# Inflection Point of Minimally Coupled Tachyonic Scalar Field

Jaskirat<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Physics, Lovely Professional University,
Phagwara, 144411, Punjab, India
Maxim Khlopov<EMAIL_ADDRESS>
Virtual Institute of Astroparticle Physics, 75018 Paris, France
Institute of Physics, Southern Federal University, Rostov on Don 344090,
Russia
National Research Nuclear University MEPHI, 115409 Moscow, Russia
Mananendra<EMAIL_ADDRESS>
Centre for Theoretical Physics and Natural Philosophy,
Nakhonsawan Studiorum for Advanced Studies, Mahidol University, Nakhonsawan,
60130, Thailand

###### Abstract

In this paper, we investigate the behaviour of a minimally coupled tachyonic
scalar field at inflection points in an accelerating universe. We consider the
different expansion factors and obtain potentials of tachyonic scalar field.
Inflection points of homogeneous tachyonic scalar field is calculated for
these potentials. We employ the tools of dynamical system analysis for the
considered potentials and obtain the stable points.

##  1 Introduction

The cosmic acceleration is now an established phenomenon. As supported by
several cosmological and astrophysical observations [1, 2, 3, 4, 5, 6, 7, 8,
9, 10] our universe is in the phase of accelerated expansion at the present
stage and generally, it is believed that behind this accelerated expansion
there is a mysterious component with negative pressure dominating $70\%$ of
the universe. One of the good candidates that can explain this acceleration is
”Dark Energy” [11, 12, 13, 14, 15] which is introduced as modified matter in
Einstein’s field equation which sits with the energy-momentum tensor. One of
the remarkable features of dark energy is its negative pressure and repulsive
gravity characteristics led by the equation of state $w_{de}<-1/3$. In
alternative gravity theories, cosmic acceleration can be accounted for by
modifications to the geometrical aspects of the field equation.
Observationally, the data from Planck-2018 CMB [7, 8, 9, 10] proposes that
$70\%$ of the energy budget in the universe is in the form of dark energy
while the remaining $30\%$ of the energy budget of the universe is attributed
to non-relativistic baryonic and non-baryonic pressureless ( $p_{m}=0$) dust
matter with equation of state(EoS) $w_{m}=0$. In recent years, various
candidates have been put forth to elucidate the nature of dark energy, with
the cosmological constant standing out as one of the simplest models,
embodying constant dark energy with an EoS of $w_{\Lambda}=-1$. Despite its
alignment with observational data to a certain accuracy, the $\Lambda$CDM
model grapples with unresolved cosmological constant and coincidence problems
[16, 17, 18, 19, 20]. In response to these limitations, dynamical dark energy
models, particularly those involving scalar fields, have gained prominence
[21, 22, 23, 24, 25, 26] and with scalar field serving as a candidate of
dynamical dark energy dubbed as $\phi$CDM having dynamical EoS $w_{\phi}$.

The nature of dark energy has been the subject of several literature; as
alternatives to the cosmological constant, dynamic dark energy models have
been developed, in which the EoS varies with time. There are many different
dynamical dark energy models like quintessence field having tracker behaviour,
late time accelerating solutions of modified Friedmann equations lead by
modification of gravity, some other models include tachyon models, holographic
dark energy models, chaplygin gas, phantom and k-essence [28, 29, 30, 31, 32,
33, 34, 35].

In our paper, we have focused on the tachyonic field as a candidate for dark
energy. A scalar field of this kind is driven by string theory as the
negative-mass mode of the open string perturbative spectrum, though the
primary use of it in the dark energy sector is phenomenological here in our
paper, we have investigated the behaviour if tachyonic scalar field near
infection point. Recently, the dark energy models with the inflection point in
the context of inflation have been studied by several authors [36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56].

To obtain a contemporary value of EoS $w_{\phi}\approx-1$ i.e. to get
accelerated expansion mechanism of slow rolling is the key ingredient so, by
imposing the approximation in which the kinetic term $\dot{\phi}^{2}/2$ of
tachyon is significantly smaller than the potential $V(\phi)$ at the present
moment i.e. $\dot{\phi}^{2}/2<<V(\phi)$ and this can be obtained by sitting
$\phi$ in an extremely flat region of the potential.We consider the late time
behavior of Tachyon scalar field in the accelerated expansion of the universe,
where few similarities with early inflation. Numerous authors have examined
the cosmic behaviour of the tachyonic scalar field [57, 58, 59], discovering
possibilities for both pure exponential growth and power law expansion. We try
to investigate the possibilities based on the superposition of these forms
because the current data have not yet been able to determine the precise shape
of the evolution of the scale factor.

