Some Aspects of the Tachyon Inflation with Superpotential in Confrontation
with Planck2018 Data
Narges<EMAIL_ADDRESS>
Department of Theoretical Physics, Faculty of Science, University of
Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN

###### Abstract

We study the tachyon inflation in the presence of the superpotential as an
inflationary potential. We study the primordial perturbations and their non-
gaussian feature in the equilateral configuration. We use the Planck2018 TT,
TE, EE+lowE+lensing+BK14+BAO joint data at $68\%$ CL and $95\%$ CL, to perform
numerical analysis on the scalar perturbations and seek for the observational
viability of the tachyon inflation with superpotential. We also check the
observational viability of the model by studying the tensor part of the
perturbations and comparing the results with Planck2018 TT, TE,
EE+lowE+lensing+BK14+BAO+ LIGO$\&$Virgo2016 joint data at $68\%$ CL and $95\%$
CL. By studying the phase space of the model’s parameters, we predict the
amplitude of the equilateral non-gaussianity in this model. The reheating
phase after inflation is another issue that is explored in this paper. We show
that, in some ranges of the model’s parameters, it is possible to have an
observationally viable tachyon model with superpotential.

##  1 Introduction

Although the simple inflation model is the one driven by a canonical single
scalar field [1, 2, 3, 4, 5, 6, 7, 8, 9], it is possible that a non-canonical
scalar field be the field responsible for driving the initial inflationary
expansion of the universe. These models, which contain the scalar field with
non-canonical kinetic energy, are usually named as $k$-inflation. The
interesting point about the $k$-inflation models is that these models predict
the scale-dependent and non-gaussian distributed primordial perturbations.
Tachyon inflation is one of the $k$-inflation models with some interesting
cosmological results which has attained a lot of attention. In fact, by slow
rolling of the tachyon field, associated with the D-branes in string theory,
down its potential, the universe experiences the smooth evolution from the
accelerating phase of the expansion to the era dominated by the non-
relativistic fluid [10, 11, 12, 13, Gar04]. Note that, although there is no
experimental evidence for the existence of the tachyon field, there are also
no indefeasible arguments against it. On the other hand, since the inflation
phase occurred around Planck scale, it seems logical that the M-String theory
inspired fields could have an important role in that phase. The tachyon field
as the field running the inflation has some interesting physical features and
cosmological consequences that expand our physical perspective.

In this regard, some authors have studied inflation in the models with the
tachyon field. For instance, the author of [14] has studied the power-law
inflation with the tachyon field and found an exact solution for the flat
universe. Also, the authors of [15] have studied the intermediate inflation in
the presence of the tachyon field. The authors of [16] have studied the
inflation in both the minimally and non-minimally coupled tachyon field and
compared the results with observational data. In [17] the authors have studied
the tachyon inflation based on the large-$N$ formalism and by comparing the
results with the observational data, they have shown the viability of the
model. To study some more interesting works on the tachyon inflation see [18,
19, 20, 21, 33].

An interesting aspect of studying the inflation model is the perturbation and
non-gaussian feature of its amplitude. The prediction of the simple single
field inflation with the canonical scalar field is the almost scale-
independent and gaussian perturbations [1, 2, 3, 4, 5, 6, 7, 8, 9]. In fact,
with the standard canonical scalar field or Higgs and with potentials such as
$\phi$, $\phi^{2}$, $\phi^{\frac{4}{3}}$, and so on, the behavior of the
tensor-to-scalar ratio versus the scalar spectral index is not consistent with
Planck2018 TT, TE, EE +lowE +lensing +BK14+BAO datasets. In recent years some
cosmologist have shown interest in the inflation models predicting the scale
dependent and large non-gaussian distributed perturbations and have presented
some interesting works on this issue [9, 22, 23, 24, 25, 26, 27, 28, 29, 30,
32, 31, 33]. In this way, by considering the nonminimal or modified gravity
models, authors have tried to propose the models of inflation leading to the
observationally viable perturbation parameters. Also, one may think about
other scalar fields. The phantom field has negative energy density and
although the negative energy density has been predicted in some realizations
of the relativistic quantum field theories, it is less favorable to consider
it. However, with a tachyon field, it is possible to have an inflation model
with observationally viable values of the scalar spectral index and tensor-to-
scalar ratio. Also, its equation of state parameter can be $-1$ (corresponding
to dark energy and early time inflationary expanding universe) and $0$
(corresponding to dust or dark matter components). In this way, the tachyonic
field can run the inflationary phase, and after generating a non-relativistic
fluid dominated phase, could be considered as the dark matter [13]. Another
interesting property of the tachyon field is that in the tachyon model, it is
possible to have the amplitude of the non-gaussianity of the order of $10^{1}$
and even more (see section 4).

It should be noticed that the current observational data also have released
somewhat scale dependence of the primordial perturbation and corresponding
amplitudes of the non-gaussianity. In fact, the scale dependence of the
perturbation, presented by the scalar spectral index, has been released as
$n_{s}=0.9658\pm 0.0038$ implied by Planck2018 TT, TE,
EE+lowE+lensing+BAO+BK14 data by considering $\Lambda CDM+r+\frac{dn_{s}}{d\ln
k}$ model [34, 35]. The mentioned data have also set a constraint on the
tensor-to-scalar ratio as $r<0.072$. Another important perturbation parameter
in studying the perturbation in the inflation models is the tensor spectral
index which shows the scale dependence of the tensor perturbations. The
observational constraint on this perturbation parameter is as
$-0.62<n_{T}<0.53$, obtained from Planck2018 TT, TE, EE
+lowE+lensing+BK14+BAO+LIGO and Virgo2016 data [34, 35]. Planck2018 team has
also released some constraints on the amplitudes of the non-gaussianity. The
released constraint on the equilateral configurations of the non-gaussianity
is as $f_{NL}^{equil}=-26\pm 47$ [36]. In this regard, by considering the
tachyon field as the field responsible for the inflation, it is possible to
check the observational viability of this inflation model.

An important phase after the end of the inflation era is the reheating
process. Inflationary expansion of the universe continues as long as
$\mid\epsilon,\eta\mid\ll 1$ (corresponding to the flat potential). As soon as
$\epsilon$ or $\eta$ reaches unity, meaning that the potential is no longer
flat, the inflationary expansion ends. Then, the inflaton reaches the minimum
of the potential, where it oscillates, loses its energy, and decays into the
plasma of the relativistic particles. By this process, the radiation component
dominates the energy density of the universe [37, 38, 39]. In fact, by
considering the reheating phase, it is possible to explain the cosmic origin
of the production of the matter component in the universe [40, 41]. It has
also been shown that, the production of the cosmic relics (for instance,
photons and neutrinos) and matter-antimatter asymmetry observed in the
universe, can be explained by considering the reheating process after
inflation [42, 43, 44, 45]. During this era, the effective equation of state
parameter (for the massive inflaton) can be $-1$, corresponding to the
domination of the potential energy over the kinetic energy, and $+1$,
corresponding to the domination of the kinetic energy over the potential
energy. Other important parameters in studying the reheating process are the
e-folds number and the temperature during this phase, which give some
information about this process. In Ref. [46] one can find a review on the
reheating.

In this paper, we study inflation with a special type of potential, named
superpotential. The idea of superpotential has been introduced in [47] (see
also [48, 49]). In Refs [16, 50, 51, 52], the authors have used the idea of
superpotential to obtain some cosmological solutions. In our previous work, we
have used the superpotential as the potential driving the inflation in a
nonminimally coupled canonical scalar field model. However, in this work, we
study the tachyon inflation with a superpotential given by

$V(\phi)=\frac{\kappa^{2}}{6}\Big{(}W(\phi)\Big{)}^{2}-\frac{1}{8}\left(\frac{dW(\phi)}{d\phi}\right)^{2}\,,$
(1)

where $W(\phi)$ is an arbitrary function of the scalar field. Note that, we
didn’t involve any supersymmetry here. However, since the form of the
potential (1) is similar to the one in the supergravity theories, this
potential has been named superpotential. In this regard, the paper is
organized as follows. In section 2, we study the inflation in the tachyon
model with superpotential and obtain the main inflation parameters in terms of
the superpotential. In section 3, we present both the linear and non-linear
perturbations. We obtain the scalar and tensor spectral indices, tensor-to-
scalar ratio and the non-linearity parameter in the equilateral configuration
of the non-gaussianity. The observational viability of the model in
confrontation with the Planck2018 data set is studied in section 4. In this
way, we obtain some constraints on the model’s parameters. We study the
reheating phase after inflation in section 5 and show that, in some ranges of
the model’s parameters, it is possible to have instantaneous reheating.

##  2 Tachyon Inflation with Superpotential

###  2.1 Tachyon Model

To study the tachyon inflation with superpotential, we start with the
following action

$\displaystyle S=\int
d^{4}x\,\sqrt{-g}\Bigg{[}\frac{1}{2\kappa^{4}}R-V(\phi)\,\sqrt{1-2\,\lambda\,X}\Bigg{]}\,.$
(2)

In the above action, $\kappa$ is the gravitational constant, $R$ is the Ricci
scalar, $V(\phi)$ is the potential of the tachyon field which we assume to be
the superpotential. The parameter $\lambda$ is the warp factor which is a
constant and also we define
$X=-\frac{1}{2}\,\partial_{\mu}\phi,\partial^{\mu}\phi$. To find the
Einstein’s field equation, we vary action (2) and get

$\displaystyle
G_{\mu\nu}=\kappa^{2}\Bigg{[}-g_{\mu\nu}V(\phi)\sqrt{1-2\,\lambda
X}+\frac{\lambda\,V(\phi)\partial_{\mu}\phi\,\partial_{\nu}\phi}{\sqrt{1-2\,\lambda
X}}\Bigg{]}\,,$ (3)

which with the FRW metric as

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,,$ (4)

leads to the following Friedmann equation for the tachyon model

$\displaystyle
3H^{2}=\frac{\kappa^{2}\,V(\phi)}{\sqrt{1-\lambda\,\dot{\phi}^{2}}}\,.$ (5)

Note that, from now on, a dot shows the derivative of the parameter with
respect to the time and a prime shows the derivative of the parameter with
respect to the tachyon field. We also find the second Friedmann equation as
follows

