Relic gravitational waves from Quintessential Inflation

TL;DR

This paper investigates relic gravitational waves produced in quintessential inflation, highlighting a kinetic regime after inflation that leads to a blue spectrum and discussing detection prospects with advanced gravitational wave observatories.

Contribution

It provides a model-independent analysis of gravitational wave production in quintessential inflation and demonstrates a concrete potential that fits observational constraints.

Findings

Kinetic regime after inflation results in a blue gravitational wave spectrum.

Instant preheating can produce the required post-inflation temperature.

Detection prospects are discussed for LIGO and LISA.

Abstract

We study relic gravitational waves in the paradigm of quintessential inflation. In this framework, irrespective of the underlying model, inflation is followed by the kinetic regime. Thereafter, the field energy density remains sub-dominant before the onset of acceleration. We carry out model independent analysis to obtain the temperature at the end of inflation and the estimate for upper bound on the Hubble parameter to circumvent the problem due to relic gravitational waves. In this process, we used Planck 2015 data to constrain the inflationary phase. We demonstrate that the required temperature can be produced by the mechanism of instant preheating. The generic feature of the scenario includes the presence of kinetic regime after inflation which results into blue spectrum of gravitational wave background at high frequencies. We discuss the prospects of detection of relic gravitational wave background in the advanced LIGO and LISA space-born gravitational wave missions. Finally we consider a concrete model to realize the paradigm of quintessential inflation and show that inflationary as well as post-inflationary evolution can successfully be described by the inflaton potential, $V(\phi) \propto Exp(-\lambda \phi^n/\Mpl^n)(n>1)$, by suitably constraining the parameters of the model.

Equations

$$\square h_{ij}=0\to {\ddot{\phi }}_k(\tau )+2\frac{\dot{a}}{a}{\phi }_k(\tau )+k^2{\phi }_k(\tau )=0;h_{ij}\sim {\phi }_k{e}^{ikx}{e}_{ij},$$

$${\phi }_{\mathrm{in}}^{(+)}(k,\tau )=(\pi {\tau }_0/4{)}^{1/2}(\tau /{\tau }_0{)}^{3/2}{H}_{3/2}^{(2)}(k\tau ).$$

$${\phi }_{\mathrm{out}}=\alpha {\phi }_{\mathrm{out}}^{(+)}+\beta {\phi }_{\mathrm{out}}^{(-)},$$

$${\phi }_{\mathrm{out}}^{(+,-)}=(\pi {\tau }_0/4{)}^{1/2}(\tau /{\tau }_0{)}^{\mu }{H}_{|\mu |}^{(2,1)}(k\tau ).$$

$${\rho }_g=<T_{00}>=\frac{1}{{\pi }^2{a}^2}\int \mathrm{d}k{k}^3|\beta {|}^2.$$

$${\rho }_g=\frac{32}{3\pi }{h}_{\mathrm{GW}}^2{\rho }_b\left(\frac{\tau }{{\tau }_{\mathrm{kin}}}\right)$$

$$\frac{{\tau }_{\mathrm{rh}}}{{\tau }_{\mathrm{kin}}}={\left(\frac{T_{\mathrm{kin}}}{T_{\mathrm{rh}}}\right)}^2$$

$${\left(\frac{{\rho }_g}{{\rho }_r}\right)}_{\mathrm{rh}}=\frac{64}{3\pi }{h}_{\mathrm{GW}}^2{\left(\frac{T_{\mathrm{end}}}{T_{\mathrm{rh}}}\right)}^2,$$

$$h_{\mathrm{GW}}^2=\frac{H_{\inf }^2}{8\pi M_{\text{Pl}}^2}=\frac{V_{\inf }}{24\pi M_{\text{Pl}}^4},$$

$${\left(\frac{{\rho }_{\varphi }}{{\rho }_r}\right)}_{\mathrm{end}}={\left(\frac{T_{\mathrm{end}}}{T_{\mathrm{rh}}}\right)}^2$$

$${\left(\frac{{\rho }_{\varphi }}{{\rho }_r}\right)}_{\mathrm{end}}=\frac{3\pi }{64}{\left(\frac{{\rho }_g}{{\rho }_r}\right)}_{\mathrm{rh}}\frac{1}{h_{\mathrm{GW}}^2}.$$

