What is needed of a scalar field if it is to unify inflation and late time acceleration?

Scalar dynamics has been used extensively in the development of models for both inflation and late-time acceleration.In this section, we shall focus on dynamics of non-minimally coupled scalar field. Let us consider the action of a minimally coupled scale field $\phi$ which populates the FLRW universe Ratra:1987rm ; Copeland:2006wr ; samrev1 ; samrev2 ,

$$\mathcal{S}=\int{d^{4}x\sqrt{-g}\left[\frac{M_{\textrm{Pl}}^{2}}{2}R+\mathcal{L}(\phi)\right]}\equiv\int{d^{4}x\sqrt{-g}\left[\frac{M_{\textrm{Pl}}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]}$$

(1)

The energy momentum tensor for the scalar field reads,

$$T_{\mu\nu}(\phi)=\partial_{\mu}\phi\partial_{\nu}\phi+g_{\mu\nu}\mathcal{L}(\phi)$$

(2)

such that the field energy density and pressure are given by,

$$T^{0}{{}_{0}}\equiv-\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi);~{}~{}T^{i}{{}_{i}}\equiv p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$$

(3)

For FLRW background dominated by the scalar field energy density, we have,

	$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}\equiv H^{2}=\left(\frac{1}{3M_{\textrm{Pl}}^{2}}\right)\rho_{\phi}$		(4)
	$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{3}{2}(1+w_{\phi})=3\frac{\dot{\phi}^{2}/{2}}{\rho_{\phi}}\Longleftrightarrow\frac{\ddot{a}}{a}=-\frac{1}{6M^{2}_{Pl}}\rho_{\phi}\left(1+3w_{\phi}\right)$		(5)

where the equation of state parameter for the field is defined as,

$$w_{\phi}\equiv\frac{P_{\phi}}{\rho_{\phi}}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)},$$

(6)

which interpolates between $1$ and $-1$. The lower (upper) limit corresponds to the potential (kinetic) energy dominated situation. The field configuration with $w_{\phi}\simeq-1$ or $-\frac{\dot{H}}{H^{2}}\ll 1$ is referred to (quasi) de Sitter for which, $H$ is approximately constant and $a(t)\sim e^{Ht}$ ²²2Expansion has character of acceleration ($\ddot{a}/a>0$ for $w_{\phi}<-1/3$), see Eq.(5).. In this case, field rolls slowly,

$$\dot{\phi}^{2}/2\ll V(\phi)\Longrightarrow\epsilon\equiv 3\frac{\dot{\phi}^{2}/2}{\rho_{\phi}}\simeq\frac{3}{2}\frac{\dot{\phi}^{2}}{V}\ll 1$$

(7)

where $\epsilon$ is one of the slow roll parameters. The (quasi) de Sitter configuration is admitted by the scalar field dynamics as a possible fixed point. Indeed, the scalar field equation that follows from (1) in the FLRW background is,

$$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=0;~{}~{}V_{\phi}\equiv\frac{dV}{d\phi}$$

(8)

where the second term with Hubble rate acts like a friction coefficient. Field equation (8) is equivalent to energy conservation,

$$\dot{\rho_{\phi}}+3H\rho_{\phi}(1+w_{\phi})=0$$

(9)

which formally integrates to,

$$\rho_{\phi}=\rho_{in}e^{-3\int{(1+w_{\phi})\frac{da}{a}}}\,,$$

(10)

with $\rho_{in}$ as a integration constant. For potential energy dominated situation, $\dot{\phi}^{2}/2\ll V$ ($w_{\phi}\simeq-1$), field energy density is approximately constant where as in the opposite case $(\dot{\phi}/2\gg V)$, dubbed kinetic regime, $\rho_{\phi}\sim 1/a^{6}$. Kinetic regime is realized in case of a steep potential where field rolls down its potential fast making the potential energy redundant giving rise to $\rho_{\phi}\sim a^{-6}$.