$\displaystyle
2\dot{H}+3H^{2}=\kappa^{2}\bigg{[}V(\phi)\,\sqrt{1-\lambda\,\dot{\phi}^{2}}\,\bigg{]}\,.$
(6)

By varying action (2) with respect to the tachyon field, $\phi$, the equation
of motion of this field is obtained as

$\frac{\lambda\,\ddot{\phi}}{1-\lambda\,\dot{\phi}^{2}}+3\,\lambda\,H\dot{\phi}+\frac{V^{\prime}(\phi)}{V(\phi)}=0\,.$
(7)

To proceed, considering that the scalar field is a function of time, we assume
the tachyon field as $\phi\equiv\phi(a)$. In this regard, the Friedmann
equations (5) and (6) take the following forms

$\displaystyle
3H^{2}=\frac{\kappa^{2}\,V(\phi)}{\sqrt{1-\lambda\,H^{2}\,\Big{(}a\frac{d\phi}{da}\Big{)}^{2}}}\,,$
(8) $\displaystyle
2H\,H^{\prime}\,a\,\frac{d\phi}{da}+3H^{2}=\kappa^{2}\bigg{[}V(\phi)\,\sqrt{1-\lambda\,H^{2}\,\Big{(}a\frac{d\phi}{da}\Big{)}^{2}}\,\bigg{]}\,.$
(9)

Also, the equation of motion of the tachyon field becomes as

$\lambda\,\Bigg{(}\ddot{a}\,\frac{d\phi}{da}+\dot{a}^{2}\,\frac{d^{2}\phi}{d\phi^{2}}\Bigg{)}+3\,\lambda\,H^{2}\,a\,\frac{d\phi}{da}\Bigg{(}1-\lambda\,H^{2}\,\Big{(}a\,\frac{d\phi}{da}\Big{)}^{2}\Bigg{)}+\frac{V^{\prime}(\phi)}{V(\phi)}\Bigg{(}1-\lambda\,H^{2}\,\Big{(}a\,\frac{d\phi}{da}\Big{)}^{2}\Bigg{)}=0\,.$
(10)

To study the inflation in this model, it is needed to calculate the slow-roll
parameters. The definition of the slow-roll parameters are given by

$\displaystyle\epsilon=-\frac{\dot{H}}{H^{2}}=-\frac{H}{H^{\prime}}\,a\,\frac{d\phi}{da}\,,$
(11)
$\displaystyle\eta=\frac{1}{H}\frac{\dot{\epsilon}}{\epsilon}=\frac{\epsilon^{\prime}}{\epsilon}\,a\,\frac{d\phi}{da}\,,$
(12) $\displaystyle
s=\frac{1}{H}\frac{\dot{c_{s}}}{c_{s}}=\frac{c^{\prime}_{s}}{c_{s}}\,a\,\frac{d\phi}{da}\,.$
(13)

In the last equation, the parameter $c_{s}$, defined as
$c_{s}^{2}\equiv\frac{P_{,X}}{\rho_{,X}}$, is the sound speed of the
perturbations where ,X shows the derivative of the parameter with respect to
$X$. Considering that in the tachyon model we have
$P=-V(\phi)\sqrt{1-2\,\lambda\,X}$, we find the following expression for the
sound speed

$\displaystyle
c_{s}=\sqrt{1-\lambda\,H^{2}\,\Big{(}a\frac{d\phi}{da}\Big{)}^{2}}\,.$ (14)

Another important parameter during inflation is the e-folds number, which in
this case is given by

$\displaystyle N=\int H\,dt=\int\frac{d\phi}{H\,a\,\frac{d\phi}{da}}\,.$ (15)

###  2.2 Superpotential

In this subsection, we consider the potential defined in equation (1) and
study the tachyon inflation. In this respect, to equations (8)-(10) be
satisfied, we should have the following expressions

$\displaystyle\left(\frac{\dot{a}}{a}\right)=\kappa^{4}\,W^{2}\,,$ (16)
$\displaystyle
a\,\frac{d\phi}{da}={\frac{\sqrt{\lambda\,W\left(5168\,{\kappa}^{4}\,{W}^{4}+24\,{W}^{2}{{\it
W^{\prime}}}^{2}{\kappa}^{2}-9\,{{\it
W^{\prime}}}^{4}\right)}}{72\,\lambda\,{W}^{3}{\kappa}^{3}}}\,.$ (17)

Now, by using the above equations, we can obtain equations (11)-(13) as
follows

$\displaystyle\epsilon=-{\frac{W^{\prime}\,\sqrt{\lambda\,W\,\Big{(}5168\,{\kappa}^{4}\,W^{4}+24\,W^{2}\,W^{\prime
2}{\kappa}^{2}-9\,W^{4}\Big{)}}}{72\,{\kappa}^{3}\,W^{4}}}\,,$ (18)
$\displaystyle\eta=\frac{\sqrt{\left(5168\,{\kappa}^{4}-9\right)W^{2}+24\,{\kappa}^{2}\,W^{\prime
2}}}{W^{\frac{5}{2}}}\Bigg{[}\sqrt{\lambda}\Big{(}10336\,W^{3}{\kappa}^{4}+96\,W^{\prime
2}\,W\,{\kappa}^{2}-18\,W^{3}\Big{)}W^{\prime\prime}$
$\displaystyle-120\,\sqrt{\lambda}{\kappa}^{2}W^{\prime
4}-15504\,\left({\kappa}^{4}-{\frac{9}{5168}}\right)\sqrt{\lambda}\,W^{2}\,W^{\prime
2}\Bigg{]}$ $\displaystyle\Bigg{[}3456\,{\kappa}^{5}W^{\prime
3}+744192\,\left({\kappa}^{4}-{\frac{9}{5168}}\right){\kappa}^{3}\,W^{2}\,W^{\prime}\Bigg{]}^{-1}\,,$
(19)

$\displaystyle
s=\frac{\Big{(}48\,W^{\prime}\,{\kappa}^{2}\,W\,W^{\prime\prime}-24\,{\kappa}^{2}\,W^{\prime
3}+W^{2}\left(5168\,{\kappa}^{4}-9\right)W^{\prime}\Big{)}{\lambda}^{\frac{5}{2}}\sqrt{24\,{\kappa}^{2}W^{\prime
2}+W^{2}\left(5168\,{\kappa}^{4}-9\right)}}{W^{\frac{5}{2}}\,\left(3456\,{\kappa}^{5}{\lambda}^{2}\,W^{\prime
2}+744192\,{\kappa}^{3}\left({\kappa}^{4}-{\frac{9}{5168}}\right){\lambda}^{2}W^{2}-746496\,{\kappa}^{5}W\right)}\,.$

We also find the following expression for the square of the sound speed of the
primordial perturbation in the tachyon model

$\displaystyle
c_{s}^{2}={\frac{1}{72}\sqrt{-{\frac{{\lambda}^{2}\left(5168\,{\kappa}^{4}\,W^{4}+24\,W^{2}\,W^{\prime
2}\,{\kappa}^{2}-9\,W^{4}\right)}{W^{3}\,{\kappa}^{2}}}+5184}}\,.$ (21)

To use these equations and perform numerical analysis on the model, we should
adopt some specific function for $W(\phi)$. In this paper, we choose the
following form for $W(\phi)$

$\displaystyle
W(\phi)=c\Bigg{(}\frac{e^{\alpha_{1}\,\phi}}{\alpha_{1}}-\frac{e^{\alpha_{2}\,\phi}}{\alpha_{2}}\Bigg{)}\,,$
(22)

where the parameters $c$, $\alpha_{1}$ and $\alpha_{2}$ are some arbitrary
constants. Note that, these parameters can be negative or positive, however,
in our model we consider the positive values of these parameters. We also
choose these parameters in the way that at $\phi\rightarrow 0$, we have
$H\rightarrow 1$. Therefore, the following relation between the constant
parameters is applied

$\displaystyle\alpha_{1}=\frac{c\,\kappa^{2}\,\alpha_{2}}{c\,\kappa^{2}+\alpha_{2}}\,.$
(23)

By having the required relations, in the next section, we study the
perturbations in the tachyon model with superpotential to seek for the
observational viability of this model.

##  3 Perturbations in the Tachyon Model with Superpotential

###  3.1 Linear Perturbations

In this subsection, by considering the following perturbed ADM line element

$\displaystyle
ds^{2}=-(1+2{\cal{R}})dt^{2}+2a(t){\cal{M}}_{i}\,dt\,dx^{i}+a^{2}(t)\left[(1-2{\Phi})\delta_{ij}+2{\Theta}_{ij}\right]dx^{i}dx^{j}\,,$
(24)

we study the linear perturbation in this setup. In perturbed metric (24), the
parameter ${\cal{M}}^{i}$ is defined as
${\cal{M}}^{i}=\delta^{ij}\partial_{j}{\cal{M}}+v^{i}$. The condition
$v^{i}_{,i}=0$ is satisfied by vector $v^{i}$. Also the parameters ${\cal{R}}$
and ${\cal{M}}$ are 3-scalars [53]. In this perturbed metric, we have used the
parameter $\Phi$ to denote the spatial curvature perturbation and the
parameter ${\Theta}_{ij}$ to show the spatial symmetric and traceless shear
3-tensor. By using the scalar part of the metric (24) as

$\displaystyle
ds^{2}=-(1+2{\cal{R}})dt^{2}+2a(t){\cal{M}}_{,i}\,dt\,dx^{i}+a^{2}(t)(1-2{\Phi})\delta_{ij}dx^{i}dx^{j}\,,$
(25)

written in the uniform-field gauge ($\delta\phi=0$) at the linear level of the
perturbations, we can explore the scalar perturbations in this setup. To this
end, and by using equation (25), we find the quadratic action as follows

$S_{2}=\int
dt\,d^{3}x\,a^{3}{\cal{Q}}\left[\bigg{(}\kappa^{2}\,W\,a\,\frac{d\Phi}{da}\bigg{)}^{2}-\frac{c_{s}^{2}}{a^{2}}(\partial{\Phi})^{2}\right],$
(26)

which is obtained by expanding action (2) up to the second-order of the small
perturbations. In the above quadratic action, the parameter ${\cal{Q}}$ is
defined as

