Circumventing the Ricci-inverse no-go theorem with
complexifiable singularities: a novel dark energy model
In memory of Prof. A.Vitturi, for his kind and always helpful soul.

Many cosmological observations require a modification to Einstein’s theory of General Relativity (GR) to explain the late-time accelerated expansion Riess et al. (1998); Abbott et al. (2019); Scolnic et al. (2018a) and the early-time cosmological inflation of our universe Starobinsky (1980); Guth (1981a); Hinshaw et al. (2013); Malik et al. (2023a).

There are many ways to construct alternative theories to GR. A possible approach in this direction consists of constructing appropriate modified gravity (MG) theories, in which an alteration of the well-known Einstein Hilbert (EH) action is considered Nojiri et al. (2017). On the other hand, most of these reformulations require troublesome assumptions in their foundations, such as the definition of dimensional coupling constants to be finely-tuned or the introduction of extra dynamical fields that have no obvious relation with Einstein’s strictly geometric formulation of GR.

Motivated by the above considerations, Amendola et al. proposed in Ref. Amendola et al. (2020) a MG theory called Ricci-inverse gravity whose gravitational Lagrangian density $\mathcal{L}_{g}(R,A)$ is constructed by combining the Ricci scalar $R$ with a new purely geometrical object called anti-curvature scalar $A=g_{\mu\nu}A^{\mu\nu}$. By construction, the anti-curvature scalar is the trace of anti-curvature tensor $A^{\mu\nu}=R_{\mu\nu}^{-1}$, which is the inverse of the Ricci tensor $R_{\mu\nu}$ defined by the relation

Here, $g_{\mu\nu}$ is the metric tensor and $\delta^{\mu}_{\nu}$ is the Kronecker delta function. For the avoidance of doubt, one must pay attention to the fact that $A\neq R^{-1}$. However, focusing only on the physical dimension, $[A]=[R]^{-1}$.

As a consequence, one can use the anti-curvature scalar to introduce new Lagrangian terms into EH action which, resulting directly from $R$, share its same physical dimension and do not invoke any new dynamical field to modify GR. Among all scale-free possibilities, we mention: (i) the pure anti-curvature term $A^{-1}$, (ii) the general power-law profile $A^{\ell}R^{\ell+1}$, and (iii) any general profile of the form $g(RA)R$. Note that (i) belongs to (ii) when $\ell=-1$, while (iii) reduces to (ii) when $g(RA)=(RA)^{\ell}$.

In recent years, Ricci-inverse gravity has been extensively studied in the currently available literature. Examples of applications include the investigation of: (i) anisotropic Shamir et al. (2023), compact Shamir and Meer (2023) and charged Malik et al. (2023b, 2024a) star structures Malik et al. (2024b, 2023a), (ii) matter-antimatter asymmetry phenomena Jawad and Sultan (2022), (iii) novel aspects related to wormhole Mustafa et al. (2024) and blackholes Ahmed and Guvendi (2024) solutions, (iv) axially symmetric spacetimes with causality violation de Souza and Santos (2023); de Souza et al. (2024), and (v) cosmic structures within a Sub-Horizon non-relativistic Weak-Field limit Scomparin (2021). Although most of these works emphasize the increasing relevance and applicability of Ricci-inverse gravity in diverse scenarios, two binding no-go theorems have been shown to rule out many cosmological realizations of this theory.