###  1.1 Dynamical set up

In this section, we present the mathematical framework on which the dynamics
are based. We choose GR to be the background theory. The cosmological
principle is encoded in the FLRW metric when expressed in terms of spherical
co-moving coordinates looks

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\equiv-
dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right).$ (1)

Here we have taken $c=1$. In the above equation, $a(t)$ is the scale factor
and the parameter $k$ is the spatial curvature having values $k=-1,0,1$
corresponding to spatially open, flat, and closed geometry respectively. For
our model, we have taken the effective four-dimensional action for the
tachyonic field which is given as [62, 63]:

$S==\frac{1}{2\kappa^{2}}\int d^{4}xf(\phi)\sqrt{-g}R+\int
d^{4}xV(\phi)\sqrt{-g}\sqrt{\left(1+g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right)},$
(2)

where $V(\phi)$ is the potential of tachyonic scalar field and
$\kappa^{2}=\frac{1}{M^{2}_{p}}=1$ set the 4D Planck scale considering $M_{p}$
is Planck mass as well as function of tachyon field $f(\phi)=1$ under minimal
coupling. One can obtain the energy-momentum tensor for this action as

$T^{(\phi)}_{\mu\nu}=\frac{V(\phi)\partial_{\mu}\phi\partial_{\nu}\phi}{\sqrt{1+g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi}}-g_{\mu\nu}V(\phi)\sqrt{1+g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi}$
(3)

Einstein’s field equation for a spatially flat universe ($k=0$) gives the
Friedmann equations as:

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\kappa^{2}}{3}\rho,$ (4)

and

$\frac{\ddot{a}}{a}=-\frac{\kappa^{2}}{2}(\rho+p),$ (5)

where $H(t)\equiv\frac{da}{dt}$ is the Hubble parameter, $\rho$ and $p$ are
the energy and the pressure respectively. In general, there could be multiple
components of matter contributing to energy-momentum tensor, each of which
with a different equation of state. However, in the post-recombination era
(after $z\longrightarrow 1000)$ and more effectively towards the present epoch
$(z\longrightarrow 0)$ matter and dark energy dominate the universe. We are
therefore driven to focus on only two components: matter and dark energy.
These two components fit together so perfectly that, if one is matter
$w_{m}\approx 0$, the other is the dark energy $(w_{de}\approx-1$). For cosmic
acceleration, we have $\rho+3p<0$. The continuity equation is given as:

$\dot{\rho}+3H(1+w)\rho=0.$ (6)

In Sec.2 investigation of the behaviour of tachyonic scalar field is carried
out near the inflection point. We have considered three different
possibilities of cosmic behaviours; first two cases of conventional power-law
expansion and pure exponential growth i.e. $a(t)=\alpha t^{n},a(t)=\gamma
e^{\beta t}$ respectively and the third case is of quasi-exponential expansion
of the scale factor $a(t)=\eta t^{n}e^{\beta t}$ for these three cases we have
calculated the respective potentials and then we investigated the behaviour of
that particular model near their respective inflection points. In Sec.3
dynamical analysis for the above two cases is carried out.

##  2 Inflection point of tachyonic scalar field:

Using Eq.(3) for flat ($K=0$) FRLW universe the energy density and pressure
for the tachyonic field are given by:

$\rho_{\phi}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^{2}}},$ (7)
$p_{\phi}=-V(\phi)\sqrt{1-\dot{\phi}^{2}}.$ (8)

Here dot signifies the derivative with respect to time. $V(\phi)$ is the
potential associated with the tachyonic field that we have calculated in the
upcoming subsections for each case. Now by using Eq.(7) and Eq.(8) the model
may be written as fluid. By using:

$\frac{\ddot{a}}{a}=H^{2}+\dot{H}^{2}$ (9)

and plugging Eq.(7) and Eq.(8) in Eq.(4) and (5) we get the following form of
the second Friedmann equation:

$\frac{\ddot{a}}{a}=\frac{\kappa^{2}V(\phi)}{3\sqrt{1-\dot{\phi}^{2}}}\left(1-\frac{3}{2}\dot{\phi}^{2}\right)$
(10)

