Towards geometric inflation: the cubic case

We present an up to cubic curvature correction to General Relativity with the following features: (i) its vacuum spectrum solely consists of a graviton and is ghost-free, (ii) it possesses well-behaved black hole solutions which coincide with those of Einsteinian cubic gravity, (iii) its cosmology is well-posed as an initial value problem and, most importantly, (iv) it entails a geometric mechanism triggering an inflationary period in the early universe (driven by radiation) with a graceful exit to a late-time cosmology arbitrarily close to $\Lambda$CDM.


Introduction
It is a well-known fact that higher curvature corrections tend to improve the renormalization properties of General Relativity (GR) [1]. It is also widely understood that GR should be seen as providing a low energy effective dynamics of the gravitational field. Therefore, higher curvature corrections are expected to be there and become relevant at somewhat higher energies. Depending upon the scale at which they become relevant, causality might be at stake pointing towards the existence of a richer structure involving higher-spin fields [2].
Several modifications of gravity have been proposed throughout the years in order to provide a proper cosmological evolution (see [3,4] and references therein). Particularly important for all these alternatives is the explanation of the accelerated periods in our universe. Some of the proposals were successful explaining late-time acceleration while others accounted for an inflationary period in the early universe. Currently, none of the proposals can both explain inflation adequately and evolve smoothly into a late time acceleration era without invoking extra degrees of freedom.
In this letter, we present an up to cubic curvature correction to General Relativity whose vacuum spectrum solely consists of a graviton and is ghost-free. While it possesses a black hole solution coinciding with that of socalled Einsteinian Cubic Gravity [5][6][7], its cosmology is significantly different. Contrary to what turns out to be a generic feature of higher curvature gravities, the set of Friedmann equations remain second order, thereby defining a well-posed initial value problem. Most interestingly, it provides a late-time cosmology arbitrarily close to ΛCDM in GR while, at the same time, giving an inflationary period in the early universe with a graceful exit.

The theory
Restricted to four space-time dimensions, it was recently shown [8] that there are four possible cubic La- * gustavo.arciniega@gmail.com, jose.edelstein@usc.es, luisa.jaime1@gmail.com grangians, which can be constructed as linear combinations of the ten possible terms, whose static spherically symmetric vacuum solutions are fully described by a single field equation -at the quadratic level, the same conditions uniquely lead to the Lanczos-Gauss-Bonnet combination, which is a boundary term-. Two of them are actually strictly a zero in disguise, corresponding to the six-dimensional Euler density, χ 6 , and The remaining two can be chosen to be Their dynamics in vacuum is free of ghosts and any kind of massive modes. It solely consists of a graviton. For this reason, the Lagrangian P -discovered earlier-has been coined Einsteinian Cubic Gravity (ECG) [5]. The inclusion of C, although identified in [8], has not been considered so far in the literature for a reason: its equations of motion are identically null when evaluated in a black hole ansatz. This led to the misleading conclusion that C is somehow trivial and can be neglected.
The situation changes drastically when one considers a cosmological scenario. If we consider a FLRW ansatz in the realm of ECG,

arXiv:1810.08166v2 [gr-qc] 23 Nov 2018
where dΩ 2 is the metric of the unit round sphere, dΩ 2 = dθ 2 + sin 2 θdφ 2 , the resulting equations of motion for a(t) are fourth-order. This is problematic from the point of view of both dealing with a well-posed initial value problem and a causal evolution. On the other hand, this is a standard issue in the realm of f (R) gravities [9], where it can be dealt with by converting it into an equivalent scalar-tensor theory; the fields doubling drives the dynamics into a system of second-order differential equations [10].
It is natural to ask whether the Lagrangian C can play a significant role to alleviate this problem -although trivial when evaluated in a black hole ansatz, by its own it also leads to fourth-order differential equations for a(t). Interestingly enough, the answer turns out to be yes! There is a single combination of P and C that -while preserving all the nice features of ECG-leads to a cosmological scenario with a well-posed initial value problem.
We consider the following single parameter cubic gravity action, The propagating physical modes and the effective gravitational coupling κ eff of such a higher curvature gravity can be obtained from the fast linearization procedure given in [11], where it is shown that, besides the graviton, there may exist two massive modes, a ghosty graviton with mass m g and a scalar mode with mass m s . The values for m g , m s and κ eff are model dependent. In the case of (4), the massive modes decouple, m g , m s → ∞, while 1/κ eff = 1/κ + 48βΛ 2 , which are the conditions for the theory to be Einsteinian; i.e., free of massive and ghost-like propagating modes. Furthermore, among all cubic curvature theories constructed from P and C, that governed by action (4) is the only one leading to a secondorder differential equation for a(t), thereby guaranteeing a well-posed initial value problem in an FLRW universe.

