# A comparison between axion-like and power law potentials in cosmological
background

Md. Wali Hossain  0000-0001-6969-8716<EMAIL_ADDRESS>Department of
Physics, Jamia Millia Islamia, New Delhi, 110025, India  Afaq Maqsood
<EMAIL_ADDRESS>Department of Physics, Jamia Millia Islamia, New
Delhi, 110025, India

###### Abstract

In this paper, we compare the scalar field dynamics in axion-like and power
law potentials for both positive and negative values of the exponents. We find
that, for positive exponents, both the potentials exhibit similar scalar field
dynamics and it can be difficult to distinguish them at least at the
background level. Even though the potentials are oscillatory in nature for
positive exponents scaling solutions can be achieved for larger values of the
exponent for which the dynamics can be different during early times. Because
of the presence of this scaling nature there is a turnaround in the values of
the scalar field equation of state as we increase the values of the exponent
in both the potentials. This indicates the deviation from the oscillatory
behaviour for the larger values of the exponent. For negative values of the
exponent, the dynamics of the scalar field is distinguishable and axion-like
potential can give rise to cosmologically viable tracker solutions unlike the
power law potentials. So, while for positive exponents we may not distinguish
the two potentials for negative exponents the dynamics of the scalar field is
distinguishable.

###### pacs:

98.80.-k, 95.36.+x, 98.80.Es

##  I Introduction

Recent cosmological observations suggest that our present universe is
expanding with an acceleration [1, 2, 3, 4]. The reason of the late time
acceleration still remains a mystery but the cosmological constant (CC),
$\Lambda$, has appeared as the simplest yet the most viable explanation
especially by the observations of cosmic microwave background (CMB)
radiation[5, 3]. However, the CC is plagued with two issues, namely the fine-
tuning problem [6] and the cosmic coincidence problem [7, 8]. This motivates
us to look for other possible solutions, e.g., making the dark energy
dynamical [9, 10] or modifying the gravity [11]. In this paper, we study the
previous case. Considering a scalar field instead of CC makes the equation of
state (EoS) dynamical. We study the cosmological dynamics of scalar field for
two specific potentials, axion-like [12, 13, 14, 15] and power law potentials
[16].

Slowly rolling scalar fields, known as the quintessence field [16, 17, 18],
can also explain the late time acceleration [9]. Scalar fields can also solve
the cosmic coincidence problem for some specific scenarios known as the
tracker models [7, 8] in which the scalar field energy density tracks the
background energy density in the past and takes over matter during the recent
past. A power law potential can give rise to tracker solutions for larger
negative values of the exponent [7, 8]. But the problem with this scenario is
that the EoS of the scalar field can be much greater than $-1$ at present for
some potentials like power law potentials [8]. But the advantage is that the
late time solution is independent of a wide range of initial conditions as it
is an attractor solution which solves the coincidence problem. There is
another class of scalar field models known as the thawing models [19, 20]. In
this scenario the scalar field behaves as the CC in the past and starts
evolving from the recent past which ensures the difference with the CC. Apart
from these two dynamics we can also have scaling behaviour for steep
exponential potential in which the scalar field scales the background energy
density [21]. It’s an attractor solution, so the scalar field scales forever
in this dynamics. To exit from attractor scaling behaviour we need to modify
the potential, e.g., double exponential potential [22] or cosh potential [23]
can give scaling behaviour followed by late time acceleration. This dynamical
behaviour is generally known as the scaling-freezing dynamics.

Even though the CC is the most suitable explanation of late time acceleration
recent tensions in cosmology [24] e.g., tension in the measurements of present
value of Hubble parameter ($H_{0}$) [25, 26] and the tension in the growth
rate known as the $S_{8}=\sigma_{8}\sqrt{\Omega_{\mathrm{m}0}/0.3}$ tension
[24, 27], where $\sigma_{8}$ is the standard deviation of matter density
fluctuations at present for linear perturbation in spheres of radius
$8h^{-1}{\rm Mpc}$ and $\Omega_{\mathrm{m}0}$ is the present value of matter
density parameter, raise the question about the validity of the standard
$\Lambda$CDM model. The $\sim 5\sigma$ tension between the local measurement
of $H_{0}$ by the SH0ES team ($H_{0}=73.04\pm 1.04~{}{\rm
km~{}sec^{-1}~{}Mpc^{-1}}$) [25] and its constraint coming from the
observation of cosmic microwave background (CMB) radiation assuming the
standard $\Lambda$CDM model ($H_{0}=67.4\pm 0.5~{}{\rm
km~{}sec^{-1}~{}Mpc^{-1}}$) [3] has opened a new challenge in cosmology. The
discrepancy between the two measurements can be due to systematic errors or
some new physics is there [26, 28, 29, 30, 31]. To explain the tension with
new physics it is very clear that we have to go beyond $\Lambda$CDM. In this
regard, scalar fields can play an interesting roll as we can evade the
tension, at least to some extent, by modifying the cosmic history either
during early times or late times [26, 30, 32, 33].

Modifying the early expansion of the universe can be done by early dark energy
(EDE) scenario [14, 15, 34, 35, 36] where the scalar field density parameter
($\Omega_{\phi}$) has a small but finite value ($<0.1$) during the matter
radiation equality. Among the different EDE models [34] two scenarios have
gained a lot of interest, one is the axion-like potential
$\sim(1-\cos\phi)^{n}$ [12, 14, 15, 34, 35, 13] and the another one is the
power law potential $\sim\phi^{2n}$ [16, 36], where $n$ is a constant. The
axion-like potential has minima and around the minima the potential behaves
like $\phi^{2n}$, i.e., power law potential for positive values of $n$ . So,
Axion-like potential has power law behaviour around the minima which should
give similar dynamics of the scalar field during the late time when the scalar
field rolls near the minima for positive values of $n$. Now, the question
remains whether the scalar field dynamics is different in the two potentials
during the early times when the scalar field is away from the minima. And if
it is, then is it significant? We address these issues in this paper. We also
explore the dynamics for negative values of $n$. Even though the axion-like
potential reduces to a power law potential around the minima for positive
values of $n$ the dynamics of the scalar field can be different for negative
values of $n$ as the axion-like potential still has a minima but the power law
is a runaway potential. This difference in the nature of the potentials can
lead to distinguishable features in the cosmological dynamics and therefor we
examine this in detail. So, in this paper, our aim is to compare the
cosmological dynamics in two widely considered potentials and try to see if
there is any degeneracy in the dynamics. In this regard, we also want to
explore the effects of the functional form and the values of the functions
$\lambda=V_{\phi}(\phi)/V(\phi)$ and
$\Gamma=V_{\phi\phi}(\phi)V(\phi)/V_{\phi}(\phi)^{2}$, where $V(\phi)$ is the
potential and the subscript $\phi$ denotes the derivative with respect to
(w.r.t.) $\phi$, on the scalar field dynamics. So, in other words, we want to
see whether the dynamics of the scalar field in the cosmological background
depends on the nature of the potential solely or the values and forms of the
functions $\lambda$ and $\Gamma$ mainly decides the the dynamics irrespective
of the explicit functional form of the potential. To study this we consider
two potentials, axion-like and power law, phenomenologically as examples. To
compare the dynamics in the two potentials we first study the fixed points and
their stability which turns out to be similar and does not give us any
distinguishable information. So, we analyse each scenario numerically. We also
compare the EDE solutions in both the potentials and we argue that the EDE
solutions are almost identical and we don’t expect anything distinguishable.

