# Original Higgs inflation and phantom crossing

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## Abstract

It is a standard practice to ignore the initial kinetic term in an inflationary analysis. Here, we overrule this practice to analyze pre-inflationary, inflationary and post-inflationary dynamics of the original Higgs-inflation model from a wider view. To this end, our calculations are not restricted to slow-roll regime. Perhaps, the most interesting result that we have found is the behavior of the field trajectories which begin their way from a phantom origin, before reaching a quasi de Sitter partial attractor. Both analytical and numerical calculations ultimately confirm that for the original Higgs inflation scenario the mentioned de Sitter path ends in an oscillatory phase which is the required condition for having (p)reheating. This achievement discriminates the current research from the previous studies about the late time phantom crossing scenarios which often end in a de Sitter future world. This confirms that a phantom energy epoch may deliver much more than has been thought earlier. In our calculations, we have employed the dynamical system method and explain the results both analytically and through the plotted numerical calculations. Beside the present results, the dynamical system approach can be used in other studies which deal with non-minimal gravity models. Reducing the number of dynamical variables to two enables us to enjoy the two dimensional phase diagram which is very elusive.

## 1 Introduction

The attempt to modify the general relativity theory has a long history and has begun almost simultaneously as the general relativity inception [1]. There are almost three categories to deal with the general relativity modifications, all are inspiring and have been remained in the mainstream of the physics research and have even become the fundamental idea for many other theoretical proposals. These categories are; assuming higher dimensions, generalizing the Hilbert–Einstein action to an arbitrary function of Ricci scalar (*f*(*R*) theories) and assuming nontrivial interaction between gravity and matter (non-minimal coupling).

Assuming higher dimensions which mostly includes new spatial dimension(s) is one of the most interesting ideas and although it has begun from the beautiful works of Kaluza (1921) [2] and Klein (1926) [3], one may find new version of it in pioneering and modern theories of the string theory, the M theory, the brane models and so on. On the other hand, adding spatial dimension(s) to our usual four dimensional space-time, immediately results in some new terms in the Einstein–Hilbert action which are known as Lovelock terms [4]. Of course, one is free to keep these new terms or accept the 4-dimensional Hilbert–Einstein Lagrangian form to outspread in higher dimensions. Disregarding the perspective one chooses to consider the higher dimensional theories, one can accommodate more ideas into the theory of gravity.

The other two modifications of the general theory of relativity (*f*(*R*) models and non-minimal coupling between mater and gravity) are somehow connected since one can make the additional scalar degree of freedom of the *f*(*R*) model to appear in the form of Brans–Dicke non-minimal coupling theory or even Einstein–Hilbert theory using conformal transformations [5].

Here, we do not aim to discuss the pros and cons of the above competitive theories. It is also out of the scope of the present work to reopen the old dispute about the physical equivalence of the conformally related Einstein and Jordan frames [6]. Instead, we focus on a typical form of Higgs-Inflation model and try to analyze it through the dynamical system approach.

It is of much interest to unify standard model Higgs field as the only available scalar degree of freedom in particle physics [7, 8, 9] with inflaton scalar field as the most attractive solution for many cosmological dilemmas [10, 11, 12]. One may even be tempted to unify the standard model symmetry breaking to the emergence of the inflation process. Unfortunately, the former idea has a serious barrier to overcome; although the symmetry breaking mechanism is suitable enough to be an inflationary candidate, both the vacuum energy level and the self coupling constant values have to be much far from the corresponding parameters in the standard model of fields and particles [13, 14, 15, 16, 17]. In this regard, the energy level in which the Higgs mechanism triggers the symmetry breaking lies considerably lower than any inflationary scenario. In spite of the above disappointing fact, there are some nontrivial models to deal with the problem [18]. The first and the most well-known idea among others has been unifying the Higgs and the inflaton field through a non-minimal interaction with curvature [17, 19, 20, 21]. We denote this non-minimal Higgs inflation scenario as the “original Higgs inflation” throughout the paper. Although the energy gap between Higgs symmetry breaking and inflation remains irreconcilably large in the original Higgs inflation model, but the theory becomes more economic since one is not required to infix a new scalar degree of freedom in the delicate structure of the standard model of fields and particles, and if this is not yet enough, the next advantage is preventing the electroweak vacuum to decay as a metastable local minimum; The quantum corrections which are the consequence of assuming non-minimal interaction between the Higgs field and the curvature compensates the running of the self coupling, preventing the model from developing another vacuum with lower energy than the electroweak vacuum. Existence of a new stable vacuum makes the higher electroweak vacuum metastable [22, 23, 24, 25], which means that the universe which is supported by such metastable vacuum finally decays by penetrating to the lower minimum through the quantum true vacuum bubble creation [26, 27]. One may propose increasing the lifetime of the metastable electroweak vacuum to exceed the lifetime of the universe as a quick remedy for the problem, but this proposition arises another question; why our universe has chosen the metastable minimum instead of the stable one during the inflation [28]? The Higgs inflation resolves the problem by adding quantum corrections to the running of the self coupling constant in the right way [23]. This hypothesis does not require any new constraint on the reheating and preheating processes [29, 30] since it is enough for the inflaton field to settle in a less than the GUT energy scale after the inflation [31, 32, 33]. This is the simplest resolution, although it seems very sensitive to yet unknown interactions. There also exist some ideas which require the standard model to be revised [34]. A very close idea is assuming inflaton as a new field which interacts with the standard Higgs field [35].