$\displaystyle{\cal{Q}}={\frac{V\,\lambda\,\dot{\phi}^{2}}{2\,H^{2}\left(1-\lambda\,\dot{\phi}^{2}\right)^{\frac{3}{2}}}}\hskip
256.0748pt$ $\displaystyle=\frac{\Big{(}-\frac{1}{8}\,W^{\prime
2}+\frac{1}{6}\,{\kappa}^{2}\,{W}^{2}\Big{)}{\lambda}^{2}\Big{(}5168\,{\kappa}^{4}\,W^{4}+24\,W^{2}\,W^{\prime
2}\,{\kappa}^{2}-9\,W^{4}\Big{)}}{10368\,\left(W\left(\phi\right)\right)^{7}{\kappa}^{10}}$
$\displaystyle\times\Bigg{[}-{\frac{{\lambda}^{2}\Big{(}5168\,{\kappa}^{4}W^{4}+24\,W^{2},W^{\prime
2}\,{\kappa}^{2}-9\,W^{4}\Big{)}}{5184\,W^{5}\,{\kappa}^{6}}}+1\Bigg{]}^{-\frac{3}{2}}\,.$
(27)

Also, the sound speed squared is given by equation (21). To find more
information about obtaining the higher-order actions, see [24, 25, 26]. One
important parameter in studying the scalar part of the perturbation is the
scalar spectral index and to obtain this parameter, we should use the
following two-point correlation function

$\langle
0|{\Phi}(0,\textbf{k}_{1}){\Phi}(0,\textbf{k}_{2})|0\rangle=(2\pi)^{3}\delta^{(3)}(\textbf{k}_{1}+\textbf{k}_{2})\frac{2\pi^{2}}{k^{3}}{\cal{A}}_{s}\,,$
(28)

with

${\cal{A}}_{s}=\frac{H^{2}}{8\pi^{2}{\cal{Q}}c_{s}^{3}}\,,$ (29)

which is named the power spectrum. The power spectrum gives the scalar
spectral index as

$n_{s}-1=\frac{d\ln{\cal{A}}_{s}}{d\ln k}\Bigg{|}_{c_{s}k=aH}\,.$ (30)

The following expression gives the scalar spectral index in terms of the slow-
roll parameters

$n_{s}=1-2\epsilon-\eta-s\,.$ (31)

The second-order action of the tensor mode is obtained as

$\displaystyle S_{T}=\int
dt\,d^{3}x\,\frac{a^{3}}{4\kappa^{2}}\Bigg{[}\dot{\Theta}_{+}^{2}-\frac{1}{a^{2}}(\partial{\Theta}_{+})^{2}+\dot{\Theta}_{\times}^{2}-\frac{1}{a^{2}}(\partial{\Theta}_{\times})^{2}\Bigg{]}\,,$
(32)

where we have used two polarization tensors $\vartheta_{ij}^{(+,\times)}$, to
rewrite the 3-tensor ${\Theta}_{ij}$ as
${\Theta}_{ij}={\Theta}_{+}\vartheta_{ij}^{+}+{\Theta}_{\times}\vartheta_{ij}^{\times}$.
The amplitude of the tensor perturbations is obtained by following the method
used in the scalar part as follows

${\cal{A}}_{T}=\frac{2\kappa^{2}H^{2}}{\pi^{2}}\,,$ (33)

giving the following tensor spectral index in this model

$n_{T}=\frac{d\ln{\cal{A}}_{T}}{d\ln k}=-2\epsilon\,.$ (34)

The ratio between the amplitude of the tensor spectral index and the amplitude
of the scalar spectral index gives another important parameter as

$r=\frac{{\cal{A}}_{T}}{{\cal{A}}_{s}}=16\,c_{s}\,\epsilon\,,$ (35)

named the tensor-to-scalar ratio. Note that, in equations (31), (33) and (35),
the slow-roll parameters are given by equations (18)-(2.2).

The perturbations parameters obtained in this subsection, are important in
studying the linear perturbations. To find more information about the
perturbations, it is useful to study the non-linear level of the perturbation.
In this regard, in the next subsection, we explore the non-linear
perturbations to get some information about the non-gaussian feature of the
primordial perturbations in the tachyon model with superpotential.

###  3.2 Non-linear Perturbations

In this subsection, we go to the non-linear level of the perturbations to
study the non-gaussianity in this model. By introducing the new parameter
$\Upsilon$ which satisfies the following relations

$\displaystyle{\cal{M}}=\frac{{\Phi}}{\kappa^{2}\,W}+\kappa^{2}a^{2}\Upsilon\,,$
(36)

and

$\partial^{2}\Upsilon={\cal{Q}}\,\kappa^{2}\,W\,a\frac{d\Phi}{da}\,,$ (37)

and expanding action (2) up to the third order in the perturbations, we get

$\displaystyle S_{3}=\int
dt\,d^{3}x\,\Bigg{\\{}\Bigg{[}\frac{3a^{3}}{\kappa^{2}c_{s}^{2}}\,\Bigg{(}1-\frac{1}{c_{s}^{2}}\Bigg{)}\epsilon\Bigg{]}{\Phi}\,\bigg{(}\kappa^{2}\,W\,a\frac{d\Phi}{da}\bigg{)}^{2}+\Bigg{[}\frac{a}{\kappa^{2}}\Bigg{(}\frac{1}{c_{s}^{2}}-1\Bigg{)}\epsilon\Bigg{]}{\Phi}\,(\partial{\Phi})^{2}$
$\displaystyle+\Bigg{[}\frac{a^{3}}{\kappa^{2}}\,\Bigg{(}\frac{1}{c_{s}^{2}\,H}\Bigg{)}\Bigg{(}\frac{1}{c_{s}^{2}}-1\Bigg{)}\epsilon\Bigg{]}\bigg{(}\kappa^{2}\,W\,a\frac{d\Phi}{da}\bigg{)}^{3}-\Bigg{[}a^{3}\,\frac{2}{c_{s}^{2}}\epsilon\,\bigg{(}\kappa^{2}\,W\,a\frac{d\Phi}{da}\bigg{)}(\partial_{i}{\Phi})(\partial_{i}\Upsilon)\Bigg{]}\Bigg{\\}}\,,$
(38)

which is dubbed the cubic action and is written up to the leading order in the
slow-roll parameters of the model. Now, we use the three-point correlation
function for the spatial curvature perturbation in the interaction picture as
[9, 54]

$\displaystyle\langle{\Phi}(\textbf{k}_{1})\,{\Phi}(\textbf{k}_{2})\,{\Phi}(\textbf{k}_{3})\rangle=(2\pi)^{3}\delta^{3}(\textbf{k}_{1}+\textbf{k}_{2}+\textbf{k}_{3}){\cal{B}}_{\Phi}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})\,,$
(39)

with following definition for ${\cal{B}}_{\Phi}$

${\cal{B}}_{\Phi}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})=\frac{(2\pi)^{4}{\cal{A}}_{s}^{2}}{\prod_{i=1}^{3}k_{i}^{3}}\,{\cal{G}}_{\Phi}(\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3})\,.$
(40)

Also, the parameter ${\cal{G}}_{\Phi}$ in equation (40) is given by the
following expression

$\displaystyle{\cal{G}}_{\Phi}=\Bigg{(}1-\frac{1}{c_{s}^{2}}\Bigg{)}\Bigg{[}\frac{3}{4}\Bigg{(}\frac{2\sum_{i>j}k_{i}^{2}\,k_{j}^{2}}{k_{1}+k_{2}+k_{3}}-\frac{\sum_{i\neq
j}k_{i}^{2}\,k_{j}^{3}}{(k_{1}+k_{2}+k_{3})^{2}}\Bigg{)}-\frac{3}{2}\Bigg{(}\frac{\left(k_{1}\,k_{2}\,k_{3}\right)^{2}}{(k_{1}+k_{2}+k_{3})^{3}}\Bigg{)}$
$\displaystyle-\frac{1}{4}\Bigg{(}\frac{2\sum_{i>j}k_{i}^{2}\,k_{j}^{2}}{k_{1}+k_{2}+k_{3}}-\frac{\sum_{i\neq
j}k_{i}^{2}\,k_{j}^{3}}{(k_{1}+k_{2}+k_{3})^{2}}+\frac{1}{2}\sum_{i}k_{i}^{3}\Bigg{)}\Bigg{]}\,.$
(41)

In studying the non-gaussian feature of the primordial perturbations, it is
useful to introduce the so-called “non-linearity parameter”. This parameter is
defined by using the parameter ${\cal{G}}_{\Phi}$ as follows

$f_{NL}=\frac{10}{3}\frac{{\cal{G}}_{\Phi}}{\sum_{i=1}^{3}k_{i}^{3}}\,.$ (42)

As it is clear from equation (42), the non-linearity parameter depends on the
values of different momenta $k_{1}$, $k_{2}$ and $k_{3}$, which lead to
different shapes of the primordial non-gaussianity with the different maximal
signal of their amplitudes. Considering that the signal of the amplitude of
the non-gaussianity in the $k$-inflation and higher-order derivative models
becomes maximal at the equilateral configuration, in this paper we consider
$k_{1}=k_{2}=k_{3}$ [55, 56, 57, 58]. By adopting the equilateral
configuration, we find

${\cal{G}}_{\Phi}^{equil}=\frac{17}{72}k^{3}\left(1-\frac{1}{c_{s}^{2}}\right)\,,$
(43)

and therefore

$f^{equil}_{NL}=\frac{85}{324}\left(1-\frac{1}{c_{s}^{2}}\right)\,,$ (44)

where $c_{s}^{2}$ is given by equation (21). Now that we have the important
parameters in both linear and non-linear levels, we can seek for the
observational viability of the tachyon inflation with superpotential. In this
regard, in the next section, we present the numerical analysis of this setup.

Figure 1: Ranges of the parameters $\alpha_{2}$ and $\lambda$ leading to the
observationally viable values of the scalar spectral index (upper-left panel),
tensor-to-scalar ratio (upper-right panel) and the tensor spectral index
(lower panel) in the tachyon model with the superpotential.

