [18], we assumed that matter behaves as radiation, as suggested by the traceless nature of the field equations. This choice led to many serious implications at a perturbative level, as discussed in Ref. [18]. In particular, as we will discuss later, there is a direct transition from the radiation era to the de Sitter era. However, due to the structure of UG, fluctuations grow strongly, even during the radiation phase, which can ensure the correct amplification of the primordial tiny fluctuations leading to a successful structure formation scenario. Other choices, besides imposing a radiative-like fluid, are possible in principle, but they generally lead to the absence of a primordial hot phase dominated by radiation, which would lead inevitably to an unrealistic cosmological model.

In what follows, we will focus on the much simpler vacuum case. This will allow us to verify to what extent the UG is, at least classically, equivalent to GR, as frequently evoked.

2.2. UG and GR Vacuum Cosmological Solutions

In the vacuum case, $\rho =p=0$. Notice, however, that $\mathsf{\Lambda }$ may remain different from zero in GR, leading to

$$\begin{array}{c}\hfill H^2=\frac{\mathsf{\Lambda }}{3}.\end{array}$$

(13)

The first remark is that only $\mathsf{\Lambda }>0$ is possible, i.e., a de Sitter solution, excluding the anti-de-Sitter case ($\mathsf{\Lambda }<0$). The Hubble function is constant, implying an exponential solution for the scale factor. Using the definition of the Ricci scalar,

$$\begin{array}{c}\hfill R=-6(\dot{H}+2H^2),\end{array}$$

(14)

the solution implies that R is constant and negative, in agreement with a maximally symmetric de Sitter space–time. Hence, the solution for the scale factor is given by

$$\begin{array}{c}\hfill a\propto e^{\pm Ht},\end{array}$$

(15)

describing either an exponentially expanding or contracting universe. Having in mind any application to the inflationary universe, only the expanding solution is used. We remark that the Minkowski case corresponds to $\mathsf{\Lambda }=0$. Hence, Minkowski and de Sitter are the possible space–times in the vacuum cosmological solutions of the GR theory in the presence of a cosmological constant, the Minkowski one being the trivial one.

For UG, the vacuum case leads to two simple equations:

$$\begin{array}{ccc}\hfill \dot{H}& =& 0,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \dot{R}& =& 0.\hfill \end{array}$$

(17)

The last one implies $R=-2{\mathsf{\Lambda }}_U$, where ${\mathsf{\Lambda }}_U$ is an integration constant that can be related to the cosmological constant in the GR context. The factor $-2$ has been introduced only to make this connection clearer. Again, the solution of (17) implies

$$\begin{array}{c}\hfill H=\pm \sqrt{\frac{{\mathsf{\Lambda }}_U}{3}}\to a\left(t\right)\propto e^{\pm \sqrt{\frac{{\mathsf{\Lambda }}_U}{3}}t}.\end{array}$$

(18)

Again, ${\mathsf{\Lambda }}_U$ must be positive or zero, the latter case leading to the trivial Minkowski space–time. The Ricci scalar leads to the equation

$$\begin{array}{c}\hfill H^2=\frac{{\mathsf{\Lambda }}_U}{3},\end{array}$$

(19)

which is the Friedmann equation in GR for the vacuum case in the presence of a cosmological constant. However, the origin of this cosmological constant is quite different: it does not come from the Lagrangian with a cosmological term, but as an integration constant. In general, it is argued that this (important) formal difference in the Friedmann equation in GR and UG alleviates the cosmological constant problem.

What happens if matter is introduced? The main point is that it is impossible to solve the UG cosmological equations in this case since there is only Equation (11) for two unknown functions, $\rho$ and a: the equation cannot be solved without an additional assumption. If the conservation of the energy–momentum tensor is imposed, we obtain the same solutions as GR in the presence of a cosmological constant, the cosmological term appearing, as in the vacuum case, as an integration constant. Another path has been followed in Ref. [18], exploring the traceless nature of the field equations in UG, which implies that the matter sector must be also traceless, indicating that a radiative fluid is the natural choice for the matter sector. In this case, the usual radiative solutions in the presence of a cosmological constant are recovered. However, the perturbative behavior strongly differs from that in the GR case.

3. Perturbative Analysis

Now, we turn to the perturbative analysis of the vacuum solution.

At a perturbative level, the differences between GR and UG become more evident. First of all, while, in GR, the general diffeomorphism invariance allows us to fix a coordinate condition or to use a gauge-invariant set of variables when the perturbative analysis is performed, in UG, the choices are much more restricted due to the invariance by the TD. This can be seen when considering the unimodular constraint

$$\begin{array}{c}\hfill \sqrt{-g}=\xi .\end{array}$$

(20)

As already discussed, $\xi$ is a fixed external quantity; hence, its perturbation is zero. We are still in the context of TD, even with the presence of the external field; see [22]. Perturbing the metric,

$$\begin{array}{c}\hfill {\tilde{g}}_{\mu \nu }=g_{\mu \nu }+h_{\mu \nu },\end{array}$$