The first no-go theorem concerns the late-time accelerated expansion phase of universe. Its original formulation states that any Lagrangian density $\mathcal{L}_{g}(R,A)$ containing terms proportional to $A^{\ell}$, with any positive or negative $\ell$, cannot smoothly join a cosmic decelerated era with the current accelerated expansion of universe Aghanim et al. (2020) (see Refs.Amendola et al. (2020); Das et al. (2022) for a detailed discussion). In fact, assuming a standard Friedmann-Lemaître-Robertson-Walker (FLRW) metric, between these two epochs $A$ must pass through both $0$ and $\pm\infty$ values, leading a generic power of $A$ to blow up due to singularities. Up to now, attempts to find loopholes to escape this no-go theorem have not proved satisfactory. Indeed, it has been shown that the shiftings introduced by spatially curved metrics or anisotropic backgrounds are too small to move the singularities sourced by $A$ outside the observational range. In the end, only well-though non-polynomial Lagrangians (i.e. respecting the property $[\mathcal{P}1]$ of being regular both for $A\rightarrow 0$ and $A\rightarrow\pm\infty$) seem promising enough to circumvent the problem, but they are very complicated and consequently have not been investigated so far Amendola et al. (2020); Das et al. (2022).

Another cosmic scenario now forbidden to Ricci-inverse gravity is the so-called inflationary era, which occurred very rapidly after the Big Bang Starobinsky (1980); Guth (1981b); Linde (1982). Although this primordial phase is not affected by singularities from anti-curvature terms, a stability analysis based on a dynamical system method showed that, even when $\mathcal{L}_{g}(R,A)$ depends on simple powers of $A$ (e.g $A$ or $A^{2}$), all isotropic inflationary solutions turn out to be unstable with respect to field perturbations Do (2021, 2022). This behavior is due to the fact that the existence domain and the stability region of the solutions do not overlap. Therefore, the impossibility to have a stable (isotropic) inflationary solutions as well as a smooth exit from inflation era (see, e.g. Ref. Das et al. (2022)) in elementary Ricci-inverse cosmologies is essentially the content of the second no-go theorem.

In the absence of effective loopholes, all these results have raised great doubts about the cosmological viability of Ricci-inverse gravity to be a suitable inflationary model or DE candidate. To the author’s knowledge, no valid proposal has been found at present to circumvent the action of the two no-go theorems. Furthermore, neither non-trivial extensions involving higher-order scalar combinations in the anti-curvature tensor¹¹1Here, $A_{\nu}^{\alpha\rho\gamma}$ is the anti-Riemann tensor, defined by the identity $A^{\alpha\rho\gamma}_{\nu}R^{\mu}_{\alpha\rho\beta}=\delta^{\mu}_{\nu}\delta^{\gamma}_{\beta}$. such as $\mathcal{L}_{g}(R,A^{\mu\nu}A_{\mu\nu})$ or $\mathcal{L}_{g}(R,A^{\alpha\beta\gamma}_{\mu}A_{\alpha\beta\gamma}^{\mu})$ seem a promising prospect for treating in a simple way the pathologies that afflict the Ricci-inverse gravity, which, in the end, is still ruled out Das et al. (2022).

At this point, one might come to the conclusion that there is no way around no-go theorems for Ricci-inverse cosmologies. Of course, every no-go theorem is based on arguments that one can try to break or reconsider. For example, in the case of the first no-go theorem, the most suitable point to be reconsidered is that $\mathcal{P}1$ is really the only property that makes non-polynomial Lagrangians capable of realizing regular equations of motion. In fact, looking for a possible workaround of the theorem in this way, the author realized that there is indeed a new property that can regularize the no-go singularities.

Generally speaking, our reasoning is as follows. Focusing on Ricci-inverse cosmologies studied so far, we noticed that their Lagrangians exhibit no-go singularities belonging only to the real axis $\mathbb{R}$. However, if such a Lagrangians could be redrawn to push the no-go singularities into the complex plane $\mathbb{C}$, we might of course expect the effects of the first no-go theorem to become inoffensive, since the cosmic trajectories are and remain purely real. We call this property $[\mathcal{P}2]$, i.e. regularizability with complexifiable singularities. It refers to the characteristic of some non-polynomial Lagrangians, under certain conditions on their free parameters, to shift their no-go singularities to the complex plane with a non-zero imaginary part.