##  4 Observational Viability

To study the observational viability of this model and find some constraints
on the model’s parameters, we use the superpotential given in the equation
(22). By this superpotential, we obtain slow-roll parameters (18)-(2.2) and
therefore the perturbations parameters (scalar spectral index, tensor spectral
index, and tensor-to-scalar ratio) in terms of the scalar fields and model’s
parameters. Also, by using equations (15)-(17), we find the value of $\phi$ in
terms of the e-folds number and model’s parameters and substitute it in the
perturbations parameters. Now, we can compare the values of these parameters
with observational data and obtain some constraints on the parameter
$\alpha_{2}$ for some values of $\lambda$. As we have mentioned in the
Introduction, from Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data and by
considering $\Lambda CDM+r+\frac{dn_{s}}{d\ln k}$ model, we have
$n_{s}=0.9658\pm 0.0038$ at $68\%$ CL [34, 35]. By using this constraint, we
have obtained the range of the parameters $\alpha_{2}$ and $\lambda$ leading
to the observationally viable values of the scalar spectral index. The result
is shown in the upper-left panel of figure 1. The upper-right panel of figure
1 shows the range of the parameters $\alpha_{2}$ and $\lambda$ which leads to
$r<0.072$, obtained from Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data at
$68\%$ CL. Also, the lower panel of figure 1 shows the range of the parameters
$\alpha_{2}$ and $\lambda$ leading to $-0.62<n_{T}<0.53$, obtained from
Planck2018 TT, TE, EE +lowE+lensing+BK14+BAO+LIGO and Virgo2016 data at $68\%$
CL [34, 35]. To plot these figures, we have adopted $N=60$, $0<\alpha_{2}\leq
0.1$ and $0<\lambda\leq 1$.

To obtain some constraints on the model’s parameters space, we have plotted
the evolution of the tensor-to-scalar ratio in the background of the
Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data. The result is shown in
figure 2. As the figure shows, the tachyon inflation with superpotential in
some ranges of the model’s parameter is consistent with observational data. We
also have plotted the evolution of the tensor-to-scalar ratio versus the
tensor spectral index in the background of the Planck2018 TT, TE, EE
+lowE+lensing+BK14+BAO+LIGO and Virgo2016 data set in figure 3. Note that, as
equation (35) and figure 3 show, the consistency relation in this setup is
satisfied. By Using the numerical analysis, we have obtained some constraints
on the parameter $\alpha_{2}$, for some sample values of $\lambda$ as $0.1$,
$0.3$, $0.6$ and $0.9$, which have been summarized in table 1. Also, by using
the constraints that Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data at
$68\%$ CL and $95\%$ CL set on the scalar spectral index and the tensor-to-
scalar ratio, it is possible to obtain the ranges of the sound speed and
$\lambda$ which fulfill the mentioned data. In this respect, figure 4 shows
the observationally viable ranges of $\lambda$ and $c_{s}$ at $68\%$ CL
(yellow region) and $95\%$ CL (magenta region).

Figure 2: Tensor-to-scalar ratio versus the scalar spectral index in the tachyon model with superpotential. The arrows show the direction in which the parameters increase. Figure 3: Tensor-to-scalar ratio versus the tensor spectral index in the tachyon model with superpotential. Table 1:  Ranges of the parameter $\alpha_{2}$ in which the tensor-to-scalar ratio, the scalar spectral index and the tensor spectral index of the tachyon inflation with superpotential are consistent with different data sets. | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE
---|---|---|---|---
| +lensing+BK14+BAO | +lensing+BK14+BAO | lensing+BK14+BAO | lensing+BK14+BAO
|  |  |  +LIGO$\&$Virgo2016 |  LIGO$\&$Virgo2016
$\lambda$ |  $68\%$ CL |  $95\%$ CL |  $68\%$ CL |  $95\%$ CL
$0.1$ | $1.41\times 10^{-2}<\alpha_{2}<1.83\times 10^{-2}$ | $1.11\times 10^{-2}<\alpha_{2}<1.98\times 10^{-2}$ | $0.65\times 10^{-2}<\alpha_{2}<2.0.2\times 10^{-2}$ | $0<\alpha_{2}<2.25\times 10^{-2}$
$0.4$ | $0.68\times 10^{-2}<\alpha_{2}<0.91\times 10^{-2}$ | $0.55\times 10^{-2}<\alpha_{2}<0.98\times 10^{-2}$ | $0.35\times 10^{-2}<\alpha_{2}<1.03\times 10^{-2}$ | $0<\alpha_{2}<1.15\times 10^{-2}$
$0.7$ | $0.48\times 10^{-2}<\alpha_{2}<0.68\times 10^{-2}$ | $0.39\times 10^{-2}<\alpha_{2}<0.73\times 10^{-2}$ | $0.28\times 10^{-2}<\alpha_{2}<0.83\times 10^{-2}$ | $0<\alpha_{2}<0.92\times 10^{-2}$
$0.9$ | $0.17\times 10^{-2}<\alpha_{2}<0.55\times 10^{-2}$ | $0.12\times 10^{-2}<\alpha_{2}<0.40\times 10^{-2}$ | $0.43\times 10^{-2}<\alpha_{2}<0.84\times 10^{-2}$ | $0<\alpha_{2}<0.90\times 10^{-2}$
Figure 4: Ranges of the parameters $\lambda$ and $c_{s}$ leading to the
observationally viable values of the scalar spectral index and the tensor-to-
scalar ratio, obtained from Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data
at $68\%$ CL (yellow region) and $95\%$ CL (magenta region).

To study the non-gaussian feature of the primordial perturbations numerically,
we consider equation (44) with the sound speed given in equation (21) and the
superpotential (22). In fact, we seek for the prediction of the model about
the amplitude of the non-gaussianity in the tachyon model with superpotential.
In this regard, in figure 6 we have plotted the phase space of the parameters
$\alpha_{2}$ and $\lambda$ and also the corresponding predicted values of the
amplitude of the equilateral configuration of the non-gaussianity. As the
figure shows, for most ranges of the parameter space, the tachyon model with
superpotential predicts small non-gaussianity. In fact, if we consider
$\lambda\leq 0.9$, for all values of $\alpha_{2}$, the amplitude of the
equilateral non-gaussianity in the tachyon model is consistent with Planck2018
TTT, EEE, TTE and EET data at $68\%$ CL. However, if we consider
$\lambda>0.9$, there would be some constraints on the parameter $\alpha_{2}$
in confrontation with Planck2018 TTT, EEE, TTE and EET data at $68\%$ CL. We
have obtained some of these constraints which have been summarized in table 2.
Also, from the constraints obtained by comparing the $r-n_{s}$ and $r-n_{T}$
behavior in the background of Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14
data, it is possible to find the observationally viable values of the sound
speed and the amplitude of the equilateral configuration of the non-
gaussianity. The results of this numerical analysis have been summarized in
table 3.

Figure 5: Space of the parameters $\alpha_{2}$ and $\lambda$ (left panel) and the corresponding predicted values of the amplitude of the equilateral non-gaussianity (right panel) in the tachyon model with superpotential. Table 2:  Ranges of the parameters $\alpha_{2}$ and $\lambda$ leading to the observationally viable values of the equilateral configuration of the non-gaussianity in the tachyon model with superpotential, in confrontation with the Planck2018 TTT, EEE, TTE and EET data at $68\%$ CL. $0<\lambda\leq 0.9$ | $\lambda=0.93$ | $\lambda=0.96$ | $\lambda=0.99$
---|---|---|---
$0<\alpha_{2}<0.1$ | $0.004<\alpha_{2}<0.1$ | $0.006<\alpha_{2}<0.1$ | $0.008<\alpha_{2}<0.1$
Table 3:  The observationally viable ranges of the sound speed and the amplitude of the equilateral configuration of the non-gaussianity in the tachyon inflation with superpotential, in confrontation with Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data at $68\%$ CL. $\lambda$ |  | $c_{s}$ |  | $f_{NL}^{equil}$
---|---|---|---|---
$0.1$ |  | $0.994<c_{s}<0.995$ |  | $0.039<f_{NL}^{equil}<0.040$
$0.4$ |  | $0.920<c_{s}<0.923$ |  | $0.727<f_{NL}^{equil}<0.759$
$0.7$ |  | $0.727<c_{s}<0.740$ |  | $3.47<f_{NL}^{equil}<3.74$
$0.9$ |  | $0.464<c_{s}<0.493$ |  | $13.0<f_{NL}^{equil}<15.2$

##  5 Reheating Phase after Inflation

After ending the inflationary expansion, the universe should be reheated for
subsequent evolution. This process is named reheating phase of the universe
which studying it gives us some more information about the model. In this
regard, following [59, 60, 61, 62, 63], in this section we obtain some
expressions for the e-folds number during the reheating, $N_{rh}$, (the
subscript “$rh$” stands for the reheating) and temperature during this era,
$T_{rh}$. Considering that these reheating parameters are expressed in terms
of the scalar spectral index, it is possible to explore their observational
viability.

We start with the following equation

$N_{hc}=\ln\left(\frac{a_{e}}{a_{hc}}\right)\,,$ (45)

defining the e-folds number between the time where physical scales cross the
horizon and the time where the inflation epoch ends. By the subscript “$e$” we
mean the end of inflation and by the subscript “$hc$” we show the horizon
crossing. Note that, the relation $\rho\sim a^{-3(1+\omega_{eff})}$ for the
energy density is satisfied during the reheating process. In this relation,
the effective equation of state, corresponding to the dominant energy density
of the universe, is shown by the parameter $\omega_{eff}$. By using this
relation between the energy density and the equation of state parameter, we
can find the e-folds number during reheating as follows

$\displaystyle
N_{rh}=\ln\left(\frac{a_{rh}}{a_{e}}\right)=-\frac{1}{3(1+\omega_{eff})}\ln\left(\frac{\rho_{rh}}{\rho_{e}}\right)\,.$
(46)

On the other hand, at the horizon crossing of the physical scales we have

$\displaystyle
0=\ln\left(\frac{k_{hc}}{a_{hc}H_{hc}}\right)=\ln\left(\frac{a_{e}}{a_{hc}}\frac{a_{rh}}{a_{e}}\frac{a_{0}}{a_{rh}}\frac{k_{hc}}{a_{0}H_{hc}}\right)\,,$
(47)

with subscript “$0$” being the current value of the corresponding parameter.
Now, equations (45)-(47) give

$\displaystyle
N_{hc}+N_{rh}+\ln\left(\frac{k_{hc}}{a_{0}H_{hc}}\right)+\ln\left(\frac{a_{0}}{a_{rh}}\right)=0\,.$
(48)

Now, we express $\frac{a_{0}}{a}$ in terms of the temperature and density. To
this end, we use the following relation between the energy density and
temperature during the reheating [61, 63]

$\rho_{rh}=\frac{\pi^{2}g_{rh}}{30}T_{rh}^{4}\,.$ (49)