We introduce the scenario with the background cosmological equations in Sec.
II. In Sec. III we study the stability of the fixed points by analysing the
dynamical system. The comparison has been done numerically in Sec. IV in which
we first study the nature of potentials in Sub Sec. IV.1. Then we study the
dynamics for positive values of $n$ in Sub Sec. IV.2. The study for negative
values of $n$ has been done in Sub Sec. IV.3. The comparison in EDE scenario
for both the potentials has been done in Sec. V. We finally summarise and
conclude in Sec. VI.

##  II The Model

We consider the following action of a minimally coupled canonical scalar field

$\displaystyle\mathcal{S}=\int\mathrm{d}^{4}x\sqrt{-\mathrm{g}}\Bigl{[}\frac{M_{\textrm{Pl}}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\Bigr{]}+\mathcal{S}_{\rm
B}\,,$ (1)

where $M_{\textrm{Pl}}=1/\sqrt{8\pi G}$ is the reduced Planck mass, $V(\phi)$
is the potential of the field and $\mathcal{S}_{\rm
B}=\mathcal{S}_{\mathrm{m}}+\mathcal{S}_{\mathrm{r}}$ with
$\mathcal{S}_{\mathrm{m}}$ and $\mathcal{S}_{\mathrm{r}}$ being the actions
for matter and radiation respectively.

Variation of the action (1) w.r.t. the metric $g_{\mu\nu}$ gives the
Einstein’s field equation

$\displaystyle
M_{\textrm{Pl}}^{2}G_{\mu\nu}=T_{(\mathrm{m})\mu\nu}+T_{(\mathrm{r})\mu\nu}+T_{(\phi)\mu\nu}\,,$
(2)

where

$\displaystyle T_{(\phi)\mu\nu}=$
$\displaystyle\phi_{;\mu}\phi_{;\nu}-\frac{1}{2}\mathrm{g}_{\mu\nu}(\nabla\phi)^{2}-\mathrm{g}_{\mu\nu}V(\phi)\,,$
(3)

is the energy-momentum tensor of the scalar field. Variation w.r.t. the scalar
field $\phi$ gives the equation of motion of the scalar field

$\displaystyle\Box\phi-\frac{\mathrm{d}V}{\mathrm{d}\phi}=0\,,$ (4)

In flat Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, with metric

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}~{},$ (5)

where $a(t)$ is the scale factor, the Friedman equations are given by

$\displaystyle 3H^{2}M_{\textrm{Pl}}^{2}$ $\displaystyle=$
$\displaystyle\rho_{\mathrm{m}}+\rho_{\mathrm{r}}+\frac{1}{2}\dot{\phi}^{2}+V(\phi)\,$
(6) $\displaystyle\left(2\dot{H}+3H^{2}\right)M_{\textrm{Pl}}^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{3}\rho_{\mathrm{r}}-\frac{1}{2}\dot{\phi}^{2}+V(\phi)\,,$
(7)

where, $H=\dot{a}/a$, is the Hubble parameter with dot representing the
derivative w.r.t. time. $\rho_{\mathrm{m}}$ and $\rho_{\mathrm{r}}$ are the
matter and radiation energy densities respectively. The energy density and
pressure of the scalar field are given by

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}+V(\phi)\,,$ (8) $\displaystyle
p_{\phi}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\dot{\phi}^{2}-V(\phi)\,.$
(9)

The equation of motion of the scalar field has the standard form

$\ddot{\phi}+3H\dot{\phi}+\frac{\mathrm{d}V}{\mathrm{d}\phi}=0\,.$ (10)

In this paper we are going to study the background cosmological dynamics of
the scalar field for an axion-like potential given by [12, 14, 15, 35]

$V(\phi)=V_{0}\left(1-\cos\left(\frac{\phi}{f_{\rm pl}}\right)\right)^{n}\,,$
(11)

where $V_{0}$, $f_{\rm pl}$ and $n$ are constants, and compare the same for
the power law potential [16]

$V(\phi)=V_{0}\left(\frac{\phi}{f_{\rm pl}}\right)^{2n}\,.$ (12)

The axion-like potential (11) is periodic in nature which repeats itself after
one full cycle or time period. So, by any simple transformation in the scalar
field we can go from one cycle to any cycle which does not affect the dynamics
of the scalar field. That means, we can study the dynamics of the scalar field
in one cycle as the dynamics will be same in each full cycle. Also, as a
periodic function the axion-like potential has maxima and minima and around
the minima the axion-like potential reduces to power law potential. In the
following discussion of this paper we compare the cosmological dynamics of the
scalar field in these two potentials for both positive and negative values of
the parameter $n$.