(21)

and preserving the unimodular constraint, we are led to the relation

$$\begin{array}{c}\hfill h=h_{\rho }^{\rho }=0.\end{array}$$

(22)

Let us now consider the general perturbed metric restricted to the scalar sector:

$$\begin{array}{c}\hfill d{s}^2=a^2\{(1+2\varphi )d{\eta }^2-2B_{,i}d{x}^{i}d\eta -[(1-2\psi ){\delta }_{ij}+2E_{,i,j}\left]d{x}^{i}d{x}^j\right\}.\end{array}$$

(23)

From now on, we follow closely the notation of Ref. [23]. The condition $h=0$ implies

$$\begin{array}{c}\hfill \varphi -3\psi -{\nabla }^{2}E=0.\end{array}$$

(24)

The Newtonian gauge is obtained by fixing $B=E=0$, implying $\varphi =3\psi$. This condition contradicts the other condition obtained from the perturbed equations when anisotropic pressure is absent, $\varphi =\psi$, leading to $\varphi =\psi =0$, and no perturbation is present. The situation with the synchronous coordinate condition is more involved, since this condition implies $\varphi =B=0$, leading to ${\nabla }^{2}E=-3\psi$, which can be re-expressed as $h_{kk}=0$. However, if the conservation of the energy–momentum tensor is imposed, $h_{kk}$ is directly related to the scalar perturbation: $h_{kk}$ being zero, there is also no matter perturbation. The situation changes when the conservation of the energy–momentum tensor is not imposed, as we will see later. The gauge-invariant formalism [23] can always be used, but with the additional condition (24).

3.1. The Perturbed Equations in the Gauge-Invariant Formalism

The perturbed field equations in GR using the gauge-invariant formalism, via the hydrodynamical approach, read [23]

$$\begin{array}{ccc}\hfill -3\mathcal{H}({\mathsf{\Psi }}^{\prime }+\mathcal{H}\mathsf{\Phi })+{\nabla }^2\mathsf{\Psi }& =& 4\pi G{a}^2\delta \overline{\rho },\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill \{{\mathsf{\Psi }}^{\prime }+\mathcal{H}\mathsf{\Phi }{\}}_{,i}& =& -4\pi G(\rho +p)a^3\delta {\overline{u}}^i,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill [{\mathsf{\Psi }}^{″}+\mathcal{H}(2{\mathsf{\Psi }}^{\prime }+{\mathsf{\Phi }}^{\prime })+(2{\mathcal{H}}^{\prime }+{\mathcal{H}}^2)\mathsf{\Psi }+\frac{1}{2}D]{\delta }_{ij}-\frac{1}{2}{D}_{,i,j}& =& 4\pi G{a}^2\delta \overline{p}{\delta }_{ij},\hfill \end{array}$$

(27)

where $D=\mathsf{\Phi }-\mathsf{\Psi }$. Moreover, $\mathcal{H}=a^{\prime }/a$, the primes indicating a derivative with respect to the conformal time.

No anisotropic pressure is considered. Hence, when analyzing the $i\ne j$ equation that we obtain, $D=0$, implying $\mathsf{\Phi }=\mathsf{\Psi }$. This leads to the equations

$$\begin{array}{ccc}\hfill -3\mathcal{H}({\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi })+{\nabla }^2\mathsf{\Phi }& =& 4\pi G{a}^2\delta \overline{\rho },\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \{{\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi }{\}}_{,i}& =& 4\pi G(\rho +p)a^3\delta {\overline{u}}^i,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^{″}+3\mathcal{H}{\mathsf{\Phi }}^{\prime }+(2{\mathcal{H}}^{\prime }+{\mathcal{H}}^2)\mathsf{\Phi }& =& 4\pi G{a}^2\delta \overline{p}.\hfill \end{array}$$

(30)

The bars over the perturbed fluid quantities indicate that we are using the gauge-invariant expressions.

In the UG, we must perturb the general field equation

$$\begin{array}{c}\hfill E_{\mu \nu }=8\pi G{\tau }_{\mu \nu },\end{array}$$

(31)

with the definitions

$$\begin{array}{ccc}\hfill E_{\mu \nu }& =& R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R,\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill {\tau }_{\mu \nu }& =& T_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }T.\hfill \end{array}$$

(33)

Using the gauge-invariant formalism, one finds the first-order perturbative UG equations (32,33) coupled to a fluid:

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^{″}+2({\mathcal{H}}^{\prime }-{\mathcal{H}}^2)\mathsf{\Phi }+{\nabla }^2\mathsf{\Phi }& =& 4\pi G{a}^2(\delta \overline{\rho }+\delta \overline{p}),\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill ({\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi }{)}_{,i}& =& -4\pi G{a}^3\delta {\overline{u}}^i.\hfill \end{array}$$

(35)

In obtaining the above expressions, we have already used the fact that $\mathsf{\Phi }=\mathsf{\Psi }$. It is worth mentioning that in the system (34,35), there are two equations for three functions to be determined, $\mathsf{\Phi }$, $\delta \tilde{\rho }=\delta \overline{\rho }+\delta \overline{p}$, and $\delta {\overline{u}}^i$. We will comment more on this issue later.