As mentioned before, we are interested in realizing the quasi de Sitter configuration. To this effect, we notice that when, friction term in Eq.(8) is large,

$$\beta\equiv-\frac{\ddot{\phi}}{3H\dot{\phi}}\ll 1\Longrightarrow~{}~{}\text{Slow roll regime}:~{}~{}3H\dot{\phi}+{V_{\phi}}\simeq 0;~{}~{}H^{2}\simeq\frac{1}{3M^{2}_{Pl}}V(\phi)$$

(11)

where $\dot{\phi}$ is small and adheres to slow roll condition, namely, $\epsilon\ll 1$, which in turn implies that the potential is nearly flat,³³3The acceleration term, $\ddot{\phi}$ in (8) can no longer be be dropped in case of a steep potential.

$$\epsilon=\frac{M^{2}_{Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}$$

(12)

In this case, even if we kick off with a huge acceleration, $\ddot{\phi}$, it is taken care of by the friction term. Thus slow roll dynamics confirms the existence of (quasi) de Sitter fixed point also known as slow roll solution. We should , however, check the consistency of (11) with the field equation (8). Taking the time derivative of slow roll equation (11) and using Eq.(5), we have,

$$\beta=-\frac{\ddot{\phi}}{3H\dot{\phi}}=+\frac{V_{\phi\phi}}{9H^{2}}-\frac{(1+w_{\phi})}{2};~{}~{}V_{\phi\phi}=\frac{d^{2}V}{d\phi^{2}}$$

(13)

Keeping $\beta$ negligible requires,

	$\displaystyle\epsilon=\frac{M^{2}_{Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}\ll 1$		(14)
	$\displaystyle\eta=M^{2}_{Pl}\frac{V_{\phi\phi}}{V}\ll 1,$		(15)

which express the self consistency of slow roll approximation with full scalar field dynamics. Thus, we are hereby led to two necessary conditions for slow roll. The first condition, tells us that the slope of the potential small and potential is shallow ($M_{Pl}|V_{\phi}/V|\ll 1$) where as the second condition is a statement about the smallness of the curvature which ensures that slow roll can be sustained for a long duration ( $\dot{\epsilon}/H\epsilon=\mathcal{O}(\eta,\epsilon$ ) needed to collect required number of e-foldings necessary to address the shortcomings of the hot big bang model. Smallness of $\eta$ is also dictated by the flatness of perturbation spectrum. Thus, the smallness of slow roll parameters ensures the required number of e-foldings and the scale-independent spectrum of adiabatic density perturbations produced during inflation.

As mentioned before, we focus on generic features of the paradigm rather than the concrete models. In what follows, we present the model independent estimates of inflation to be used later. We recall that the tensor-to-scalar ratio, $r$ is defined as the ratio of amplitudes of the tensor and scalar perturbations,

$\displaystyle r\equiv\frac{P_{T}}{P_{S}}$

(16)

which is expressed through $\epsilon$ in the case of canonical scalar field as $r=16\epsilon$. Observations Planck:2018jri put a bound on the tensor to scalar ratio of perturbations, namely, $r\lesssim 0.032$ Tristram:2021tvh which translates into a restriction on the slope of the potential, $\lambda_{s}\equiv-M_{\textrm{Pl}}\frac{V,\phi}{V}\lesssim 0.06$ which is a more tighter bound on the slope than the one imposed by slow roll condition. The scalar power spectrum,

$\displaystyle P_{S}=\left.{\frac{1}{8\pi^{2}}\frac{1}{\epsilon}\frac{H^{2}}{M_{\textrm{Pl}}^{2}}}\right|_{k=aH}=\frac{2}{\pi}\,.\frac{1}{r}\frac{H_{\rm inf}^{2}}{M_{\textrm{Pl}}^{2}}$

(17)

is bound due to CMB observations and adhering to recent findingsPlanck:2018jri ; Tristram:2021tvh , we have $H_{\rm inf}\simeq 5.8\times 10^{-5}\,r^{1/2}M_{\textrm{Pl}}=1.4\times 10^{14}r^{1/2}\rm GeV$ or equivalently, $V_{\rm inf}^{1/4}=0.01\times r^{1/2}M_{\textrm{Pl}}$. At the end of inflation, $\epsilon=1$ which implies that $\dot{\phi}_{\rm end}=V_{\rm end}^{1/2}$ and we have the Hubble parameter at the end of inflation,

$\displaystyle H_{\rm end}=\frac{1}{\sqrt{2}M_{\textrm{Pl}}}V_{\rm end}^{1/2}.$

(18)

Since inflation is a quasi de-Sitter phase of expansion it is expected H does not change much from commencement to the end of inflation. The argument is supported by the observational upper-bound on tensor-to-scalar ratio $r\lesssim.032$ such that the field excursion during inflation is not much. This is supported by numerical calculation for generalized exponential potential which shows that not only $H_{inf}\&H_{end}$ but $H_{inf},H_{end}\&H_{kin}$ are within the same order of magnitude Ahmad:2017itq .

