# Cosmological perturbations in a class of fully covariant modified theories: application to models with the same background as standard LQC

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## Abstract

Bouncing cosmologies are obtained by adding to the Einstein–Hilbert action a term of the form \(\sqrt{-\,g}f(\chi )\), with \(\chi \) a scalar depending on the Hubble parameter only, not on its derivatives, and which is here shown to arise from the divergence of the unitary time-like eigenvector of the stress tensor. At background level, the dynamical equations for a given *f*-theory are calculated, showing that the simplest bouncing cosmology resulting leads to exactly the same equations as those for holonomy corrected Loop Quantum Cosmology (LQC). When dealing with perturbations, the equation for tensor ones is the same as in General Relativity (GR); for scalar perturbations, when one uses the *f*-theory which leads to the same background as the standard version of holonomy corrected LQC, one obtains similar equations (although a bit more elaborated) as those coming from LQC in the so-called *deformed algebra approach*.

## 1 Introduction

One of the most simple bouncing backgrounds (see [1, 2, 3] for recent reviews about bounces) is obtained from holonomy corrected Loop Quantum Cosmology (LQC), where the corresponding Friedmann equation depicts an ellipse on the plane \((H, \rho )\) [4, 5, 6, 7, 8, 9], being *H* the Hubble parameter and \(\rho \) the energy density. As shown in several papers, this simple background can be mimicked by modifying the Einstein–Hilbert action through the introduction of a term of the form \(\sqrt{-g}f(\chi )\) where *g* is the determinant of the metric, *f* is a well-known function [10, 11, 12, 13] and \(\chi \) is a scalar, only depending on *H* but not on its derivatives. The problem with this method is to actually find a scalar that for synchronous observers in the Friedmann–Lemaître–Robertsont–Walker (FLRW) spacetime will only depend on the Hubble parameter.

From our viewpoint the simplest scalar is the extrinsic curvature [14], which appears in a natural way when using the ADM formalism [15]. Disappointingly, this is not a covariant theory because the extrinsic curvature is not a true scalar in the sense that it depends on the slicing chosen and, thus, its use is only justified if there exists a preferred foliation of the spacetime. Following the spirit of *Weyl’s principle* (see [16] for a historical review), one could choose a preferred slicing as follows. The time-like eigenvector of the stress tensor, which always exists for realistic matter (see pages 89–90 of [17]), generates a preferred non-crossing family of world-lines and one can construct, at any given time *t*, a family of hypersurfaces orthogonal to these world-lines, obtaining in this way the so-called *co-moving slicing*.

Another simple scalar could be obtained by working in the Weitzenböck spacetime (the usual Levi-Civita connection is replaced by the Weitzenböck one) [18], where torsion does not vanish. In this spacetime, the scalar torsion for synchronous observers in the flat FLRW geometry is equal to minus three times the Hubble parameter, thus satisfying the required property. Unfortunately, as has been shown in [19], this theory is not locally Lorentz invariant.

For this reason, people have kept looking for a really covariant invariant, and have dealt with the Carminati–McLenaghan invariants [20, 21], with a general function of the Ricci and Gauss–Bonnet scalars or with second derivatives of the Riemann tensor [22]. The problem with these purely gravitational invariants is that they are quite involved and lead to very complicated equations for cosmological perturbations. Moreover, in our approach – not using the principle of the *limiting curvature hypothesis* considered in [22] – they can easily lead to Ostrogradski or gradient instabilities, as well as to the appearance of ghost fields. All these problems led specialists to explore other ways, such as modified mimetic gravity [13, 23, 24], so as to find out such scalar.

Following these arguments, guided by Weyl’s principle and taking into account that for co-moving observers in flat FLRW spacetime the divergence of a unitary time-like vector is equal to \(-\,3H\), in the present paper we propose as our fully covariant scalar the divergence of the unitary time-like eigenvector of the stress tensor, which, in the case of a universe filled up with a scalar field \(\phi \) minimally coupled to gravity, is equal to \(u_{\mu }=\frac{\phi _{,\mu }}{\sqrt{\phi _{, \nu }\phi ^{, \nu }}}\) (throughout the work we will use the notation: \(\phi _{,\mu }\equiv \partial _{\mu }\phi =\nabla _{\mu }\phi \)).