Cubic cosmology
Let us thoroughly analyze the resulting cosmology. The field equations resulting from action (4) are where G µν is the Einstein tensor (including the cosmological term), while E P µν and E C µν result from the variation of P and C. After considering k = 0 for a flat spatial curvature in the metric (3), and plugin it into the equations of motion (5), we obtain a modified Friedmann dynamics: involving up to second order derivatives of a(t). Let us now consider an ideal fluid as a source of matter; in natural units, Then, in terms of the Hubble parameter, H ≡ȧ/a, the field equations (6) and (7), can be rewritten as We can recover the standard Friedmann equations by setting β = 0. It is worth noticing at this point that we are bound to have a different evolution for H(t) than the standard one, without the need of adding any kind of scalar field or effective energy-momentum tensor whatsoever. By substituting (9) in (10) we obtain For the inflationary period and for late time acceleration,ä > 0. In the GR framework, we need to invoke some field such that P < 0 in order to obtain a positive acceleration at early times [12]. In the same sense works the addition of the cosmological constant for the late time acceleration: it is necessary in order to prevent the deceleration of the universe.
In the present proposal, the evolution of the acceleration depends on the factor 96βκH 4 , which modules the evolution of κ(P + ρ) and allows H 2 to dominate for some periods, making it possible to obtain inflation and late time acceleration in purely geometric terms. If β were negative we would reach a singularity sometime in the past [13]. Thereby, we choose β > 0.
In alternative theories of gravity, it is a widespread habit to define an effective energy-momentum tensor, in such a way that the corrections can be seen as the addition of some kind of fluid. Such fluid should be tested in order to avoid ill behaviors (e.g., ghosts or massive modes). In our proposal, this strategy is not necessary: with the simple re-definition of the critical density, accordingly to (9), we avoid the inclusion of extra components. In order to explore the capability of this theory to produce a viable universe and even to explore if it can provide both an inflationary period and late time acceleration, we numerically integrate the field equations. As usual, the evolution of the matter content goes like ρ m = ρ 0 m a −3 with P m = 0 and radiation goes as ρ r = ρ 0 r a −4 with P r = 1/3ρ r .