##  III Dynamical system analysis

To compare the dynamics of the scalar field in the axion-like (11) and power
law (12) potentials we first study the dynamical system for the axion-like
potential and analyse the nature of stability of the fixed points. This gives
us the idea about the stability of the background cosmological solutions for
the axion-like potential (for power law potential see Ref. [37]). So, in this
section, we examine whether we get any specific fixed points which can
distinguish the dynamics of the scalar field for the axion-like potential from
the power law potential. To form the dynamical system we define the following
dimensionless variables

$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}}{\sqrt{6}HM_{\textrm{Pl}}}\;,$ (13)
$\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{\sqrt{V}}{\sqrt{3}HM_{\textrm{Pl}}}\;,$ (14)
$\displaystyle\Omega_{\mathrm{r}}$ $\displaystyle=$
$\displaystyle\frac{\rho_{r}}{3H^{2}M_{\textrm{Pl}}^{2}}\ ;,$ (15)
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle-
M_{\textrm{Pl}}\frac{V_{\phi}(\phi)}{V}\;,$ (16) $\displaystyle\Gamma$
$\displaystyle=$
$\displaystyle\frac{V_{\phi\phi}(\phi)V(\phi)}{V_{\phi}(\phi)^{2}}\;.$ (17)

While the variable $\lambda$ signifies the slope of the potential the variable
$\Gamma$ represents the nature of the potential, e.g., $\Gamma=1$ for an
exponential potential of constant slope $\lambda$. For the axion-like
potential (11),

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-\frac{n}{f}\cot\left(\frac{\phi}{2f_{\rm pl}}\right)\,,$ (18)
$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2n}-\frac{n}{2f^{2}\lambda^{2}}\,,$ (19)

where, $f=f_{\rm pl}/M_{\textrm{Pl}}$. For power law potential (12)

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-\frac{2n}{(\phi/M_{\textrm{Pl}})}\,,$ (20)
$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle 1-\frac{1}{2n}\,.$ (21)

As the function $\Gamma$ represents the nature of the potential and from Eqs.
(19) and (21) we can see that the difference in the functional form of
$\Gamma$ for the two potentials is the last term of Eq. (19). This term can be
negligible for large values of the slope $\lambda$ which makes both the
$\Gamma$s identical. So, the scalar field dynamics is expected to be similar
for both the potentials though it will also depend on the values of $\lambda$.
Now it would also be interesting to see whether the last term of Eq. (19)
makes any significant difference in the dynamics of the scalar field. In this
section we look for these answers by doing fixed points analysis. In the next
section we discuss the effects of the nature and values of the functions
$\lambda$ and $\Gamma$ on the dynamics of the scalar field for both the
potentials.

From Eq. (6) we see that the dimensionless variables follow the constraint
equation $x^{2}+y^{2}+\Omega_{\mathrm{r}}+\Omega_{\mathrm{m}}=1$, where
$\Omega_{\mathrm{m}}=\rho_{m}/3H^{2}M_{\textrm{Pl}}^{2}$ is the density
parameter of matter. The density parameter of the scalar field is defined as
$\Omega_{\phi}=x^{2}+y^{2}$ while $\Omega_{\mathrm{r}}$ is that for radiation.
Effective EoS and the EoS of the scalar field, in terms of the dimensionless
variables, are given by

$\displaystyle w_{\text{eff}}$ $\displaystyle=$
$\displaystyle-\left(1+\frac{2}{3}\frac{\dot{H}}{H^{2}}\right)=x^{2}-y^{2}+\frac{1}{3}\Omega_{\mathrm{r}}\;,$
(22) $\displaystyle w_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)}=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\;.$
(23)

Table 1: Fixed points of the autonomous system (29)-(32) with their nature of
stability and the values of different cosmological parameters at the fixed
points are listed.

Point | $x$ | $y$ | $\Omega_{\mathrm{r}}$ | $\xi$ | Eigenvalues | Existence | Hyperbolicity | Stability | $\Omega_{\mathrm{m}}$ | $\Omega_{\phi}$ | $w_{\phi}$ | $w_{\rm eff}$
---|---|---|---|---|---|---|---|---|---|---|---|---
A | 0 | 0 | 1 | Any | 0, $2(1-\xi)$, $1-\xi$, | All $n$ | Non-hyperbolic | Saddle for $\xi\neq 1$ | 0 | 0 | Indeterminate | 1/3
|  |  |  |  | $-(1-\xi)$ |  |  |  |  |  |  |
B | 0 | 0 | 0 | Any | 0, $\frac{3}{2}(1-\xi)$, $-(1-\xi)$, | All $n$ | Non-hyperbolic | Saddle for $\xi\neq 1$ | 1 | 0 | Indeterminate | 0
|  |  |  |  | $-\frac{3}{2}(1-\xi)$ |  |  |  |  |  |  |
C | $x$ | 0 | $\Omega_{\mathrm{r}}$ | 1 | 0, 0, $-\frac{1}{n}\sqrt{\frac{3}{2}}x$, | Finite $n$ | Non-hyperbolic | Stable for $n>0$ | $1-x^{2}-\Omega_{\mathrm{r}}$ | $x^{2}$ | 1 | $x^{2}+\frac{1}{3}\Omega_{\mathrm{r}}$
|  |  |  |  | $-\sqrt{\frac{3}{2}}x$ |  |  | and $x>0$ (Fig. 1) |  |  |  |
|  |  |  |  |  |  |  | Saddle for $x<0$ |  |  |  |
|  |  |  |  |  |  |  | or $n<0$ and $x>0$ |  |  |  |
|  |  |  |  |  |  |  | Unstable for $n>0$ |  |  |  |
|  |  |  |  |  |  |  |  and $x<0$ |  |  |  |
$D_{\pm}$ | 0 | $\pm 1$ | 0 | 0 | -4, -3, $-\frac{3}{2}(1+\sqrt{1+\frac{2}{3}n})$, | All $n$ | Hyperbolic | Stable for $n<0$ | 0 | 1 | -1 | -1
|  |  |  |  | $-\frac{3}{2}(1-\sqrt{1+\frac{2}{3}n})$ |  |  | Saddle for $n>0$ |  |  |  |
|  |  |  |  |  |  |  | (Fig. 2) |  |  |  |

Evolution of the universe is represented by the equations (7) and (10), which
can be recast in terms of the evolution equations of the dimensionless
variables (13)-(16) by the following autonomous system,

$\displaystyle\frac{dx}{dN}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}x(1-x^{2}+y^{2}-\frac{1}{3}\Omega_{r})+\sqrt{\frac{3}{2}}y^{2}\lambda\;,$
(24) $\displaystyle\frac{dy}{dN}$ $\displaystyle=$
$\displaystyle\frac{3}{2}y(1+x^{2}-y^{2}+\frac{1}{3}\Omega_{r})-\sqrt{\frac{3}{2}}xy\lambda\;,$
(25) $\displaystyle\frac{d\Omega_{\mathrm{r}}}{dN}$ $\displaystyle=$
$\displaystyle-\Omega_{\mathrm{r}}(1-3x^{2}+3y^{2}-\Omega_{\mathrm{r}})\;,$
(26) $\displaystyle\frac{d\lambda}{dN}$ $\displaystyle=$
$\displaystyle\sqrt{6}x\lambda^{2}(1-\Gamma)=\sqrt{\frac{3}{2}}x\left(\frac{\lambda^{2}}{n}+\frac{n}{f^{2}}\right)\;.$
(27)

where $N=\ln a$. Here we should mention that the autonomous system (24)-(27)
is valid for $n\neq 0$ as apparently it seems that the Eq. (27) diverges as
$n\to 0$. For $n=0$ the potential (11) reduces to a constant potential i.e.,
$\lambda=0$. So, the autonomous system will not contain the $\lambda$ variable
for $n=0$.