Now as we know for the accelerated expansion of the universe we have
$\ddot{a}>0$ thus using Eq.10; $\dot{\phi}^{2}<\frac{2}{3}$ and the EoS
(Equation of state) for $\phi$ is given by:

$w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=(\dot{\phi}^{2}-1)$ (11)

For cosmic acceleration the possible range of the values of $w_{\phi}$ is
$-1\leq w_{\phi}<-\frac{1}{3}$ and $0<\dot{\phi}^{2}<\frac{2}{3}$.
Using Eq.(7) and (8) one can easily find the equation of motion for this
tachyonic model as:

$\frac{\ddot{\phi}}{(1-\dot{\phi}^{2})}+3H\dot{\phi}+\frac{1}{V}\frac{dV}{d\phi}=0.$
(12)

Now using Eq.(4) and Eq.(5); $\phi(t)$ and $V(\phi)$ can easily be written in
the from of $H$ and $\dot{H}$ as following :

$\phi(t)=\int{\left(dt\sqrt{-\frac{2\dot{H}}{3H^{2}}}\right)}$ (13)
$V(t)=\frac{3H^{2}}{\kappa^{2}}\sqrt{1+\frac{2\dot{H}}{3H^{2}}}$ (14)

Now using above two equations we will find the potential and their respective
inflection point behaviour for the three cases to be discussed below.

###  2.1 For Case-I: $a(t)=\alpha t^{n}$

In this subsection we consider the simplest model for describing the expansion
of the universe in certain cosmological scenarios, namely, the power law form
of scale factor suggesting that the universe’s expansion follows a simple
power-law relationship with time. i.e. $a(t)=\alpha t^{n}$. Here $\alpha$ is a
constant describing the overall scale of the universe and the exponent $n$
determines the rate of expansion and is greater than $0$ for the universe to
experience the accelerated expansion. For this kind of model the Hubble
parameter is of the following form:

$H=\frac{n}{t}$ (15)

Now on substituting this value of $H$ in Eq.(14) and using Eq.(13) we found
the following form of the potential:

$V(\phi)=V_{0}(\phi-\phi_{0})^{-2}$ (16)

here $V_{0}=\frac{2n}{\kappa^{2}}\sqrt{1-\frac{2}{3n}}$ and $n>2/3$ for the
expansion of the universe.

Figure 1: The plot shows the variation of $\phi$ with potential $V(\phi)$ for
different values of the exponent factor $n$ ($n=0.7$ (red solid line) and
$n=0.9$ (blue dotted line))

From above Eq.(16) we found there is no such inflection point for this
particular model and we can also see that from the Fig. (1) as well that the
curve show no inflection but one can observe that the scalar field is rolling
slowly along the potential. Now using Eq.(12) and on applying the slow roll
approximation i.e. $\dot{\phi}\approx constant\implies\ddot{\phi}\approx 0$ we
found the following evolution of scalar field with respect to time:

$\phi(t)=\sqrt{\frac{2}{3n}(t^{2}-t_{0}^{2})}+\phi_{0}$ (17)

here $t=t_{0}$ when $\phi=\phi_{0}$

Figure 2: The plot shows the evolution of $\phi$ with time $t$ for different
values of the exponent factor $n$

Fig.(2) shows the evolution of $\phi$ for different values of $n.$

###  2.2 For Case-II: $a(t)=\gamma e^{\beta t}$

For this particular case we have taken a simplest exponential form of the
scale factor $a(t)=\gamma e^{\beta t}$ where $\gamma$ and $\beta$ are the
constants. $\beta$ determines the overall rate at which the scale factor
changes and can have two possibilities for the values of first is $\beta>0$
implies the exponential growth of the universe i.e. expanding universe over
time and the second possibility is $\beta<0$ implies the exponential decay
i.e. contracting universe. As we know during early epoch the universe
experiences expansion so we can discard the second possibility. Now the second
constant $\gamma$ is a normalization factor that sets the overall scale of the
universe at a particular time $t$ and $H$ for this exponentially expanding
universe is proportional to the constant:

$H=\beta$ (18)

Now again using this value of $H$ and Eq.(13) we hot the following form of
potential:

$V(\phi)=\frac{3\beta^{2}}{\kappa^{2}}$ (19)

therefore for this particular model there is no inflection point because the
potential is coming out to be constant and is consistent with the case that of
quintessence model. From Eq.(12), (19) and after applying the slow roll
approximation the scalar field $\phi$ is coming out to be a constant and same
result can be obtained from Eq.(10).