With this covariant scalar, we will show how to construct *f*-theories leading to bouncing backgrounds and calculate the perturbed equations for scalar and tensor perturbations for a given *f*-theory. When dealing with perturbations and working in the longitudinal gauge – where the Newtonian potential, namely \(\Phi \), and the variation of the scalar field, namely \(\delta \phi \), are the dynamical variables – the corresponding dynamical system turns out to be a coupled one. This is an essential difference with respect to theories such as General Relativity (GR) or LQC in the *deformed algebra approach*, where the dynamical equation for the potential \(\Phi \) decouples (\(\delta \phi \) does not appear, see for instance [30, 31]).

Once these equations are obtained, we study some characteristics of the matter-ekpyrotic bounce scenario [34, 35] as the calculation of some spectral quantities and the reheating temperature via the gravitational particle production of massless particles, in the contracting regime, during the phase transition from matter domination to the ekpyrotic regime.

The manuscript is organized as follows: In Sect. 2 we present our class of modified gravitational theories and obtain the corresponding dynamical equations. We study them at the background level and find which is the model that leads to the simple bounce predicted by holonomy corrected LQC. A Hamiltonian analysis of our theory is performed in Sect. 3, which leads to the conclusion that the present theory, as in the case of mimetic gravity, has one more degree of freedom than GR. In Sect. 4, we study scalar and tensor perturbations. For scalar perturbations, working in the longitudinal gauge, we obtain the equations for the Newtonian potential and for the perturbed part of the scalar field, showing that they are coupled. Moreover, we derive the corresponding Mukhanov–Sasaki equations for our theory. Then, dealing with tensor perturbations, we show that, since the modification of our theory is performed on the matter sector, the equations must be the same as for GR. Section 5 is devoted to the comparison, at the perturbative level, of our model which leads to the same background as holonomy corrected LQC, with other theories which also lead to the same background, as: LQC in the *deformed algebra approach* [31, 32, 33], teleparallel LQC [36, 37], extrinsic curvature LQC [14] and mimetic LQC [12, 13, 24]. In Sect. 6, we study the matter-ekpyrotic scenario applied to the model that leads the same background as LQC. We calculate the spectral index and its running and show that they match the most recent observational data. Moreover, we study the reheating process via gravitational massless particle production. Finally, the last Section is devoted to conclusions.

- 1.
\(\varphi _{,\mu }\equiv \partial _{\mu }\varphi =\nabla _{\mu }\varphi \) for a given scalar \(\varphi \).

- 2.
\(\bar{g}\) is the unperturbed part of

*g*. - 3.
\(g_{\chi }\) means derivative of

*g*with respect to \(\chi \) and \(g_{\phi }\) derivative of*g*with respect to \(\phi \), for a given function*g*.

## 2 A class of modified gravitational theories

*f*is a given function which, in order to recover GR, vanishes at low energy densities, while \(\chi \) is a fully covariant scalar built from the scalar field that fills the whole universe and whose value in any co-moving slicing of the flat FLRW spacetime is proportional to the Hubble parameter. More precisely, we consider

*R*being the scalar curvature. We have assumed that the matter sector of the universe is described by a scalar field \(\phi \) with a potential, \(V(\phi )\) which is minimally coupled to gravity and whose Lagrangian is given by

### 2.1 The background

*N*(

*t*) is the lapse function. Hence, after integration by parts, the action becomes

*N*and taking at the end \(N=1\), one obtains the modified Friedmann equation for synchronous observers

### Remark 2.1

A different way to obtain such equations is to directly consider the metric for synchronous observers \(ds^2=-dt^2+a^2\delta _{ij}dx^idx^j\) and use the equations \(0-0\) and \(i-i\), which also leads to the modified Friedmann and Raychaudhuri equations.

