# Mutated hilltop inflation revisited

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## Abstract

In this work we re-investigate pros and cons of mutated hilltop inflation. Applying Hamilton–Jacobi formalism we solve inflationary dynamics and find that inflation goes on along the \({\mathscr {W}}_{-1}\) branch of the Lambert function. Depending on the model parameter mutated hilltop model renders two types of inflationary solutions: one corresponds to small inflaton excursion during observable inflation and the other describes large field inflation. The inflationary observables from curvature perturbation are in tune with the current data for a wide range of the model parameter. The small field branch predicts negligible amount of tensor to scalar ratio \(r\sim \mathscr {O}(10^{-4})\), while the large field sector is capable of generating high amplitude for tensor perturbations, \(r\sim \mathscr {O}(10^{-1})\). Also, the spectral index is almost independent of the model parameter along with a very small negative amount of scalar running. Finally we find that the mutated hilltop inflation closely resembles the \(\alpha \)-attractor class of inflationary models in the limit of \(\alpha \phi \gg 1\).

## 1 Introduction

The standard model of hot Big-Bang scenario is instrumental in explaining the nucleosynthesis, expanding universe along with the formation of cosmic microwave background (CMB henceforth). But there are few limitations in the likes of *flatness problem, homogeneity problem* etc., which can not be answered within the limit of Big-Bang cosmology. In order to overcome these shortcomings an early phase of accelerated expansion – *cosmic inflation* was proposed [1, 2, 3, 4, 5]. Big-Bang theory is incomplete without inflation and turns into brawny when combined with the paradigm of inflation. Though inflation was initiated to solve the cosmological puzzles, but the most impressive impact of inflation happens to be its ability to provide persuasive mechanism for the origin of cosmological fluctuations observed in the large scale structure and CMB. Nowadays inflation is the best bet for the origin of primordial perturbations.

Since its inception, almost four decades ago, inflation has remained the most powerful tool to explain the early universe when combined with big-bang scenario. It is still a paradigm due to the elusive nature of the scalar field(s), *inflaton*, responsible for inflation and the unknown shape of the potential involved. That the potential should be sufficiently flat to render almost scale invariant curvature perturbation along with tensor perturbation [6, 7, 8, 9, 10, 11, 12] has been only understood so far. As a result there are many inflationary models in the literature. With the advent of highly precise observational data from various probes [13, 14, 15, 16], the window has become thinner, but still allowing numerous models to pass through [17, 18]. The recent detection of astronomical gravity waves by LIGO [19, 20] has made the grudging cosmologists waiting for primordial gravity waves which are believed to be produced during inflation through tensor perturbation. The upcoming stage-IV CMB experiments are expected to constrain the inflationary models further [21] by detecting primordial gravity waves.

The most efficient method for studying inflation is the slow-roll approximation [22, 23], where the kinetic energy is assumed to be very small compared to the potential energy. But this is not the only way for successful implementation of inflation and solutions outside slow-roll approximation have been found [24]. In order to study inflationary paradigm irrespective of slow-roll approximation Hamilton–Jacobi formalism [25, 26] has turned out to be very handy. Here the inflaton itself is treated as the evolution parameter instead of time, and the Friedmann equation becomes first order which is easy to extract underlying physics from. Another interesting class of inflationary models has been introduced very recently, *constant-roll inflation* [27, 28, 29], where the inflaton rolls at a constant rate.

Here we would like to study single field mutated hilltop model (MHI henceforth) of inflation [30, 31] using Hamilton–Jacobi formalism. In MHI observable inflation occurs as the scalar field rolls down towards the potential minimum. So MHI does not correspond to usual hilltop inflation [32, 33] directly, but the shape of the inflaton potential is somewhat similar to the mutated hilltop in hybrid inflation and hence the name. We shall see that for a wide range of values of the model parameter MHI provides inflationary solution consistent with recent observations. Our analysis also reveals that MHI has two different branches of inflationary solutions: one corresponds to small field inflation and the other represents large field inflation. In earlier studies [30, 31] we have reported that MHI can only produce a negligible amount of tensor to scalar ratio, \(r\sim 10^{-4}\). But, we shall see here that it is capable of generating *r* as large as \({\mathscr {O}}(10^{-1})\) depending on the model parameter. Consequently a wide range of *r*, \(10^{-4}\lesssim r\lesssim 10^{-1}\), can be addressed by MHI. Recent data from Planck [15, 16, 34] has reported an upper bound \(r_{0.002}<0.07\) and upcoming CMB-S4 experiments are expected to survey tensor to scalar ratio up to \(r\sim 2\times 10^{-3}\) [21]. So sooner or later the model can be tested with the observations. The prediction for inflationary observables from MHI are in tune with recent observations. Further, MHI predicts spectral index which is almost independent of model parameter along with small negative scalar running consistent with current data. It has been also found that MHI closely resembles the \(\alpha \)-attractor class of inflationary models [35, 36] in the limit of \(\alpha \phi \gg 1\).