In the above equation, the parameter $g_{rh}$ is the effective number of the
relativistic species at the reheating era. Also, the conservation of the
entropy gives the following equation [61, 63]

$\frac{a_{0}}{a_{rh}}=\left(\frac{43}{11g_{rh}}\right)^{-\frac{1}{3}}\frac{T_{rh}}{T_{0}}\,.$
(50)

From equations (49) and (50) we get

$\displaystyle\frac{a_{0}}{a_{rh}}=\left(\frac{43}{11g_{rh}}\right)^{-\frac{1}{3}}T_{0}^{-1}\left(\frac{\pi^{2}g_{rh}}{30\rho_{rh}}\right)^{-\frac{1}{4}}\,.$
(51)

In the tachyon model with superpotential, the energy density in terms of the
slow-roll parameter $\epsilon$ is given by

$\displaystyle\rho=\frac{\frac{\kappa^{2}}{6}W^{2}-\frac{1}{8}W^{\prime
2}}{\sqrt{1-\frac{2}{3}\epsilon}}\,.$ (52)

At the end of inflation, where we have $\epsilon=1$, the energy density takes
the following form

$\rho_{e}=\sqrt{3}\,\bigg{(}\frac{\kappa^{2}}{6}W_{e}^{2}-\frac{1}{8}W_{e}^{\prime
2}\bigg{)}\,.$ (53)

The energy density during the reheating era is obtained from equations (46)
and (53) as follows

$\displaystyle\rho_{rh}=\sqrt{3}\,\bigg{(}\frac{\kappa^{2}}{6}W_{e}^{2}-\frac{1}{8}W_{e}^{\prime
2}\bigg{)}\,\exp\Big{[}-3N_{rh}(1+\omega_{eff})\Big{]}.$ (54)

Now, equations (51) and (53) give

$\displaystyle\ln\left(\frac{a_{0}}{a_{rh}}\right)=-\frac{1}{3}\ln\left(\frac{43}{11g_{rh}}\right)-\frac{1}{4}\ln\left(\frac{\pi^{2}g_{rh}}{30\rho_{rh}}\right)-\ln
T_{0}+\frac{1}{4}\ln\Bigg{(}\sqrt{3}\,\bigg{(}\frac{\kappa^{2}}{6}W_{e}^{2}-\frac{1}{8}W_{e}^{\prime
2}\bigg{)}\Bigg{)}$ $\displaystyle-\frac{3}{4}N_{rh}(1+\omega_{eff})\,.$ (55)

By using equations (29), (48) and (5), we find the e-folds number during
reheating as follows

$\displaystyle
N_{rh}=\frac{4}{1-3\omega_{eff}}\Bigg{[}-N_{hc}-\ln\Big{(}\frac{k_{hc}}{a_{0}T_{0}}\Big{)}-\frac{1}{4}\ln\Big{(}\frac{40}{\pi^{2}g_{rh}}\Big{)}-\frac{1}{3}\ln\Big{(}\frac{11g_{rh}}{43}\Big{)}$
$\displaystyle+\frac{1}{2}\ln\Big{(}8\pi^{2}{\cal{A}}_{s}{\cal{Q}}_{s}c_{s}^{3}\Big{)}-\frac{1}{4}\ln\Bigg{(}\sqrt{3}\,\bigg{(}\frac{\kappa^{2}}{6}W_{e}^{2}-\frac{1}{8}W_{e}^{\prime
2}\bigg{)}\Bigg{)}\Bigg{]}.$ (56)

We can also find the temperature during reheating from equations (46), (50)
and (53) as follows

$T_{rh}=\bigg{(}\frac{30}{\pi^{2}g_{rh}}\bigg{)}^{\frac{1}{4}}\,\bigg{[}\sqrt{3}\,\bigg{(}\frac{\kappa^{2}}{6}W_{e}^{2}-\frac{1}{8}W_{e}^{\prime
2}\bigg{)}\bigg{]}^{\frac{1}{4}}\,\exp\bigg{[}-\frac{3}{4}N_{rh}(1+\omega_{eff})\bigg{]}\,.$
(57)

Now, we study the reheating process numerically. To this end, by using
equation (2), we obtain equations (5) and (57) in terms of the tachyon field
$\phi$. After that, we use equations (18), (2.2), (2.2) and (31) to find the
value of the tachyon field in terms of the scalar spectral index. In this way,
we get the e-folds number and the temperature during the reheating phase in
terms of $n_{s}$. By using the observational constraint on the scalar spectral
index, obtained from Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO joint data,
we can find some constraints on $n_{rh}$ and $T_{rh}$. In this regard, in
figure 6 we have plotted the e-fold number during reheating versus the
effective equation of state. This figure has been plotted for some sample
values of the parameters $\lambda$ and $\alpha_{2}$ which have been adopted
from the constraints presented in table 1. Also, to plot this figure we have
used the constraint on the scalar spectral index as $n_{s}=0.9658\pm 0.0038$.
The range used for the effective equation of state is $-1\leq\omega_{eff}\leq
1$. In fact, at the end of the inflationary expansion, we have
$\omega_{eff}=-\frac{1}{3}$ and at the beginning of the radiation dominated
era $\omega_{eff}=\frac{1}{3}$. Therefore, the adopted range for the effective
equation of state seems suitable for consideration. As the figure shows, the
value of e-folds number during reheating increases by changing the value of
the effective equation of state from $-1$ to $\frac{1}{3}$. Also, our
numerical analysis shows that for $\lambda<0.2$ and $\alpha_{2}<1.53\times
10^{-2}$, it is possible to have instantaneous reheating. However, for
$\lambda\geq 0.2$ and $\alpha_{2}\geq 1.53\times 10^{-2}$, a few e-folds
number are needed for the reheating process to be completed. We have also
plotted the behavior of $N_{rh}$ versus $n_{s}$, for some sample values of
$\lambda$ and $\alpha_{2}$, in figure 7. In this figure, we have also adopted
three values of the effective equation of state as
$\omega_{eff}=-1,-\frac{1}{3}$ and $0$. The behavior of the temperature during
the reheating versus the scalar spectral index is shown in figure 8. This
figure has been plotted for the same values of the parameters as figure 7. By
performing these numerical analyses, we have obtained some constraints on the
important reheating parameters $N_{rh}$ and $T_{rh}$, summarized in tables 4
and 5.

Figure 6: Ranges of the e-folds number and the effective equation of state
parameter during the reheating phase for the tachyon model with
superpotential, leading to the observationally viable values of the scalar
spectral index.

Figure 7: The behavior of the e-folds number during the reheating phase versus
the scalar spectral index, for the tachyon model with superpotential. The red
region shows the values of the scalar spectral index released by Planck2018
TT, TE, EE+lowE+lensing+BK14+BAO joint data at $68\%$ CL.

Figure 8: The behavior of the temperature during the reheating phase versus the scalar spectral index, for the tachyon model with superpotential. The dark green region shows the temperatures below the electroweak scale, $T<100$ GeV, and the light green region corresponds to the temperatures below the big bang nucleosynthesis scale, $T<10$ MeV. Table 4:  Constraints on the e-folds number during reheating in the tachyon model with superpotentail, obtained from Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO joint data at $68\%$ CL. $\lambda$ | $\alpha_{2}$ | $\omega=-1$ | $\omega=-\frac{1}{3}$ | $\omega=0$
---|---|---|---|---
$0.1$ | $1.41\times 10^{-2}<\alpha_{2}<1.83\times 10^{-2}$ | $0\leq N_{rh}\leq 0.134$ | $0\leq N_{rh}\leq 0.181$ | $0.012\leq N_{rh}\leq 0.513$
$0.4$ | $0.68\times 10^{-2}<\alpha_{2}<0.91\times 10^{-2}$ | $0.010\leq N_{rh}\leq 0.883$ | $0.197\leq N_{rh}\leq 1.22$ | $0.514\leq N_{rh}\leq 4.31$
$0.7$ | $0.48\times 10^{-2}<\alpha_{2}<0.68\times 10^{-2}$ | $0.201\leq N_{rh}\leq 3.32$ | $0.862\leq N_{rh}\leq 6.67$ | $1.72\leq N_{rh}\leq 15.6$
$0.9$ | $0.17\times 10^{-2}<\alpha_{2}<0.33\times 10^{-2}$ | $0.541\leq N_{rh}\leq 5.81$ | $1.404\leq N_{rh}\leq 7.66$ | $2.89\leq N_{rh}\leq 22.2$
Table 5:  Constraints on the temperature during reheating in the tachyon model with superpotentail, obtained from Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO joint data at $68\%$ CL. $\lambda$ | $\alpha_{2}$ | $\omega=-1$ | $\omega=-\frac{1}{3}$ | $\omega=0$
---|---|---|---|---
$0.1$ | $1.41\times 10^{-2}<\alpha_{2}<1.83\times 10^{-2}$ | $14.6\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)$ | $13.3\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)$ | $9.91\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)$
$0.4$ | $0.68\times 10^{-2}<\alpha_{2}<0.91\times 10^{-2}$ | $14.3\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 14.9$ | $12.2\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 14.6$ | $8.69\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 14.2$
$0.7$ | $0.48\times 10^{-2}<\alpha_{2}<0.68\times 10^{-2}$ | $12.3\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 14.5$ | $9.43\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 13.9$ | $4.03\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 13.1$
$0.9$ | $0.17\times 10^{-2}<\alpha_{2}<0.33\times 10^{-2}$ | $11.5\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 14.3$ | $6.32\leq\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 12.9$ | $\log_{10}\left(\frac{T_{rh}}{GeV}\right)\leq 10.8$

##  6 A Short Discussion on the Case with Quantum Effects

In this section, we study the consequence of presenting the quantum effects on
the tachyon’s dynamics in the early universe. To this end, following Ref.
[64], we include the quantum effect by considering the conformal anomaly. By
considering ${\cal{N}}$ scalar, ${\cal{N}}_{1}$ vector fields, ${\cal{N}}_{2}$
gravitons, ${\cal{N}}_{\frac{1}{2}}$ spinor and ${\cal{N}}_{hd}$ higher-
derivative conformal scalars, we have the anomalous trace of the stress tensor
as [65, 66, 64]