Using Eq. (18) we have the estimation for $V_{\rm end}$,

$\displaystyle V_{\rm end}\approx 2H_{\rm inf}^{2}M_{\textrm{Pl}}^{2}\Rightarrow\rho_{\phi,\rm end}=\frac{3}{2}V_{\rm end}\approx 3H_{\rm inf}^{2}M_{\textrm{Pl}}^{2}\,$

(19)

In the kinetic epoch followed by inflation, the energy densities of the filed and radiation evolve as

	$\displaystyle\rho_{\phi}$	$\displaystyle=$	$\displaystyle\rho_{\phi,\rm end}\left(\frac{a_{\rm end}}{a}\right)^{6}$		(20)
	$\displaystyle\rho_{r}$	$\displaystyle=$	$\displaystyle\rho_{r,\rm end}\left(\frac{a_{\rm end}}{a}\right)^{4}$		(21)

Thus, the ratio of the scale factors at the end of inflation to the commencement of radiation epoch or that of the respective temperatures are obtained by using $\rho_{\phi}(a_{r})=\rho_{r}(a_{r})$ which gives

$\displaystyle\frac{a_{r}}{a_{\rm end}}=\frac{T_{\rm end}}{T_{r}}=\left[\frac{\rho_{\phi,\rm end}}{\rho_{r,\rm end}}\right]^{1/2}\,.$

(22)

The length of kinetic regime depends upon the ratio of field energy density to the radiation energy density at the end of inflation. Efficient reheating process implies smaller value of this ratio corresponding to shorter kinetic regime. Owing to the steep nature of scalar field potential in the post-inflationary era, field runs down its potential fast after inflation ends and in a short while, enters the kinetic. In fact, it is found numerically that for the potential (69) ${H_{\rm inf}}~{},{H_{\rm end}}~{}\&~{}H_{\rm kin}$ are within the same order of magnitude Ahmad:2017itq . We shall use, ${H_{\rm inf}}\simeq{H_{\rm end}}\simeq~{}H_{\rm kin}$ in our estimations for model independent predictions.

Scalar field dynamics in presence of background matter (cold matter/radiation) exhibits distinguished features. Under specific conditions scalar field might mimic background matter $\hat{\rm a}$ la scaling solution which plays an important in model building for dark energy and quintessential inflation. In order to check for scaling behaviour of $\rho_{\phi},$, let us consider evolution equations in the presence of background matterCopeland:2006wr ; samrev1 ; Copeland:1997et ; Liddle:1998xm ; Skugoreva:2019blk

	$\displaystyle H^{2}=\frac{1}{3M^{2}_{Pl}}\left(\rho_{b}+\rho_{\phi}\right)$		(23)
	$\displaystyle\dot{H}=-\frac{1}{2M^{2}_{Pl}}\left((1+w_{B})\rho_{b}+\dot{\phi}^{2}\right)$		(24)
	$\displaystyle\dot{\rho}_{b}+3H\rho_{b}(1+w_{b})=0$		(25)

where $\rho_{b}$ and $w_{b}$ designate energy density and equation of state parameter for background matter respectively. As for the scalar field equation, background dependence is reflected there through Hubble parameter. It follows from continuity equation for background matter that $\rho_{B}\sim a^{-3(1+w_{b})}$. Scaling solution is defined requiring that scalar field evolves exactly as the background matter,

$$\text{ Scaling solution}:~{}~{}w_{\phi}=w_{b};~{}~{}\rho_{\phi}\sim a^{-3(1+w_{b})}$$

(26)

Scaling solution is a specific dynamical arrangement which preserves the ratio of kinetic to potential energy of the field,

$$w_{\phi}\equiv\frac{P_{\phi}}{\rho_{\phi}}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)}=w_{b}\Rightarrow\dot{\phi}^{2}(1-w_{b})=2V(1+w_{b})$$

(27)

We recall from earlier discussion that the interplay between the friction term and the gradient of potential reflects the nature of the solution of the field equation; the scaling solution should impose certain restrictions on their ratio. Indeed, differentiating, (27) left right and using the field equation (8), we have,

$$(3H\dot{\phi}+V_{\phi})\dot{\phi}(1-w_{b})=-(1+w_{b})V_{\phi}\dot{\phi}\Rightarrow-3(1-w_{b})H\dot{\phi}^{2}=2V_{\phi}\dot{\phi}$$

(28)

Eqs (27) and (28) yield two important properties of scaling solutions, namely Skugoreva:2019blk ,

	$\displaystyle X\equiv\frac{\dot{\phi}^{2}}{2V}=\frac{1+w_{b}}{1-w_{b}}$		(29)
	$\displaystyle Y\equiv\frac{V_{\phi}}{\dot{\phi}H}=\frac{3}{2}(w-1)$		(30)

Let us also note that for scaling solution.

