One fundamental aspect of any physical theory is the agreement that it must hold with experiment. However, it has been known from the beginning that any approach to a quantum theory of gravity would have severe issues with this aspect and string theory is not an exception. Decades have passed since the first string theory was proposed. In that amount of time the theory has been improved vastly leading to a framework with a vast scenario of possible realizations. One of the most controversial issues that string theory possesses in this regard is the choice of a vacuum for the theory, since the physical constants on any string theory come from vacuum expectation values for scalar fields, the choice of such a vacuum is fundamental. However, at present none mechanism to choose dynamically such a vacuum exists. Instead, there is a very big number of possible vacua that can be compatible, the so called Landscape of string theory. If we have a full quantum theory of gravity, any model constructed within it will lead to an effective theory in the low energy limit. However the opposite is not in general true, and thus the idea that it should be possible to distinguish an effective theory that can be completed to a correct quantum theory of gravity in the ultraviolet (UV) from those which cannot is an important subject. The theories that can be completed in this sense are said to belong to the mentioned Landscape. In contrast if an effective theory in the infrared (IR) cannot be completed in this sense is said to belong to the Swampland. In order to pursue conditions on an effective field theory containing gravity to be completed in the UV a set of Swampland conjectures have been proposed recently [1,2,3,4,5,6]. They describe some behaviours found generically on string theoretic realizations that seem to be key elements of a quantum gravity theory. Although features derived from string theory models are used as guidelines, these conjectures are expected to be valid in a general sense independently of string theory.

There is a vast number of constructions in string theory aiming to describe a cosmological scenario compatible with observation. For example from the inflationary point of view string theory have many proposals for the inflaton field (see for example [7]). However, it has been found very difficult to obtain a dS background with a complete construction on the higher dimensional theory. This seemingly impossibility led to the dS conjecture that now claims that it is not possible to obtain such a background [8,9,10,11,12]. There are other Swampland conjectures in this sense such as the Distance conjecture, the Transplanckian Censorship conjecture, etc. but in this article we will focus exclusively on the dS conjecture. When we only have one scalar field coupled to gravity as standard in inflationary models, this conjecture is written as inequalities on the scalar field potential. However, it is known that the dS conjecture is not compatible with the standard cosmological picture of an inflationary universe driven by a scalar field that obeys the slow-roll conditions on General Relativity (GR) [13,14,15], since those conditions are expressed in terms of the scalar field potential and they are the opposite of the dS conjecture among other problems. There are various ways out of this inconsistency such as considering Warm Inflation[16], Multi-field inflation [17] or variations of the gravitational theory such as $f\left(R\right)$ theories [18,19,20] among other proposals [21,22,23,24,25]. In this work we will be interested on studying the compatibility of this conjecture with another modification of the gravitational theory known as Hořava-Lifshitz $F\left(\overline{R}\right)$ theories.

It is known that GR must be the theory of gravity on the low energy regime. String theory of course fulfils this condition. However, instead of modifying the theory of gravity at the fundamental level to discover its quantum description, there is also the possibility of improving GR to obtain such description in a closer way to the original theory. In this sense, inspired on the Lifshitz scaling of condensed matter systems, Petr Hořava proposed to modify the Einstein-Hilbert action by incorporating spatial higher-derivative terms compatible with an anisotropic scaling of space and time variables [26,27,28,29,30]. These terms break the Lorentz symmetry in the UV but makes the theory power counting renormalizable and thus it represents a better candidate to a quantum UV completion of gravity. The IR limit of this theory was found to be troublesome but it led to new relevant cosmological behaviours such as bouncing universes [31,32,33] or the appearance of a dark energy term as a constant of integration [34] to name a few of them. This theory has been generalized in a number of ways in order to improve the issues in the IR limit that the original theory has. For example in [35,36] an extra $U\left(1\right)$ field was added. It has also been proposed a generalization of such a theory in the same way as to $f\left(R\right)$ theories are a generalization of the standard Einstein-Hilbert action, this has led to the Hořava-Lifshitz $F\left(\overline{R}\right)$ theories [37,38]. These theories describe new scenarios that do not appear in standard $f\left(R\right)$ theories and thus they are relevant to be further explored. In the present article we perform an analysis of the compatibility of these theories with the Swampland conjectures. In particular we are going to focus on the dS conjecture as explained before.

The outline of this article is as follows. In Section 2 we are going to summarize the way in which standard $f\left(R\right)$ theories are studied in the light of the dS conjecture. We will particularly focus on the way in which this conjecture can be made independent of R. In Section 3 we present the $F\left(\overline{R}\right)$ theories and in Section 4 we perform the full analysis of their compatibility with the dS conjecture. Finally in Section 5 we present our final remarks section.