${\cal{T}}=b\left(\Upsilon+\frac{2}{3}\Box R\right)+b^{\prime}\,G\,,$ (58)

wehere

$b=\frac{{\cal{N}}+6{\cal{N}}_{\frac{1}{2}}+12{\cal{N}}_{1}+611{\cal{N}}_{2}-8{\cal{N}}_{hd}}{120(4\pi)^{2}}\,,$
(59)

and

$b^{\prime}=-\frac{{\cal{N}}+11{\cal{N}}_{\frac{1}{2}}+62{\cal{N}}_{1}+1411{\cal{N}}_{2}-28{\cal{N}}_{hd}}{360(4\pi)^{2}}\,.$
(60)

If we consider a warped de-Sitter space-time with a scale factor as
$a=e^{\frac{t}{L}}$ ($L$ is the curvature radius of the space-time), the
energy density and pressure corresponding to the quantum effects are given by
(see Refs. [66, 67])

$\rho_{q}=-p_{q}=-\frac{6b^{\prime}}{L^{4}}\,.$ (61)

Therefore, by considering the quantum effects and the tachyon field, the
Friedmann equations (5) and (6) take the following forms

$\displaystyle\frac{3}{L^{2}}=\frac{\kappa^{2}\,V(\phi)}{\sqrt{1-\lambda\,\dot{\phi}^{2}}}-\frac{6\kappa^{2}\,b^{\prime}}{L^{4}}\,,$
(62)
$\displaystyle\frac{3}{L^{2}}=\kappa^{2}\,V(\phi)\,\sqrt{1-\lambda\,\dot{\phi}^{2}}-\frac{6\kappa^{2}\,b^{\prime}}{L^{4}}\,.$
(63)

Equations (62) and (63) are satisfied if $\dot{\phi}=0$ or
$\phi=\phi_{0}=constant$. Also, from equation of motion (7) we find
$V^{\prime}=0$. This means that in the presence of the quantum effects, the
potential of the tachyon field becomes constant. We name this constant
potential as $V_{0}$. Now, from equation (62) we have

$\displaystyle\frac{1}{L^{2}}=-{\frac{1}{{4\,b^{\prime}}\,{\kappa}^{2}}}\pm\sqrt{{\frac{V_{0}}{6b^{\prime}}}+{\frac{1}{16\,{b^{\prime}}^{2}{\kappa}^{4}}}}\,.$
(64)

Considering that $b^{\prime}$ is constant, equation (64) give us constant
Hubble parameter. The constant Hubble parameter means $\epsilon=\eta=0$. Even
we consider small potential (as the case in Ref. [64]) and consider the
quantum effect as the running of inflation, there is still a problem. With a
constant Hubble parameter, the inflation phase would never have a graceful
exit. This problem remains even we assume the tachyon/phantom filed or a
canonical scalar field. If we insist to account for the quantum effect in the
inflation epoch, we may consider models with two scalar fields with quantum
effects (as has been done in Ref. [64]) or maybe nonminimal coupling models.

##  7 Summary

In this paper, we have studied the tachyon inflation with the superpotential
as an inflationary potential. In this regard, without including any
supersymmetry, we have used a form of the potential which is similar to the
one in the supergravity. By studying the inflation in this model, we have
obtained the slow-roll parameters in terms of the superpotential. We have also
explored the perturbations in this model at both the linear and non-linear
levels. At the linear level, we have obtained the important perturbation
parameters, such as the scalar spectral index, the tensor spectral index and
the tensor-to-scalar ratio. At the non-linear level, we have obtained a
parameter named non-linearity parameter which gives the amplitude of the non-
gaussianity. Considering that the amplitude of the non-gaussianity in the
tachyon model has a peak at the equilateral configuration, we have obtained
the non-linearity parameter in this configuration.

After that, we have sought for the observational viability of the tachyon
model with superpotential. In this regard, we have focused on the model’s
parameters $\lambda$ and $\alpha_{2}$. By using the observational constraints
on the perturbations parameters, obtained from Planck2018 TT, TE,
EE+lowE+lensing+BAO+BK14 and Planck2018 TT, TE, EE +lowE+lensing+BK14+BAO+LIGO
and Virgo2016 data at $68\%$ CL, we have obtained the observationally viable
regions of the parameters $\lambda$ and $\alpha_{2}$. We have also studied the
behavior of $r-n_{s}$ and $r-n_{T}$, in the background of Planck2018 TT, TE,
EE+lowE+lensing+BAO+BK14 and Planck2018 TT, TE, EE +lowE+lensing+BK14+BAO+LIGO
and Virgo2016 data, respectively, at $68\%$ CL and $95\%$ CL. Also, by using
the observational constraints on the scalar spectral index at $68\%$ CL and
$95\%$ CL, we have plotted the observationally viable ranges of the sound
speed and $\lambda$ which fulfill the constraints. By these numerical
analyses, we have obtained some constraints on the parameter $\alpha_{2}$ for
some sample values of $\lambda$ as $\lambda=0.1,0.4,0.7$ and $0.9$. Our
numerical analysis on the perturbations parameters shows that, depending on
the values of $\lambda$, the tachyon model with superpotential is
observationally viable if $0.040\times 10^{-2}\leq\alpha_{2}<0.1$ at $68\%$ CL
and $0.021\times 10^{-2}\leq\alpha_{2}<0.1$ at $95\%$ CL. We have also studied
the non-gaussian feature of the primordial perturbations numerically. By
plotting the phase space of the parameters $\lambda$ and $\alpha_{2}$, we have
presented the prediction of the tachyon model with superpotential for the
amplitude of the equilateral non-gaussianity. We have shown that, for
$\lambda>0.9$, to the amplitude of the primordial non-gaussianity be
consistent with Planck2018 TTT, EEE, TTE and EET data, there are some
constraints on the parameter $\alpha_{2}$. Note that, for $\lambda\leq 0.9$,
all values of $\alpha_{2}$ lead to the observationally viable values of
$f_{NL}^{equil}$.

Another issue that has been studied in this paper, is the reheating process
after inflation by assuming the superpotential. To study this process, we have
found the e-folds number and temperature during the reheating era in terms of
the superpotential. Then, given the relation between these parameters and
scalar spectral, by using the observational constraints on $n_{s}$, we have
studied the parameters $N_{rh}$ and $T_{rh}$ numerically. We have shown that,
by increasing the values of the e-folds number during the reheating phase, the
effective equation of state in the tachyon model with superpotential increases
from $-1$ to $\frac{1}{3}$ (corresponding to the radiation dominated
universe). We have also, studied the behavior of the parameters $N_{rh}$ and
$T_{rh}$ versus $n_{s}$ for some sample values of $\lambda$ and $\alpha_{2}$.
Our numerical analysis shows that, in the tachyon model with superpotential,
the reheating is instantaneous for $\lambda<0.2$ and $\alpha_{2}<1.53\times
10^{-2}$. In other cases, the reheating process needs a few e-folds number to
be completed. We have also shortly discussed the case with presenting the
quantum effects. It seems that if we consider the models with single scalar
fields, the presence of quantum corrections makes the inflation phase
permanent. To have a graceful exit from inflation, we may consider the quantum
effects in two-field or nonminimal models.

Acknowledgement
I thank the referee for the very insightful comments that have improved the
quality of the paper considerably.

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#  JWST and ALMA imaging of dust-obscured, massive substructures in a typical
$\it z\sim 3$ star-forming disk galaxy

Wiphu Rujopakarn11affiliationmark:  22affiliationmark:  , Christina C.
Williams33affiliationmark:  44affiliationmark:  , Emanuele
Daddi55affiliationmark:  , Malte Schramm66affiliationmark:  , Fengwu
Sun44affiliationmark:  ,
Stacey Alberts44affiliationmark:  , George H. Rieke44affiliationmark:  , Qing-
Hua Tan (谈清华)77affiliationmark:  , Sandro Tacchella88affiliationmark:
99affiliationmark:  ,
Mauro Giavalisco1010affiliationmark:  , and John D.
Silverman1111affiliationmark:  1National Astronomical Research Institute of
Thailand, Don Kaeo, Mae Rim, Chiang Mai 50180, Thailand<EMAIL_ADDRESS>
2Department of Physics, Faculty of Science, Chulalongkorn University, 254
Phayathai Road, Pathumwan, Bangkok 10330, Thailand
3NSF’s National Optical-Infrared Astronomy Research Laboratory, 950 North
Cherry Avenue, Tucson, AZ 85719, USA
4Steward Observatory, The University of Arizona, Tucson, AZ 85721, USA
5Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France
6Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany
7Purple Mountain Observatory, Chinese Academy of Sciences, 10 Yuanhua Road,
Nanjing 210023, People’s Republic of China
8Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge, CB3 0HA, UK
9Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue,
Cambridge, CB3 0HE, UK
10Department of Astronomy, University of Massachusetts, B 619E 710 North
Pleasant Street Amherst, MA, USA
11Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583,
Japan

###### Abstract

We present an identification of dust-attenuated star-forming galactic-disk
substructures in a typical star-forming galaxy (SFG), UDF2, at $z=2.696$. To
date, substructures containing significant buildup of stellar mass and
actively forming stars have yet to be found in typical (i.e., main-sequence)
SFGs at $z>2$. This is due to the strong dust attenuation common in massive
galaxies at the epoch and the scarcity of high-resolution, high-sensitivity
extinction-independent imaging. To search for disk substructures, we
subtracted the central stellar-mass disk from the JWST/NIRCam rest-frame 1.2
µm image ($0\farcs 13$ resolution) and subtracted, in the visibility plane,
the central starburst disk from ALMA rest-frame 240 µm observations ($0\farcs
03$ resolution). The residual images revealed substructures at rest-frame 1.2
µm co-located with those found at rest-frame 240 µm, $\simeq 2$ kpc away from
the galactic center. The largest substructure contains $\simeq 20$% of the
total stellar mass and $\simeq 5$% of the total SFR of the galaxy. While UDF2
exhibits a kinematically-ordered velocity field of molecular gas consistent
with a secularly evolving disk, more sensitive observations are required to
characterize the nature and the origin of this substructure (spiral arms,
minor merger, or other types of disk instabilities). UDF2 resides in an
overdense region ($N\geqslant 4$ massive galaxies within 70 kpc projected
distance at $z=2.690-2.697$) and the substructures may be associated with
interaction-induced instabilities. Importantly, a statistical sample of such
substructures identified with JWST and ALMA could play a key role in bridging
the gap between the bulge-forming starburst and the rest of the galaxy.