	$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{3}{2}(1+w_{b})$		(31)
	$\displaystyle\frac{\ddot{\phi}}{\dot{\phi}H}=-\frac{3}{2}(1+w_{b})\,.$		(32)

These equation readily yield the behaviour of scalar field in the scaling regime Skugoreva:2019blk ; Haro:2019peq ,

$$H(t)=\frac{1}{3(1+w_{b})t};~{}~{}\dot{\phi}\propto t^{-1}\Rightarrow\phi(t)\propto\ln(t)$$

(33)

So far, we did not need any information on the field potential that would support the scaling behaviour. It is interesting to compute the $\Omega_{\phi}$ for scaling solution that would tell us about the fraction of total energy, $\rho_{\phi}$ constitutes and also the nature of the potential. Using Eqs.(29) $\&$ and (30), we find,

$$\Omega_{\phi}=\frac{\rho_{\phi}}{3M^{2}_{Pl}H^{2}}=\frac{1}{3M_{\textrm{Pl}}^{2}}\left(\frac{XY^{2}}{1-w_{b}}\right)\left(\frac{V}{V_{\phi}}\right)^{2}=\frac{3(1+w_{b})}{4\lambda^{2}_{s}};~{}~{}\lambda_{s}\equiv-M_{Pl}\frac{V_{\phi}}{V}$$

(34)

where $\lambda_{s}$ is the slope of the potential. Since $\rho_{\phi}$ scales as the energy density of background matter in the scaling regime, $\Omega_{\phi}=Const$, which in turn implies that the slope of the potential should be constant and consequently, the exponential potential is singled out,

$$V(\phi)=V_{0}e^{-\alpha\phi/M_{Pl}}~{}~{}~{};~{}\lambda_{s}=\alpha$$

(35)

However, an important information is yet missing in our analysis. For instance, if the slope is small, namely, $\alpha^{2}<2$, slow roll condition applies and we have $w_{\phi}<-1/3\to\dot{\phi}^{2}/2<V/2$ and no scaling solution in this case. This brings us to the question of the existence and stability of the scaling solutions we are interested in, which is addressed by dynamical analysis using the autonomous form of evolution equations. Using the following notations Copeland:1997et ⁴⁴4It might look more natural to use the variable $X\&Y$ as we know their values for scaling solution. However, in this case, one requires one more variable $A$ which is directly linked to the field $\phi$. But the equation for this variable decouples from the system and we can analyse equations for $X\&Y$ without bothering about the third equation for the variable $A$. These variables are useful in the analysis of asymptotic scaling solutions which occur in case of steep potentials of variable slope where we have system of three coupled equations for $X,Y$ and $A$.,

$$x\equiv\frac{\dot{\phi}}{\sqrt{6}M_{Pl}H};~{}~{}~{}y\equiv\frac{\sqrt{V}}{\sqrt{3}M_{Pl}H},$$

(36)

one can cast evolution equations, with exponential potential, in the autonomous form,

$$x^{\prime}=f(x,y);~{}~{}~{}y^{\prime}=g(x,y)$$

(37)

along with the Friedman constraint equation,

$$\frac{\rho_{B}}{3M^{2}_{Pl}H^{2}}+x^{2}+y^{2}=1$$

(38)

where prime denotes the derivative with respect to $N\equiv\ln(a)$; $f$ and $g$ are functions of $x$ $\&$ $y$ whose functional forms are not required here. Fixed point of the dynamical system is the one for which, $x^{\prime},y^{\prime}=0$ whose stability is checked looking at the sign of eigen values of the perturbed matrix. Using this analysis, one finds Copeland:1997et ; Liddle:1998xm ; Haro:2019peq ,

$$\text{Scaling solution}:~{}~{}~{}~{}~{}~{}w_{\phi}=w_{m};~{}~{}~{}\Omega_{\phi}=\frac{3(1+w_{b})}{\alpha^{2}}~{}~{}~{}\lambda^{2}_{s}=\alpha^{2}>3$$

(39)

which is an attractor of the dynamical system and this completes the story. In the scaling regime, $\rho_{\phi}$ constitutes a constant fraction of the total energy density. The fraction is smaller for the larger value of the slope of the potential. In fact, lower bound on the slope is fixed by the nucleosynthesis constraint. Exponential potential with slope larger than $\sqrt{3}$ is termed as steep otherwise shallow which also serves as yardstick for an arbitrary potential. Let us note that field dominated solution ($\Omega_{\phi}=1$) is also an attractor for $\alpha^{2}<{3}$ that we are not interested in here. The demarcation is clear: For $\alpha^{2}>3$, system settles into a scaling regime, otherwise a field dominated solution, which is accelerating if $\alpha^{2}<2$.