As we have already constructed the autonomous system we can calculate the
fixed points by equating the equations (24)-(27) to zero. Before that, we
should notice from the Eq. (18) that $0\leq|\lambda|<\infty$, and for
$\lambda\to\infty$ the Eq. (27) diverges. So the autonomous system (24)-(27)
can give us the fixed points as long as $\lambda$ is either zero or finite. To
avoid this divergence problem we define a new variable corresponding to
$\lambda$ as [38, 37]

$\displaystyle\xi=\lambda/(1+\lambda)$ (28)

and rewrite the autonomous system. We can see that $0\leq\xi\leq 1$ for
$0\leq\lambda\leq\infty$ i.e., $\xi$ remains finite even if
$\lambda\to\infty$.

In terms of $\xi$ the last terms of Eqs. (24) and (25) will have factor
$\xi/(1-\xi)$ which blows up for $\xi\to 1$. To avoid this divergence in the
autonomous system we use the transformation $\mathrm{d}N\to(1-\xi)\mathrm{d}N$
[37] which does not change the dynamical behaviour of the system as there will
be an overall multiplication of the term $(1-\xi)$ in the autonomous system
but makes the dynamical system consistent around $\xi=1$ [37]. So the final
dynamical system, consistent around $\xi=1$, becomes

$\displaystyle\frac{dx}{dN}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}x(1-\xi)(1-x^{2}+y^{2}-\frac{1}{3}\Omega_{r})+\sqrt{\frac{3}{2}}y^{2}\xi\;,~{}~{}~{}~{}$
(29) $\displaystyle\frac{dy}{dN}$ $\displaystyle=$
$\displaystyle\frac{3}{2}y(1-\xi)(1+x^{2}-y^{2}+\frac{1}{3}\Omega_{r})-\sqrt{\frac{3}{2}}xy\xi\;,$
(30) $\displaystyle\frac{d\Omega_{\mathrm{r}}}{dN}$ $\displaystyle=$
$\displaystyle-\Omega_{\mathrm{r}}(1-\xi)(1-3x^{2}+3y^{2}-\Omega_{\mathrm{r}})\;,$
(31) $\displaystyle\frac{d\xi}{dN}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}x(1-\xi)\left(\frac{\xi^{2}}{n}+\frac{n(1-\xi)^{2}}{f^{2}}\right)\;.$
(32)

Figure 1: Nature of stability of the point $C$ for $n,x>0$ in the $y-\xi$
plane. The gray dot represents the point $C$ i.e., $(y,\xi)\equiv(0,1)$. We
have considered $n=2$, $x=0.02$ and $\Omega_{\mathrm{r}}=0$.

The fixed points of the autonomous system (29)-(32) with their nature of
stability are listed in the Tab. 1. We can see that the fixed points and their
stability are similar same as the power law potential (12) [37]. The
difference comes in the points $D_{\pm}$ as for the power law potential these
points are non-hyperbolic but for the potential (11) we have hyperbolic points
with same condition on $n$ for the nature of stability. From Tab. 1 we can see
that in the cosmological dynamics of axion-like potential we can have three
stable points same as power law potential [37]. Point $C$ is stable for
$n,x>0$. We have shown this in Fig. 1 in the $y-\xi$ plane by considering
fixed values of $x$ and $\Omega_{\mathrm{r}}$. We can see that if we perturb
the system from the fixed point $(0,1)$ in $y-\xi$ plane then all the
trajectories are moving towards the fixed point $C$ for $n,x>0$ which ensures
the stable nature of the point $C$. Similarly, in the upper figure of Fig. 2
we have shown the stable nature of the fixed points $D_{\pm}$ for $n=-2<0$ in
the $x-\xi$ plane. In this figure all the trajectories are moving towards the
fixed points $D_{\pm}$. On the other hand, the lower figure of Fig. 2 shows
that some of the trajectories are moving towards the fixed points $D_{\pm}$
and some of them are going away from it for $n=2>0$ in the $x-\xi$ plane. This
shows the saddle nature of the fixed points $D_{\pm}$ for positive values of
$n$.

Figure 2: Nature of stability of the point $D$ for $n=-2$ (upper) and $n=2$
(lower) has been shown in the $x-\xi$ plane. The gray dot represents the the
points $D_{\pm}$ i.e., $(x,\xi)\equiv(0,0)$. We have considered $y=1$. Here we
have also shown the region $\xi>1$ to show the nature of the trajectories
properly.

From Tab. 1 we can see, from the fixed point analysis, only $D_{\pm}$ points
can give rise to late time acceleration with $w_{\rm eff}=w_{\phi}=-1$ and
$\Omega_{\phi}=1$ only for $n<0$. For $n>0$ the only stable point is the point
$C$ which can not give rise to late time acceleration. This point, in fact,
can not give any viable cosmological solution as it is an attarctor solution
also in the case of power law potential [37]. Which means, for $n>0$,
apparently, we do not have any late time acceleration solution. But that is
not the case as from the fixed point analysis we do not have the full dynamics
of the system. This can be considered as a limitation of the dynamical system
analysis. This will be clearer in the next section where we discuss the
dynamics with the numerical results by analysing the nature of the potentials
as well as the corresponding functions of $\lambda$ and $\Gamma$.

##  IV Numerical study of the dynamics

In this section we discuss the dynamics of the scalar field qualitatively and
point out the missing information about the dynamics in the fixed point
analysis.