$$V_{\inf }^{1/4}=0.013r^{1/4}{M}_{\text{Pl}},$$

$${\left(\frac{{\rho }_{\varphi }}{{\rho }_r}\right)}_{\mathrm{end}}=\frac{9{\pi }^2}{8}{\left(\frac{{\rho }_g}{{\rho }_r}\right)}_{\mathrm{rh}}\frac{M_{\mathrm{Pl}}^4}{V_{\inf }}\lesssim \frac{3.88\times 10^6}{r}$$

$$\frac{k}{a_0{H}_0}=\frac{a_{\inf }}{a_{\mathrm{end}}}\frac{a_{\mathrm{end}}}{a_0}\frac{H_{\inf }}{H_0}\Rightarrow e^{-\mathcal{N}}\frac{T_0}{T_{\mathrm{end}}}\frac{H_{\inf }}{H_0}=1.$$

$$T_{\mathrm{end}}=1.99\times 10^{15}{r}^{1/2}\text{GeV}.$$

$$H_{\mathrm{end}}\lesssim 1.85\times 10^{15}{r}^{1/2}\text{GeV}.$$

$${\rho }_{\mathrm{r},\mathrm{end}}\simeq 0.01g_p{H}_{\mathrm{end}}^4$$

$${\left(\frac{{\rho }_{\varphi }}{{\rho }_r}\right)}_{\mathrm{end}}\simeq \frac{{\rho }_{\varphi ,\mathrm{end}}}{0.01g_p{H}_{\mathrm{end}}^4}\gtrsim \frac{2.1\times 10^{10}{g}_p^{-1}}{r}.$$

$${\Omega }_{\mathrm{GW}}(k)=\frac{{\tilde{\rho }}_{\mathrm{g}}(k)}{{\rho }_{\mathrm{c}}},$$

$${\tilde{\rho }}_{\mathrm{g}}(k)\propto k^{1-2|\mu |};\mu =\frac{3}{2}\left(\frac{w-1}{3w+1}\right).$$

$${\Omega }_{\mathrm{GW}}^{(\mathrm{MD})}=\frac{3}{8{\pi }^3}{h}_{\mathrm{GW}}^2{\Omega }_{\mathrm{m}0}{\left(\frac{\lambda }{{\lambda }_{\mathrm{h}}}\right)}^2,{\lambda }_{\mathrm{MD}}<\lambda \le {\lambda }_{\mathrm{h}},$$

$${\Omega }_{\mathrm{GW}}^{(\mathrm{RD})}(\lambda )=\frac{1}{6\pi }{h}_{\mathrm{GW}}^2{\Omega }_{\mathrm{r}0},{\lambda }_{\mathrm{RD}}<\lambda \le {\lambda }_{\mathrm{MD}},$$

$${\Omega }_{\mathrm{GW}}^{(\mathrm{kin})}(\lambda )={\Omega }_{\mathrm{GW}}^{(\mathrm{RD})}\left(\frac{{\lambda }_{\mathrm{RD}}}{\lambda }\right),{\lambda }_{\mathrm{kin}}<\lambda \le {\lambda }_{\mathrm{RD}},$$

$${\lambda }_{\mathrm{h}}$$

$${\lambda }_{\mathrm{MD}}$$

$${\lambda }_{\mathrm{RD}}$$

$${\lambda }_{\mathrm{kin}}$$

$${\mathcal{L}}_{\mathrm{int}}=-\frac{1}{2}{g}^2{\varphi }^2{\chi }^2-h\bar{\psi }\psi \chi ,$$

$$|\dot{m_{\chi }}|\gtrsim m_{\chi }^2\Rightarrow |\dot{\varphi }|\gtrsim g{\varphi }^2.$$

$$|\varphi |\lesssim |{\varphi }_p|=\sqrt{\frac{{\dot{\varphi }}_{\mathrm{end}}}{g}}.$$

$${\omega }_{\varphi }=\frac{{\dot{\varphi }}^2-2V}{{\dot{\varphi }}^2+2V}.$$

$${\dot{\varphi }}_{\mathrm{end}}=\sqrt{V_{\mathrm{end}}}.$$

$$\frac{{\dot{\varphi }}_{\mathrm{end}}}{g}\lesssim M_{\text{Pl}}^2\to g\gg \frac{1}{M_{\text{Pl}}^2}\sqrt{V_{\mathrm{end}}}.$$