In Sect. 2 we have briefly reviewed Hamilton–Jacobi formalism. In the next Sect. 3 we have discussed about the MHI in Hamilton–Jacobi formalism and have shown resemblance with \(\alpha -\)Attractor class of inflationary models in Sect. 4. Finally we conclude in Sect. 5.

## 2 Quick look at Hamilton Jacobi formalism

*N*in such a way that at the end of inflation \(N=0\) and

*N*increases as we go back in time. The observable parameters are generally evaluated when there are 55–65 e-foldings still left before the end of inflation. It is customary to define another parameter by

*potential slow-roll*parameters.

## 3 Mutated Hilltop inflation: the model

In Fig.2 we have plotted the solutions of \(\epsilon _\mathrm{H}=1\), \(|\mathrm \eta _{_H}|=1\) and \(|\zeta _{_H}|=1\) using the Hubble parameter as given by Eq. (11). From the figure we see that \(|\eta _{_H}|\) becomes order of unity before \(\epsilon _\mathrm{H}\), but \(|\zeta _{_H}|\) remains small.

From now on all the results that we shall present in this article are based on the approximated form of the Hubble parameter and without considering the effect of higher order slow-roll parameters.

### 3.1 Number of e-foldings

*N*.

### 3.2 The Lyth bound for MHI

*r*, where \(\Delta \phi \ge \ \mathrm{m_P}\) due to the higher energy scale required for successfully explaining the observable parameters. For the model under consideration we have found

In Fig. 4 we have shown the variation of \(\mathrm Log_{10} r\) with \(\Delta \phi \). From the figure we see that small field MHI may give rise to negligible amount of tensor to scalar ratio, \(r\sim \mathscr {O}(10^{-4})\), on the other hand for large \(\Delta \phi \), *r* can be as large as \(\mathscr {O}(10^{-1})\).

### 3.3 Inflationary observables in the slow-roll limit

In Fig.6 we have shown how the scalar spectral index changes with the model parameter. We also see that spectral index is almost constant in both the large and small field sector of MHI. The current bound on \(n_{_S}\) from Planck 2015 has also been plotted.

Here in Fig. 7 logarithmic variation of the absolute value of scalar running with \(\alpha \) has been plotted. From the figure it is clear that MHI predicts very small running of the spectral index. The maximum amount of scalar running that can achieved in MHI is \(|n_\mathrm{S}'|\sim 10^{-3}\).

## 4 Similarity with \(\alpha \)-attractor class of inflationary models

*r*, may be expressed as

*e*-foldings,

*N*, is approximately given by Eq. (16), which in the limit \(\alpha \phi \gg 1\) may be written as

In Fig. 10 we have shown the variation of \(N(1-n_{_S})/2\) with the model parameter \(\alpha \) and the number of *e*-foldings, N. From the figure it is clear that for large values of the model parameter MHI belongs to the \(\alpha \)-attractor class of models and deviation occurs for the small values of the model parameter. In Fig. 11 we have shown the variation of \(N_\mathrm{CMB}(1-n_{_S})/2\) with the model parameter \(\alpha \) for two different values of \(N_\mathrm{CMB}\).

In Fig. 12 we have shown the variation of \(\alpha ^2M_P^2\ r \ N^2/8\) with the model parameter \(\alpha \) and the number of *e*-foldings, N. From the figure it is clear that for large values of the model parameter MHI indeed belongs to the \(\alpha \)-attractor class of models and small deviation occurs for the small values of the model parameter. In Fig. 13 we have plotted \(\alpha ^2M_P^2\ r \ N^2_\mathrm{CMB}/8\) for two different values of \(N_\mathrm{CMB}\).

So, from the above results it is quite transparent that MHI indeed falls into the category of \(\alpha \)-attractor class of inflationary models in the limit of \(\alpha \phi \gg 1\).

## 5 Conclusion

In this article we have revisited mutated hilltop inflation driven by a hyperbolic potential. Employing Hamilton–Jacobi formulation we found that inflation ends naturally through the violation slow-roll approximation. More interestingly, MHI has two different branches of inflationary solution. One corresponds to large field variation and the other represents small change in inflaton during the observable inflation depending on the model parameter.

Observable parameters as derived from this model are in tune with the latest observations for a wide range of the model parameter, \(\alpha \mathrm{M_P}\gtrsim 0.094561\). The scalar spectral index is found to be independent of the model parameter with a small negative running. We have also found that MHI can address a broad range of the tensor to scalar ratio, \(0.0001\lesssim r <0.07\). In a nutshell, MHI though does not belong to the usual hilltop inflation is extremely attractive with only one model parameter consistent with recent observations. Not only that, in the limit of \(\alpha \phi \gg 1\), MHI closely resembles the \(\alpha \)-attractor class of inflationary models.

## Notes

### Acknowledgements

I would like to thank Supratik Pal for useful discussions and constructive suggestions. I would also like to thank IUCAA, Pune for giving me the opportunity to carry on research work through their Associateship Program. Finally author would like to sincerely thank anonymous reviewers for their critical and constructive suggestions on the first version of this work.

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