Here, we try to employ the powerful dynamical system tools to provide a wider view from inflationary and pre/post-inflationary dynamic. In some parts, we look back to the original equations to resolve technical stiffness and modify the proposed dynamical system calculations. This is important because our analysis is not restricted to slow-roll conditions and therefore gives a panorama of what has been previously considered in a much tighter frame. We also keep the kinetic term and by this promote the scalar field from an auxiliary one to a physical one and break the explicit duality relation between the Higgs-Inflation and the Starobinsky *f*(*R*) model [10, 36, 37]. It is interesting to follow the pre-inflationary behavior of the variables where it seems that de Sitter attractor pulls inward the phantom trajectories [38, 39, 40, 41]. Following observations that implied the idea of slight shift of the cosmological equation of state into the phantom region (\(w<-1\)), some pioneering researches showed the capability of scalar-tensor theories to yield late time attractive property of the de Sitter expansion for the phantom trajectories [42, 43, 44, 45, 46] which are partly inline with the current research. Phantom crossing at present epoch was investigated numerically by Motohashi et al. [45, 46] Although we will keep the mentioned surveys as the cornerstones for our work and even dedicate one section to compare their approaches with ours in more details, what makes this paper taste differently, besides trying to deliver exact results, is that we have shown that for a phantom originated path, the de Sitter trajectory may behave as a partial attractor which ends in an oscillatory phase. This bestows the entire procedure the possibility of being an inflationary scenario. All the above propositions will be supported by analytical and numerical calculations and the corresponding plots.

The outline of the paper is as follows; first, in the next section, we introduce a more general Lagrangian which is able to provide the Higgs-Inflation scenario. We start from writing the Friedmann equations and derive the corresponding equation of state. In Sect. 3, we introduce a suitable set of dynamical variables and find the dynamical equations. To see the consequences of the proposed dynamical structure, we have wait until Sect. 4, where by fixing the model to its final form we close the equations with respect to the variables. Section 5 contains a detailed discussion about the oscillatory phase. Having the appropriate dynamical variables and equations, in Sect. 6, we proceed by finding fixed points and eigenvalues and anticipating the flow behavior. In Sect. 7, we explain the mathematical results in a physical perspective where we focus on crossing the phantom borders toward de Sitter attractor. In Sect. 8, we transform back to the original variables (\(\phi \) and \({\dot{\phi }}\)), in order to compare our results with those of others. Section 9 is devoted to comparison between our results and some pre-existing works. Finally, in the last section, we summarize the methods and the achievements of the present work.

## 2 \(f(R,\phi )\) as a generalized gravity theory

*i*. The additive property of isotropic pressures and homogeneous energy density allows one to write

*C*, because, one expects by setting \({\mathcal {F}}=1\) to recover minimal gravity-scalar theory with \(\rho _{eff}={\mathcal {G}}+\frac{1}{2}\dot{\phi }^2\) and \(\rho _{eff}+p_{eff}=\dot{\phi }^2\), therefore,

*C*has to be normalized to unity. The final relations for effective density and pressure become

## 3 Dynamical system approach

*R*is equal to zero. To derive the right hand side of (19) and (21), we use (3) and (4) to obtain

## 4 Higgs-inflation: the unitary case

*H*stands for the standard model Higgs doublet. We choose to work in the unitary gauge \(H=h/\sqrt{2}\) [33, 37] and assume \(h\gg \nu \) for the infant universe. Then (26) simplifies as

