# Deformed Starobinsky model in gravity’s rainbow

## Abstract

In the context of gravity’s rainbow, we study the deformed Starobinsky model in which the deformations take the form \(f(R)\sim R^{2(1-\alpha )}\), with *R* and \(\alpha \) being the Ricci scalar and a positive parameter, respectively. We show that the spectral index of curvature perturbation and the tensor-to-scalar ratio can be written in terms of \(N,\,\lambda \) and \(\alpha \), with *N* being the number of *e*-foldings, \(\lambda \) a rainbow parameter. We compare the predictions of our models with Planck data. With the sizeable number of *e*-foldings and proper choices of parameters, we discover that the predictions of the model are in excellent agreement with the Planck analysis. Interestingly, we obtain the upper limit and the lower limit of a rainbow parameter \(\lambda \) and a positive constant \(\alpha \), respectively.

## 1 Introduction

The prediction of a minimal measurable length of the order of Planck length in various theories of quantum gravity restricts the maximum energy that any particle can attain to the Planck energy. This could imply a modification of linear momentum and also quantum commutation relations and results to modified dispersion relation. Moreover, as an effective theory of gravity, Einstein’s general theory of gravity is valid in the low energy (IR) regime, while at very high energy regime (UV) the Einstein theory could in principle be improved.

One of the interesting approaches that naturally deals with modified dispersion relations is called doubly special relativity [1, 2, 3]. Then Magueijo and Smolin [4] generalized this idea by including curvature. The modification of the dispersion relation occurs when replacing the standard one, i.e. \(\epsilon ^{2}-p^{2}=m^{2}\), with the form \(\epsilon ^{2}{{\tilde{f}}}^{2}(\epsilon ) - p^{2}{{\tilde{g}}}^{2}(\epsilon )=m^{2}\) where functions \({{\tilde{f}}}(\epsilon )\) and \({{\tilde{g}}}(\epsilon )\) are commonly known as the rainbow functions. It is worth noting that the rainbow functions are chosen in such a way that they are satisfied, at a low-energy IR limit, i.e. \(\epsilon /M \rightarrow 0\), the standard energy-momentum relation and they are required to satisfy \({{\tilde{f}}}(\epsilon ) \rightarrow 1\) and \({{\tilde{g}}}(\epsilon )\rightarrow 1\) where *M* is the energy scale that quantum effects of gravity become important.

*a*(

*t*) is a scale factor. For convenience, we choose \({{\tilde{g}}}(\epsilon )=1\) and only focus on the spatially flat case. As suggested in Ref. [19], this formalism can be generalized to study semi-classical effects of relativistic particles on the background metric during a longtime process. For the very early universe, we consider the evolution of the probe’s energy with cosmic time, denoted as \(\epsilon (t)\). Hence the rainbow functions \({{\tilde{f}}}(\epsilon )\) depends on time implicitly through the energy of particles.

In recent years, gravity’s rainbow has attracted a lot of attentions and became the subject of much interest in the literature. In the context of such gravity, the various physical properties of the black holes are investigated, see e.g. [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. In addition, the effects of the rainbow functions have also been discussed in several other scenarios, see for instance [39, 40, 41, 42, 43]. Moreover, the gravity’s rainbow was investigated in Gauss-Bonnet gravity [44], massive gravity [45, 46] and *f*(*R*) gravity [47]. More specifically, the gravity’s rainbow has also been used for analyzing the effects of rainbow functions on the Starobinsky model of *f*(*R*) gravity [11] and other cosmological scenarios [48, 49, 50, 51].

One of the intriguing features of the Starobinsky model [12, 13] is that gravity itself is directly responsible for the inflationary period of the universe without resorting to the introduction of new *ad hoc* scalar fields. The authors of Ref. [14] studied quantum-induced marginal deformations of the Starobinsky gravitational action of the form \(R^{2(1-\alpha )}\), with *R* and \(\alpha \) being the Ricci scalar and a positive parameter, respectively. The latter is assumed to be smaller than one half. The model predicted sizable primordial tensor modes. In the present work, we consider the model proposed by Ref. [14] in the context of gravity’s rainbow.

This paper is organized as follows: in Sect. 2, we revisit the formalism in *f*(*R*) theory [7, 8] in the framework of gravity’s rainbow [11]^{1}. We then focus on the deformed Starobinsky’s model [14] in which *f*(*R*) takes the form \(f(R)\sim R^{2(1-\alpha )}\). Notice that the deformations of *f*(*R*) inflation has been also considered in Refs. [15, 16, 17, 18]. We take a short recap of a cosmological linear perturbation in the context of the gravity’s rainbow generated during inflation and calculate the spectral index of scalar perturbation and the tensor-to-scalar ratio of the model in Sect. 3. In Sect. 4, we compare the predicted results with Planck data. We conclude our findings in the last section.

## 2 *f*(*R*) theories with gravity’s rainbow effect

*f*(

*R*) theories where the Einstein–Hilbert term in the action is replaced by a generic function of the Ricci scalar. We start with the traditionally 4-dimensional action in

*f*(

*R*) gravity [7, 8].