###  IV.1 Nature of axion-like and power law potentials

Figure 3: Blue (solid), red (dot-dashed) and green (dashed) lines represent
the potential (11) normalised by $V_{0}$, its slope (16) and corresponding
$\Gamma-1$ (19), respectively for $n=2$ (upper) and $n=-2$ (lower) with $f=1$.

Figure 4: Blue (solid) and red (dashed) lines represent the potential (12)
normalised by $V_{0}$ and corresponding slope $\lambda$ (20) for $n=1$ (upper)
and $n=-1$ (lower) with $f=1$.

In Fig. 3 we have shown the nature of the axion-like potential (11) and its
corresponding slope $\lambda$ (18) and $\Gamma-1$ (19) for $n=2$ (upper) and
$n=-2$ (lower). Fig. 4 shows the nature of the power law potential (12) and
its slope $\lambda$ for both positive and negative values of $n$. For axion-
like potential (Fig. 3), while the potential is continuous for positive values
of $n$ it has periodic discontinuities for negative values of $n$. But within
two discontinuities the potential is continuous for negative values of $n$
which can give us interesting cosmological solutions. Now, note that, for
axion-like potential (11), the points of maxima of the potential with positive
values of $n$ become the points of minima for negative values of $n$ and the
points of minima for the positive values of $n$ become the points of
discontinuity for negative values of $n$. So, for both the cases the potential
has minima. Now, any minima at $\phi=\phi_{0}$ can be shifted to the origin by
the transformation $\phi\to\phi\pm\phi_{0}$. For positive values of $n$,
around the minima we can approximate the axion-like potential (11) as the
power law potential (12). If we compare the upper figures of Fig. 3 and 4 we
can see that in one cycle, around $\phi=0$, the axion-like potential (11) has
similar behaviour as the power law potential (12) including the nature of
their slopes. For $\phi>0$, in the half cycle of axion-like potential (11),
the slope varies from zero (at the maxima) to $-\infty$ (at $\phi=0$) which is
similar to the power potential for $\phi>0$. The same similarity is there in
the nature of the slopes for $\phi<0$. So, even though these two potentials,
apparently, seem different as the axion-like potential (11) is periodic in
nature but the power law potential (12) is not, they have similarities in one
cycle of the axion-like potential for positive values of $n$. These
similarities in the nature lead to the similarity in the cosmological dynamics
of the scalar field in both the potentials.

On the other hand, for negative values of $n$, the nature of the potentials
and the corresponding functions of $\lambda$ and $\Gamma$ are different. From
the lower figure of Fig. 3 we can see that the axion-like potential has minima
but from the lower figure of Fig. 4 we see that the power law potential is a
runaway potential. These figures also show the difference in the slope of the
potentials. But the important difference comes in the form of the function
$\Gamma$ as it becomes very large near the minima and becomes smaller as we go
away from the minima for the axion-like potential (lower figure of Fig. 3)
while it is a constant for power law potential (Eq. (21)). This gives a major
difference between the two potentials and we shall see that this nature of the
function $\Gamma$ can lead to viable tracker solutions for axion-like
potential unlike the power law potential [7, 8]. In this regard, the slope of
the potential also has an important role to paly which we shall also discuss.

###  IV.2 Positive $n$ and scaling solutions

Figure 5: Matter (long dashed green), radiation (short dashed red) and scalar
field (solid blue (for $n=3$) and dashed purple (for $n=20$)) energy
densities, normalised with the present value of critical density $\rho_{\rm
c0}$, are shown for axion-like potential (11) (left) and power law potential
(12) (lower). Same initial conditions have been considered by fixing the
initial value of the field at $\phi_{i}/M_{\textrm{Pl}}=\pi f-0.004$ which is
very close to the point of maxima for axion-like potential (11) and
$\dot{\phi}_{i}/M_{\textrm{Pl}}=10^{-5}$ with $V_{0}=10^{6}\rho_{\rm c0}$ and
$f=0.1$.

In both the potentials, for positive values of $n$, the scalar field will
oscillate around the minima. During this oscillation we can show that for a
canonical scalar field $<\frac{1}{2}\dot{\phi}^{2}>=n<V(\phi)>$, where the
symbol $<\dots>$ represents average value over one cycle [39, 40]. So, the
average EoS of the scalar field becomes

$<w_{\phi}>=\frac{n-1}{n+1}\,.$ (33)

For, $n=1$ we have $<w_{\phi}>=0$, i.e., $\rho_{\phi}\sim a^{-3}$ like matter
while for $n=2$ we have $<w_{\phi}>=1/3$ giving rise to $\rho_{\phi}\sim
a^{-4}$ like radiation. When $n\ll 1$ we can have $<w_{\phi}>\approx-1$ which
may give rise to accelerated phase of the universe. So, we can have different
scalar field dynamics depending upon the values of $n$.

The function $\Gamma$, which tells us about the nature of the potential, plays
an important role to determine the scalar field dynamics in the cosmological
background. $\Gamma=1$ for exponential potential with constant slope which
gives rise to scaling solution for $\lambda>\sqrt{3}$ [21]. For Tracker
dynamics we need $\Gamma>1$ [8]. From the upper figure of Fig. 3 we see that,
for positive $n$, around the minima of the potential, $|\Gamma-1|$ is also
minimum and as we increase the value of $n$ it gets closer to zero around the
minima which can be understood from Eq. (19). So, as we increase the value of
positive $n$ there is a possibility to get scaling solution near the minima as
in this case $\Gamma$ will be close to 1 and $\lambda$ is very large which
means the potential can behave like a steep exponential one giving rise to
scaling behaviour. This same argument is also valid for the power law
potential (12). From Eq. (21) we see that $\Gamma\approx 1$ for large positive
values of $n$ and from the upper figure of Fig. 4 we see that the slope
$\lambda$ is large around the minima. So, also in power law potential we can
get scaling solutions for larger positive values of $n$. In Fig. 5 we have
shown the evolution of the scalar field energy density $\rho_{\phi}$ along
with matter and radiation energy densities for $n=3$ (dashed purple line) and
$n=20$ (solid blue line) for both the potentials (upper one for axion-like and
lower one for power law potential). We can see the similarity in the evolution
of $\rho_{\phi}$ in both the potentials for $n=3$. This can easily be
understood from Eq. (19) which tells us that when $\lambda$ is very large the
function $\Gamma$ becomes same for both the potentials. So, the axion-like
potential basically reduces to a power law potential and gives rise to similar
dynamics. We can also see that during the late times $\rho_{\phi}$ decays
faster than radiation which follows Eq. (33).