In this section we briefly overview the situation of the Swampland dS conjectures in the context of $f\left(R\right)$ theories. Our aim is not to be exhaustive but just introduce conventions and notation which will be necessary in the further sections.

We consider the theory of gravity coupled to a scalar field, the dS conjecture was written originally in the following form (first dS conjecture) [8,9,10,11,12] where V is the scalar field potential, $V_{\varphi }$ is the derivative of the potential with respect to the scalar field which we are considering to be unique and c is an order 1 constant. Later this conjecture was refined to incorporate another possibility which is written as (second dS conjecture) where $\tilde{c}$ is another order 1 constant. Thus the dS conjecture is expressed now as the statement that either one of the two inequalities, the first or the second dS conjecture must hold.

Let us give a general picture of how can we study the Swampland conjectures in the light of $f\left(R\right)$ theories [18]. These theories are a generalization of GR by considering the gravitational action in the Jordan frame as1 where $f\left(R\right)$ is some well behaved function of the curvature scalar R and no matter component has been considered. In this scenario it is thought that the action for GR is just the first term on a Taylor expansion of the f function around a small curvature. These theories are of great interest since they lead to many new interesting cosmological behaviours [39]. In order to transform this action into a form suitable to investigate its compatibility with the conjectures we consider the equivalent action [39,40] where $\psi =\frac{df}{dR}=f_R$ is just an auxiliary field and $U=R\psi -f\left(R\right)$. Then we can perform a conformal transformation in order to describe the dynamics in the Einstein frame in the form with $\phi =\frac{1}{2}ln{f}_R$. We then obtain the following action which represents just the action of gravity plus a scalar field where the scalar field is given by with potential

Let us remark that in the Einstein frame we have the action of gravity plus a canonically coupled scalar field although in the Jordan frame we have not considered any matter contribution. This scalar field originated in the process of getting to the Einstein frame by a conformal transformation and thus the field and its potential are entirely defined by geometry. However since the dS conjecture is expected to hold for any scalar field that is canonically coupled to gravity, we can apply the conjecture in this scenario as well. We will not consider any matter content in the Jordan frame since as we will see by considering only gravity we can use the conjectures to constraint the parameters of the gravitational theory and thus the allowed solutions.2 We note from (1) that if the potential is negative the conjecture is satisfied immediately. Therefore we are interested in the scenario where the potential is positive. In this case from (8) we see therefore that we have to impose the condition

And from (7) we also note that we need $f_R>0$. Using these definitions we can obtain the derivative of the potential with respect to the scalar field as

With this setup the first dS conjecture (1) takes the general form and the second dS conjecture (2) is written as

From these expressions the conjectures were studied in a general way in [18] finding the constraints on the f function that makes it compatible with the conjectures. In [19] particular attention was put in a function of the form $f\left(R\right)=R^{1+ϵ}$ where $ϵ$ is a small number, it was found that the conjectures were compatible with the region of $ϵ$ of phenomenological interest. Finally in [20] the first dS conjecture was found to be in good agreement with numerical data.

We expect that in general (11) will lead to inequalities involving R. However since we want to use the conjectures to constraint the theory in this article we will be interested in a scenario where the conjecture could be made independent of R and thus valid for all values of R. We can achieve this by imposing the condition where A is any constant that should be positive in order to fulfil (9). Let us remark that this is an ansatz and therefore it is not the only way to fulfil (11), but it is the only way to fulfil it independently of R. We have therefore two possibilities regarding the sign of the term inside the absolute value. If $2f>R{f}_R$ (13) leads to where $\gamma =\frac{A+2}{A+1}$ with $1<\gamma <2$. In order to have a well-defined scalar field and not ghostly gravitons we require that $f_R>0$ and in order to have positive mass of the curvature fluctuations we also require $f_{RR}>0$ [39]. The expression (14) fulfils both conditions for $\beta >0$. Substituting this form in the first dS conjecture we obtain that $\gamma$ is restricted to be

In particular for $c=1$ this leads to $\gamma \lesssim 1.45$. Thus the conjecture leads to a restriction on the exponent of the f function that can not be the GR value but it is restricted to be close.

For the case $2f<R{f}_R$ we obtain the same form but in this case $\gamma =\frac{A-2}{A-1}$ and thus $\gamma >2$. This form can also fulfil the two conditions $f_R>0$ and $f_{RR}>0$ for $\beta >0$. However in this range of values $\gamma$ the first dS conjecture can not be fulfilled.

On the other hand, substituting the form (14) into (2) we obtain that the second dS conjecture is written as which is independent of R but cannot be fulfilled for any value of $\gamma$. Therefore this form for the f function is incompatible with the second dS conjecture. This is not a great deal since we only need to fulfil one of the two versions of the conjecture.