*f*(

*R*) models in the absence of matter sources there is an elegant way to reduce the equations to one first-order equation for

*H*as a function of R in the Jordan frame [52]. But considering the field kinetic term, it is not possible for us to transform the equations to the matter free

*f*(

*R*) gravity, although, we still have a chance to eliminate one of the variables thanks to (18). Eliminating \(\chi _3\) is more reasonable since \(\chi _1=0\) requires \(\dot{\phi }=0\) and \(\chi _2=0\) requires \(\phi =0\) while \(\chi _3=0\) happens to be true for both mentioned conditions separately or together which makes the analysis more complicated. In this regard, the ultimate equations will be

## 5 An oscillatory fate; the vindication of an inflationary scenario

The requirement that the original Higgs inflation model should be followed by a transient oscillatory era after the inflationary period is an old and well-known practice. On the other hand, there have been suggestions that there might be phantom crossings before reaching the de Sitter destiny of the late universe [42, 43, 45, 46]. In this regard, it does not seem likely to reconcile these two different ideas in a manner that a phantom originated path reaches a de Sitter partial attractor where enough inflationary e-folds are accomplished before falling into an oscillatory phase. Figure 7 demonstrates the complete evolution of \(\chi _1\) and \(\chi _2\) in which the motion stars from phantom era and after approximately 1000 e-folding ends in an oscillatory phase. The evolution of \(w_{eff}\) for this simulation has been plotted in Fig. 6 where one can recognize the smooth phantom crossing, the de Sitter inflationary period and the oscillatory ultimate from the plots. It is already known that phantom originated trajectories may exhibit many oscillations around the de Sitter boundary, flipping between phantom and ordinary equations of state, while these oscillations never provide an end to the de Sitter main trajectory [45, 46]. Therefore, one may exclude any inflationary hypothesis which has a phantom origin since it seems impossible to find a required end of de Sitter expansion for them. In other words, Higgs inflation like other inflationary scenarios must possess an ordinary (non-phantom) initial condition. Here, we intend to show that this conclusion is not generally true and at least for the original Higgs inflation scenario, a phantom origin can evolve to an ordinary inflation with a desired oscillatory end. Numerical simulations also confirm this behavior (Figs. 2, 3, 5, 6) and one can recognize the \(w_{eff}\) evolution from phantom region to de Sitter expansion and follow its way towards \(\bar{w_{eff}}=1/3\) during ultimate oscillations around vacuum (Figs. 3, 6). The entire scenario has been plotted in Fig. 6 where one can follow the history from a phantom-like pre-inflationary start point to a quasi de Sitter partial attractor which ends at an oscillatory phase. We have already shown that for a typical *f*(*R*) inflationary theory, the post-inflationary oscillations occur around \(R=0\) which in FLRW background \(R=6{\dot{H}}+12 H^2\) can be written as \({\dot{H}}/H^2=-2\) or equivalently \(w_{eff}=1/3\) [53, 54, 55]. Of course, being in the Jordan frame makes the oscillatory interpretation slightly weird but one can easily infer that during an oscillation the point \((\chi _1=0,\chi _2=1)\) marks the return point of the Higgs field where \(\dot{\phi }\) vanishes and \((\chi _1=1,\chi _2=0)\) indicates the central point of the Higgs potential. Without assuming higher powers of Ricci scalar in the Lagrangian, one may not expect the system to exhibit explicit oscillations around \(R=0\) which is the overall behavior of an extra scalar degree of freedom (scalaron) in *f*(*R*) theories. In Sect. 8, we will see that such oscillations around \(R=0\) exist and become quasi sinusoidal as they lose their energy due to expansion (Fig. 10). We also see that the oscillations occur around the Higgs field minimum. Therefore, we witness a twofold oscillation. As we have stated earlier, numerical calculations confirm that \(w_{eff}\) shows flattering behavior around the average 1 / 3 value (see Fig. 3). Any complete oscillation needs to meet the minimum at \(\phi =0\) \((\chi _1=1,\chi _2=0)\) as well as the return point \(\dot{\phi }=0\) at \((\chi _1=0,\chi _2=1)\). In other words, our choice of variables restrict the minimum of the \(\chi _1\) and \(\chi _2\) oscillatory amplitude to be unity. It may seem weird that in an expanding universe \(\chi _1\) and \(\chi _2\) oscillations do not decay out and asymptotically reach a unit amplitude. But one has to remember that \(\chi _1\) and \(\chi _2\) have nontrivial relations with *R*, \(\phi \) and \(\dot{\phi }\) and as we will show, \(\phi \) and \(\dot{\phi }\) oscillations continue to decay with time even when \(\chi _1\) and \(\chi _2\) oscillate with an approximately constant amplitude (Fig. 11).