We have already argued that larger positive values of $n$ give rise to scaling
solutions. Here, the solid blue line of Fig. 5, for $n=20$, shows the scaling
behaviour of the scalar field dynamics for both the potentials with same
initial conditions. In both the figures, $\rho_{\phi}$ follows the matter
during lower redshifts which gives $w_{\phi}\approx 0$. This behaviour is an
attractor solution. This tells us that the behaviour of the scalar field
dynamics during the lower redshifts is same in both the potentials except that
the power law potential reaches the attractor solution earlier than the axion-
like potential for same initial conditions. The reason behind this can be
understood from Eqs. (19) and (21) which tell us that the difference between
the function $\Gamma$ of the two potentials is the last term of Eq. (19). For
finite values of $n$, this term is negligible only when $\lambda\to\infty$. As
around the maxima of axion-like potential $\lambda$ is very small and the last
term of Eq. (19) remains non-negligible which makes $\Gamma<1$ i.e. away from
behaving like exponential potential which means away from scaling solutions.
So, if we fix the initial conditions around the maxima, which is the case for
Fig. 5, the scalar field will take some time to reach the scaling solutions
until the slope $\lambda$ becomes sufficiently large. Until then it will
follow the dynamics which will be sensitive to the initial conditions. And, if
we consider very large values of $n$ e.g., $n=20$, before reaching the scaling
solutions, $\rho_{\phi}\sim a^{-6}$ which makes $\rho_{\phi}$ much smaller
than the background energy density very fast which causes large Hubble
damping. Because of this damping the scalar field freezes to evolve causing it
to behave like a CC. This CC like behaviour increases its energy density and
it again starts evolving. Now, if $\lambda$ is very large and $\Gamma\neq 1$,
$\rho_{\phi}$ can repeat this behaviour and this is what is happening in the
evolution of $\rho_{\phi}$ in the upper figure of Fig. 5 for $n=20$ before
reaching the attractor scaling solutions, i.e., at the higher redshifts. This
difference in the dynamics at higher redshifts comes only for very large
values of $n$. For $n=3$, Fig. 5 shows that the dynamics is very similar in
both the potentials. Also, we can reduce this difference even for large values
of $n$ by choosing suitable initial conditions. These similarities in the
dynamics suggest us that rather than the actual functional form of the
potential the form and the values of the functions $\lambda$ and $\Gamma$
determine the dynamics of the scalar field irrespective of the potential. So,
there can be degeneracy in the dynamics of the scalar field for different
potentials as we have seen in this case.

Figure 6: Trajectories in the $\phi-\phi^{\prime}$ plane are shown for the
axion-like potential with $n=2,\;3,\;5\;{\rm and}\;6$. Same initial conditions
($\phi_{i}=(\pi f-0.1)M_{\textrm{Pl}}$ and
$\phi^{\prime}=10^{-5}M_{\textrm{Pl}}$), $f=0.1$ and $V_{0}=10^{6}\rho_{\rm
c0}$ have been considered for all the figures.

Here, we have another interesting thing to notice that the Eq. (33) suggests
that for large positive values of $n$, $w_{\phi}\to 1$. But, for $n=20$,
instead of having $w_{\phi}\approx 1$ we have $w_{\phi}\approx 0$ which does
not follow Eq. (33). This suggests that for larger positive values of $n$ the
oscillatory nature of the potentials may not be valid. In fact, numerically,
we have seen that as we increase the positive values of $n$ the scalar field
EoS follows the Eq. (33) and increases towards 1 initially. But after some
particular value of $n$, $w_{\phi}$ starts returning towards value
$w_{\phi}=0$. So, there is a turning point in $w_{\phi}$ and we have seen that
for $n>3$, $w_{\phi}$ starts returning towards 0. In Fig. 6 we have shown the
trajectories in the $\phi-\phi^{\prime}$ plane, where prime represents
derivative w.r.t. $\ln(1+z)$, for increasing values of $n$, from $n=2$ to
$n=6$ for axion-like potential. We can see that as we increase the values of
$n$ the oscillations reduce and the oscillations seizes to exist for around
$n>5$. This clearly shows that for larger values of $n$ the axion-like or
power law potential becomes non-oscillatory. In Ref. [36] also this nature of
turnaround in $w_{\phi}$ has been shown for power law potential but this
nature is also present in the axion-like potential. The physical reason behind
this is that for larger positive values of $n$ the minima of both the
potentials become wider which makes it less oscillatory.

###  IV.3 Negative $n$ and tracker solutions

Figure 7: (Left:) Matter (short dashed green), radiation (long dashed red) and
scalar field (dotted purple (for $n=-1$) and solid blue (for $n=-6$)) energy
densities, normalised with the present value of critical density $\rho_{\rm
c0}$, are shown. (lower:) Scalar field EoS is shown for $n=-1$ (dotted purple)
and $n=-6$ (dashed blue). Initial conditions are fixed by considering the
present value of matter density $\Omega_{m0}\approx 0.3$ for $f=1$. Both the
figures are for axion-like law potential (11).

Figure 8: (Left:) Matter (long dashed green), radiation (short dashed red) and
scalar field (dotted purple (for $n=-1$) and solid blue (for $n=-6$)) energy
densities, normalised with the present value of critical density $\rho_{\rm
c0}$, are shown. (lower:) Scalar field EoS is shown for $n=-1$ (dotted purple)
and $n=-6$ (solid blue). Initial conditions are fixed by considering the
present value of matter density $\Omega_{m0}\approx 0.3$ and $f=1$. Both the
figures are for power law potential (12).