Full text

I Introduction

Accelerated expansion has played an important role in the history of our universe. It is a common belief that Universe has gone through a phase of rapid expansion dubbed inflation Starobinsky:1980te ; Starobinsky:1982ee ; Guth:1980zm ; Linde:1983gd at early times and it started accelerating once again around the present epoch Copeland:2006wr ; Sahni:2006pa ; Peebles:2002gy ; Sami:2009dk ; Sami:2009jx ; Sami:2013ssa . The standard model of Universe, therefore, should be complemented by two phases of acceleration. Inflation is a beautiful paradigm that not only resolves the inconsistencies of hot big bang model but also provides with a mechanism of generation of primordial perturbations required for structure formation. Late-time cosmic acceleration is needed to address the age problem in the standard model. The phenomenon was detected in 1998 by supernovae observations Riess:1998cb ; Perlmutter:1998np which was supported by other probes indirectly thereafter Bennett:2003bz ; Lange:2000iq ; Ade:2015tva . Similar confirmation is yet awaited for cosmological inflation.

Cosmic acceleration is a generic phenomenon in our Universe which manifests at early and late times keeping the thermal history intact. Often, these two phases are described independently. As for late-time acceleration, a variety of scalar field models has been investigated in the literature since 1998. At the background level, cosmology community seems to converge on the cosmological constant as the underlying cause of the said phenomenon. Even if acceleration is caused by a slowly rolling scalar field, the latter is not distinguished from cosmological constant at the background level. As for perturbations, the study of large scale structure might reveal in case there is something beyond $\Lambda$CDM. The current observational constraints related to inflation are tight such that many popular models are on the verge of being ruled out. We should emphasize that in a successful model, inflation should be followed by an efficient reheating also.

It is interesting to ask whether a successful model of inflation can also describe late-time acceleration without interfering with the thermal history of Universe or can inflaton be dark energy. Unification of inflation with late-time acceleration or dark energy is termed as quintessential inflation Spokoiny:1993kt ; Peebles:1999fz ; Peloso:1999dm ; Dimopoulos:2000md ; Copeland:2000hn ; Sahni:2001qp ; Dimopoulos:2001ix ; Majumdar:2001mm ; Giovannini:2003jw ; Sami:2004xk ; Sami:2004ic ; Rosenfeld:2005mt ; Hossain:2014xha ; Hossain:2014coa ; Hossain:2014zma ; Dimopoulos:2017zvq . It is indeed challenging to describe inflation and late-time acceleration using a single scalar field. At the onset, it sounds possible if the field potential is shallow at early and late stages and steep for most of the history of Universe. First, the model should comply with all the observational constraints related to inflation and should give rise to successful reheating which is itself a difficult task. As we pointed out, the current observational constraints are quite tight putting many known models in tension. Secondly, conventional reheating mechanism Dolgov:1982th ; Abbott:1982hn ; Ford:1986sy ; Spokoiny:1993kt ; Kofman:1994rk ; Shtanov:1994ce ; Kofman:1997yn ; Campos:2002yk is not applicable to this class of models as the field potential is typically run-away type. However, in this case, one can invoke an alternative mechanism dubbed instant preheating Felder:1998vq ; Felder:1999pv ; Campos:2004nc ; Tashiro:2003qp . It is desirable that we have scaling regime Steinhardt:1999nw after inflation allowing inflaton to go into hiding till late times which is necessary to keep the thermal history intact. To what extent, the field should be hidden is decided by the nucleosynthesis constraint. In this case, unlike the thawing picture, evolution, to great extent, is independent of initial conditions.

The generic feature of quintessential inflation includes the presence of kinetic regime Sahni:2001qp ; Sami:2004xk ; Tashiro:2003qp that follows inflation. The duration of the regime depends upon the temperature at the end of inflation. The energy density of relic gravitational waves  Grishchuk:1974ny ; Starobinsky:1979ty ; Sahni:1990tx ; Souradeep:1992sm ; Gasperini:1992dp ; Gasperini:1992pa ; Giovannini:1999bh ; Giovannini:1999qj ; Langlois:2000ns ; Tashiro:2003qp as compared to field energy density enhances during kinetic regime and might challenge the nucleosynthesis constraint at the commencement of radiative regime. The distinguished signature of the unification, irrespective of the underlying model, includes the production of relic gravitational waves with blue spectrum. Clearly, an alternative reheating mechanism is needed in this case to circumvent the problem.