3.2. Vacuum Case: Perturbations

For a vacuum, $\delta \overline{\rho }$, $\delta \overline{p}$, and $\delta {\overline{u}}^i$ are absent. The GR perturbed equations become

$$\begin{array}{ccc}\hfill -3\mathcal{H}({\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi })+{\nabla }^2\mathsf{\Phi }& =& 0,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill \{{\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi }{\}}_{,i}& =& 0,\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^{″}+3\mathcal{H}{\mathsf{\Phi }}^{\prime }+(2{\mathcal{H}}^{\prime }+{\mathcal{H}}^2)\mathsf{\Phi }& =& 0.\hfill \end{array}$$

(38)

On the other hand, the corresponding UG equations are

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^{″}+2({\mathcal{H}}^{\prime }-{\mathcal{H}}^2)\mathsf{\Phi }+{\nabla }^2\mathsf{\Phi }& =& 0,\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill ({\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi }{)}_{,i}& =& 0.\hfill \end{array}$$

(40)

Let us first consider the de Sitter solution, for which, in the conformal time, $a\propto \frac{1}{\eta }$. Equations (37) and (40) are the same, and this is satisfied in two cases: either $\mathsf{\Phi }\propto 1/a$ or the perturbed quantities are spatially independent. Both hypotheses are consistent with each other. Hence, in both the GR and UG cases, the solutions of the perturbed equations are

$$\begin{array}{c}\hfill \mathsf{\Phi }=\frac{{\mathsf{\Phi }}_0}{a},\end{array}$$

(41)

with ${\mathsf{\Phi }}_0$ being a constant. The metric perturbation decreases as the universe expands, in agreement with the structure of the de Sitter space–time.

If, now, the Minkowski vacuum solution is inserted into the perturbed equations, ${\mathsf{\Phi }}^{″}=0$, leading to $\mathsf{\Phi }\propto \eta$. This happens in both GR and UG. Hence, this case represents a growing solution.

3.3. Introducing Matter Fields

When matter is present, many new features appear. First of all, many aspects of the problem depend on whether the conservation of the energy–momentum tensor is imposed. If the energy–momentum tensor is conserved, as in GR, one of the first consequences is that the synchronous coordinate condition cannot be used. The reason is the following. The unimodular constraint implies

$$\begin{array}{c}\hfill h_{\rho }^{\rho }=0.\end{array}$$

(42)

If the synchronous coordinate condition $h_{\mu 0}=0$ is imposed, the unimodular constraint reduces to $h_{kk}=0$ (a sum on the index k is understood). Using the conservation of the energy–momentum tensor, the UG equations reduce to the GR in the presence of a cosmological constant. The introduction of perturbed quantities in the field equations leads to [24]

$$\begin{array}{c}\hfill \ddot{h}+2H\dot{h}=8\pi G\delta \rho ,\end{array}$$

(43)

with $h=h_{kk}/a^2$. If $h_{kk}=0$, then $\delta \rho =0$ and no perturbation is present.

Due to this property, a possibility is to use the gauge-invariant formalism. This has been done in Ref. [22]. There, the authors found essentially the same equations of GR but with a new component, a relation between the perturbed quantities due to the unimodular constraint. Hence, at a perturbative level, even imposing the conservation of the energy–momentum tensor, UG has some distinguishing features.

If the conservation of the energy–momentum tensor is not imposed, the situation becomes more complex. The restriction to the use of the synchronous coordinate condition does not exist any longer, but, even so, $h_{kk}=0$. However, the density perturbation becomes connected to another metric perturbation $f=h_{ik,i,k}/a^2$. One important remark is that, now, there is no residual coordinate freedom associated with the synchronous coordinate condition. In fact, in GR, the synchronous coordinate condition does not fix completely the coordinate system, and a residual, non-physical mode remains. This is reflected in the third-order (instead of a second-order) differential equation for the density perturbation. However, in UG, the unimodular condition eliminates this non-physical mode, and we obtain second-order differential equations.

Of course, the gauge-invariant formalism can always be used in UG, even with the modified conservation laws. However, there is a technical issue. As we can see from Equations (34) and (35), in the perturbed UG field equations, there are two equations for three unknown functions, i.e., there is the need for a new, independent equation. This new equation comes from the modified conservation law [18]. The use of the gauge-invariant formalism to determine this equation is a highly technical issue, while this complexity is somewhat reduced when using the synchronous coordinate condition. This has been shown in Ref. [18], leading to the equation

$$\begin{array}{c}\hfill \ddot{f}+3H\dot{f}-\frac{k^2}{3a^2}f=0.\end{array}$$

(44)

In this equation, the scalar perturbation f is proportional to the enthalpy perturbation, i.e., $f\propto (\delta \rho +\delta p)$ (see [18] for details) and k is the wavenumber associated with the perturbation.