To summarize, scaling solution follows the background matter being sub-dominant. During radiation era, field energy density scales as radiation energy density ($\rho_{r}$), namely, $\rho_{\phi}\sim a^{-4}$ which adjusts itself to $\rho_{m}$ (energy density of cold matter) after matter domination is established and follows it for ever. Scaling regime is characterized by a fixed kinetic to potential energy ratio, for instance, $\dot{\phi}^{2}/2=V$ in radiation era whereas, $\dot{\phi}^{2}/2=2V$ for matter domination. Last but not least, since scaling solutions are non-accelerating, one requires an additional feature in the potential which would allow exit to late time acceleration. In this framework, $\Gamma=1$ and $w_{\phi}=w_{b}$ for most of the history of Universe and only at late times, transition to acceleration would take place $\hat{\rm a}$ la a perfect tracker In the forthcoming discussion, we shall mention exit mechanism discussed in the literature.

One of the major successes of hot big bang includes the synthesis of light elements, Deuterium (${}^{2}\rm{H}$), Tritium (${}^{3}\rm{H}$) and Helium (${}^{4}\rm{He}$) in the early Universe. When temperature in the Universe was around $1~{}\rm{TeV}$, all the standard model degrees of freedom were in equilibrium. All of them were relativistic and contributed to radiation energy density Kolb:1990vq ; Husdal:2016haj ,

$\displaystyle\rho_{r}=\frac{\pi^{2}}{30}g_{*}(T)T^{4}$

(40)

where $g_{*}(T)$ denotes the effective number of relativistic degrees of freedom in the universe at temperature T. Relativistic species including both the bosons and the fermions contribute to $g_{*}(T)$,

$$g_{*}(T)=\sum_{Bosons}{g_{B}{\left(\frac{T_{i}}{T}\right)^{4}}}+\frac{7}{8}\sum_{Fermions}{g_{F}{\left(\frac{T_{i}}{T}\right)^{4}}},$$

(41)

where $T$ is the photon gas temperature and $g_{B}~{}\&~{}g_{F}$ denote bosonic and fermionic degrees of freedom with mass, $m_{i}\ll T$ ($m_{i}$ is the mass of a particular species). The effective degrees of freedom is defined relative to the photon gas. When temperature dropped to around $1$ $MeV$, the relativistic species included $\gamma$, $e,e^{+}$ and three generations of neutrinos $\nu$ such that, $g_{*}=10.75$. Weak processes such as, $p+e^{-}\leftrightarrow n+\nu_{e}$ then froze out leaving behind a definite ratio of $n/p$ densities. Freezing temperature can be estimated comparing the reaction rate $\Gamma_{int}=G_{F}^{2}T^{5}_{f}$ with the Hubble parameter $H(T)$ Cyburt:2015mya ,

$$G_{F}^{2}T_{f}^{5}=\frac{g_{*}^{1/2}}{\sqrt{3}M_{\textrm{Pl}}}T_{f}^{2}\Rightarrow T_{f}=\left(\frac{g_{*}^{1/2}}{\sqrt{3}G^{2}_{F}M_{\textrm{Pl}}}\right)^{1/3}\simeq 1~{}MeV$$

(42)

where $G_{F}$ is Fermi constant and $T_{f}$ designates the freezing temperature. It should be noted that the $n/p$ density ratio is temperature sensitive, as $n/p\sim Exp(-Q/T_{f}$), where $Q$ denotes the neutron and proton mass difference. Since $T_{f}$ is less than the binding energy of Deuterium, its synthesis should naively begin. However, there is more than a billion photons per proton ($\eta=n_{B}/n_{\gamma}\simeq 6\times 10^{-10}$, $n_{B}$ designated baryon number density) in the universe such that the number of photons whose energy exceeds $T_{f}$ is large and Deuterium does not form due to photo-dissociation. This process is delayed and occurs at a lower temperature, $T_{BBN}\simeq 0.1eV$, which then initiates the chain reaction for Helium formation. Helium abundance in Universe has been measured accurately; it is sensitive to $T_{f}$, for obvious reasons, it also depends upon $\eta$. We should emphasize that $\eta$ is an important input here required for a successful nucleosynthesis. It is further important to note that bringing in any extra relativistic degree of freedom like a scalar field would affect the numerical value of $T_{f}$ and consequently the Helium abundance. The latter should, therefore,impose a constraint on any relativistic degree of freedom in the Universe over and above the standard model of particle physics $-$ nucleosynthesis constraint which, in particular, applies to often used quintessence as well as relic gravity waves. Indeed, for temperatures under consideration,