## 6 Fixed points and their behavior

## 7 Where does the inflation happen?

*w*is smaller than \(-1\) which corresponds to phantom energy domination while outside them \(w>-1\) . For large \(\xi \), the line \(\chi _2=1+24\xi \chi _1\) almost matches with the adjacent flow curves.The other line (\(\chi _2=1+\frac{\chi _1}{2}\)) exhibits a constant 1 / 2 slope. The more interesting fact is that the field flow moves from the ordinary region toward the \(w<-1\) area crossing the \(\chi _2=1+\frac{\chi _1}{2}\) line. One has to note that since there is no fixed point for de Sitter expansion, the inflation happens when the field flow asymptotically moves near one of the two \(w=-1\) boundaries. Besides the numerical portrait, it is easy to prove that the field flow could not move tangential to \(\chi _2=1+\frac{\chi _1}{2}\), a simple explanation will be derived from

*Note 1;* Without considering the electroweak vacuum expectation value, the potential minimum becomes negative since the central local maximum is zero (\(V(\phi =0)=0\)). This Point, of course, reduces the level of generality for the calculations and the plotted simulations. But since we are interested in high energy behavior of the field dynamics and concentrate on the effect of non trivial kinetic energy, all the discussions remain valid with very good accuracy.

*Note 2*; In our model, we expect the big bang singularity (if any) to occur at \(\chi _1=\chi _2=0\), since at BB, \(a\rightarrow 0\) and \(H\rightarrow \infty \). Numerical results (especially Fig. 1) show that \(\chi _1=\chi _2=0\) is an unstable fixed point and the behavior of trajectories is more consistent with a bouncing universe. This point is essentially unreachable in our approach since it requires \(\chi _1=\chi _2=\chi _3=0\) which is in contradiction to the \(\chi _1+\chi _2+\chi _3=1\) constraint. The repelling characteristic of the \(\chi _1=\chi _2=0\) point physically stems from tachyonic instability at the origin (Fig. 6).

## 8 Return to the original variables

*R*is oscillating around \(R=0\). Second: the oscillations imitate a damping sinusoidal as time passes. Both the above results are in complete accord with the analysis of the scalar-tensor inflationary models in low energy limit ([53, 54, 55]).