The final solution in terms of the conformal time ($\eta \propto t^{1/2}$) reads

$$\begin{array}{c}\hfill f=A\frac{sinh\frac{k}{\sqrt{3}}\eta }{k\eta }+B\frac{cosh\frac{k}{\sqrt{3}}\eta }{k\eta }.\end{array}$$

(45)

This solution reveals the exponential growth in the perturbations, even if the background corresponds to the radiative phase. This is due to the “wrong” sign with the k-dependent term in (44), which is related to the Laplacian operator. We have already remarked that in the gauge-invariant formalism, such a “wrong sign” of the Laplacian operator also appears. See (34), where similar behavior can be expected.

Ref. [18] has shown that a viable cosmological model can be obtained in UG even in the absence of the usual conservation law. This model must be refined in many ways, but, in general, most of the main elements of a viable cosmological scenario, such as the current age of the universe, the cosmic microwave background radiation, the present accelerated expansion phase, and the origin of the structures from a homogeneous and isotropic universe, can be predicted by this model.

4. Gauge-Invariant Perturbations with Matter

The complete set of equations for the perturbations in the unimodular theory in the presence of matter with the generalized conservation laws using the gauge-invariant formalism is given by

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^{″}+2({\mathcal{H}}^{\prime }-{\mathcal{H}}^2)\mathsf{\Phi }+{\nabla }^2\mathsf{\Phi }& =& 4\pi G{a}^2\delta \tilde{\rho },\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill a^{-3}{\nabla }^2({\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi })& =& -4\pi G(\rho +p)\overline{\theta },\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill 8\pi G\{\delta {\tilde{\rho }}^{\prime }+4\mathcal{H}\delta \tilde{\rho }+4(\rho +p)a\overline{\theta }-4(\rho +p){\mathsf{\Phi }}^{\prime }\}& =& -\frac{1}{3}\delta R^{\prime },\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill \frac{32\pi G}{9}\{{\left[(\rho +p)a\overline{\theta }\right]}^{\prime }+4\mathcal{H}(\rho +p)a\overline{\theta }+\frac{1}{4}\nabla \delta \tilde{\rho }+(\rho +p){\nabla }^2\mathsf{\Phi }\}& =& \frac{1}{3}{\nabla }^2\delta R,\hfill \end{array}$$

(49)

with

$$\begin{array}{c}\hfill \delta R=-\frac{6}{a^2}\{{\mathsf{\Phi }}^{″}+4\mathcal{H}{\mathsf{\Phi }}^{\prime }+2({\mathcal{H}}^{\prime }+{\mathcal{H}}^2)\mathsf{\Phi }-\frac{{\nabla }^2\mathsf{\Phi }}{3}\}.\end{array}$$

(50)

We have noted in these expressions

$$\begin{array}{c}\hfill \delta \tilde{\rho }=(\delta \overline{\rho }+\delta \overline{p}),\end{array}$$

(51)

where $\overline{\theta }$, $\delta \overline{\rho }$ and $\delta \overline{p}$ are the gauge-invariant versions of $\delta u_{,k}^k$, $\delta \rho$ and $\delta p$, respectively. In (46)–(49), all perturbed functions are gauge-invariant.

Equations (46) and (47) come from the field equations, while Equations (48) and (49) come from the perturbations of the generalized conservation laws

$$\begin{array}{c}\hfill \frac{R_{;\nu }}{4}=8\pi G\{{T_{\nu }^{\mu }}_{;\mu }-\frac{T_{;\nu }}{4}\}.\end{array}$$

(52)

Differing from the GR case, there are only two equations arising from the field equations, since the $0-0$ and $i-j$ components lead to the same Equation (46). Hence, in principle, we are obliged to use the conservation law in order to obtain an additional constraining equation. It is time-consuming but we can directly show that the use of (46) and (47) in (48) and (49) provides an identity. It can be argued that this result is expected from the beginning, since the conservation laws appear from the application of the divergence to the field equations. However, it is worth noting that the unimodular tensor $E_{\mu \nu }$ (32) does not obey the Bianchi identities, in contrast to GR. In using the synchronous coordinate condition, for example, this trivial identity relation for the perturbed conservation law is not verified.

Hence, the use of the gauge-invariant formalism leads to an underdetermined system of equations, as occurs with the background and in contrast to the synchronous coordinate condition case. This feature shows the specificity of the unimodular theory when a generalized conservation law is adopted. We remark that if the usual conservation laws are used, the above equations become

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^{″}+2({\mathcal{H}}^{\prime }-{\mathcal{H}}^2)\mathsf{\Phi }+{\nabla }^2\mathsf{\Phi }& =& 4\pi G{a}^2(\delta \overline{\rho }+\delta \overline{p}),\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill {\nabla }^2({\mathsf{\Phi }}^{\prime }+\mathcal{H}\mathsf{\Phi })& =& -4\pi G{a}^3(\rho +p)\overline{\theta },\hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill \delta {\overline{\rho }}^{\prime }+3\mathcal{H}\delta \overline{\rho }+(\rho +p)a\overline{\theta }-3(\rho +p){\mathsf{\Phi }}^{\prime }& =& 0,\hfill \end{array}$$