$$g_{*}=2+\frac{7}{8}\left(4+2N_{\nu}\right)$$

(43)

where the first term is due to photon, first term in the brackets is attributed to electron and positron; $N_{\nu}=3$ in the standard model. Any extra radiation present in the Universe, can be parametrized through $N_{\nu}$Caprini:2018mtu

$$\Delta\rho_{r}=\frac{\pi^{2}}{30}\frac{7}{4}\Delta N_{\nu}T^{4}$$

(44)

where $\Delta N_{\nu}$ is anything over and above the standard model value of $N_{\nu}$ ($N_{\nu}=3$) which should be constrained from the observed primordial abundances and other data. Let us now assume an extra relativistic degree of freedom in the Universe dubbed ”X” with energy density $\rho_{X}$ which should not exceed $\Delta\rho_{r}$ as a necessary requirement for not disturbing the Helium abundance Caprini:2018mtu ,

$$\left(\frac{\rho_{X}}{\rho_{\gamma}}\right)_{1~{}MeV}\leq\frac{7}{8}\Delta N_{\nu}$$

(45)

where $\rho_{\gamma}=(\pi^{2}/15)T^{4}$ is the energy density for photon gas⁵⁵5Given that $\Delta N_{\nu}$ is constrained using primordial abundances and other data, bound (45) is referred to as a ”nucleosynthesis” constraint or ”BBN” bound.. Let us note that Helium abundance depends upon two parameters, $T_{f}$ and $\eta$ which can be used to constrain $\Delta N_{\nu}$. The constraint can be improved considerably by considering, in addition, the Deuterium abundance and CMB data Cyburt:2004yc . However, in this case, we should estimate the the ratio (45) below the $e^{+}e^{-}$ annihilation temperature, $T_{e^{+}e^{-}}$. After electron positron annihilation, $T_{\nu}=(4/11)^{4/3}T$ such that,

$$\left(\frac{\rho_{X}}{\rho_{\gamma}}\right)_{T_{e^{+}e^{-}}}\leq\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Delta N_{\nu}$$

(46)

Observational constraint on $\Delta N_{\nu}$ should put a restriction on the extra degree of freedom not the part of standard model. Remember that in scalar field models with tracking behavior, $\rho_{\phi}\propto a^{4}$during the radiation era, and the fraction of total energy that $\rho_{\phi}$ comprises is determined by the slope of the steep exponential potential responsible for scaling behavior. Using combined data, gives $N_{\nu}<3.2$ at $95\%$ confidence level which implies a bound on the slope of the potentialCaprini:2018mtu ,

$$\left(\frac{\rho_{X}}{\rho_{\gamma}}\right)_{T_{e^{+}e^{-}}}\simeq\Omega_{\phi}|_{T_{e^{+}e^{-}}}=\frac{4}{\alpha^{2}}\lesssim\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\times 0.2\Rightarrow\alpha\gtrsim 10.$$

(47)

It may be noted that in case of thawing models, $\Omega_{\phi}\ll 1$ and the $BBN$ constraint is trivially satisfied. Similarly, for relic gravity waves, $\rho_{GW}$ should satisfy the following bound,

$$\left(\frac{\rho_{GW}}{\rho_{\gamma}}\right)_{T_{e^{+}e^{-}}}\leq\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Delta N_{\nu}$$

(48)

which can be translated into a bound on $\Omega_{GW,0}$,

$$\Omega_{GW,0}=\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Omega_{\gamma,0}\Delta N_{\nu}\simeq 5.6\times 10^{-6}\Delta N_{\nu}$$

(49)

where we used the fact that $\rho_{GW}\sim\rho_{\gamma}\sim a^{-4}$ and also used the observed value of fractional energy density of radiation at the present epoch, $\Omega_{\gamma,0}\simeq 2.47\times 10^{-5}$ Using the combined abundances of Helium and Deuterium with the CMB data $(\Delta N_{\nu}\lesssim 0.2$), we finally have,

$$\Omega_{GW,0}\lesssim 1.12\times 10^{-6}$$

(50)

which puts an upper bound on the kinetic regime or the lower bound on reheating temperature in models of quintessential inflation, see section V for details.