Let us note that the generalized exponential potential $V(\phi)\propto Exp(-\lambda\phi^{n}/M_{\textrm{Pl}}^{n}),(n>1)$ Geng:2015fla can successfully realize the paradigm of quintessential inflation. This potential has a remarkable property: it is shallow around $\phi=0$ and steep thereafter. Thanks to an additional parameter $n$ compared to standard exponential, the model can comply with observational constraints on inflation. Secondly, the model shares the property of a simple steep exponential potential Hossain:2014xha ; Sahni:2001qp ; Copeland:2000hn ; Lucchin:1984yf ; Ratra:1989uz ; Sami:2002fs at late times, namely, the approximate scaling regime is a late-time attractor in this case as $\Gamma=V^{\prime\prime}V/V^{\prime 2}\to 1$ for large values of the field.

Last but not least, we need to exit from scaling regime around the present epoch. This can be achieved by invoking non-minimal coupling to massive neutrino matter (see Wetterich:2013aca ; Wetterich:2013jsa ; Wetterich:2013wza ; Wetterich:2014eaa ; Wetterich:2014bma ; Amendola:2007yx ; Wetterich:2007kr ; Hossain:2014zma ; Geng:2015fla and references therein for details). The coupling appears at late stages as neutrinos turn non-relativistic. As a result, the effective potential for the field acquires minimum where the field can settle causing exit from scaling regime to late-time cosmic acceleration.

In this paper, we first carry out a model independent analysis to obtain the temperature at the end of inflation and the estimate on the upper bound of the Hubble parameter to circumvent the problem due to relic gravitational waves. We also investigate an alternative reheating mechanism suitable to the scenario under consideration. Finally, we discuss a model of quintessential inflation which can successfully realize the paradigm. Throughout this paper, $M_{\textrm{Pl}}$ denotes the reduced Planck mass.

II Relic gravitational wave spectrum

In this section, we shall study relic gravitational waves in the scenario of quintessential inflation. The evolution of tensor perturbations, $h_{ij}$, is given by the Klein-Gordon equation

[TABLE]

where $e_{ij}$ is the polarization tensor, $\tau$ (${\rm d}\tau={\rm d}t/a$) is the conformal time and $k$ is the comoving wave number. For simplicity, we shall consider the exponential inflation , namely, $a=\tau_{0}/\tau$ and $H_{\rm in}=-1/\tau_{0}$. The positive frequency solution of Eq. (1) in the adiabatic vacuum, corresponds to the “in” state, $\varphi_{\rm in}^{(+)}(k,\tau)$

[TABLE]

where $H^{(2)}_{3/2}$ is the Hankel function of second kind.

In a scenario of quintessential inflation, Universe enters into the kinetic phase with stiff equation of state parameter soon after the inflation endsSahni:2001qp ; Sami:2004xk . This transition involves the non-adiabatic change of space-time geometry. Assuming for simplicity that the post-inflationary dynamics is described by the power law expansion, i.e., $a=(t/t_{0})^{p}\equiv(\tau/\tau_{0})^{1/2-\mu}$ where $\mu\equiv(3(w-1)/2(3w+1))$ with $w$ being the post-inflationary equation of state parameter. Let us note that $\mu=0$ in the kinetic regime ($w=1$). As for the “out” state, it contains both positive and negative frequency solutions of (1),

[TABLE]

where $\alpha$ and $\beta$ designate the Bogoliubov coefficients Sahni:2001qp and

[TABLE]

We then estimate the energy density of relic gravitational waves Sahni:2001qp ; Sahni:1990tx ,

[TABLE]

During kinetic regime, $|\beta_{\rm kin}|^{2}\sim(k\tau_{\rm kin})^{-3}$, using then Eq. (5) and the fact that $H_{\rm in}=-1/\tau_{0}$, we obtain,

[TABLE]

where $\rho_{b}$ denotes the background energy density and contains radiation and stiff scalar matter in the kinetic regime. We have hereby assumed that the generation of radiation took place at the end of inflation thank to some mechanism to be specified in the subsequent section.

Since, at the equality of radiation and scalar field energy densities ($\tau=\tau_{\rm rh}$),

[TABLE]

and $\rho_{b}=2\rho_{r}$, we find using Eq. (6),

[TABLE]

where $h_{\rm GW}$ is the dimensionless amplitude of the gravitational waves given by,

[TABLE]

where $V_{\rm{inf}}$ is the value of the inflationary potential at the time the cosmological scales exit the horizon and can be fixed by imposing the COBE normalization Bunn:1996da ; Bunn:1996py .