Driven by this idea, the aim of this paper is twofold: (i) to review the known aspects about the first no-go theorem in Ricci-inverse gravity in the form of $\mathcal{L}_{g}(R,A)$, and (ii) to present new original results on this topic. In particular, using $[\mathcal{P}2]$, we present a novel singularity-free DE model whose cosmic trajectories smoothly connect a cosmic decelerated era with the current accelerated expansion of the universe, which in turn emerges as a stable attractor solution alongside a de Sitter phase.

The outline of our work is as follows. In Section II we begin by briefly introducing the full Ricci-inverse theory $\mathcal{L}_{g}(R,A)$ and discussing the related covariant field equations. In Section III we introduce the Ricci and anti-curvature scalars in a flat-space FLRW universe and, contextually, in Subsection III.1 we review the state of the art on the first no-go theorem. Section IV is devoted to the introduction of a novel class of Ricci-inverse models that might be expected to violate the first no-go theorem. Such models depend on two free-parameters and in Subsection IV.1 we explain how to implement our property $[\mathcal{P}2]$ to tune them and obtain healthy Ricci-inverse cosmological solutions. After finding the related modified Friedmann equations in Section V, in Subsection V.1 and Subsection V.2 we provide proofs of stable de Sitter and power-law solutions (compatible with the current accelerated phase). Finally, in Section VI we summarise our results and identify future avenues of investigation..

In this work, we use the metric signature $(-,+,+,+)$ and Einstein’s convention on repeated dummy indices is assumed.

Let us consider the full Ricci-inverse theory described by the basic action

where $g$ is the determinant of the metric $g_{\mu\nu}$ and $\kappa\equiv 8\pi G/c^{4}$ is a constant that depends on the gravitational parameter $G$ and the speed of light $c$. The first term $\mathcal{L}_{g}=\mathcal{L}_{g}(R,A)$ corresponds to the MG Lagrangian density, which is an a priori arbitrary function of the Ricci and anti-curvature scalars. The second term $\mathcal{L}_{m}$ is the matter Lagrangian density, that we assume minimally coupled with the metric tensor only.

It is immediate to check that if $\mathcal{L}_{g}=R$, then Eq. (2) reduces to the usual EH action for GR.

Varying the above action with respect to the metric, the field equation is $\delta S=0$ Amendola et al. (2020); Lee (2021). By differentiating definition (1) and using integration by parts we obtain

where we introduced the modified Einstein tensor

and the matter energy-momentum tensor

Since equation (3) must apply for any variation $\delta g_{\mu\nu}$, this implies that the tensor field equation becomes

In our notation, the $\nabla_{\mu}$ symbol is understood as the covariant derivative. Additionally, unless otherwise specified, we use the symbols $\partial_{R}\equiv\partial/\partial R$ and $\partial_{A}\equiv\partial/\partial A$ to denote derivative with respect to $R$ or $A$, respectively.

We recall that many functional forms for $\mathcal{L}_{g}$ have been proposed to obtain cosmic solutions within the Ricci-inverse theory Amendola et al. (2020); Das et al. (2022).

However, as we are going to show in the next Section III, such models generally fail to explain the late-time acceleration phase of the Universe since their trajectories encounter no-go singularities.

On sufficiently large scales, the Cosmological Principle states that the universe is uniformly isotropic and homogeneous. The most general space-time compatible with such assumptions is parametrized in terms of the FLRW line-element, which in polar coordinates is given by

where $a(t)$ is the scale factor and $d\Omega_{2}^{2}=d\theta^{2}+\sin^{2}\!\theta\,d\varphi^{2}$ is the tridimensional solid-angle element.

We point out that in (7) we assume the flat-space case. In fact, being our study focused on DE (and inflationary) models, observations show that in these casuistics the space-curvature density parameter $\Omega_{k}$ is very close to zero Aghanim et al. (2020); Hergt et al. (2022). Therefore the space-curvature interferes little or nothing on the actual realization of the no-go theorems Amendola et al. (2020); Do (2022).

Now, the expression for the Ricci scalar calculated from the metric (7) is given by

where the Hubble parameter $H\equiv\dot{a}/a$ quantifies the rate of time-evolution of the scale factor, and the overdot denotes derivative with respect to cosmic time $t$.