(55)

$$\begin{array}{c}\hfill {\left[(\rho +p)a\overline{\theta }\right]}^{\prime }+4\mathcal{H}(\rho +p)a\overline{\theta }+\nabla \delta \overline{p}+(\rho +p)\mathsf{\Phi }=0.\end{array}$$

(56)

Equation (53) is the combination of the equations $0-0$ and $i-j$ of the GR field equations. Using Equation (54) in (56), we obtain

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^{″}+3\mathcal{H}{\mathsf{\Phi }}^{\prime }+(2{\mathcal{H}}^{\prime }+{\mathcal{H}}^2)\mathsf{\Phi }=4\pi G{a}^2\delta p,\end{array}$$

(57)

which is the usual equation for metric perturbations in the GR theory with the gauge-invariant formalism. Inserting this equation in (53), the equation for the density perturbation in terms of $\mathsf{\Phi }$ and its derivatives is obtained. Hence, the usual GR structure for the perturbative equations in the gauge-invariant formalism can be recovered [22].

5. An Extension of UG: Including Scalar Fields

The inclusion of scalar fields is the most direct extension of GR. It can be introduced as a self-interacting field representing the matter sector. In this case, we modify only the right-hand side of the field equations. However, this can also be implemented via a non-trivial coupling with the geometric sector, and, in this case, the implications are more profound. A well-known example is the Brans–Dicke (BD) theory [25], whose field equations, in the presence of a cosmological constant, are given by

$$\begin{array}{ccc}\hfill R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R& =& \frac{8\pi }{\varphi }{T}_{\mu \nu }+\frac{\omega }{{\varphi }^2}({\varphi }_{;\mu }{\varphi }_{;\nu }-\frac{1}{2}{g}_{\mu \nu }{\varphi }_{;\rho }{\varphi }^{;\rho }\left)+\frac{1}{\varphi }\right({\varphi }_{;\mu \nu }-g_{\mu \nu }\square \varphi )+g_{\mu \nu }\mathsf{\Lambda },\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill \square \varphi & =& \frac{8\pi T}{3+2\omega }+\frac{4}{3+2\omega }\mathsf{\Lambda },\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill {T^{\mu \nu }}_{;\mu }& =& 0.\hfill \end{array}$$

(60)

In these equations, $\omega$ is a free coupling parameter. GR is recovered when $\omega \to \infty$. The present estimations indicate a very high value for $\omega$. Nonetheless, BD remains an important object of study, and it can be connected to many other fundamental theories, such as string theories [26].

The unimodular version of the Brans–Dicke (UBF) theory has been proposed in Ref. [19]. The deduction of the field equations follows closely the GR case, introducing the unimodular constraint through Lagrangian multipliers. The final equations read

$$\begin{array}{ccc}\hfill R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R& =& \frac{8\pi }{\varphi }(T_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }T)+\frac{\omega }{{\varphi }^2}({\varphi }_{;\mu }{\varphi }_{;\nu }-\frac{1}{4}{g}_{\mu \nu }{\varphi }_{;\rho }{\varphi }^{;\rho })\hfill \end{array}$$

$$\begin{array}{ccc}& +& \frac{1}{\varphi }({\varphi }_{;\mu \nu }-\frac{1}{4}{g}_{\mu \nu }\square \varphi ),\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill \square \varphi & =& \frac{1}{2}\frac{{\varphi }_{;\rho }{\varphi }^{;\rho }}{\varphi }-\frac{\varphi }{2\omega }R,\hfill \end{array}$$

(62)

$$\begin{array}{ccc}\hfill {\left(\varphi R\right)}^{;\nu }& =& \omega \left(\frac{{\varphi }_{;\rho }{\varphi }^{;\rho }}{\varphi }\right){}^{;\nu }+32\pi ({T^{\mu \nu }}_{;\mu }-\frac{1}{4}{T}^{;\nu })+3{(\square \varphi )}^{;\nu }.\hfill \end{array}$$

(63)

In this case, as in the GR one, the usual conservation of the energy–momentum tensor has not been imposed. If the usual conservation laws are introduced, the GR equations in the presence of a cosmological constant are recovered.

UBD has many new features in comparison with the traditional BD theory. We will comment on only one of them. In Ref. [27], an extensive perturbative analysis of cosmological models obtained from the BD theory was carried out. The vacuum cosmological solutions in UBD coincide with the BD cosmological solutions in the presence of a cosmological constant, as occurs with the corresponding solutions in GR. In Ref. [27], the vacuum solutions in the presence of a cosmological term in the BD were shown to be stable. However, in the UBD case, the vacuum solutions are unstable in the interval $-1/2<\omega <3/2$. This is an important difference, pointing out that, even if the background UBD solutions can be mapped in the BD solutions in the presence of a cosmological constant (which we could expect from our experience with GR and UG), the perturbative behavior is different and, in the BD and UBD context, much stronger than in the GR and UG context, due to the presence of the scalar field non-minimally coupled to the gravity sector.

