Mutated hilltop inflation revisited
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 branch of the Lambert function. Depending on the model parameter mutated hilltop model renders two types of inflationary solution: 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 , while the large field sector is capable of generating high amplitude for tensor perturbations, . Further we see that the spectral index is almost independent of the model parameter along with a very small negative amount of scalar running.
a]Barun Kumar Pal
\affiliation[a]Department of Mathematics
Netaji Nagar College for Women, Kolkata 700 092, INDIA \emailAddterminatorbarun@gmail.com
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]. 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 [4, 5] 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 [6, 7, 8, 9], the window has become thinner, but still allowing numerous models to pass through [10, 11]. The recent detection of astronomical gravity waves by LIGO [12, 13] 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  by detecting primordial gravity waves.
The most efficient method for studying inflation is the slow-roll approximation , 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 . In order to study inflationary paradigm irrespective of slow-roll approximation Hamilton-Jacobi formalism [17, 18] 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.
Here we would like to study single field mutated hilltop model (MHI henceforth) of inflation [19, 20] 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 [21, 22] 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 [19, 20] we have reported that MHI can only produce a negligible amount of tensor to scalar ratio, . But, we shall see here that it is capable of generating as large as depending on the model parameter. Consequently a wide range of , , can be addressed by MHI. Recent data from Planck [8, 9] has reported an upper bound and upcoming CMB-S4 experiments are expected to survey tensor to scalar ratio up to . So sooner or later the model can be tested with the observations. The prediction for inflationary observables from MHI are in very good agreement with recent observational bound. Further, MHI predicts spectral index which is almost independent of model parameter along with small negative scalar running consistent with current data.
2 Quick Look at Hamilton Jacobi Formalism
where prime and dot denote derivatives with respect to the scalar field and time respectively, and is the reduced Planck mass. The associated inflationary potential can then be found by rearranging the terms of Eqn.(1)
where has been defined as
We further have
Therefore accelerated expansion takes place when and ends exactly at . The evolution of the scale factor turns out to be
The amount of inflation is expressed in terms of number of e-foldings and defined as
We have defined in such a way that at the end of inflation and increases as we go back in time. The observable parameters are generally evaluated when there are e-foldings still left before the end of inflation. It is customary to define another parameter by
It is worthwhile to mention here that the parameters and are not the usual slow-roll parameters. But in the slow-roll limit and , and being usual potential slow-roll parameters.
3 Mutated Hilltop Inflation: The Model
where is the typical energy scale of inflation and is a parameter having dimension of inverse Planck mass. The potential under consideration does not actually represent typical hilltop inflation [21, 22], but the form of the potential is somewhat similar to mutated hilltop inflation in hybrid scenario and hence the name. Accelerated expansion takes place as the inflaton rolls towards the potential minimum. Not only that, from Eq.(9) it is obvious that which is significantly different from the usual hilltop potential.
The associated Hubble parameter may be written as
The value of the constants can be fixed from the conditions for successful inflation and the observational bounds.
3.1 End of Inflation
Two parameters and in the Hamilton Jacobi formalism now take the form
The inflation ends naturally when at , which is the root of the following equation
Eq.(12) can be solved analytically and the relevant solution turns out to be
where . On the other hand absolute value of becomes order of unity at
In Fig.1 we have shown the variation of the with and the solution of . From the figure it is obvious that the occurs well before the actual end of inflation for , where represents the value of for the simultaneous occurrence of , which can be found analytically from Eq.(13) and Eq.(14), and corresponding value of the inflaton is given by . Consequently, slow-roll approximation becomes poor towards the end of inflation for . But this is not so problematic as we are interested at the value of inflaton when cosmologically relevant scales leave the horizon and there slow-roll is a very good approximation. Also this happens only for negligible period of time as shown in Fig.2. On the other hand for the situation is somewhat different, precedes . Here we would like to mention that for , always remains below the unity. Therefore slow-roll approximation is valid till the end of inflation. It is possible to derive a bound on the model parameter , by demanding that inflation ends through the violation of the slow-roll approximation and for that we need to be a real number. Imposing that restriction we find . For the rest of the article we shall adhere to this range of only.
3.2 Number of e-foldings
The number of e-foldings for MHI is found to have the following form
The above Eq.(15) can be analytically inverted to get the scalar field as a function of e-foldings as follows
where we have defined and is the Lambert function. From the above Eq.(16), one can see that mutated hilltop inflation occurs along the branch of the Lambert function, first pointed out in Ref.. The value of the inflaton when cosmological scale leaves the horizon, , is then given by
The slow-roll parameters now can be expressed as a function of the e-foldings
This makes life simpler as now all the inflationary observable parameters when derived in the slow-roll limit can be expressed as a function of .
3.3 The Lyth Bound for MHI
The fluctuations in the tensor modes solely depends on the Hubble parameter whereas curvature perturbation is a function of the Hubble parameter and inflaton. Consequently, tensor to scalar ratio determines excursion of the inflaton during observable inflation, first shown in Ref. and known as Lyth bound
where is the actual Planck mass. corresponds to large field model and small field models. One expects to get larger tensor to scalar ratio, , where due to the higher energy scale required for successfully explaining the observable parameters.
For the model under consideration we have found
In Fig.3 we have shown the variation of the scalar field excursion in the unit of , with the model parameter . From the figure it is obvious that the mutated hilltop model of inflation has small excursion of the inflaton for and large field excursion for , where is the solution of Eq.(21) for with . So this model is capable of addressing both the large and small field inflationary scenarios for suitable values of the model parameter.
In Fig.4 we have shown the variation of with . From the figure we see that small field MHI may give rise to negligible amount of tensor to scalar ratio, , on the other hand for large , can be as large as .
3.4 Inflationary Observables in the Slow-Roll Limit
The inflationary observable parameters can be found analytically in the slow-roll approximation. The power spectrum of the curvature perturbation turns out to be
In Fig.5 we have shown the variation of the typical energy scale associated with MHI for different values of the model parameter. From the figure it is clear that maximum energy scale that can be achieved is GeV. To determine this energy scale we have used from Planck 2015 result . The scale dependence of the spectrum of curvature perturbation is described by spectral index. In MHI we have found
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 from Planck 2015 has also been shown.
The scale dependence of the spectral index itself is estimated from the scalar running and we have
Here in Fig.7 logarithmic variation of the absolute value of scalar running with 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 .
Finally, the tensor to scalar ratio is found to have the following form
In Fig.8 we have plotted the tensor to scalar ratio (in scale) with for three different values of . We see that MHI can address wide range of values of tensor to scalar ratio, depending on the model parameter . But has to be discarded which is observationally forbidden which determines a lower bound on the model parameters, .
In Fig.9 we exhibit variation of the MHI energy scale with the tensor to scalar ratio. So in order to achieve we need an energy scale GeV.
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, . 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, . 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.
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.
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