###### Subject headings:

galaxies: evolution — galaxies: structure

##  1\. Introduction

The groundbreaking angular resolution and sensitivity of far-IR/submillimeter
(submm) observations from the Atacama Large Millimeter/submillimeter Array
(ALMA) and the subsequent flurry of blank-field interferometric submm surveys
have enabled capturing an unbiased sample of typical111Typical star-forming
galaxies are defined here as those with SFR within a factor of four of the
main-sequence (e.g., Rodighiero et al., 2011) at their corresponding redshift
on all of the following main-sequence parameterizations: Whitaker et al.
(2012); Speagle et al. (2014), and Schreiber et al. (2015), hereafter ‘typical
SFGs’. star-forming galaxies (SFGs) at $z\sim 1-4$ that represents the
assembly sites of most stellar mass in the Universe independent of their dust
attenuation (e.g., Dunlop et al., 2017; Elbaz et al., 2018).

Sub-arcsecond ALMA observations of typical SFGs at this epoch have revealed a
common pattern: they contain a compact region of intense star formation (SF)
at their centers. These central star-forming regions usually harbor star-
formation rate (SFR) of $\sim 100-300$ $M_{\odot}$yr-1 within $r_{\rm SF}\sim
0.5-1$ kpc, which is $\sim 2-5$ times more compact than the extent of their
stellar buildups virtually independently of the way they are selected, e.g.,
far-IR/submm blank field surveys (Dunlop et al., 2017; Fujimoto et al., 2018;
Elbaz et al., 2018), H$\alpha$-selected galaxies (Tadaki et al., 2017), or
optically-selected massive compact SFGs (Barro et al., 2016).

Figure 1.— JWST images of UDF2. The left panel shows JWST color composite; the
four sub-panels on the right show F182M image with the cyan dotted circle
indicating the position of the brightest feature previously seen in HST/F160W
image (Figure 1 of Rujopakarn et al., 2019) that is now resolved into multiple
point-like features; F210M image with the yellow dotted arc indicating the
semicircular feature around the center of stellar mass distribution that is
clearly visible in the F430M and F480M images. Each cutout is
$3^{\prime\prime}\times 3^{\prime\prime}$; north is up, east is to the left.

Zooming in further, high-S/N sub-0$\farcs$1 ALMA observations that can confirm
or rule out substructures of SF are still scarce, but early results indicate
that these central star-forming regions are strikingly smooth (Rujopakarn et
al., 2019; Ivison et al., 2020). There has been no identification of off-
center structures containing significant SFR in typical SFGs thus far — ALMA
identification of SF substructures at $z>2$ so far are in luminous submm-
selected galaxies (e.g., Swinbank et al., 2015; Iono et al., 2016; Tadaki et
al., 2018; Hodge et al., 2019). Also, the UV-bright clumps commonly found in
typical SFGs at $z\gtrsim 1$ contain $\lesssim$1% of the total SF and
molecular gas. That is, the UV-bright clumps are not the signposts for the
‘tip of the iceberg’ beneath which the bulk of SF is occurring (Cibinel et
al., 2017; Rujopakarn et al., 2019).

For typical SFGs at $z>2$, the connection between their compact core observed
at far-IR/submm and their extended optical morphologies remains unclear.
Bridging this gap requires finding off-center substructures that are detected
in both far-IR/submm and optical bands, which represent the sites of SF and
stellar mass buildups, respectively. Identification of off-center
substructures with substantial presence of stellar mass and harboring
significant SFR could indicate, e.g., spiral arms or other types of disk
instability. Notably, rest-frame optical images of typical SFGs often exhibit
strong dust attenuation in the proximity of the far-IR/submm emission (Nelson
et al., 2019; Rujopakarn et al., 2016, 2019), which suggest that JWST
observations will play a key role in bridging the gap between optical and far-
IR/submm pictures.

In this Letter, we report the first identification of off-center massive
substructures in a typical SFG at $z\sim 3$ that is detected in both
JWST/NIRCam and ALMA observations. We adopt a $\Lambda$CDM cosmology with
$\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$, $H_{0}=70~{}{\rm km\,s}^{-1}{\rm
Mpc}^{-1}$, where $1^{\prime\prime}$ at $z=2.7$ corresponds to 7.9 kpc, and
the Salpeter IMF.

Figure 2.— To search for substructures in UDF2’s star-forming regions, we
subtract from the ALMA rest-frame 240 µm continuum visibilities a Spergel
profile (an approximation of the Sersic profile with analytical Fourier
transform). From left to right is the dirty ALMA image, the residual map with
the main dust disk modeled by a single Spergel profile subtracted, and the
residual after all components are subtracted, respectively. The top row is the
native resolution image, whereas the bottom row is tapered at 1000 meters to
enhance the sensitivity to extended features. Each cutout is
$1^{\prime\prime}\times 1^{\prime\prime}$; north is up, east is to the left;
contours are $[-1.5,1.5,1.5^{1.5},1.5^{2},...]\times\sigma$.

##  2\. Target and Observations

Our target (“UDF2”) was identified as a main-sequence SFG via an ALMA blank-
field survey (Dunlop et al., 2017) of the Hubble Ultra Deep Field (HUDF). It
has a total SFR of 260 $M_{\odot}$yr-1 and log($M_{*}$/$M_{\odot}$) $=10.9\pm
0.2$ (Rujopakarn et al., 2019) at $z=2.6961$ (Kaasinen et al., 2020). HST
imaging of UDF2 at $0.6-1.6$ µm (rest-frame $0.2-0.4$ µm) show disturbed
morphology extending up to 2$\farcs$5 ($\simeq 20$ kpc) whereas the dust
emission observed by ALMA originates from within the central 3 kpc region
(Rujopakarn et al., 2016).

UDF2 is a subject of a sensitive search for SF substructure employing ALMA
rest-frame 240 µm dust continuum observations at $42\times 30$ mas resolution
and 9.6 $\mu$Jy beam-1 rms sensitivity (Rujopakarn et al., 2019), which finds
that the cold dust is distributed in a smooth disk with no substructures at
the 200 pc scale. This data set remains one of the highest angular resolution
imaging at the highest S/N of a typical SFG at $z>2$ to date.

Kaasinen et al. (2020) further revealed by means of CO($3-2$) observations at
0$\farcs$88 $\times$ 0$\farcs$61 resolution that this disk is kinematically-
ordered, well-described by a single rotating component with maximum rotational
velocity, $V_{\rm rot,max}=349^{+19}_{-23}$ km s-1 and velocity dispersion,
$\sigma=77^{+10}_{-11}$ km s-1. The corresponding $V/\sigma$ ratio of $4.5\pm
0.7$ is considerably more ‘dynamically cold’ than disk galaxies at its epoch
(cf. compilation by Rizzo et al., 2020). While a kinematically-ordered
velocity field does not necessarily preclude mergers (e.g., Simons et al.,
2019), these observations lend credence to a picture of UDF2 being an isolated
rotating disk, representative of typical SFGs at $z\sim 3$.

UDF2 was observed with JWST/NIRCam as a part of the JWST Extragalactic Medium-
band Survey (JEMS; Program ID: 1963; PI: Williams). Observations were taken on
2022 October 11 in five filters, namely F182M, F210M, F430M, F460M, and F480M.
The integration time was 28 ks in each filter (except F430M and F460M which
were 14 ks each). The data reduction process is described in detail in
Williams et al. (2023). The final mosaics are sensitive to 29.3, 29.2, 28.5,
28.3, and 28.6 AB mag (5$\sigma$, point source) in the five filters in the
ascending order of wavelength and are resampled to 30 mas pixel size. We
independently inspect the astrometry of the final mosaic with 30-mas ALMA
images of the nearby UDF1, 3, and 7 (whose astrometry is tied to the ICRF to
$\simeq 2$ mas; Rujopakarn et al., 2019) and find that the offsets between
brightest features in ALMA and JWST images are sub-pixel (i.e., $<30$ mas).

Figure 3.— To search for substructures in the stellar mass distribution, we
first model all components of the F430M emission (a), whose subtraction from
the original image affords a flat residual (b), then only subtract the Primary
Sérsic model of the stellar mass disk (c). The residuals in (d) indicate the
presence of stellar mass substructures northeast and southwest of the galaxy
center. Cutouts in panels (a) to (d) are $3^{\prime\prime}\times
3^{\prime\prime}$ and with a log scaling. Panel (e) shows the central
$1^{\prime\prime}\times 1^{\prime\prime}$ of the residuals in panel (d) with a
linear scaling; the tapered ALMA residuals overplotted as orange contours
starting from 3$\sigma$ in 1$\sigma$ increment. The grey contours show the
pre-subtraction ALMA continuum, tapered to the same scale, starting from
$3\sigma$ in 5$\sigma$ increment. The substructure in the northeast harboring
SF and significant stellar mass buildup, dubbed the “northern ridge” is
clearly visible.

##  3\. Identification of Galactic-Disk Substructures

To search for galactic-disk substructures, we model and subtract the central
component from ALMA and JWST images to search for co-locating residuals in
both bands. We note that the galaxy ‘center’ refers to the centroid of the
dust emission from the ALMA observations, whose position is localized within
$\simeq$ 2 mas.

Figure 4.— The star-forming substructures (orange contours starting from
3$\sigma$ in 1$\sigma$ increment), in the context of the rest of the galaxy.
Notably, the HST/F606W image (rest-frame 160 Å) shows the spatial dislocation
between unobscured SF and the northern ridge; the JWST/F182M and F210M images
show that the northern ridge appears to be a part of the semicircular arc
around the central stellar mass distribution observed at F430M. The red
contour delineates the 10$\sigma$ isophotal extent of unsubtracted ALMA
emission (with the red dot marking the peak).

###  3.1. Rest-frame optical/near-IR morphology from JWST

The JWST images of UDF2 are shown in Figure 1. The most striking feature is
the presence of a single stellar mass component, seen in all long-wavelength
filters (F430M-F480M; rest-frame $1.2-1.3$ µm), but not obvious at shorter
wavelength bands. The long-wavelength images show an isolated disk with no
major morphological disturbance. This represents the true stellar mass
distribution, an observation that was not previously measurable in HST/F160W
image because of the dust attenuation. The centroid and position angle of the
stellar mass component is in close alignment with that of the cold dust disk
and the rotation plane of the molecular gas disk (Figure 7 of Kaasinen et al.,
2020).