## 9 Comparison with some pioneering researches

The idea that scalar-tensor gravity modifications may realize the smooth phantom crossing without the necessity of ghosts originated from an inspiring research published in 2000 [42]. The main motivation to hypothesize such an idea stemmed from finding a justification for the clues of the late time (low redshift) phantom behavior of the dominant constituent of the energy of the universe. As a late time universe idea, the authors have tried to establish the conditions that scalar–tensor theories respect the post-Newtonian solar system and cosmological tests. In the more detailed paper [43] by recasting the dynamical equations through the redshift *z* and considering the solar system measurement and the dust-like perturbations in longitudinal gauge and adiabatic scale, the authors have provided general conditions for constructing such late time scalar–tensor models. Although we employ the same idea about the smooth phantom crossing capability of scalar–tensor models, we mostly concentrate on early time, high energy and particularly the original Higgs inflation theory. Therefore, the constraints we have respected also stem from the standard model of fields and particles as well as the inflationary footprints on the CMB. A very interesting point explored in [43] is that while the vanishing of \({\mathcal {G}}(\phi )\) (see Eq. 1) is excluded due to the singular behavior at low redshift (\(z<0.66\)), the constancy of \({\mathcal {G}}\) is also ruled out because of producing singularity for larger redshifts (although it works well for \(z\le 2\)). It seems that for obtaining a non-singular behavior which satisfies the phantom behavior in low redshifts, the potential has to possess implicit redshift (time) dependency (\({\mathcal {G}}(\phi (t))\)). The mentioned results for small redshifts have been generally based on the viability of small *z* expansion of the cosmic scale parameters. We can not use redshift functionality in our work since this method loses its applicability beyond the last scattering surface. But the Higgs inflation automatically admits the required conditions and remains well-defined in all energy intervals of our interest. There is another work about phantom crossing in scalar–tensor theories which raises more interests with respect to our research. The 2011 paper [45, 46] has proposed the idea of late time oscillations around de Sitter attractor. The authors have analyzed their proposal by perturbing the equations around de Sitter fixed point concluding that the oscillations may be separated to decaying and oscillatory parts and the latter may lead to several phantom crossings. The above idea has assumed the de Sitter stability and they have also mentioned that this is not a general behavior. Our research, at first glance, may demonstrate a sort of periodic phantom-passing motion as has already been predicted by [45, 46]. Although, the present research exploits the smooth phantom crossing (Figs. 5, 6), The difference between our results and phantom crossings due to oscillations around de Sitter boundary is that we are finding an oscillatory regime after a transient de Sitter phase which can be an appropriate condition for (p)reheating. Our claim is supported in two ways: analytically, we have shown that the trajectories, finally, leave de Sitter partial asymptote towards the Higgs field minimum outside the phantom region and begin to oscillate. Numerically, the time average of \(w_{eff}\) during this post-inflationary stage is approximately 1 / 3 which is an indicator of the oscillatory phase. As it has been mentioned earlier, \(\chi _1=0\) and \(\chi _2=1\) marks the return point of the scalar field oscillations which qualitatively resembles bounded oscillation under viral conditions. Therefore, we recognize an oscillatory phase which one expects to proceed by (p)reheating and this is why our approach may become suitable for an early universe possibility of phantom crossing. Moreover, the oscillations also show flipping between phantom and ordinary regions crossing the de Sitter boundary, but as has been stated earlier, this is not a bounce around the de Sitter attractor since the oscillations contain the scalar field minimum and \(\bar{w_{eff}\approx 1/3}\) during the oscillations. The other difference between this research and the late time studies is that for late time approaches it is necessary to add a matter (or radiation) term to the Lagrangian and let the classical evolution of the cosmic perfect fluid govern the scene. Instead, the matter source in pre-inflationary and inflationary phase is usually ignored. In fact, this has been the original reason to hypothesize more complicated quantum procedures for (p)reheating [29, 30].

## 10 Conclusion

We managed to reduce the dynamical equations of the original Higgs-Inflation scenario to two first order, coupled equations. The supportive numerical calculations have been delivered in the form of elusive two dimensional phase portrait plots. We then used the dynamical system approach to analyze the original Higgs-Inflation scenario in a broader scope. This mathematical tool lets us to be more accurate and extend the analysis to pre-inflationary and post-inflationary periods, where one can not use the slow-roll approximation. We included the kinetic term in our calculation. By this, we have followed two goals; first considering how the kinetic term breaks the auxiliary field relation between Higgs-Inflation and the Starobinsky model and second, although one expects the kinetic term to be trivially small during inflation, but if this smallness appears as a kind of attractor path, the picture becomes more compelling. This is indeed the case in our survey; the inflationary period emerges as an attractor which makes the process more natural. Perhaps, the more interesting achievement is that this de Sitter partial attractor pulls inward the phantom trajectories. The phantom trajectories which are trapped by de Sitter attraction may evolve a very smooth and familiar path after leaving the de Sitter attractor path which is an indication of the oscillations around the minimum, where the reheating or preheating processes take place. Therefore, one may talk about phantom region as the birthplace for the universes like what we live in. If one combines the results of this research with the fact that our universe is probably falling into a phantom era [57, 58, 59, 60, 61, 62, 63], then it is appealing to talk about the birth and the fate of the universe both in phantom era. This revives the cyclic universe hypothesis in an astonishing manner [64].