In UBD, the scalar perturbations in the vacuum case, using the synchronous coordinate condition (which is allowed here), read

$$\begin{array}{c}\hfill (3-2\omega )\delta \ddot{\varphi }-[3(1+2\omega )H-8\omega \frac{\dot{\varphi }}{\varphi }\left]\delta \dot{\varphi }+\right[12(\dot{H}+H^2)-4\omega \frac{{\dot{\varphi }}^2}{{\varphi }^2}]\delta \varphi +\frac{1+2\omega }{a^2}{\nabla }^2\delta \varphi =0.\end{array}$$

(64)

The background solution implies $H=(1+2\omega )/t$. The instability in the UBD case can be directly seen from (64) by inspecting the relative sign of the second derivative and the Laplacian terms of $\delta \varphi$. This is connected to the perturbative speed of sound, which, in (64), becomes imaginary for $-1/2<\omega <3/2$.

The latest results warrant some comments. We employed the synchronous coordinate condition, which, in the unimodular framework, results in a second-order differential equation, unlike the case of GR (or BD), where it leads to a third-order differential equation. This fact indicates that the residual coordinate freedom, which is a characteristic of the synchronous coordinate condition in GR, is absent in UGR (or UBD). Consequently, the previous result is not connected with a gauge artifact, as can occur in the GR and BD cases for this choice of coordinate condition. The results of Section 4 corroborate this reasoning, as they demonstrate that the gauge-invariant formalism produces the same outcomes as those obtained using the synchronous gauge. There is another aspect of the problem in the BD context. By means of a conformal transformation, the BD (or UBD) theory can be reformulated in the framework of GR with a minimally coupled scalar field, even if a non-trivial coupling emerges with the matter sector. However, this is not relevant here, since we consider the vacuum configuration. The sound speed of a canonical scalar field is always equal to 1 (in units of c). However, in the unimodular framework, the traceless condition introduces a non-canonical structure for the scalar field coupled to gravity, and, as a result, the sound speed of the scalar field may deviate from the canonical case. This deviation could depend on the parameter $\omega$, which may explain the instability observed in the UBD context within a specific range of $\omega$. An extensive analysis of the sound speed for canonical and non-canonical scalar fields has been performed in [28]. The unimodular case may present, however, some subtle aspects due to the presence of the Lagrangian constraint.

6. Unimodular $\mathit{f}\left(\mathit{R}\right)$ Class of Theories

Among the modified theories of gravity, the class of $f\left(R\right)$ theories occupies an important position. One of the reasons is that $f\left(R\right)$ generalizations of general relativity, despite containing higher derivatives, are free of the Ostrogradsky instabilities that plague almost all other theories with derivatives higher than the second order. There is a connection between $f\left(R\right)$ theories and quantum gravity proposals, and the very successful Starobinsky inflationary model is a remarkable example of the application of these theories in a cosmological context. For a review of $f\left(R\right)$ theories, refer to Ref. [29].

The usual $f\left(R\right)$ theories are defined by the Lagrangian

$$\begin{array}{c}\hfill \mathcal{L}=\sqrt{-g}\frac{f\left(R\right)}{16\pi G}+{\mathcal{L}}_m,\end{array}$$

(65)

where ${\mathcal{L}}_m$, as usual, denotes the matter sector. The corresponding field equations are

$$\begin{array}{c}\hfill R_{\mu \nu }{f}_R-\frac{1}{2}{g}_{\mu \nu }f-{\nabla }_{\mu }{\nabla }_{\nu }{f}_R+g_{\mu \nu }\square f_R=8\pi G{T}_{\mu \nu }.\end{array}$$

(66)

The index R denotes a derivative with respect to the argument of the function $f\left(R\right)$ that remains, until this stage, arbitrary.

One important aspect of these theories is that we can define $f_R=G\varphi$, where $\varphi$ is now a scalar field, leading to the structure

$$\begin{array}{ccc}\hfill R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R& =& \frac{8\pi }{\varphi }{T}_{\mu \nu }+\frac{1}{\varphi }\{{\varphi }_{;\mu ;\nu }-g_{\mu \nu }\square \varphi \}+g_{\mu \nu }\frac{V\left(\varphi \right)}{\varphi },\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill \square \varphi & =& \frac{8\pi }{3}T-V_{\varphi },\hfill \end{array}$$

(68)

with

$$\begin{array}{c}\hfill V=\frac{f-R{f}_R}{2G}.\end{array}$$

(69)

Hence, $f\left(R\right)$ theories can be mapped into the scalar–tensor Brans–Dicke-like theory with the Brans-Dicke parameter $\omega =0$ and a potential given by (69). This map reflects the fact that the $f\left(R\right)$ theories are free of the Ostrogradsky instabilities.

The construction of a $f\left(R\right)$ unimodular theory has been proposed in Ref. [30]. Using the same procedure as before, we find the equations

$$\begin{array}{c}\hfill (R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R\left)f_R-{\nabla }_{\mu }{\nabla }_{\nu }{f}_R+\frac{1}{4}{g}_{\mu \nu }\square f_R=8\pi G\right\{T_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }T\}.\end{array}$$

(70)

By performing the same transformation indicated above, the unimodular $f\left(R\right)$ theory can be mapped to the Brans–Dicke unimodular theory discussed in Ref. [19]. It is important to remark that in the unimodular Brans–Dicke theory, the potential appears only in the Klein–Gordon equation for the scalar field $\varphi$.