While we get similar behaviour in the cosmological dynamics of the scalar
field in the axion-like (11) and power law (12) potentials for positive values
of $n$, we shall see, as we discuss further, for negative values of $n$ the
dynamics is very distinguishable in terms of the tracker solutions. In fact,
the axion-like potential has the ability to make the tracker solutions more
viable than the power law potential. For negative values of $n$, from Eqs.
(19) and (21), we can see that $\Gamma-1$ will be symmetrically opposite to
the one for positive $n$ for both the potentials. From the lower figure of
Fig. (3) we can see that, for axion-like potential, $\Gamma-1$ is minimum near
the point of discontinuity and becomes larger as we approach the minima of the
potential where the slope $\lambda$ is also very small. This nature in the
functions $\lambda$ and $\Gamma$ can give rise to interesting dynamics of the
scalar field. As the scalar field rolls down the potential it rolls down from
larger slope to much smaller slope towards the minima which can result to late
time acceleration. At the same time, $\Gamma$ increases and eventually it
becomes $>1$ which can give rise to tracker behavior [8, 7]. Initially, while
rolling down, the value of $\Gamma-1$ was small and if it is close to zero the
potential will behave like an exponential potential with large slope giving
rise to scaling-like behaviour. Towards the bottom of the potential since
$\Gamma-1$ becomes large and $\lambda$ becomes small the scaling-like
behaviour is followed by a slow roll of scalar field which may result to late
time acceleration. As $\Gamma-1$ eventually increases, in this case, we can
not get a perfect scaling behaviour rather we get tracker behaviour in the
scalar field dynamics where the scalar field almost scales the background in
the past and eventually exits from the scaling-like nature and takes over
matter in the recent past (upper figure of Fig. 7). Figs. 7 and 8 show the
evolution of the energy densities and scalar field EoS for axion-like and
power law potentials respectively for $n=-1$ and $n=-6$ with $f=1$ and
$\Omega_{\mathrm{m}0}\approx 0.3$. From both the figures we can see that the
scalar field dynamics possesses the tracker behaviour for $n=-6$. For $n=-1$
we don’t have tracker solutions but the dynamics is more close to thawing
dynamics [19, 20] which gives rise to viable cosmology but can be very
sensitive to initial conditions. From Fig. 7 we see that as we increase the
negative values of $n$ the tracker dynamics becomes more prevalent e.g., for
$n=-6$, same as the power law potential [7] (Fig. 8). But from the lower
figure of Fig. 8 we see that for larger negative values of $n$ the present
value of scalar field EoS ($w_{\phi 0}$) becomes larger than $-1$ which can be
disfavoured by the data [3]. On the other hand, from the lower figure of Fig.
7, we can see that for axion-like potential we get $w_{\phi 0}\approx-1$ even
for larger negative values of $n$ as $\lambda$ becomes very small around the
minima and at some point it becomes close to zero while for power law
potential the slope becomes negligibly small only asymptotically. So, While
for the power law potential the tracker solutions may not be viable for the
axion-like potential the scenario changes drastically which has been depicted
in Fig. 9. Fig. 9 compares the tracker solution for both the potentials for
$n=-6$ and $f=1$ by considering $\Omega_{\mathrm{m}0}\approx 0.3$. Upper
figure of Fig. 9 shows the evolution of the energy densities which shows the
tracker behavior of the scalar field in both the potentials. The lower figure
of Fig. 9 shows the evolution of the scalar field EoS in both the potentials.
It can easily be seen from the lower figure that the EoS of the scalar field
is close the $-1$ at present for axion-like potential while it is away from
$-1$ for power law potential for $n=-6$. If we increase the value of $n$, for
axion-like potential we can still get $w_{\phi}$ close to $-1$ at present but
the for power law potential the difference with $-1$ increases even further.
So, for negative values of $n$ we have different behaviour in the scalar field
dynamics which makes the two potentials distinguishable by making the tracker
dynamics more cosmologically viable for the axion-like potential.

Figure 9: (Left:) Matter (short dashed green), radiation (long dashed red) and
scalar field (dotted purple (for axion-like potential (11)) and solid blue
(for power law potential (12))) energy densities, normalised with the present
value of critical density $\rho_{\rm c0}$, are shown. (lower:) Scalar field
EoS is shown for axion-like potential (11) (dashed purple) and power law
potential (12) (solid blue). Initial conditions are fixed by considering the
present value of matter density $\Omega_{m0}\approx 0.3$ for $n=-6$ and $f=1$.

##  V Early dark energy

Figure 10: Matter (long dashed green), radiation (short dashed red) and scalar
field (solid blue (for $\phi_{i}^{\prime}=0.01$) and dotted purple (for
$\phi_{i}^{\prime}\approx 0$)) energy densities, normalised with the present
value of critical density $\rho_{\rm c0}$, are shown for the axion-like
potential (11) (upper) and power law potential (12) (lower). Both the figures
are for $\phi_{i}=\pi f-0.01$, $n=3$, $f=0.1$ and $V_{0}=10^{6}\rho_{\rm c0}$.
In the figure $\phi_{i}$ and $\phi_{i}^{\prime}$ are normalised with
$M_{\textrm{Pl}}$.

Figure 11: scalar field density parameter for axion-like potential (solid
blue) and power law potential (dashed purple) are shown. upper figure is for
the same initial values of $\phi_{i}=\pi f-0.01$ and $\phi_{i}^{\prime}\approx
0$ with $V_{0}=10^{6}\rho_{\rm c0}$. lower figure is for $V_{0}=2.5\times
10^{4}\rho_{\rm c0}$ and $f=0.3$ with $\phi=\pi f-0.01$ for the power law
potential while these values are same as the upper figure for the axion-like
potential.

EDE solutions can alleviate $H_{0}$ tension to some extent [34] by injecting a
fraction of dark energy around the matter-radiation equality (MReq). This can
be achieved with scalar fields if the scalar field freezes to evolve before
the MReq which leads to a CC like behaviour in the scalar field dynamics and
increases the value of $\rho_{\phi}$. Now, for viable cosmology, the scalar
field can’t remain frozen as this will result to cosmic acceleration much
earlier. So, $\rho_{\phi}$ has to decay and to avoid interfering with the
structure formation it has to decay fast around MReq which has been shown in
the Fig. 10. This nature in the scalar field dynamics can be seen in steep
potentials [41] for which the scalar field eventually exhibits attractor
scaling behaviour as discussed above. In fact, the specific requirements in
the EDE dynamics can be possible with scaling-freezing dynamics [42, 43].