## References

- 1.For a modern review consult: modifications of Einstein’s theory of gravity at large distance. In
*Lecture Notes in Physics*, vol. 892, ed. by E. Papantonopoulos (Springer, New York, 2015)Google Scholar - 2.T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 966–972 (1921)Google Scholar
- 3.O. Klein, Z. Phys. A
**37**(12), 895–906 (1926)CrossRefGoogle Scholar - 4.T. Padmanabhan, D. Kothawala, Phys. Rep.
**531**(3), 115–171 (2013)ADSMathSciNetCrossRefGoogle Scholar - 5.Y. Fujii, K. Maeda,
*The Scalar Tensor Theory of Gravitation*(Cambridge University Press, Cambridge, 2004)Google Scholar - 6.V. Faraoni, E. Gunzig, Int. J. Theor. Phys.
**38**, 217–225 (1999)CrossRefGoogle Scholar - 7.P. Higgs, Phys. Rev. Lett.
**13**(16), 508–509 (1964)ADSMathSciNetCrossRefGoogle Scholar - 8.F. Englert, R. Brout, Phys. Rev. Lett.
**13**(9), 321–23 (1964)ADSMathSciNetCrossRefGoogle Scholar - 9.R. Brout, F. Englert, arXiv:hep-th/9802142 (1998)
- 10.A.A. Starobinsky, Phys. Lett. B
**91**, 99 (1980)ADSCrossRefGoogle Scholar - 11.A. H. Guth, The above initial. Phys. Rev. D.
**23**(2), 347–356 (1981)Google Scholar - 12.A. Linde, Lect. Notes Phys.
**738**, 1–54 (2008) (inflationary cosmology)Google Scholar - 13.Planck Collaboration, Astron. Astrophys. A
**22**, 571 (2014)Google Scholar - 14.Planck Collaboration, Astron. Astrophys. A
**20**, 594 (2016)Google Scholar - 15.Sh Assyyaee, N. Riazi, Ann. Phys.
**376**, 460–483 (2017)ADSCrossRefGoogle Scholar - 16.F.L. Bezrukov, M. Shaposhnikov, Phys. Lett. B
**659**, 703 (2008)ADSCrossRefGoogle Scholar - 17.R. Allahverdi, R. Brandenberger, F. Y. Cyr-Racine, A. Mazumdar, Annu. Rev. Nucl. Part. Sci.
**60**, 27–51 (2010)Google Scholar - 18.K. Kamada, T. Kobayashi, T. Takahashi, M. Yamaguchi, J. Yokoyama, Phys. Rev. D
**86**(2), 023504Google Scholar - 19.J.L. Cervantes-Cota, H. Dehnen, Nucl. Phys. B
**442**, 391412 (1995)CrossRefGoogle Scholar - 20.A.O. Barvinsky, A. Yu, Kamenshchik, A.A. Starobinsky, JCAP
**0811**, 021 (2008)Google Scholar - 21.F.L. Bezrukov, M.E. Shaposhnikov, Phys. Lett. B
**659**, 703–706 (2008)ADSCrossRefGoogle Scholar - 22.A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner, O.L. Veretin, Phys. Rev. Lett.
**115**, 201802 (2015)ADSCrossRefGoogle Scholar - 23.F. Bezrukov, J. Rubio, M. Shaposhnikov, Phys. Rev. D
**D92**, 083512 (2015)ADSCrossRefGoogle Scholar - 24.S. Alekhin, A. Djouadi, S. Moch, Phys. Lett. B
**716**, 214–219 (2012)ADSCrossRefGoogle Scholar - 25.M.S. Turner, F. Wilczek, Nature
**298**, 633–634 (1982)ADSCrossRefGoogle Scholar - 26.S. Coleman, F. De Luccia, Phys. Rev. D
**D21**(12), 33053315 (1980)Google Scholar - 27.J. Ellis, J.R. Espinosa, G.F. Giudice, A. Hoecker, A. Riotto, Phys. Lett. B.
**679**, 369375 (2009)CrossRefGoogle Scholar - 28.J. Kearney, H. Yoo, K.M. Zurek, Phys. Rev. Lett.
**115**, 201802 (2015)CrossRefGoogle Scholar - 29.L. Kofman, A. Linde, A. Starobinsky, Phys. Rev. Lett.
**73**, 3195 (1994)ADSCrossRefGoogle Scholar - 30.L. Kofman, A. Linde, A. Starobinsky, Phys. Rev. D
**56**, 3258 (1997)ADSCrossRefGoogle Scholar - 31.K. Kohri, H. Mastui, Phys. Rev. D
**94**, 103506 (2016)Google Scholar - 32.A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett.
**48**(17), 12201223 (1982)CrossRefGoogle Scholar - 33.J. Rubio, J. Phys. Conf. Ser.
**631**(conference 1) (2015)Google Scholar - 34.M.P. Hertzberg, Adv. High Energy Phys.
**2017**, 6295927 (2017)CrossRefGoogle Scholar - 35.Y. Ema, M. Karciauskas, O. Lebedev, S. Rusak, M. Zatta, arXiv:1711.10554 [hep-ph] (2017)
- 36.A.D. Felice, Sh Tsujikawa, Living Rev. Rel.