*f*-theory is given by

*f*-theory, the modified Friedmann and Raychaudhuri equations read

A final remark is in order. Since in our case the matter sector is depicted by a non-phantom scalar field \(\bar{\phi }\) (see Eq. (10)), we will have infinitely many backgrounds because the conservation equation \(\ddot{\bar{\phi }}+3H\dot{{\bar{\phi }}}+V_{\phi }({\bar{\phi }})=0\) is a second order differential equation. Effectively, for any initial condition \(\bar{\phi }(0)=\alpha _0\) and \(\dot{\bar{\phi }}(0)=\alpha _1\), one has a different background. Therefore, if we choose a curve \(\rho =g({\bar{\chi }})\) in the plane \((H,\rho )\) containing a point of the form \((0, \tilde{\rho })\) with \(\tilde{\rho }>0\) and initial conditions satisfying \( \frac{\alpha _1^2}{2}+V(\alpha _0)=\tilde{\rho }\), its corresponding solution will lead to a bouncing background provided that \(V_{\phi }(\alpha _0)\not = 0\).

Moreover, since we are dealing with theories beyond GR, in order to have a bouncing background it is not needed to violate the *null energy condition* \(\rho +P\ge 0\) near the bounce using quintom or Lee–Wick matter (see [44] and references therein) because the bounce occurs when the value of the energy density is strictly positive. For example, looking at (19) one can see that at the bouncing time \(t=0\) the energy density is given by \(\rho _c\) and the pressure, which could be obtained from the Raychaudhuri equation (17), is zero. So, the null energy condition is fulfilled at the bounce.

## 3 Hamiltonian analysis

*D*the induced Levi-Civita connection in the slicing \(\Sigma _t\).

Now we examine the stability of the constraint \(P_{\chi }\). Looking for the only term in the Hamiltonian where \(\chi \) appears and for the term where it appears in the action (24), i.e. \(-N\sqrt{\gamma }(f(\chi )+\beta \chi )\), one has \(\dot{P}_{\chi }=\{P_{\chi },H\}=-N\sqrt{\gamma }(f'(\chi )+\beta )\), which leads to the constraint \(C_{\chi }\equiv f'(\chi )+\beta \approx 0\). Finally, since \(\dot{C}_{\chi }=\{C_{\chi },H\}=N\sqrt{\gamma }f''(\chi )\alpha _{\chi }\), the stability of \(C_{\chi }\) is ensured by fixing the Lagrange multiplier \(\alpha _{\chi }\) as \(\alpha _{\chi }=0\).

Summing up, we have obtained the constraints \({{\mathcal {H}}}\cong 0\), \({{\mathcal {H}}}_i\cong 0\), \(P_{\chi }\cong 0\) and \(C_{\chi }\cong 0\), and the canonical pairs \((q_{ij}, P^{ij})\), \((\phi , P_{\phi })\), \((\chi , P_{\chi })\) and \((\beta , P_{\beta })\). Then, from the constraints \(P_{\chi }\cong 0\) and \(C_{\chi }\cong 0\) one may remove two variables, for example \(P_{\chi }\cong 0\) and \(\beta \approx -f'(\chi )\), thus obtaining, as in mimetic gravity [12], one more degree of freedom than in the case of GR. However, as we will show in next Section, when dealing with perturbations in longitudinal gauge, the degrees of freedom are the Newtonian potential \(\Phi \) and the perturbation of the scalar field \(\delta \phi \) for scalar perturbations and two degrees for the tensor ones, exactly the same as in GR. This is the same as what happens in mimetic gravity [45], where the degrees of freedom are the perturbed part of the mimetic field and \(\delta \phi \).

## 4 Perturbations

In this section we will calculate, for a given *f*-theory, the scalar and tensor perturbations using for scalar perturbations the longitudinal gauge (see for a review of the used setup [28, 29]).