As discussed in the preceding sub-section, scaling solutions are specific to exponential potential, which are either suited to field dominated situations or scaling behaviour but not to both simultaneously. Thus, we might use a potential suitable for inflation that subsequently changes to exponential form soon after inflation ends or think of a non-exponential type of potential capable of addressing the inflationary requirements at an early time and dynamically mimicking the exponential behavior in the post inflationary era.

Exponential potential has a distinguished feature, its slope, $\lambda_{s}\equiv-M_{Pl}V_{\phi}/V$, is constant. In this case, we have two dynamical variable, $x$ and $y$ and correspondingly two autonomous equations. For a potential which not an exponential, slope becomes a variable quantity and one needs an additional dynamical equations for $\lambda_{s}$Copeland:2006wr ; gammascale1 ; Nunes:2000yc ; Ng:2001hs ,

	$\displaystyle x^{\prime}=f(x,y,\lambda_{s});~{}~{}~{}y^{\prime}=g(x,y,\lambda_{s})$		(51)
	$\displaystyle\lambda^{\prime}_{s}=-\sqrt{6}\lambda^{2}_{s}(\Gamma-1)x$		(52)
	$\displaystyle\Gamma=\frac{V_{\phi\phi}V}{V^{2}_{\phi}}$		(53)

where $\Gamma$ is an important field construct which controls the dynamics of the system with a general type of potential. For exponential function, $\Gamma=1$, corresponding to constant slope of the potential making Eq.(52) redundant. In this case, $\rho_{\phi}$ scales as background energy density $\rho_{b}$ giving rise to scaling solution which is an attractor of dynamics. Without the loss of generality, we may consider a class of potentials in which field $\phi$ rolls down the potential from smaller to larger values ($\phi\to\infty$) such that $x>0$. In case of potentials with $\Gamma>1$, for instance,

$$V=V_{0}\frac{M^{n}_{Pl}}{\phi^{n}};~{}~{}\Gamma=1+1/n~{}(n>1),$$

(54)

slope decreases with evolution, see Eq.(52), eventually, giving rise to slow roll and forcing the field to evolve to quasi de Sitter ($\rho_{\phi}\simeq\textrm{const.},w_{\phi}\simeq-1$) which is an attractorNunes:2000yc . While the field is rolling down the steep part of the potential ($\lambda_{s}$ is large), $\rho_{\phi}$ closely follows the background Steinhardt:1999nw ,

$$w_{\phi}=\frac{2\Gamma}{2\Gamma-1}=w_{b}+\frac{2}{n}(1-w_{b})$$

(55)

It should be noted that for generic values of $n$, $w_{\phi}\simeq w_{b}$ and $\Gamma\simeq 1$⁶⁶6Notice that, $w_{\phi}=w_{b}$ for $n\to\infty$ which is not surprising as power law corresponds to exponential in this limit. . However, in this case ($\Gamma>1$), and slope ($\lambda_{s}=n/\phi$ ) decreases with evolution such that the field enters the slow roll regime at late stages,

$$w_{\phi}=-1+\frac{\lambda^{2}_{s}}{3}\to-1,~{}~{}\lambda_{s}\to 0$$

(56)

and can account for late time acceleration. Scalar field solution, that mimics the background in the post inflationary era and only at late stages overtakes it to account for the observed late time acceleration, has been assigned a nomenclature as tracker, see Fig.2(a). Obviously, the class of potentials with $\Gamma>1$ are of interest and have been widely investigated in the literature.

On the other hand, if $\Gamma<1$, slope of the potential increases with evolution such that field runs down its potential faster and faster making the potential energy redundant and naturally ends up in the kinetic regime with, $\rho_{\phi}\sim a^{-6}$. In this case, it follows from (53) that slope increases,

$$\frac{1}{\lambda_{s}}=\int{x(\Gamma-1)}dN$$

(57)

and in process of evolution as $\lambda_{s}$ becomes large and $\Gamma\to 1$. In this case, system would eventually join the scaling track in the asymptotic regime. We shall demonstrate it in an example. On the other hand, during evolution, $\lambda_{s}\to 0$ if we begin from $\Gamma>1$ ( $\Gamma\neq Constant$), and in this case $\Gamma$ increases towards infinity. In a sense, ”$\Gamma$” defines the type of steepness of the potential: Slope tells us about the steepness of the potential, whereas $\Gamma$ expresses whether, with evolution, steepness is unchanged ($\Gamma=1$), increases ($\Gamma<1$), or decreases ($\Gamma>1$). Potentials with $\Gamma<1$ ($\Gamma>1)$ are sometimes referred to in the literature as steeper (shallower) than the exponential potential for brevity. It is interesting to think of slope $\lambda_{s}(\phi)$ as defined over the smooth set of functions $\{V(\phi\}$. Since,