Eqs. (7) and (8) implies that longer kinetic regime, i.e., large $\tau_{\rm rh}$, would corresponds to smaller $T_{\rm rh}$ and hence the larger value of $\rho_{g}/\rho_{r}$ at equality. As given in Sahni:2001qp , it can be shown from (5) that for $w>1/3$, $\rho_{g}\propto 1/a^{4}$ and for $w<1/3$, $\rho_{g}\propto\rho_{b}$; during radiation era also $\rho_{g}$ approximately tracks the background. For $w=1/3$ (radiation), log factor appears 111Precisely, $\rho_{g}\propto a^{-4}\log(\tau/\tau_{0})\propto a^{-4}\log{(a)}$ but we have ignored the logarithmic factor as its contribution is negligible during evolution..

For simplicity 222It is supported by numerical simulation., we hereby assume that kinetic regime ($\rho_{\phi}\propto 1/a^{6}$) immediately follows inflation, thus we can use the approximation, $H_{\rm end}\simeq H_{\rm kin}$ and $T_{\rm end}\simeq T_{\rm kin}$. Since $T_{\rm rh}\sim T_{\rm end}(a_{\rm end}/a_{\rm rh})$, we find,

[TABLE]

and the following important relation

[TABLE]

From COBE normalisation, we find that

[TABLE]

where $r$ is the scalar-to-tensor ratio. Combining this with Eq. (9) and inserting into the Eq. (11), we find

[TABLE]

where we have used the constraint on the ratio, $(\rho_{g}/\rho_{r})_{\rm rh}\lesssim 0.01$ as imposed by nucleosynthesis. We can compute temperature at the end of inflation by using the fact that the cosmological scales, $k$ that crosses the horizon during inflation and the scales that re-enters it at a later time satisfies the relation, $k=a_{\rm inf}H_{\rm inf}=a_{0}H_{0}$, which implies that

[TABLE]

For $\mathcal{N}=65$, we find that

[TABLE]

Now, the upper bound on the value of Hubble parameter at the end of inflation, $H_{\rm end}$, can be obtained from Eq. (13) and using the fact that $\rho_{\rm{r,end}}=T_{\rm end}^{4}$,

[TABLE]

However, the bound (16) should be consistent with the estimate for $H_{\rm inf}$ obtained after using COBE normalization, namely, $H_{\rm inf}\simeq 2.37\times 10^{14}r^{1/2}$GeV. In general, the ratio $H_{\rm inf}/H_{\rm end}$ is more than few, in the realistic scenarios with graceful exit. Clearly, $H_{\rm inf}=H_{\rm end}$ in case of de-Sitter. Thus we can take $H_{\rm end}\lesssim H_{\rm inf}$ for estimates which is clearly a more conservative bound than (16). The bound on $H_{\rm end}$ in turn gives us the bound on $V_{\rm end}$ which is $V_{\rm end}\lesssim 1.15\times 10^{-6}r~{}M_{\textrm{Pl}}^{4}$. Again, we know that $V_{\rm end}$ can at most take the value of $V_{\rm inf}$.

Since the potential in the model of quintessential inflation is generically run-away type, the conventional reheating mechanism does not operate in this case. Gravitational particle production is a known possibility. After inflation, the geometry of space-time undergoes a non-adiabatic process which gives rise to particle production. Let us note that gravitational particle production gives rise to the energy density Ford:1986sy ; Spokoiny:1993kt

[TABLE]

where $g_{p}$ is the number of different particles produced which is typically $\sim 10-100$ and since $H_{\rm end}\lesssim H_{\rm inf}$, we find that

[TABLE]

In the aforesaid model independent discussion, we have used $H_{\rm end}\simeq H_{\rm inf}$. For an exact relation between these quantities, we need a concrete model. However, in models with $r\ll 1$ such as Starobinsky inflation model Starobinsky:1980te , $H_{\rm inf}$ is close to $H_{\rm end}$. Thus, in general, the respective energy density produced due to gravitational particle production misses the nucleosynthesis constraint imposed by relic gravitational waves (13) by at least two orders of magnitude which is related to the fact that temperature of radiation due to gravitational particle production $T_{\rm end}^{g}$ is some what less than $T_{\rm end}$ (15), namely, $T_{\rm end}^{g}\simeq 2.63\times 10^{14}r^{1/2}$GeV. In the scenario under consideration, due to the steep post-inflationary behavior of the potential, the field enters the kinetic regime soon after inflation ends and remains there for a long time in view of the aforesaid. Clearly, the problem of relic gravitational waves in quintessential inflation is associated with the longevity of kinetic regime. During this time the energy density in gravitational waves enhances compared to field energy density challenging the nucleosynthesis constraint at the commencement of radiative regime. We naturally require more efficient process to circumvent the problem, one should then look for a suitable reheating mechanism.