For convenience, we have also introduced the new variable $\xi\equiv H^{\prime}/H=\ddot{a}\,a/\dot{a}^{2}-1$, where the prime symbol stands for ${}^{\prime}=d/d\ln a$.

According to these results, using definition (1), the corresponding anti-curvature scalar has expression

We end up this section by pointing out: (i) that our results (8) and (9) confirm once again that $A\neq R^{-1}$, and (ii) that in the Minkowski limit $H\!\rightarrow\!0$ the anti-curvature scalar $A$ becomes singular while $R$ and $A^{-1}$ do not.

In this Section we introduce the notion of regularizability with complexifiable singularities $[\mathcal{P}2]$ and show explicitly how it allows us to get rid of the no-singularities. In mathematical terms, this property refers to the characteristic of certain non-polynomial functions to be able to shift, under certain conditions on their free parameters, their (real) singularities from the real axis $\mathbb{R}$ to the complex plane $\mathbb{C}$.

With this idea in mind, we can argue as follows. In the case of Ricci-inverse gravity we observe that $\xi_{\mbox{\tiny c,i}}\in\mathbb{R}$, $\forall\,i=1,2,3,4$. Hence, if we able to designing a Lagrangian density $\mathcal{L}_{g}$ that satisfies $[\mathcal{P}2]$, then we can transform any troublesome singularity $\xi_{\mbox{\tiny c,i}}\in\mathbb{R}\rightarrow\xi^{\prime}_{\mbox{\tiny c,i}}=\mbox{Re}\xi^{\prime}_{\mbox{\tiny c,i}}+j\mbox{Imm}\xi^{\prime}_{\mbox{\tiny c,i}}\in\mathbb{C}$. As a consequence, if $\mbox{Imm}\xi^{\prime}_{\mbox{\tiny c,i}}\neq 0$ (see, e.g. Fig. 1), then the effects of the first no-go theorem become inoffensive, since cosmic trajectories are physical, i.e. they only evolve along the real axis. In particular, by virtue of this fact, the undesirable singularities that lie between the decelerated and accelerated phases of our universe can be eliminated, resurrecting the Ricci-inverse theory as a suitable DE alternative.

Essential to this scenario is the observation that the design of Lagrangian densities satisfying the $[\mathcal{P}2]$ property is by no means straightforward. For example, the classes of (ruled-out) theories $\mathcal{L}_{g}=R-\alpha A^{-1}$ (see, e.g. Refs. Amendola et al. (2020); Das et al. (2022)) or $\mathcal{L}_{g}=R-\alpha A$ (see, e.g. de Souza et al. (2024)), with $\alpha\in\mathbb{R}$, do not satisfy $[\mathcal{P}2]$. Indeed, there is no value of $\alpha$ able to influence the position of the $\xi_{\mbox{\tiny c,1}}=-1$ and $\xi_{\mbox{\tiny c,4}}=-6/5$ singularities. This fact is easily visible by substituting (8) and (9) in the above Lagrangian densities and then making the common denominator.

The preceding Subsection IV.1 was dedicated to introduce a novel Lagrangian density (11) that in principle, by imposing condition (15), we expect to regulate the no-go singularities plaguing Ricci-inverse cosmologies. Now we have knowledge of the $\mathcal{L}_{g}$ profile, we want to find the evolution of the scale factor and other quantities to verify that this is exactly what happens.

To achieve this, we start with the energy-momentum tensor (5), which we assume to be that of a perfect fluid. According to the literature, it can be written as

where $\rho=\sum_{i}\rho_{(i)}$ and $P=\sum_{i}P_{(i)}$ are the total mass-energy density and the total hydrostatic pressure of a mixture of two or more non-interacting perfect fluids, each labelled with the index $i\in I$. It is well known that the set $I$ can include a variety of fluids, such as: radiation $(r)$ for photons, pressureless non-relativistic matter $(m)$ for baryonic matter and dark matter, and the cosmological constant $(\Lambda)$ for non-dynamical dark energy.