### 4.1 Scalar perturbations

*f*given in (16), differs from the corresponding equation of LQC in the

*deformed algebra approach*[31, 32] only in the right hand side term, which in the last approach vanishes.

*curvature fluctuation in co-moving coordinates*, one obtains the following M–S equations

*v*, after taking the Laplacian from the second equation and using the first one, becomes

*v*, which encodes the scalar perturbations (it depends on \(\Phi \) and \(\delta \phi \)), is not independent and one needs another equation in order to calculate the evolution of the scalar perturbations. Fortunately in the matter (or matter-ekpyrotic) scenario, in the contracting phase, the pivot scale leaves the Hubble radius at rather low energy densities, as compared to the Planck one, so the corrections due to

*f*can be safely disregarded and, thus,

*v*satisfies approximately the usual equation \(v''-\Delta v-\frac{z''}{z}v=0\). This finally means that the calculation of the spectral quantities, such as the spectral index, its running, and the ratio of tensor to scalar perturbations, can safely be done using GR in the contracting phase, as we will show in next section.

*u*and \(\delta \sigma \equiv a\delta \phi \), with the corresponding dynamical equations being

A final remark is in order. In a bouncing scenario, as for instance the matter-ekpyrotic one, at early times GR holds, meaning that \(\bar{f}_{\chi \chi }\) could be disregarded and thus obtaining for the variable *v*, in Fourier space, the classical equation \(v_k''+(k^2-\frac{a''}{a})v_k=0\). Then, when the pivot scale leaves the Hubble radius, i.e. in the long wavelength approximation (\(k^2\ll a^2 H^2\)), one can safely disregard the Laplacian terms which appear on the right hand side of Eq. (44) (see the end of section 4 in [50]). And maybe the same happens with the right hand side of our Eq. (44) because it contains a Laplacian. If so, Eq. (44), in the long wavelength approximation, will become, as usual, \(v_k''-\frac{z''}{z}v_k=0\) and, for the *f*-theory given by (16), we will recover the results obtained in LQC using the *deformed algebra approach* [34, 35, 49]. Anyway, this has to be properly checked by solving numerically the system of Eq. (45).

*u*is, in Fourier space,

*I*refers to an early time where GR holds, and the coefficients \(C_1(k)\) and \(C_2(k)\) are obtained by matching both expressions of \(u_k\) in (48) and (50) when \(k^2\tau ^2\ll 1\).

*method of variation of constants*for second-order differential equations (see for example Chapts. 13–17 of [51]). When one has \(u_k^{(1)}\) one calculates \(\Phi _k^{(1)}\) and, inserting the result in (51), one gets \(\delta \sigma _k^{(1)}\). And, thus, the new iterations are iteratively obtained and a fully fledged method is constructed.

*v*and Eq. \(i-0\) to find \(\delta \sigma _k^{(1)}\), and so continue successively to obtain the next iteration.

### 4.2 Tensor perturbations

*h*and introducing the variable \(v_T=a h\), in Fourier space this equation becomes

## 5 Comparison with other models

*deformed algebra approach*, the teleparallel LQC approach, the intrinsic curvature LQC approach and the mimetic LQC approach. First, we shall review the dynamical equations for each of these theories:

- 1.
LQC in the

*deformed algebra approach*[31, 32].In this case the equation for the potential \(\Phi \) decouples. It is given byThe variable M–S variable$$\begin{aligned}&{\ddot{\Phi }}-\frac{\Omega }{a^2}\Delta \Phi +\left( H-2\frac{\ddot{{\bar{\phi }}}}{\dot{{\bar{\phi }}}}-\frac{\dot{\Omega }}{\Omega }\right) {\dot{\Phi }}\nonumber \\&\quad +\left( 2\left( \dot{H}-H\frac{\ddot{{\bar{\phi }}}}{\dot{{\bar{\phi }}}}\right) -H\frac{\dot{\Omega }}{\Omega } \right) {\Phi }=0. \end{aligned}$$(65)*v*adquires the usual form \(v=a(\delta \phi +\frac{{\bar{\phi }'}}{{{\mathcal {H}}}}\Phi ) \) and the M–S equations decouple in the simple formIn particular the equation \(v''-\Omega \Delta v-\frac{z''}{z}v=0\) can be solved when the pivot scale is well inside and outside of the Hubble radius, obtaining the whole evolution of the variable$$\begin{aligned} u''-\Omega \Delta u-\frac{\theta ''}{\theta }u=0, \quad v''-\Omega \Delta v-\frac{z''}{z}v=0. \end{aligned}$$(66)*v*, which encodes all the information about scalar perturbations and, thus, the knowledge of the power spectrum for the*curvature fluctuation in co-moving coordinates*.On the other hand, for tensor perturbation the corresponding M–S equation is [33]where \(z_T\equiv \frac{a}{\sqrt{\Omega }}\).$$\begin{aligned} h''-\Omega \Delta h-\frac{z_T''}{z_T}h=0, \end{aligned}$$(67) - 2.Teleparallel LQC [36, 37]. In teleparallel LQC the equation for the Newtonian potential also decoupleswhere the square of the velocity of sound is \(c_s^2={\Omega }\sqrt{\frac{\rho _c}{12 H^2}}\arcsin \left( \sqrt{\frac{12H^2}{\rho _c}} \right) \), which, contrary to what occurs in LQC in the$$\begin{aligned}&{\ddot{\Phi }}-\frac{c_s^2}{a^2}\Delta \Phi +\left( H-2\frac{\ddot{{\bar{\phi }}}}{\dot{{\bar{\phi }}}}-\frac{\dot{\Omega }}{\Omega }\right) {\dot{\Phi }}\nonumber \\&\quad +\left( 2\left( \dot{H}-H\frac{\ddot{{\bar{\phi }}}}{\dot{{\bar{\phi }}}}\right) -H\frac{\dot{\Omega }}{\Omega } \right) {\Phi }=0, \end{aligned}$$(68)
*deformed algebra approach*, is always positive, meaning that in this approach there are no gradient instabilities. In the same way, the M–S also decouple and the only difference with the ones of LQC in the*deformed algebra approach*is that the velocity of sound is the same that appears in (68). As in LQC in the*deformed algebra approach*, the equation for the variable*v*decouples, which allows us to know the power spectrum of the*curvature fluctuations in co-moving coordinates*.For tensor perturbations the velocity of sound is equal to 1, and the M–S equation is given bywhere \(z_T\equiv a \sqrt{ \sqrt{\frac{\rho _c}{12 H^2}}\arcsin \left( \sqrt{\frac{12H^2}{\rho _c}} \right) } \).$$\begin{aligned} h''-\Delta h-\frac{z_T''}{z_T}h=0, \end{aligned}$$(69) - 3.
Intrinsic curvature LQC [14].

In this approach the equations for scalar perturbations are the same as in LQC in the*deformed algebra approach*. However, for tensor perturbations, the corresponding M–S equation iswhere \(z_T\) is the same as in teleparallel LQC, and in this case the square of the velocity of sound is given by \(c_T^2= \frac{\sqrt{\frac{12H^2}{\rho _c}}}{\arcsin \left( \sqrt{\frac{12H^2}{\rho _c}} \right) }\).$$\begin{aligned} h''-c_T^2\Delta h-\frac{z_T''}{z_T}h=0, \end{aligned}$$(70) - 4.
Mimetic LQC [45].