We are now in position to discuss the gravitational wave spectrum. The fractional spectral energy density parameter of the relic gravitational wave is defined as follows,

[TABLE]

where $\rho_{\rm c}$ denotes the critical energy density (see Ref. Sahni:2001qp for details) and $\tilde{\rho}_{\mathrm{g}}(k)$ is the spectral energy density

[TABLE]

From the above equation, we see that during the kinetic regime, i.e., for $w=1$, $\rho_{\mathrm{g}}(k)\propto k$ giving rise to a blue spectrum in the wave spectrum. This is a generic feature of any model of quintessential inflation where the scalar field enters into the kinetic regime after inflation.

We also have the following relations,

[TABLE]

with

[TABLE]

where “MD”, “RD” and “kin” designate matter, radiation and kinetic energy dominated epochs, $H_{0}$, $\Omega_{\mathrm{m}0}$ and $\Omega_{\mathrm{r}0}$ are the present values of the Hubble parameter, dimensionless density parameters corresponding to matter energy density and radiation energy density, and $T_{\rm end}$ and $H_{\rm end}$ denote the temperature and Hubble parameter at the end of inflation, respectively. We note that in Eq.(22), we again ignore the insignificant logarithmic correction.

III Instant particle production and preheating

We hereby emphasize once again that the gravitational particle production is quite inefficient for circumventing the problem due to relic gravitational waves, one therefore should look for an alternative mechanism. One of the alternatives suitable to the class of models similar to the one we are going to discuss in the next section, is based upon instant particle production dubbed instant preheating which proceeds as follows. We assume that the scalar field, $\phi$, interacts with some other scalar field, $\chi$, which interacts with the Fermion field, $\psi$,

[TABLE]

where g and h denote the coupling constants, assumed to be positive for convenience and $g,h<1$ for perturbation treatment to be applicable. In this case, $\chi$ does not have a bare mass. However, the $\chi$ field has an effective mass which grows with $\phi$ as $m_{\chi}=g|\phi|$. The Lagrangian is specially design to give rise to this feature. As inflation ends, the field $\phi$ enters the kinetic regime as the potential is very steep in the post-inflationary regime. Consequently, the field $\phi$ rolls down its potential fast soon after inflation ends. Since, mass of $\chi$ depends upon $\phi$, we can shift the field as $\phi\to\phi^{\prime}=\phi-\phi_{\rm end}$ such that the effective mass of $\chi$ vanishes at the end of inflation. However, the Lagrangian (28) does not obey shift symmetry, we thus need to assume an enhanced symmetry in the Lagrangian which can be achieved by adding a suitable counter term to (28). It is important to check how $m_{\chi}$ changes with time around the transition from inflation to kinetic phase. The non-adiabatic change in $m_{\chi}$ is crucial for particle production. Indeed, the production of $\chi$ particles take place when the adiabaticity condition is violated ,i.e.,

[TABLE]

The above condition is satisfied if

[TABLE]

The equation of state parameter for the inflaton field is given by

[TABLE]

Now using the fact that inflation ends when $\omega_{\phi}$ increases to -1/3, we find that

[TABLE]

Considering the fact that $\phi_{p}\lesssim M_{\textrm{Pl}}$, Eq. (30) gives us the bound on the coupling g

[TABLE]

In addition, the production time of $\chi$ particles is given by

[TABLE]

The uncertainty relation allows us to obtain the momentum, $k_{p}\simeq(\delta p)^{-1}\simeq\sqrt{g\dot{\phi}_{\rm end}}$. Following the Ref. Felder:1999pv ; Kofman:1997yn , the occupation number of $\chi$ is given by

[TABLE]

which allows us to estimate the number density of $\chi$ particles

[TABLE]

and their total energy density

[TABLE]

Assuming that at the end of inflation, the produced energy is thermalized instantaneously, we obtain

[TABLE]

The above equation combined with Eq. (13) gives the lower bound on the coupling $g$

[TABLE]

Now, since

[TABLE]

the expansion is negligible during the particle production. It should be noted that since $\phi_{p}\lesssim 0.13M_{\textrm{Pl}}$, the production of particles commences immediately after the inflation ends. In the post-inflationary regime, the scalar field $\phi$ rolls down its potential faster and hence the effective mass $g|\phi|$ of the scalar field $\chi$ grows which facilitates the decay of $\chi$ particles into fermions. The decay rate given by

[TABLE]