Thus we conclude that in $f\left(R\right)$ theories the only way to fulfil the first dS conjecture independently of R is to take f as a power term of R. The conjecture restricts the exponent of this function to be close to the GR value and the second dS conjecture is never valid.

In the following sections we will describe the $F\left(\overline{R}\right)$ Hořava-Lifshitz theories and investigate if a similar procedure as the one just described can be applied to them to look for the viability of the Swampland conjectures.

This section will be devoted to briefly overview Hořava-Lifshitz theory of gravity in the context of modified gravity. We will consider the ADM decomposition of spacetime in which we write the metric in the general form where N is the lapse function, $N^i$ the shift functions and $g_{ij}^{\left(3\right)}$ the three-metric. We note that the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric has this form with vanishing shift functions and it is always possible to choose the lapse function to be equal to one.

In the Hořava-Lifshitz theory of gravity the metric is naturally written in an ADM form and the action for the gravitational part can be written as [26,27,28,29,30] where $K_{ij}$ is the extrinsic curvature, $\lambda$ is a parameter and $\mathcal{L}\left(g_{ij}^{\left(3\right)}\right)$ is a potential term for gravity. In the general scenario this term will contain the scalar curvature of the three-metric $R^{\left(3\right)}$ and seven constants accompanying higher spatial derivative terms written in terms of the Ricci scalar of the spatial three-metric. If the detailed balanced condition is proposed this term is derived from a geometrical action and the number of constants is reduced. However, for the flat FLRW metric which will be of interest to us this term will always be zero since in that case the three-metric is just the scale factor times the three dimensional euclidean flat metric. Thus in the following we will not be concerned with the specific form for this term. As can be easily seen, the IR limit will be obtained in the limit $\lambda \to 1$ and it is obtained $\mathcal{L}\left(g_{ij}^{\left(3\right)}\right)\to R^{\left(3\right)}$. The lapse function appearing in the action can lead to different versions of the theory depending on the variables in which it may depend on. If the projectability condition is used, the lapse function only depends on the time variable and thus the hamiltonian constraint is a global constraint contrary to the case in which the lapse function is allowed to depend also in the spatial variables. Since we will deal with the FLRW metric we will choose $N=1$ but in general the global nature of the hamiltonian constraint leads to an integration constant and thus a new term in the equations of motion absent in the other scenario [28,34]. Thus even though in both cases we can choose the lapse function to be 1 the equations of motion will be different. This will be an important remark later on.

In [41] the Swampland conjectures have been studied when a nonlinear dispersion relation is taken into account. Such feature comes naturally from HL gravity and its possible extensions (such as [35]). However, for an FLRW flat metric the equations of motion are almost identical in all types of HL theories as the ones obtained in GR and thus the problem of the incompatibility of the slow roll criteria is still present. Therefore in the following we will consider a generalization of HL theory in the same line as $f\left(R\right)$ theories and investigate the compatibility with the Swampland conjectures.

As we can see from the form of the action (18) it is natural to generalize the HL theories in the same way as the generalization leading to the $f\left(R\right)$ theories described in the previous section. In this case it has been proposed a more general action in the form where now $\overline{R}$ can be understood as a generalization of R which includes the new terms of spatial derivatives which is proposed to have the form [37,38] where $\mu$ is a constant. The term containing $\mu$ is usually omitted in the standard $f\left(R\right)$ proposal since it turns out to be a total derivative term. However it is necessary for these theories. As we can see for the above definitions the limit $\lambda \to 1$, $\mu \to 1$ of this theory will lead to the standard $f\left(R\right)$ theory used in the previous section. This theory was originally proposed in a series of papers [37,38] using the detailed balanced condition. The resulting cosmological scenarios were found to describe new and interesting behaviours for the flat FLRW metric such as solutions with two periods of accelerating expansion. In [42] the Ekpyrotic scenario was studied for these theories. In [43] the detailed balanced condition was abandoned. However for the flat FLRW metric, as we have pointed out earlier, the equations of motion are the same. A hamiltonian analysis for these theories was performed in [44] and the relation to general scalar tensor HL theories was investigated in [45]. An extension to include a $U\left(1\right)$ field was performed in [46]. However as in the latter case the equations of motion found for the FLRW flat metric is the same as in the original scenario with or without the detailed balanced condition. Thus, if we consider only the flat FLRW metric at the level of equations of motion, all the versions proposed for the $F\left(\overline{R}\right)$ metric will lead to the same results. Therefore the following analysis will be valid in all these scenarios.

Let us consider then a flat FLRW metric in the form

With this metric we obtain from (20) the following result where $H=\frac{1}{a}\frac{da}{dt}$ is the Hubble parameter. In contrast the standard curvature scalar in this case is and thus the difference between both theories for this metric are just the appearance of the two parameters $\mu$ and $\lambda$ and thus at least for the flat FLRW metric the interpretation of $\overline{R}$ can be considered as related to the curvature. However we will see that although both expressions differ only by constants they will lead to very important differences.