Now, let us take the divergence of Equation (70), using the commutation of the covariant derivatives, the Bianchi identities, and the chain rule for the derivative of a given scalar function. We then obtain

$$\begin{array}{c}\hfill \frac{1}{4}{\nabla }_{\nu }\{2f-R{f}_R-\frac{3}{4}\square f_R\left\}=8\pi G\right\{{T_{\nu }^{\mu }}_{;\mu }-\frac{1}{4}{T}_{;\nu }\}.\end{array}$$

(71)

If we impose the conservation of the energy–momentum tensor, we can directly obtain again Equation (66), with a cosmological constant that appears as an arbitrary integration constant:

$$\begin{array}{c}\hfill R_{\mu \nu }{f}_R-\frac{1}{2}{g}_{\mu \nu }f-{\nabla }_{\mu }{\nabla }_{\nu }{f}_R+g_{\mu \nu }\square f_R=8\pi G{T}_{\mu \nu }+g_{\mu \nu }\mathsf{\Lambda }.\end{array}$$

(72)

If the conservation of the energy–momentum tensor is not imposed, we find the same problem as before: the system of equations is underdetermined. In fact, for example, the flat FLRW background is determined by only one equation. It reads

$$\begin{array}{c}\hfill \dot{H}=-4\pi G(\rho +p)-\frac{1}{2}({\ddot{f}}_R-H{\dot{f}}_R).\end{array}$$

(73)

The background and perturbative studies of $f\left(R\right)$ unimodular gravity bring an additional difficulty compared to the usual case. Since there is only one equation for two variables, an additional ansatz must be imposed, similar to the cases of general relativity (GR) and Brans–Dicke (BD). However, the specific form of the function $f\left(R\right)$ needs to be chosen. In principle, we can utilize the results from Ref. [19] since $f\left(R\right)$ is equivalent to the Brans–Dicke scalar–tensor theory with $\omega =0$ and a potential. The inclusion of a potential implies a specific choice for the function $f\left(R\right)$, in addition to the aforementioned problem of selecting an ansatz to close the set of equations. These complexities necessitate a separate analysis, which is beyond the scope of this discussion. Nevertheless, some general considerations can be made to differentiate the usual $f\left(R\right)$ theories from their unimodular counterparts at the perturbative level. We will illustrate this using the scalar–tensor version of the unimodular $f\left(R\right)$ theory.

The Brans–Dicke unimodular theory with a non-vanishing potential and $\omega =0$ is given by the following equations [19]:

$$\begin{array}{ccc}\hfill R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R& =& \frac{8\pi }{\varphi }(T_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }T\left)+\frac{1}{\varphi }\right({\varphi }_{;\mu ;\nu }-\frac{1}{4}{g}_{\mu \nu }\square \varphi ),\hfill \end{array}$$

(74)

$$\begin{array}{ccc}\hfill R& =& -2V_{\varphi }.\hfill \end{array}$$

(75)

We remark that, due to the condition $\omega =0$, the Ricci scalar is directly connected with the derivative of the potential. We will consider, from now on, the vacuum case, $\rho =p=0$.

The corresponding equations of motion for a flat FLRW universe are given by

$$\begin{array}{ccc}\hfill \dot{H}& =& -\frac{1}{2}(\frac{\ddot{\varphi }}{\varphi }-H\frac{\dot{\varphi }}{\varphi }),\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill \dot{H}+2H^2& =& \frac{V_{\varphi }}{3}.\hfill \end{array}$$

(77)

Let us seek exponential solutions under the form

$$\begin{array}{ccc}\hfill a& =& a_0{e}^{rt}\to H=r=\mathrm{constant},\hfill \end{array}$$

(78)

$$\begin{array}{ccc}\hfill \varphi & =& {\varphi }_0{e}^{st}.\hfill \end{array}$$

(79)

Upon inspecting the equations of motion, it becomes evident that the only non-trivial solution is $s=r$ accompanied by a linear potential $V\propto \varphi$. Furthermore, there exists another solution where $s=0$, again with a linear potential. However, this solution essentially reduces to the de Sitter case in unimodular gravity (UGR), as the scalar field remains constant. It is important to note that in the case of the unimodular Brans–Dicke (UBD) model with $\omega =0$ and no potential term, the resulting solution corresponds to the trivial Minkowski space. On the other hand, the traditional Brans–Dicke (BD) theory only allows for power law solutions.