Around the end of inflation, $\rho_{\phi}\propto 1/a^{2}$ and diminishes slower than $\rho_{\chi}$ with the expansion of universe333Immediately, after inflation ends ($\omega_{\phi}=-1/3$), $\rho_{\phi}\sim 1/a^{2}$, thereafter it redshifts faster and enters the kinetic regime. Since kinetic regime establishes fast after inflation, the estimates change insignificantly if we identify the end of inflation with the commencement of kinetic regime. . We now should arrange the decay rate of $\chi$ into matter fields such that the decay takes place before the back reaction of $\chi$ on $\phi$ evolution becomes important which implies that

[TABLE]

For $\phi\lesssim M_{\textrm{Pl}}$, the above condition gives us the lower bound on the coupling h, namely, $h\gtrsim 0.13g^{-1/2}r^{1/4}$. In Fig. (1), we depict the allowed values of $g$ and $h$ which shows that we have a wide parameter space for an efficient preheating to occur.

Fig. (2) illustrates the spectral energy density ($\Omega_{\rm GW}$) of the relic gravitational wave background for the temperature $T_{\rm{end}}=6.29\times 10^{13}$GeV which corresponds to $g=1.96\times 10^{-3}$ (for $r\simeq 10^{-3}$) of the coupling, along with the sensitivity curve of AdvLIGO LIGO ; aLIGO and LISA LISA ; LISA1 . Figure shows that spectrum is blue at high frequencies.

IV Successful model of quintessential inflation

We shall now check whether we can realize the aforesaid requirement in a particular model of quintessential inflation. To this effect, we shall consider a model based upon the following generalized exponential potential with two parameters $n$ $\&$ $\lambda$,

[TABLE]

Using the expressions of slow-roll parameters,

[TABLE]

we have the standard expressions for the scalar spectral index ($n_{s}$) and tensor-to-scalar ratio ($r$) as

[TABLE]

In flat FRW background, the slow-roll parameter, $\epsilon$, for the generalized exponential potential (43) is given by:

[TABLE]

which shows that slow-roll takes place for $\phi/M_{\textrm{Pl}}\ll 1$ whereas potential is steep for large values of the field. Using the condition for the end of inflation, namely, $\epsilon\lvert_{\phi=\phi_{\rm{end}}}=1$ and the expression for the number of the e-folding

[TABLE]

we estimate the numerical value of $\phi$ at the commencement of inflation,

[TABLE]

We can then eliminate $\phi$ from slow-roll parameter, $\epsilon$, in favor of model parameters $n,\lambda$ and the number of e-folds by inserting the above expression for $\phi$ as

[TABLE]

We shall use the relation (51) in the subsequent discussion for the estimation of $V_{\rm{end}}$. Current observations, namely, the Planck 2015 results, impose constraints on the model parameters $n$ $\&$ $\lambda$. Indeed, the predictions of the model are within $2\sigma$ bound provided that $n\gtrsim 5$ and $\lambda\lesssim 10^{-4}$, see Refs Geng:2015fla ; Geng:2017mic .

The radiation energy produced due to the production of particle at the end of inflation Ford:1986sy ; Spokoiny:1993kt ,

[TABLE]

can now be calculated exactly. Choosing, $r=10^{-3}$, $\mathcal{N}=65$, $n=6$ and making use of Eqs. (45) and (51), we find that $\lambda=8.08\times 10^{-6}$. In addition, using COBE normalization, we can estimate $V_{0}=3.4\times 10^{-8}rM_{\textrm{Pl}}^{4}$ and $V_{\rm end}=5.39\times 10^{-9}rM_{\textrm{Pl}}^{4}$ so that

[TABLE]

Clearly, the radiation energy density is very low in this case, only one part in $10^{12}$ (assuming $g_{p}\simeq 100$ and $r\simeq 10^{-3}$) as compared to field energy density at the end of inflation and violates the constraint on $\left(\rho_{\phi}/\rho_{r}\right)_{\rm{end}}$ put by nucleosynthesis (13). Also note that for this model, the ratio $H_{\rm inf}/H_{\rm end}$, for $r\simeq 10^{-3}$, is found to be 1.87, which implies that $H_{\rm inf}$ is very close to $H_{\rm end}$ for $r\ll 1$ as we have mentioned in section II. This ratio is larger for larger value of $r$. For the model under consideration, one can invoke instant preheating mechanism as explained in the previous section. In this case, the lower bound on the temperature at the end of inflation can be calculated using Eq. (11) together with the constraint on ratio, $(\rho_{g}/\rho_{r})_{\rm{rh}}\lesssim 0.01$, due to nucleosynthesis as,

[TABLE]

which is consistent with the model independent estimate obtained earlier. Now that we have a lower bound on $T_{\rm end}$, using Eq. (14), we can obtain a bound on the number of e-foldings in this model, namely, $\mathcal{N}\lesssim 66.4$. Note that the number of e-foldings depend on the temperature at the end of inflation and would be even more than this for the temperature of radiation due to gravitational particle production.