This is also a fully covariant approach where, at the perturbative level, the dynamical variables are the perturbed mimetic field, namely \(\delta \varphi \), and the perturbed scalar field \(\delta \phi \). In the Newtonian gauge, the potential is related with the mimetic field, as follows \(\Phi =\delta \dot{\varphi }\), and the dynamical equation, as in our approach, becomes coupled [45]where the square of the velocity of sound is given, as in [46], by$$\begin{aligned} \left\{ \begin{array}{l} \delta \ddot{\varphi }+H\delta \dot{\varphi }-\frac{c_s^2}{a^2}\Delta \delta \varphi =\frac{\Omega }{2}\dot{{\bar{\phi }}}\delta \phi \\ \delta \ddot{\phi }+3H\delta \dot{\phi }-\frac{1}{a^2}\Delta \delta \phi +(V_{\phi \phi }-2{\dot{{\bar{\phi }}}}^2{{}\Omega })\delta \phi \\ \quad = \frac{4{{}\dot{{\bar{\phi }}}}c_s^2}{a^2}\Delta \delta \varphi -2(2{{}H\dot{{\bar{\phi }}}}+V_{\phi })\delta \dot{\varphi }, \end{array}\right. \end{aligned}$$(71)which exhibits the well-known gradient instability of the mimetic gravity case [46, 47] (see also [48] for the study of perturbations in specific mimetic matter models).$$\begin{aligned} c_s^2=\frac{\Omega }{2}\bar{f}_{\chi \chi }=\frac{\frac{1}{2}\bar{f}_{\chi \chi }}{1-\frac{3}{2}\bar{f}_{\chi \chi }}, \end{aligned}$$(72)Dealing with tensor perturbations, since the mimetic field does not alter the gravitational sector, as in our approach, the equations for tensor perturbations are the same as in GR.

*deformed algebra approach*, teleparallel LQC, or intrinsic curvature LQC, the equations for scalar perturbations decouple, which actually simplifies the theory a lot, allowing us to calculate the corresponding power spectrum. On the contrary, for the fully covariant theories, i.e. our approach and mimetic LQC, the equations of scalar perturbations do not decople, which makes their analytic study more difficult and only a numerical analysis seems to be viable in order to understand their evolution. However, the clear advantage of the covariant theories is that the equation for tensor perturbations is the simplest one because it coincides with the one for tensor perturbations in GR.

## 6 Reheating and the calculation of the spectral parameters in the bouncing matter-ekpyrotic scenario

In this section we consider the background given by holonomy corrected LQC. In other words, we consider the function *f* given by (16), which means that the universe bounces when its energy density is \(\rho _c\). We will show how to calculate the reheating temperature of the universe via gravitational particle production due to a phase transition from the matter domination to an ekpyrotic era, and how the theoretical values of the spectral index and its running match well with the corresponding observational data.

*w*is the effective Equation of State (EoS) parameter and the “star” means that the quantities are evaluated when the pivot scale leaves the Hubble radius.

*k*is a positive parameter and \(\lambda \) and \(\bar{\lambda }\) satisfy

### 6.1 Reheating

Here we will consider a reheating process due to the gravitational particle production of massless particles minimally coupled with gravity during a phase transition from a matter dominated regime to an ekpyrotic one in the contracting phase. Recall that in our model (77) we have an ekpyrotic phase with EoS parameter \(w=2\). Let \(H_E\) be the value of the Hubble parameter at the beginning of this phase. Then, from the relations \(V=3H^2+\dot{H}\) and \(\dot{H}=-\frac{9}{2}H^2\), we obtain \(\lambda = \frac{3}{2}H_E^2\). On the other hand, in holonomy corrected LQC a viable value of \(H_E\) is approximately \(-10^{-3}\) [34, 35], which justifies our choice \(|\bar{\lambda }|\ll \rho _c\).

### Remark 6.1

Equation (80) was obtained considering a phase transition, in the expanding phase, from the de Sitter phase to another one with constant EoS parameter \(w>1/3\) [57, 58]. In the case we consider a phase transition in the contracting phase from the matter domination phase to another one with constant EoS parameter \(w>1\), this formula is also valid due to the duality, pointed out in [59], between the de Sitter regime in the expanding phase and the matter domination in the contracting one.