###  2.3 For Case-III: $a(t)=\eta t^{n}e^{\beta t}$

For this case we consider the universe in Quasi-exponential phase with the
scale factor given by $a(t)=\eta t^{n}e^{\beta t}$ i.e. product of power law
and exponential, here $\eta$, n and $\beta$ are the constants. For this quasi-
exponential model the Hubble parameter is coming out to be:

$H=\frac{n}{t}+\beta$ (20)

Now using Eq.(13), (14) and (20), one can found the following form of
potential in terms of $t$ given as follow:

$V(t)=\frac{3}{\kappa^{2}}\sqrt{\left(\frac{n+\beta
t}{t}\right)^{4}-\frac{2n}{3}\left(\frac{n+\beta t}{t^{2}}\right)^{2}}$ (21)

and

$t=-\frac{n}{\beta}+\frac{n}{\beta}\exp{\left(\beta\sqrt{\frac{3}{2n}}(\phi-\phi_{0})\right)}$
(22)

On substituting the Eq. (22) in (21) we got the following form of potential
$V(\phi)$:

$V(\phi)=\frac{2n}{\kappa^{2}(\phi-\phi_{0})^{2}}\sqrt{\left(1+\sqrt{\frac{3}{2n}}\beta(\phi-\phi_{0})\right)^{2}-\frac{2}{3n}}$
(23)

On solving the above equation the inflection points for this particular
potential are: $\phi_{0}=\frac{3\beta\phi\pm 2+6\sqrt{6n}}{3\beta}$
On substituting the value of potential (23) in equation of motion for this
quasi-exponential tachyonic model, we get the following form of the evolution
of scalar field:

$\phi(t)=\sqrt{2-2\left(\frac{\exp{(\frac{n+\beta
t}{3\beta^{2}}+\frac{2-(\phi-\phi_{0})^{2}}{2})}}{(n+\beta
t)^{n/3\beta^{3}}}\right)^{2}}+\phi_{0}$ (24)

Figure 3: The plot shows the evolution of $\phi$ with time $t$ for the first
inflection point with different values of the exponent factor $n$ and $\beta$
Figure 4: The plot shows the evolution of $\phi$ with time $t$ for the second
inflection point for the different values of the exponent factor $n$ and
$\beta$

##  3 Dynamical Analysis:

In this section we performs the dynamical analysis of our inflection point
tachyon model. Dynamical analysis is the study of the evolution of the
universe, its behaviour over the time and allow us to analyse the various
components of the universe like dark matter, dark energy, ordinary matter. The
background equations are rewritten as an independent system of first order
ODEs so that the stability at equilibrium points may be examined. Setting the
dimensionless variables as:

$X=\dot{\phi}$ (25a) $Y=\frac{\kappa\sqrt{V}}{\sqrt{3}H}$ (25b)
$\Gamma=\frac{V(d^{2}V/d\phi^{2})}{(dV/d\phi)^{2}}$ (25c)
$\lambda=-\frac{dV/d\phi}{\kappa\sqrt{V^{3}}}$ (25d)

Here we have obtained an autonomous 2 dimensional system which has been
studied by [59] and other two parameters ((25c)) and ((25d)) still has $\phi$
dependency and on differentiating above equations with respect to $\eta=loga$
we get the following set of 2 dimensional autonomous equations:

$X^{\prime}=\frac{dX}{d\eta}=(X^{2}-1)(3X-\lambda\sqrt{3}Y)$ (26a)
$Y^{\prime}=\frac{dY}{d\eta}=\frac{-Y}{2}\left(\frac{3Y^{2}}{\sqrt{1-X^{2}}}(w_{m}-X^{2}+1)-3(w_{m}+1)+\sqrt{3}\lambda
XY\right)$ (26b)
$\lambda^{\prime}=\frac{d\lambda}{d\eta}=-\left(\frac{\sqrt{27}}{2}-\sqrt{3}\Gamma\right)XY\lambda^{2}$
(26c)

With a constraint equation:

$\frac{Y^{2}}{\sqrt{1-X^{2}}}+\Omega_{m}=1$ (27)

The equation of state and the energy density of the tachyonic scale field in
terms of these dimensionless parameters can be written as:

$w_{\phi}=X^{2}-1$ (28) $\Omega_{\phi}=\frac{Y^{2}}{\sqrt{1-X^{2}}}$ (29)

and the effective equation of motion can be written as:

$w_{eff}=w_{m}\left(1-\frac{Y^{2}}{\sqrt{1-X^{2}}}\right)-Y^{2}\sqrt{1-X^{2}}\\\
$ (30)

The critical points for the Eq.((25a)) and ((25b)) are discussed below and are
listed in the table(1):

$X$ | $Y$ | $w_{eff}$ | $w_{\phi}$ | $\Omega_{\phi}$ | Stability
---|---|---|---|---|---
$0$ | $0$ | $w_{m}$ | $-1$ | $0$ | Saddle point
$\pm 1$ | $0$ | $0$ | $0$ | $-$ | Unstable Nodes
$1/5$ | $\frac{\sqrt{3}}{5\lambda}$ | $-$ | $\approx-0.96$ | $\approx 1$ | Stable node
Table 1: Above table shows the critical points of the system (25a)-(30) with
stability properties.

For the inflection point there are four critical points as shown in fig(5) and
are discussed below:

(a.) Point O: For this critical point $X=0$ and $Y=0$; the energy density of
tachyonic scalar field is coming out to be $0$ signifying the matter dominated
epoch. From the Friedmann constraint ((27)) we can see that $\Omega_{m}=1$ and
$w_{eff}=w_{m}$ again signifying the matter dominated epoch, thus this origin
point of phase space refers to the saddle point having following eigenvalues:

$\nu_{1}=-3<0$ (31)

and

$\nu_{2}=\frac{3}{2}(w_{m}+1)>0$ (32)

This saddle point has an attractor behaviour towards X-axis as shown in fig(5)
and at this point the value of $w_{\phi}=-1$ imitating the cosmological
constant behaviour.

Figure 5: phase portait of the dynamical system with $\lambda=1$ and $w=0$
near inflection point. Here point $A$ in the phase space shows the region
where universe undergoes accelerated expansion  Figure 6: shows the phase
portait for the above mentioned set of dynamical equations with $\lambda=1$
and $w=0$.

(b.) Point B$\\_$ : Here for this point we got the following critical points
$X=-1$ and $Y=0$. At this point the tachyonic equation of state vanishes i.e.
$w_{\phi}=0$. At this critical point we have following eigenvalues:

$\nu_{1}=6>0$ (33a)

and

$\nu_{2}=\frac{3}{2}(w_{m}+1)>0$ (33b)

signifying this critical point B$\\_$ is an unstable node.

(c.) Point B+: (X=1,Y=0) this point also represent unstable node having same
eigenvalues as mentioned above in Eq.33

(d.) Point A: $(X=\frac{1}{5},Y=\frac{\sqrt{3}}{5\lambda})$ is the stable node
having eigen values:

$\nu_{1,2}=\frac{1}{2}\left(-4.8-1.9w_{m}\pm\sqrt{(0.7-4.3w_{m}-3.9w_{m}^{2})}\right)<0$
(34)

Fig.(3) and (4) shows the phase portait of the derived set of autonomous
equations and the region near the point A depict the accelerated expansion of
the universe.

##  4 Conclusion

In this paper the evolution of minimally coupled Tachyonic scalar field at the
inflection point has been investigated in an accelerating universe. The
inflection point signifies a transition between regions of stability and
instability. Near this point, the field’s dynamics has been investigated for
different expansion factors. We have mainly considered the three categories of
scale (expansion) factor namely: (a) Power law expansion ($a(t)=\alpha
t^{n}$), (b) Exponential form ($a(t)=\gamma e^{\beta t}$) and (c) Quasi-
Exponential expansion ($a(t)=\eta t^{n}e^{\beta t}$); for all these
categories, the potentials and the corresponding possible inflection points
have been calculated and the dynamical system of equations for the model has
been obtained. Fig.(3) and (4) represents the critical points and the
stability of the derived dynamical system of equations and the region near
point A shows the accelerated expansion of the universe.

In addition to what is being done, it is important to study the inflection
point of scalar field models in the presence of quantum gravity effects. In
particular, scalar field dynamics plays a significant role in the post bounce
scenario of loop quantum cosmology as it gives rise to inflationary solution
[64, 65]. We leave it as our future pursuit to investigate how inflection
points are effected due to quantum geometric correction for scalar fields.

## Acknowledgement

The research by M.K. was carried out in Southern Federal University with
financial support from the Ministry of Science and Higher Education of the
Russian Federation (State contract GZ0110/23-10-IF).

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