Also, the lower bound on the coupling constant $h$, for $n=6$, is found to be $h\gtrsim 0.12g^{-1/2}r^{1/4}$. Clearly, the model under consideration successfully implements the paradigm of quintessential inflation. Last but not least, we should emphasize that it is challenging to find a model that can implement all the requirements of the scenario listed in the introduction.

V Conclusions

In scenario of quintessential inflation, field enters the steep region of the potential soon after inflation ends, such that $\rho_{\phi}\propto a^{-6}$ ${\it a~{}la}$ the kinetic regime. The duration of this phase, which generally follows inflation in the scenario of the quintessential inflation, irrespective of the field potential, depends upon the temperature at the end of inflation. A longer kinetic regime or the lower value of $T_{\rm end}$ implies the enhancement of energy density in relic gravitational waves compared to field energy density and might challenge nucleosynthesis constraint at the commencement of the radiative regime. The temperature at the end of inflation can be estimated from $H_{\rm inf}$ whose value is fixed using COBE normalization. The estimate on the temperature is found to be $T_{\rm{end}}\simeq 1.99r^{1/2}\times 10^{15}$GeV. We certainly need a reheating mechanism other than the standard one which is not operative in case of quintessential inflation. The gravitational particle production is a quantum mechanical process of particle creation from vacuum. The leading contribution to energy density of created particles in this process comes from the epoch when the transition from acceleration to deceleration takes place, namely, from the end of inflation. Using the estimate for $H_{\rm end}\simeq H_{\rm inf}$, we estimated the temperature of radiation created during this process. The radiation temperature so estimated turns out to be less than $T_{\rm end}$ challenging the nucleosynthesis constraint.

In order to set the goal, we implemented the instant preheating which is based upon the assumption that $\phi$ interacts with an auxiliary scalar field $\chi$ with the coupling $g$ which then interacts with matter field with the coupling strength $h$. We have found a wide range in the parameter space, $(g,h)$ which allows to obtain the desired temperature (see Fig. (1)).

The generic feature of the paradigm of quintessential inflation includes relic gravitational wave background with blue spectrum produced during the transition from inflation to kinetic regime. In Fig. (2), we depict spectral energy density of relic gravitational waves for the temperature at the end of inflation which we have estimated model independently as well as for the temperatures we have calculated from the model we have considered. We have also plotted the proposed sensitivity curves for advanced LIGO and LISA. As seen in the figure, the blue spectrum appears at high frequencies which clearly distinguishes the paradigm of quintessential inflation from conventional inflation.

We show that the aforesaid requirements of quintessential inflation, which are model independent, can be successfully met by a model based upon the inflaton potential, $V\propto Exp(-\lambda\phi^{n}/M_{\textrm{Pl}}^{n}),(n>1)$. This potential has an interesting property, namely, its slope goes as $\phi^{n-1}$ giving rise to slow-roll for small $\phi$ whereas it exhibits steep behavior at late stages for large values of the field. As demonstrated in Refs. Geng:2015fla ; Geng:2017mic , the model under consideration leads to a viable post-inflationary dynamics. In particular, scaling solution is an attractor of the dynamics, namely, an approximate scaling solution, despite $n>1$ in (43). In this case, the late-time exit from scaling regime to cosmic acceleration may be successfully realized by invoking non-minimal coupling of the field with massive neutrino matter Geng:2015fla ; Geng:2017mic . The present scenario, therefore, is of great interest; it provides with a successful unification of inflation and late-time cosmic acceleration.

ACKNOWLEDGMENTS

We thank Md. Wali Hossain, C. Q. Geng, C. C. Lee and E. N. Saridakis for useful discussions. S. Ahmad acknowledges DST, Govt. of India for financial support through Inspire Fellowship (DST/INSPIRE FELLOWSHIP/2012/614). MS thanks the Eurasian International Center for Theoretical Physics, Eurasian National University, Astana for hospitality where the work was initiated. MS also thanks Maulana Azad National Urdu University, Gachibowli, Hyderabad where part of the work was completed.