Now, let us delve into the perturbative analysis. By utilizing the relationships presented in Ref. [19], it is possible to confirm that the scalar perturbations are identically zero. This finding is not particularly surprising given the high symmetry of the de Sitter space–time, the linear potential, and the nature of unimodular gravity. However, significant effects arise concerning gravitational waves. The equation governing gravitational waves in unimodular Brans–Dicke (UBD) gravity is given by

$$\begin{array}{c}\hfill {\ddot{h}}_{ij}-(H-\frac{\dot{\varphi }}{\varphi }){\dot{h}}_{ij}+[\frac{k^2}{a^2}-(2\dot{H}+2H^2+2H\frac{\dot{\varphi }}{\varphi }\left)\right]h_{ij}=0.\end{array}$$

(80)

Inserting the background solution with $r=s$, this equation reduces to

$$\begin{array}{c}\hfill {\ddot{h}}_{ij}+[\frac{k^2}{a^2}-4r^2]h_{ij}=0.\end{array}$$

(81)

As a experiences exponential growth, the gravitational wave modes oscillate but rapidly exhibit exponential growth. This behavior is in significant contrast to the typical scenarios observed in the RG and BD theories, primarily due to the presence of a non-zero friction term.

However, the linear potential case somewhat weakens the connection to the $f\left(R\right)$ theories, as discussed in Ref. [29]. It has been presented above merely as an illustration of the specific characteristics that arise due to the UBD framework in the presence of a potential term.

Nevertheless, we can utilize the findings of Ref. [19] by setting $\omega =0$ to derive a single equation governing the fluctuations of the scalar field when the potential term is included, without imposing a fixed value for it initially. The result is as follows:

$$\begin{array}{c}\hfill \delta \ddot{\varphi }-H\delta \dot{\varphi }-\{\frac{k^2}{3a^2}-2\dot{H}+\frac{2}{3}{V}_{\varphi \varphi }\varphi \}\delta \varphi =0.\end{array}$$

(82)

Exponential instabilities reappear at small scales ($k\to \infty$) due to the “wrong” sign preceding the term involving $k^2$. This stands in stark contrast to the typical $f\left(R\right)$ theories, similar to the case of GR in relation to UGR [29,31]. It is worth noting that Ref. [19] identifies instabilities in the absence of a potential for the range $-\frac{1}{2}<\omega <\frac{3}{2}$, which includes the case $\omega =0$. The presence of a potential does not alter this result, with the exception of certain cases where $V_{\varphi \varphi }$ could be negative, potentially leading to a negative effective mass squared (tachyons) for the scalar field. Such a condition may arise, for example, in the inflaton potential under certain conditions. However, even with these considerations, the instabilities at very small scales are not suppressed. Nevertheless, it is crucial to evaluate the specific features of the perturbations in these unimodular $f\left(R\right)$ theories on a case-by-case basis.

7. Conclusions

Unimodular gravity has been extensively discussed in the literature due to its potential to provide new insights into some of the most significant issues in general relativity. In UG, the cosmological constant is somewhat concealed within the overall structure of the theory, and when the usual energy–momentum tensor conservation is imposed, it appears explicitly as an integration constant. This is generally considered as, at the very least, a partial resolution to the cosmological constant problem that affects GR. Additionally, it has been argued that UG could shed light on quantum-level issues present in GR [4].

The question of the equivalence between UG (without a cosmological constant) and GR (with a cosmological constant) has been debated, as evidenced by studies such as [4,5,12]. At the cosmological background level, the equivalence seems to be well established. However, the situation becomes less clear when considering cosmological perturbations. In our view, the key aspect to emphasize is the invariance of UG under the more restricted transverse diffeomorphic transformations, while GR is invariant under the full diffeomorphism group.

We have discussed the background and perturbative issues in UG, comparing them to GR (always with a cosmological constant). In a vacuum, UG yields the same results as GR, even at the perturbative level, albeit following a different approach. However, in the presence of matter, the situation becomes much more complex, particularly depending on whether the usual conservation laws are retained. If they are not retained, the configuration is clearly different, as exemplified by the necessity of using only the synchronous gauge, which leads to the exponential growth of matter perturbations. Even when the conservation laws are preserved, new features emerge [18]. These differences become even more pronounced when a non-minimally coupled scalar field is introduced. By doing so, we can extend the Brans–Dicke theory to the unimodular Brans–Dicke theory. An example of this significant difference is the appearance of unstable modes in UBD, which do not exist in the BD case. We have also demonstrated that by introducing a potential term in UBD, we can establish a connection with the class of $f\left(R\right)$ theories, showing that unstable modes are still present within the $f\left(R\right)$ context.

Furthermore, we have shown that when matter is present, the conservation of the canonical energy–momentum tensor is not retained when applying a fully gauge-invariant formalism. However, our current results seem to support the idea that the equivalence between UG and RG, even at the classical level, is incomplete, especially when considering perturbations.

The main outcome of this study can be summarized as follows: the differentiation between GR and UG concerning cosmological scalar perturbations depends on whether or not the usual conservation laws are adopted. The difference found in Ref. [22] arises when the energy–momentum tensor has a vanishing divergence, which is a specific case within the broader UG formalism. Here, on the other hand, we have found that employing the generalized conservation law (5) does not allow us to reach the same conclusion.

The perturbative analysis presented here needs to be extended when employing the generalized conservation law. Additional information should be imposed in order to close the set of equations at a background level. For instance, the implementation of the holographic principle [32,33,34] may offer the potential to obtain new results at both the background and perturbative levels by providing a self-contained set of equations.