### Remark 6.2

### Remark 6.3

It is possible to obtain a lower reheating temperature by increasing the EoS parameter in the ekpyrotic phase. For example, if in the ekpyrotic phase one takes \(w=5\), then the reheating temperature is reduced by one order.

Once we have calculated the reheating temperature we can show that the pivot scale, in the contracting phase, leaves the Hubble scale when GR holds. The pivot scale is related with its physical value by \(k_*=a_0 k_{phys}(t_0)\), where the sub-index 0 means present time, and we choose, as usual, \(k_{phys}(t_0)\sim 10^2 H_0\sim 10^{-59}\).

On the other hand, as we have showed at the reheating time, i.e., when both energy densities are of the same order, one has \(a_{reh}\sim \sqrt{\frac{\rho _c\left( \frac{3H_E^2}{\rho _c} \right) ^{\frac{2}{3}}}{R H_E^4}}a_E\sim 5\times 10^4 a_E\). Now, from the conservation of the entropy we have the adiabatic relation \(a_0\sim \frac{T_{reh}}{T_0}a_{reh}\) [60] and using that the current and reheating temperature are respectively \(T_0\sim 8\times 10^{-32}\) and \(T_{reh}\sim 8\times 10^{-10}\), one gets \(a_0\sim 5\times 10^{26} a_E\). As a consequence, \(k_*\sim 5\times 10^{-33}a_E\), which means, since \(|H_E|\sim 10^{-3}\), that \(k_*\ll |H_E|a_E\). In other words, the pivot scale leaves in the contracting phase the Hubble radius well after the phase transition, more precisely when \(H_*\sim 5\times 10^{-33}\frac{a_E}{a_*}\le 5\times 10^{-33}\) (in the contracting phase \(a_E<a_*\)) and, thus, since \(\rho _c\cong 252\), one can safely disregard the effects of the *f* theory when the pivot scale leaves the Hubble radius.

## 7 Conclusions

We have constructed a class of modified gravitational theories based on the addition to the Einstein–Hilbert action of a function *f*, which depends on the divergence of the unitary time-like eigenvector of the stress tensor. We have obtained in this way a fully covariant theory, which, as in the case of mimetic gravity, has one more degree of freedom than GR. The main advantage, at the background level, of our class of models is that, for the FLRW geometry, this divergence is minus three times the Hubble parameter, which allows, by choosing the function *f* appropriately, to obtain very simple bouncing backgrounds, as the one resulting in holonomy corrected LQC.

At the level of cosmological perturbations, working in the Newtonian gauge, the equations for scalar perturbations exhibit some of the same interesting features as those appearing in LQC in the so-called *deformed algebra approach*. However, they are not exactly the same, owing to the fact that our theory is fully covariant, in contrast with LQC in the *deformed algebra approach* [61, 62, 63]. In fact, contrary to what happens with LQC and other non-covariant approaches such as teleparallelism or modified theories using the extrinsic curvature, in our fully covariant approach the equations do not decouple, which is an added difficulty and translates into the fact that, in practice, the equations cannot be solved analytically but only by standard numerical methods.

A very positive feature is, however, that for tensor perturbations our model leads to the same equations as GR because the modification of the action does not affect the gravity sector.

Finally, we have studied the matter-ekpyrotic bouncing scenario for the LQC background, when reheating is a consequence of the production of massless particles minimally coupled with gravity, during the phase transition from matter domination to the ekpyrotic regime in the contracting phase. We have obtained a viable reheating temperature of the order of \(10^9\) GeV and have shown that the observable modes leave the Hubble radius in the contracting phase, when the holonomy correction can be disregarded. This permits a very simple calculation of the theoretical values of the spectral index and of its running, both of which, as it turns out, perfectly match the current observational data at the \(2\sigma \) C.L.

## Notes

### Acknowledgements

This investigation has been supported in part by MINECO (Spain), Projects MTM2017-84214-C2-1-P and FIS2016-76363-P, by the CPAN Consolider Ingenio 2010 Project, and by the Catalan Government 2017-SGR-247.

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