**13**, 3 (2010)CrossRefGoogle Scholar - 37.A. Kehagias, A. Moradinezhad Dizgah, A. Riotto, Phys. Rev. D
**89**, 043527 (2014)Google Scholar - 38.R.R. Caldwell, Phys. Lett. B
**545**, 23 (2002)ADSCrossRefGoogle Scholar - 39.R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. Lett.
**91**, 071301 (2003)ADSCrossRefGoogle Scholar - 40.S.M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D
**68**, 023509 (2003)ADSCrossRefGoogle Scholar - 41.P. Singh, M. Sami, N. Dadhich, Phys. Rev. D
**68**, 023522 (2003)ADSCrossRefGoogle Scholar - 42.B. Boisseau, G. Esposito-Farse, D. Polarski, A.A. Starobinsky, Phys. Rev. Lett.
**85**, 2236 (2000)ADSCrossRefGoogle Scholar - 43.R. Gannouji, D. Polarski, A. Ranquet, A.A. Starobinsky, JCAP
**0609**, 016 (2006)ADSCrossRefGoogle Scholar - 44.M. He, A.A. Starobinsky, J. Yokoyama, JCAP
**1805**, 064 (2018)ADSCrossRefGoogle Scholar - 45.H. Motohashi, A.A. Starobinsky, J. Yokoyama, Prog. Theor. Phys.
**123**, 887 (2010)Google Scholar - 46.H. Motohashi, A.A. Starobinsky, J. Yokoyama, JCAP
**1106**, 006 (2011)ADSCrossRefGoogle Scholar - 47.T. Chiba, M. Yamaguchi, JCAP
**0901**, 019 (2009)ADSCrossRefGoogle Scholar - 48.A. Stabile, S. Capozziello, Phys. Rev. D
**87**, 064002 (2013)ADSCrossRefGoogle Scholar - 49.T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys.
**82**, 451–497 (2010)ADSCrossRefGoogle Scholar - 50.D. Lyth,
*Cosmology for Physicists*(CRC Press, London, 2016)Google Scholar - 51.L. Amendola, R. Gannouji, D. Polarski, S. Tsujikawa, Phys. Rev. D
**75**, 083504 (2007)ADSCrossRefGoogle Scholar - 52.H. Motohashi, A.A. Starobinsky, EPJC
**77**, 538 (2017)ADSCrossRefGoogle Scholar - 53.A.A. Starobinsky, Nonsingular model of the Universe with the quantum-gravitational de Sitter stage and its observational consequences, in
*Quantum Gravity*, Proceedings of the 2nd Seminar on Quantum Gravity, Moscow, 13–15 Oct 1981, p. 5872 (INR Press, Moscow, 1982). Reprinted in:*Quantum Gravity*ed. by M.A. Markov, P.C. West (Plenum Press, New York, 1984), p. 103128Google Scholar - 54.A. Vilenkin, Phys. Rev. D
**32**, 25112521 (1985)MathSciNetCrossRefGoogle Scholar - 55.M.B. Mijic, M.S. Morris, W.M. Suen, Phys. Rev. D
**34**, 29342946 (1986)CrossRefGoogle Scholar - 56.M.W. Hirsch, S. Smale, R.L. Devaney,
*Differential Equations, Dynamical Systems, and an Introduction to Chaos*, 3rd edn (Academic Press, New York, 2012)Google Scholar - 57.Planck Collaboration, Astron. Astrophys.
**594**, A13 (2016)CrossRefGoogle Scholar - 58.Planck Collaboration, Astron. Astrophys.
**571**, A22 (2014)CrossRefGoogle Scholar - 59.D.L. Shafer, D. Huterer, Phys. Rev. D
**89**, 063510 (2014)ADSCrossRefGoogle Scholar - 60.Alexander Vikman, Phys. Rev. D
**71**, 023515 (2005)ADSCrossRefGoogle Scholar - 61.A.A. Costa, A. Xiao-Dong Xu, B. Wang, E. Abdallaa, JCAP
**01**, 028 (2017)Google Scholar - 62.A. Melchiorri, L. Mersini, C.J. Odman, M. Trodden, Phys. Rev. D
**68**, 043509 (2003)ADSCrossRefGoogle Scholar - 63.J.S. Alcaniz, Phys. Rev. D
**69**, 083521 (2004)ADSCrossRefGoogle Scholar - 64.A. Einstein,
*Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse*, vol. 235237 (1931)Google Scholar

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