$$\frac{d\lambda_{s}}{d\phi}=-\lambda^{2}_{s}(\Gamma-1)=0\to\Gamma=1,$$

(58)

slope, $\lambda_{s}$, is extremum for an exponential function. The second derivative,

$$\frac{d^{2}\lambda_{s}}{d\phi^{2}}=-\frac{1}{V^{4}_{\phi}}\left(V^{2}_{\phi}(V_{\phi\phi\phi}V+V_{\phi\phi}V_{\phi})-2V^{2}_{\phi\phi}V_{\phi}V\right)$$

(59)

computed for exponential potentials also vanishes. Thus $\Gamma=1$ is saddle point in the functional space which explains the mentioned features of the dynamics.

It is desirable that for most of the history of Universe, field follows the background (scaling behaviour) and only at late stages, it exits the scaling regime, takes over the background to become the dominant component and might mimics dark energy. In what follows, we discuss as how to incorporate this feature in the dynamics.

The formalism of slow roll parameters can be applied to quintessence, keeping in mind that the Hubble rate is not solely defined by scalar field energy density, background matter density also contributes to it.

To be fair, unlike in the case of inflation, slow roll parameters might be useful here only for a broad perspective. An interesting feature of field dynamics is noticed for field rolling down a steep potential when its energy density is negligible compared to background energy density. In this case, field freezes on its potential and waits there for conducive situation to resume motion. Indeed, in case of slow roll⁸⁸8It should be noted that, in the present situation, $-\dot{H}/H^{2}=(1+w_{b})/2$, is not related to slow roll parameter due to the presence of background matter. In case of slowly rolling quintessence, the friction term need not to be large and $\beta$ may not be negligible. For tracker models, $|\beta|\ll 1$, however, for thawing quintessence, $\beta=\mathcal{O}(1)$ and $\beta$ is nearly constant, $\dot{\beta}/\beta H\ll 1$. Unlike the thawing case, the consistency of slow roll gives rise to, $\eta=3(1+w_{b})/2=3/2$ in case of the trackers.

$$\dot{\phi}^{2}/2\ll V\Longrightarrow\epsilon=3\frac{\dot{\phi}^{2}/2}{\rho_{\phi}}\simeq\frac{3}{2}\frac{\dot{\phi}^{2}}{V}=\frac{1}{6}\frac{V^{2}_{\phi}}{VH^{2}},$$

(71)

where we used the slow roll value of $\dot{\phi}$. In what follows, we shall be interested in the case when $\rho_{\phi}\ll\rho_{b}$ such that, $H^{2}\simeq\rho_{b}/3M^{2}_{Pl}$,

	$\displaystyle\epsilon=\frac{M^{2}_{Pl}}{2}\left(\frac{V_{\phi}}{V}\right)^{2}\left(\frac{\rho_{\phi}}{\rho_{B}}\right)\equiv\epsilon_{0}\left(\frac{\rho_{\phi}}{\rho_{B}}\right)$		(72)
	$\displaystyle\epsilon=\frac{\lambda^{2}_{s}}{2}\left(\frac{\rho_{\phi}}{\rho_{b}}\right)$		(73)

Since, we are dealing with a steep potential, $\epsilon_{0}>1$, but interestingly, $\epsilon\ll 1$ as $\rho_{b}\gg V(\phi)$ $-$Hubble damping. Thus, unlike the case of inflation, slow roll becomes operative here due to Hubble damping. However, this is not useful for dark energy as $\rho_{\phi}\ll\rho_{b}$ in this case, but it is certainly useful for the understanding of scalar field evolution in a steep potential⁹⁹9Let us note that this feature is central to thawing models where field is frozen on a shallow potential such that field begins slow roll after it covers from Hubble damping and accounts for late time acceleration. Initial conditions are set specially or tuned allowing it to happen around the present epoch and model parameters are chosen to comply with observation. As a result of Hubble damping, field gets frozen on its potential and waits there till the ratio, $\rho_{\phi}/\rho_{b}$ acquires a specific numerical value ($\rho_{b}\sim a^{-3(1+w_{B})}$, $\rho_{\phi}=Const$) given by Eq.(73) when slow roll is violated ($\epsilon=1$). Field evolution then commences but, hereafter, field dynamics, crucially depends upon the type of steepness of the potential. We shall come back to this point in the sub-section on post inflationary evolution.