The axion-like and the power law potentials are two widely studied potentials
as EDE solutions. EDE solutions can be achieved for the positive values of $n$
in both the potentials [14, 15, 36]. As we have already argued that for
positive values of $n$ the dynamics in both the potentials is almost same we
expect that the EDE solutions will also be same in both the potentials. In
Fig. 10, the EDE solutions in both the potentials have been shown for same
initial conditions and parameter values. We can see the similarity in the
evolution of $\rho_{\phi}$ in both potentials except in the values of
$\Omega_{\rm EDE}$ which is clear from the upper figure of Fig. 11 where the
evolution of EDE density parameter ($\Omega_{\rm EDE}$) in both the scenario
has been shown. Note that, $\phi=\pi f$ is one of the maxima of the axion-like
potential and we have chosen the initial value of the field, $\phi_{i}=\pi
f-0.01$ which is very close to the maxima. So, this can be nearly maximum
possible difference we can have in the values of $\Omega_{\rm EDE}$ for same
initial conditions and parameter values. The upper figure of Fig. 11 shows the
evolution of $\Omega_{\rm EDE}$ for same initial conditions and parameters and
we can see that the maximum value of $\Omega_{\rm EDE}$ is larger in axion-
like potential which also shows the deviation of the axion-like potential from
the power law behaviour. But as we have already argue that we can tune the
initial conditions and the parameters to achieve similar behaviour in the
$\Omega_{\rm EDE}$ in both the potentials which has been shown in the lower
figure of Fig. 11. In his figure the values of $V_{0}$ and $f$ are different
in two potentials. While $V_{0}$ sets the energy scale $f$ sets the initial
value of $\phi$. So, by changing the value of $V_{0}$ we have achieved the EDE
at the same energy scale and by changing the value of $f$ we change the
maximum value of $\Omega_{\rm EDE}$ as $\Omega_{\rm EDE,max}\approx
V(\phi_{\rm c})$ where $\phi_{\rm c}$ is the field value when field is frozen.
So, in both the potentials we not only have same amount of EDE but their
evolution is also same. And even if it differs than that difference will have
insignificant effect on the background cosmological evolution.

##  VI Discussions and Conclusion

In this paper, scalar field dynamics in the cosmological background has been
studied for axion-like (11) and power law (12) potentials. Both positive and
negative values of the exponents have been considered. These potentials can
give rise to EDE solutions and thus have the potential to reduce the $H_{0}$
tension [34]. This study aims to study the similarities between the two
potentials in scalar field dynamics in a cosmological background. In this
regard, first, we have studied the stability of the background cosmology using
fixed point analysis method. The fixed points and their stability are similar
for both the potentials except for the points $D_{\pm}$ which are hyperbolic
points for axion-like potential unlike the power law potential for which these
points are non-hyperbolic. So, the fixed point analysis does not give us any
new information about the dynamics of the scalar field and points towards the
similarity between the the axion-like and the power law potentials. This
motivates us to study the scenario for both the potentials numerically.

We then numerically studied the background cosmology for both the potentials.
Possible scaling and tracker solutions, for both the potentials, have been
explored. It has been found that for positive values of the exponent $n$ both
the potentials can exhibit scaling solutions for larger values of $n$ and the
dynamics is very similar. We do not have any distinguishable signature in any
potential to differentiate. The reason behind is that, for both the
potentials, for larger values of $n$, $\Gamma$ approaches towards 1 while the
slope $\lambda$ approaches very large values (see Figs. 3, 4). This scenario
can be thought as an exponential potential with large slope
($\lambda>\sqrt{3}$) which gives scaling solution. Fig. 5 shows the scaling
behaviour of the scalar field for the axion-like and power law potentials
respectively. We can see that, for the, initial conditions they can reach the
scaling regime at different time. This can be understood from the expressions
of $\Gamma$. From Eq. (21) we see that for power law potential $\Gamma$ is
close to one for large values of $n$. So, the difference is in the last term
of Eq. (19) which depends on $\lambda$. Now in Fig. 5 the scalar field’s
evolution is frozen twice and it freezes for the first time at high redshifts
for which $n/\lambda^{2}$ is finite. So, even for large values of $n$,
$\Gamma$ may not be close to 1 at high redshifts for axion-like potentials as
the last term of Eq. (19) can still be finite. So, it can repeat the dynamics
for steep potentials [41] until $\lambda$ becomes very large and $\Gamma\to
1$. But by tuning the initial conditions and choosing suitable parameter
values we can make $\lambda$ large even at high redshifts and the dynamics of
the scalar field can be almost non-distinguishable in both the potentials. So,
as long as the background cosmological dynamics is concerned we can say that
the axion-like and the power law potentials can give rise to similar dynamics
of the scalar field for positive values of $n$. In this regard we should also
mention that the two potentials have been studied as EDE solutions [14, 15,
36] and we expect the solutions to be almost degenerate which can also be seen
in ref. [35] where the authors have studied the the oscillatory potentials and
the results for axion-like and power law potentials are very similar.

Unlike the positive values of the exponent $n$ for negative values of $n$ the
background solutions are very distinct. In this scenario both the potentials
can give rise to tracker solutions (see Figs. 7 and 8). For the power law
potential, in tracker dynamics, the background cosmology may not be viable as
the EoS of the scalar field, at present, becomes much larger than $-1$ (lower
figure of Fig. 8). But, interestingly, for axion-like potential the scenario
changes drastically as we get viable cosmology with tracker solution where EoS
of the scalar field is close to $-1$ at present (lower figures of Figs. 7 and
9). This is a distinguishable feature of the axion-like potential for negative
values of $n$. This discrepancy in the cosmological dynamics in the two
potentials arises because of the nature of the $\Gamma$ function. From Eq.
(19) and (21) we can see that the difference in these equations is in the last
term of Eq. (19) which plays a crucial role for negative values of $n$ but
remains almost insignificant for positive values of $n$. As we can see from
the lower figure of Fig. 3 that $\Gamma$ becomes much larger than one at the
bottom of the potential where the slope $\lambda$ becomes very small giving
rise viable tracker solutions for axion-like potential. But for power law
potential $\Gamma$ remains constant and can be close to one for larger values
of $n$ (Eq. (21)) while its slope $\lambda$ (20) becomes close to zero only
asymptotically. In summary, we can conclude that considering the background
cosmological dynamics of the scalar field, for positive $n$, we have
degeneracy among the axion-like (11) and power law (12) potentials but the
degeneracy is broken for negative values of $n$ for which we get viable
tracker dynamics for axion-like potential.

###### Acknowledgements.

MWH acknowledges the financial support from SERB, Govt of India under the
Start-up Research Grant (SRG), file no: SRG/2022/002234.

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