The mass-energy densities and the hydrostatic pressures that appear in Eq. (16) are all measured in the fluid’s (comoving) rest frame, where the fluid’s four-velocity $u^{\mu}$ has expression $u^{\mu}=\!^{\top}\!(c,0,0,0)$. Hence, we can then rewrite the above expression as

In this context, it is useful recalling that the relationship between energy density and pressure is generally described by means of an Equation of State (EoS). Accordingly, henceforth we use assume the following $P(\rho)$ barotropic EoS

which applies for each $i$-component of the fluid. Here, $w_{(i)}\in\mathbb{R}$ are constant parameters. In particular, $w_{(r)}=1/3$ for a radiation density, $w_{(m)}=0$ for a pressureless non-relativistic matter density, and $w_{(\Lambda)}=-1$ for a cosmological constant density.

Having introduced the necessary background, we can now proceed to derive the modified Friedmann equations.

Using the profiles (7) (11) and expressions (17) (18) and then applying them to the field equation (6), the following cosmological equations for the density parameter $\Omega_{(i)}\equiv\tfrac{8}{3}\pi G\rho_{(i)}H^{-2}$ can be obtained after somewhat tedious algebra

Strictly speaking, Eqs. (19a) and (19b) are derived directly from the $(t,t)$ and the $(r,r)$ components of the field equation (6), respectively. Using these, we then also derived the matter conservation equation (19c), which emerges because the left-hand side of Eq. (6) is divergenceless, i.e. it satisfies the Bianchi identity, when equipped with our profile (11). However, only two of the three Eqs. (19) are functionally independent.

Interestingly, since Eq. (19c) depends at most on $\xi^{\prime\prime}$ and being $\xi\sim\ddot{a}$, our model (as the whole Ricci-inverse framework) is a type of fourth-order gravity. Our system (19) depends on a set of seven $\xi$-dependent functions, which have the following expression

and

and

Finally, by defining the anti-curvature density parameter $\Omega_{(A)}=1-\psi_{0}+\psi_{1}\xi^{\prime}$, then Eq. (19a) can be rewritten as

Now we are ready to examine how various cosmological scenarios can be realized within our gravitation model. This topic will be addressed in the next section.

In the present work, a novel MG theory called Ricci-inverse gravity was investigated. This theory is based on the introduction of a new, purely geometric object called the anti-curvature tensor, defined as the inverse of the well-known Ricci tensor. From this foundation, it is possible to modify the EH action without introducing new fields or requiring a fine-tuning of the coupling constants.

Although many studies emphasize the increasing relevance and applicability of Ricci-inverse gravity in diverse scenarios, some analyses have fed the idea that Ricci-inverse gravity is not a promising theory to describe the current accelerated expansion of the universe. Foremost among them, later translated into terms of a blocking no-go theorem, is the presence of singularities along the path of cosmic trajectories connecting a decelerated (matter-dominated) epoch to the current accelerated expansion of the universe. By requiring the Lagrangian density of such theories to satisfy a new property that we have called regularization via complexification of singularities ($\mathcal{P}2$), we have shown how it is possible to evade the effects of the aforementioned no-go theorem and reconsider cosmologies based on anti-curvature.

In addition to providing a formal introduction to $\mathcal{P}2$, in this paper we have designed and analyzed for the first time a new scale-free Ricci-inverse gravity model that, by implementing $\mathcal{P}2$, (i) is not rent by no-go singularities and (ii) brings out the current observed expansion as a stable attractor solution. In particular, after obtaining the modified Friedmann equations, we approached the study of their solutions from both an analytical and numerical point of view, giving evidence that they are regular up to the present observed values.

We trust that, with this analysis, we can once again reopen the discussion aimed at reconsidering gravity theories based on anti-curvature as good dark energy models.

We leave for future investigation the analysis of ghost or possible other instabilities.