*g*is the determinant of the metric \(g_{\mu \nu }\), and the matter field Lagrangian \({{\mathcal {L}}}_{M}\) depends on \(g_{\mu \nu }\) and matter fields \(\Psi _{M}\). We can derive the field equation by varying the action (1) with respect to \(g_{\mu \nu }\) to obtain [7, 8]

*i*,

*j*)-component of Eq. (4) reads

*f*(

*R*) takes the following form [14]:

*h*is a dimensionless parameter. It is worth noting that the Starobinsky model is recovered when \(\alpha =0\). Note that in the standard Starobinsky scenario the \(R^{2}\) plays a key role in the very early universe instead of relativistic matter. In the scalar field framework, the Starobinsky theory can be equivalent to the system of one scalar field (an inflaton). It is reasonable if we assume here that the inflaton dominates the very early universe and hence in what follows we can neglect the contributions from matter and radiation, i.e. \(\rho _{M}=0\) and \(P_{M}=0\).

*N*is thus given by to the first order of \(\alpha \):

## 3 Cosmological perturbation in gravity’s rainbow revisited

*k*is a comoving wave number and \(Q_s\) is defined by

*r*can be obtained as

*e*-fold from \(t=t_k\) to the end of the inflation can be estimated as \(N_k \simeq 1/2\epsilon _1(t_k)\). We also find from Eq. (54) to the leading order of \(\alpha \) that

*r*can be rewritten in terms of the number of e-foldings as

## 4 Confrontation with observational data

*N*are arbitrary but keep \(\lambda \) and \(\alpha \) fixed. From Fig. 2, we observe that in order for the predictions to be satisfied the Planck data at one sigma level values of \(\lambda \) can not be greater than 4.0 and 3.6 with \(\alpha =0.01\) and 0.0001, respectively. The 2018 recent release of the Planck cosmic microwave background (CMB) anisotropy measurements [6] determines the spectral index of scalar perturbations to be \(n_{s}=0.9649\pm 0.0042\) at 68% CL and the 95% CL upper limit on the tensor-to-scalar ratio is further tightened by combining with the BICEP2/Keck Array BK14 data to obtain \(r_{0.002} < 0.064\). We use these updated parameters to constrain our model parameters. Let us consider Eq. (58) and then we obtain the upper limit of a parameter \(\lambda =\lambda _{*}\) as

*N*but keep \(\alpha \) fixed with Planck’15 results displayed in Fig. 3. We find that using \(\alpha =0.006\) and \(\lambda =3.38\) the predictions in the \((r-n_{s})\) plane lie in the one sigma confidence level only when \(N=[60,70]\). Using parameters of the base \(\Lambda \)CDM cosmology reported by Planck 2018 [5] for \(P_{{\mathcal {R}}}\) at the scale \(k=0.05\) \(\hbox {Mpc}^{-1}\), we find from Eq. (56) that the mass \({{\tilde{M}}}\) is constrained to be

*h*to obtain

*h*is in general scale-dependent. The explicit computations via heat kernel methods [20] shows that a logarithmic form of

*h*can be induced by leading order quantum fluctuations. The RG improved treatment of

*h*can be found in Ref. [14].

## 5 Conclusion

In this work, we studied the deformed Starobinsky model in which the deformations take the form \(R^{2(1-\alpha )}\), with *R* and \(\alpha \) being the Ricci scalar and a positive parameter, respectively [14]. We started by revisiting the formalism in *f*(*R*) theory [7, 8] in the framework of gravity’s rainbow [11]. We took a short recap of a cosmological linear perturbation in the context of the gravity’s rainbow generated during inflation and calculated the spectral index of scalar perturbation and the tensor-to-scalar ratio predicted by the model. We compared the predicted results with Planck data. With the sizeable number of e-foldings and proper choices of parameters, we discovered that the predictions of the model are in excellent agreement with the Planck analysis. Interestingly, we obtained the upper limit of a rainbow parameter \(\lambda < 1.33\times 10^{-2} \Big (5.48\,N-75\Big )\) and found the lower limit of a positive constant \(\alpha > 2.65\times 10^{-2}\Big (1.33\times 10^{-2}(5.48 N-75)+1.00\Big )^{-1}\).

Regarding our present work, the study the cosmological dynamics of isotropic and anisotropic universe in *f*(*R*) gravity, see e.g. [8, 55, 56] and references therein, via the dynamical system technique can be further studied. Interestingly, the swampland criteria in the deformed Starobinsky model can be worth investigating by following the work done by Ref. [52]. The reheating process in the present work is an interesting topic to be investigated [53, 54]. Moreover, the effects of rainbow functions on the structure of compact objects are also worth investigating, see e.g. [57, 58].

## Footnotes

## Notes

### Acknowledgements

The author thanks Vicharit Yingcharoenrat for his early-state collaboration in the present work. Valuable comments and intuitive suggestions from the referee are also acknowledged.

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