Considering only gravity the first equation of motion obtained through the hamiltonian constraint is [37] where $\dot{H}=\frac{dH}{dt}$, $F^{\prime }=\frac{dF}{d\overline{R}}$ and where $C\ne 0$ is an integration constant in the projectable version of the theory which can be considered to represent dark matter for the way in which it appears in the equations of motion when it takes positive values [34]. In the non-projectable version of the theory we have $C=0$. This is the only difference between both versions of the theory that will be of importance to us. Varying with respect the three-metric the second equation of motion is obtained [37]

As stated before we note from these equations and (20) that the limit in which we can recover the standard $f\left(R\right)$ theories is $\mu ,\lambda \to 1$. If we put an ansatz of the form $H=H_0$ in the above equations and squared or cubic polynomials form for the $F\left(\overline{R}\right)$ function the cosmological solutions describe exponential acceleration for two different periods [37,38]. This behaviour is exclusive of these theories since it is removed in the mentioned limit. This is one of the reasons that shows the importance of these kind of theories.

Before continue let us remark an important point. The Swampland conjectures were proposed originally in theories with Lorentz invariance. However HL theories, as well as their generalizations, break this invariance in the UV so one may worry about the applicability of the conjectures to these kind of theories. If we apply the conjectures and find that they are incompatible we may conclude that Lorentz invariance is a key requirement of the conjectures. However in the next section we will discover that the conjectures are indeed compatible with these theories and they actually lead to some expected behaviours. Thus we can support the idea that the conjectures may be applicable to more scenarios that originally thought considering our results as one indication a posteriori supporting this idea. In [25] the compatibility of the conjectures with a cosmologcal setup with explicit breaking of Lorentz invariance was studied with positive results as well.

In the same way that we need to perform a conformal transformation to the $f\left(R\right)$ theory in order to obtain an action of GR plus a scalar field in the Einstein frame, we will perform a conformal transformation in the three-metric of the above theories and then we will be able to study the compatibility with the Swampland conjectures. Let us start by considering the action with $U=\overline{R}{F}^{\prime }\left(\overline{R}\right)-F\left(\overline{R}\right)$. This action is equivalent to (19) with an auxiliary field. Then we perform a conformal transformation only on the three-metric of the form and choose $N=1$ and $N^i=0$ as in the flat FLRW metric, which leads to the action in the Einstein frame [38,43] given by with

In order to eliminate the term that combines the metric with the scalar field and to obtain a canonically coupled scalar field (that is that it appears as in standard scalar theories) we choose

Thus we perform the redefinition of the scalar field as where then we finally obtain in the Einstein frame the action in the following form

Moreover in the Einstein frame we obtain an action of gravity plus a canonically coupled scalar field. We note that we do not obtain the HL theory of gravity since in this case there is no higher derivative terms of the scalar field, however since the general form is the one of gravity plus a scalar field we can use this action to investigate the compatibility with the Swampland conjectures of the $F\left(\overline{R}\right)$ theories. We do not worry that the gravity part of the action is not GR since we only need to describe a theory of gravity. For example in [23] the conjectures are studied with a Gauss-Bonnet term in the action. We note that since we made the choice (30), we can no longer recover the $f\left(R\right)$ theories in any limit. We first study this case since this simplification allows a complete analytic analysis. The general scenario where this special choice is not required will be studied in Subsection 4.4 .

With this set up and using (10) with $\overline{R}$, the first dS conjecture (1) is given in the general form which can be written as and the condition for the scalar field to be positive is

On the other hand the second dS conjecture takes the general form

We note that similarly as what happened with the $f\left(R\right)$ theories the way in which the above expressions are to be independent of $\overline{R}$ is by considering $F\left(\overline{R}\right)$ as some power of $\overline{R}$. However, since these theories have not been studied as extensively as the $f\left(R\right)$ theories we will proceed in a different way. Instead of choosing some form for the F function, we will use the system of equations of motion in the Jordan frame with an ansatz for the Hubble parameter which is of cosmological interest and then construct the $F\left(\overline{R}\right)$ function. Finally we will check if this function can fulfil the conjectures at the same time that leads to cosmological interesting solutions.

Choosing (30) the system of equations of motion (24) and (25) takes the form

With this choice we also obtain

From (38) and (39) we can obtain

Thus, the general procedure will be to use this system of equations to construct the $F\left(\overline{R}\right)$ function with an interesting cosmological behaviour. Since the constant C will change the possible solutions we will consider separately the two different scenarios, namely the case in which it can be eliminated, that is the non-projectable case and the case in which it has to be different from zero which is the projectable case.