Cosmological exploration
In the present letter we want to explore robust features of our proposal leaving for the future a detailed exploration of the full parameter space [14]. Therefore, we will consider and compare just three cases that give us enough pieces of evidence to reach generic and interesting conclusions. They are: A. General Relativity in the ΛCDM framework.
We performed a numerical integration of the field equations (10) and analyzed the following observables: the age of the universe, its early and late time acceleration, the evolution of the matter, radiation and dark energy densities, and the evolution of the Hubble parameter for low values of the redshift z. The results are plotted and discussed below. age of the universe, under the present proposal, is affected in an interesting direction: in both scenarios, B and C, a(t) displays a characteristic shape that generically makes this theory able to produce an accelerated epoch at early times; a(t) ∼ t 3/2 . This is confirmed in Figure 2, where we analyze the acceleration given by (11).
We display the acceleration at late and early (interior frame) times. For late time we have plotted the three cases. For case A, the acceleration comes from negative values until the cosmological constant starts to dominate and thenä/a goes slightly positive. In case B, acceleration comes from positive to negative values and then, The interior frame of Figure 2 shows the evolution at early times for B and C. In both cases the acceleration goes likeä ∼ a −1/3 during several e-folds. This evolution is limited on physical grounds just by the Planck length, l P , otherwise the singularity a = 0 would be apparently reached. The exit of what can be considered as the inflationary period in this theory transits smoothly to the standard behavior in GR, solving in this way the graceful exit problem.
We show the evolution of the densities Ω m , Ω r and Ω Λ for the three cases in Figure 3. We split them into two  figure). Instead, when we consider the case C, the evolution of Ω m , Ω r and Ω Λ go differently: even when matter domination is well behaved, the matter-radiation equality time is reached at lower values of z. Even when this can be seen as a reason to rule out this latter model, a thorough analysis of the parameter space seems necessary before reaching a definitive conclusion.
In order to have the insight to compare our predictions, at the background level, with observational data for late time cosmology, we compute H(z)/H 0 . Figure  4 shows the evolution of the Hubble parameter for the three cases. As expected, the evolution of A and B are arbitrarily close at late times, thereby making possible to match for instance with supernovae IA observations [16,17]. For higher values of z (in the figure, z ∼ 4, but this value certainly depends on the choice of β), the evolution of H(z)/H 0 in case B goes slower than the corresponding one in case A. Close to the CMB the values of H(z)/H 0 for models B and C are very different from the ones obtained with GR. In case C, H(z)/H 0 behaves similarly to GR for very low values of z and rapidly departs, staying down from A and B.

Discussion
In this letter we have presented a novel cubic gravitational theory, free of massive and ghost-like modes, which, in the case of an isotropic and homogeneous fourdimensional universe, has second-order field equations for the scale factor. The modification depends on a single parameter; it is purely geometric and has the potential to provide both an accelerated inflationary epoch at early times as well as a late time evolution that is arbitrarily close to GR in the ΛCDM framework. Both the early and late time evolutions are phenomenologically viable provided the parameter β is small. In the past, the universe appears to have displayed accelerated expansion, a(t) ∼ t 3/2 ; a sufficient number of e-folds can be obtained from the instant when the scale factor was given by the Planck length, l P . The inflationary period has a graceful exit.
We have also briefly discussed a case in which the cosmological term vanishes, Λ = 0, in order to show that it leads to late time acceleration; albeit not enough to be comparable to the one provided by a cosmological constant. The matter-radiation equality time is also reached at lower values of z than those customarily expected, although these results depend on the choice of parameters -including the matter, radiation and dark energy densities-and a full analysis is required before arriving at categorical conclusions.
In comparison with the usual modifications of gravity proposed to solve cosmological controversies, the value of β in this theory is not just a parameter to be fixed by observations; in our case, it can be seen as a new fundamental gravitational constant. It is natural to expect that it originates from a UV complete theory of gravity as the result of an effective Wilsonian low-energy integration.
Further analysis should be performed in order to constrain β. Current observations of late time cosmological tracers such as supernovae [18], cosmic chronometers [19] and baryonic acoustic oscillations [20,21], as well as the Cosmic Microwave Background [15], can be used to explore the best-fitted values of the parameters Ω 0 m , Ω 0 r , Ω Λ and β [14].
It is worth mentioning that the evolution of H(z) predicted in this theory might help in reducing the conflicting difference of ∼ 3.5σ between the value of the Hubble constant, H 0 , predicted by Planck 2015 [15] and the one found when more local sources are analyzed (see, for example, [21]). On the other hand, a forecast could be cooked up in order to have ready-to-test predictions in the light of the releases of EUCLID [22] or DESI [23] experiments.
Finally, we must mention that different scenarios should be explored in order to pass other gravitational tests [24], and further constrain the value of β in an independent way. For instance, the existence of compact objects has been an issue in some other modifications of gravity [9], and it is certainly a problem to be addressed in our currently proposed cubic theory [25].