The F182M and F210M images (rest-frame $0.5-0.6$ µm) show a more complex
morphology including clumpy substructures surrounding the location of the
stellar mass buildup. The F182M image contains a bright region 0$\farcs$2
northeast of the center, which corresponds to a prominent point-like feature
in HST/F160W image (owing to the lower resolution). This bright region becomes
less pronounced in F210M, where it appears to be a part of a semicircular
feature (yellow arc in Figure 1) surrounding the location of the stellar mass
buildup.

###  3.2. Identification of star-forming substructures

To search for off-center cold dust substructures that could indicate galactic-
disk SF, we re-analyze the ALMA rest-frame 240 µm observations of UDF2 using
the visibility-based modeling technique described in Rujopakarn et al. (2019).
In the previous analysis, we modeled the SF disk using superpositioned
Gaussians to represent the Sérsic profile (Hogg & Lang, 2013) because the
latter lacks the analytical Fourier transform necessary for the visibility-
based analysis. The best fit model was a double concentric Gaussian that
affords an approximation of a Sérsic. A region approximately 0$\farcs$2
northeast of the center contains some residual flux peaking at $\simeq
2.9\sigma$, though the lack of definitively associable HST counterpart at its
position (see Figures 1 and 2 of Rujopakarn et al., 2019) precludes the
recognition of its significance over other sub-$3\sigma$ noise peaks.

In this analysis, we employ the newly implemented Spergel (2010) profile in
GILDAS version nov22a, which offers an accurate representation of the Sérsic
profile without relying on multiple components and featuring an analytical
Fourier transform. The Spergel profile is characterized by the Spergel index
$\nu$, flux, and half-light radius (Spergel, 2010), where Spergel $\nu$ plays
a similar role to the Sérsic $n$. Within a range of $\nu$, the Spergel
profiles resemble Sérsic profiles. For example, at $\nu=0.5$, the Spergel
profile is identical to an exponential profile ($n=1$). The robustness of the
GILDAS implementation of Spergel fitting was tested using simulations by
Kalita et al. (2022); a robust mapping between the Sérsic $n$ and Spergel
$\nu$ will be presented in Q. Tan et al., in prep.

We fit all three galaxies in the ALMA beam (UDF1, 2, and 7) using the Spergel
profile simultaneously using the GILDAS uv_fit task. The best-fit Spergel
profile of UDF2 has major and minor axes of $135\pm 3$ mas and $65\pm 2$ mas,
respectively, $\nu=-0.28\pm 0.04$, and a flux of $2885\pm 106$ $\mu$Jy. After
subtracting the best-fit Spergel profile in the $uv$-plane, two regions of
residual remain: one at the northeast and southwest of the galaxy center
(Figure 2). The northeast one, in particular, peaks at $5.5\sigma$ when
Gaussian tapering is applied to baselines longer than 1000 meter (which
results in a synthesized beam of $0\farcs 16\times 0\farcs 13$ — similar to
that of JWST/F430M). We measure the fluxes of these substructures by fitting —
in the $uv$-plane — elliptical Gaussians centering at the peaks of the
substructure in the tapered image. The residual maps after subtracting the
best-fit models, from which we have measured the substructure fluxes are shown
in the ‘All subtracted’ column in Figure 2. The total flux in the northeast
and southwest substructures are $172\pm 28$ and $148\pm 42$ $\mu$Jy,
respectively, each being $\simeq$ 5% of the total flux of the entire galaxy.

###  3.3. Identification of stellar-mass substructures

To search for off-center stellar mass buildups that serve as a proxy for
galactic disk substructures, we perform a GALFIT decomposition on the F430M
image by modeling the source following the same procedure performed on the HST
images in Rujopakarn et al. (2019). The F430M image is chosen for this
analysis because it is the shortest band of the three long-wavelength module
images, thereby affording the best spatial sampling of the substructures. The
PSF model was derived from a median combination of non-saturating stars in the
field.

The best-fit model (Figure 3) indicates a primary Sérsic (containing 66% of
the total F430M emission) with $n=0.98$, indicating that the central stellar
mass buildup is well-described by an exponential disk; the half-light extent
of the disk is $3.3\times 2.3$ kpc with a position angle of 11∘
(counterclockwise from the north). Subtracting this exponential disk reveals
two off-center substructures in the northeast and southwest of the center,
which contain 23% and 10% of the total F430M flux, respectively; these three
components altogether account for 99% of the total model flux. The northeast
substructure (which we will refer to as the ‘northern ridge’ hereafter) is a
part of (and has an elongated shape along) the semicircular feature observed
at F210M.

Most significantly, the ALMA substructures are co-located with these F430M
substructures, as shown in panel (e) of Figure 3, suggesting that they are
off-center stellar mass substructures that harbor ongoing SF. Furthermore, the
northern ridge appears to be partially overlapping with point-like features
observed at F182M and F210M, which could represent the relatively less dust-
attenuated (or patchy) emission from young stars (Figures 3 and 4).

###  3.4. Physical properties of the off-center substructures

Having established the existence of off-center substructures containing both
stellar mass and SF, we put constraints on their physical properties in order
to aid the interpretation of their nature. We will focus on the northern ridge
in this Letter and follow up with a more comprehensive study of UDF2 in the
future when deeper JWST/NIRCam and MIRI imaging from the ongoing programs in
Cycle 1 becomes available.

The northern ridge, shown in panel (e) of Figure 3, forms an arc spanning the
position angle of $\simeq 0-50$ deg (counterclockwise from the north) of, and
at a distance of $0\farcs 20-0\farcs 35$ ($1.6-2.8$ kpc) from the center. The
length and width of the ridge are $\simeq 2.5\times 1$ kpc.

The ALMA rest-frame 240 µm residual flux in the northern ridge is $\simeq 5\%$
of the total flux. If we assume that dust temperature and IMF do not vary
spatially across the galaxy and that SFR varies linearly with the far-IR flux,
the inferred SFR within the northern ridge is $\simeq 13$ $M_{\odot}$yr-1. We
note that the unobscured SFR inferred from rest-frame UV emission is $<1$
$M_{\odot}$yr-1 (Dunlop et al., 2017); its contribution to the SFR of the
northern ridge remains small, i.e., even the extended SF in the outer region
of the galaxy is strongly dust-obscured. Likewise, the JWST rest-frame 1.2 µm
residual flux in the northern ridge is 23% of the total flux. If we assume a
constant IMF and that the stellar mass varies linearly with the 1.2 µm flux,
the stellar mass in the northern ridge is about log($M_{*}$/$M_{\odot}$) of
$10.2\pm 0.2$. This stellar mass estimate should be taken as an upper limit
considering the potential contribution from young stars to the rest-frame 1.2
µm emission.

In summary, we have identified, within a typical SFG at $z=2.7$, two off-
center substructures containing $\simeq 10-20$% of total stellar mass and
$\simeq$5% of total SF. We highlight the fact that the central star-forming
region is where $\simeq 90\%$ of SF is located and also contains $\simeq$70%
of the total stellar mass, which implies that the substructures have $\sim
2-4\times$ lower specific SFR (sSFR) compared to that of the center.

##  4\. Discussions

There are notable parallels between the substructures identified in this work
and those observed at $z\sim 1-4$ using JWST. For example, spiral arms have
been identified optically by Wu et al. (2023) using NIRCam observations of
A2744-DSG-$z$3, a typical SFGs at $z=3.06$, but the angular resolution and
sensitivity of their ALMA observations are insufficient to resolve the spiral
structures observed by JWST. Similarly, using NIRCam observations of seven
sub-mm-selected typical SFGs at $z\sim 2$, Chen et al. (2022) found spiral
arms and bars and also nearby companions that suggest ongoing dynamical
interactions (their IR/radio observations are $\sim$few arcsecond resolution,
still not sufficient to associate the stellar mass and SF structures). Cheng
et al. (2023) observed 19 ALMA-selected cluster-lensed and field galaxies at
$z\sim 1-4.5$ spanning both main-sequence and quiescent sSFR ranges using
NIRCam and found substructures such as spiral arms to be common. Their ALMA
observations were at $0\farcs 3-2\farcs 1$ angular resolution and start to
reveal potential cases of SF within substructures and suggest that definitive
associations of SF within off-center stellar-mass substructures could be
commonplace with greater availability of high-resolution, high-S/N ALMA
observations of NIRCam targets.

In both the Chen et al. (2022) and Wu et al. (2023) samples, companion and/or
group environment appears to be a common attribute. We also find that UDF2
also resides in a potentially overdense region with four massive galaxies
(including UDF2) identified within a projected distance of 70 kpc,
spectroscopically confirmed (F. Sun, private communication) to be at
$z=2.690-2.697$ (Figure 5). These findings allude to a picture that the
stellar-mass substructures with ongoing SF identified in UDF2 could represent
a forming spiral arm or other types of disk instability resulting from minor
merger or interactions in an overdense environment (despite the overall cold
gas velocity field appearing to be kinematically-ordered; Kaasinen et al.,
2020).

Figure 5.— Three other massive galaxies spectroscopically confirmed to be at
$z=2.690-2.697$ are within a projected distance of 70 kpc from UDF2,
suggesting a potentially overdense region that may have implications on the
origin of the substructures identified.

On the contrary, local spiral arms contain stellar masses $\sim 1-5\%$ of the
disk (e.g., Pichardo et al., 2003), significantly smaller than the
substructures we have identified. The $2\times$ asymmetry of the stellar mass
between the northern and southern ridges of UDF2 is another departure from
classical spiral arms and bears a resemblance to the lopsided stellar disk
reported by Kalita et al. (2022), who proposed that mass lopsidedness in the
stellar disk could be a remnant indicating the ‘impact point’ of the cold gas
accretion stream (also in a cluster environment). Another similarity to the
Kalita et al. substructure, which is quiescent, is that the northern ridge has
$4\times$ lower sSFR compared to the core, and hence could represent an
evolutionary stage prior to the Kalita et al. picture. The fueling and highly
unstable disk (e.g., a local analog of $z\sim 1-2$ typical SFGs studied in
detail by Puschnig et al., 2023) that has led to a structure like that of the
northern ridge and the lopsided disks could be characteristic of typical SFGs
at $z\sim 3$.

Simulations of gas-rich galactic disks have long predicted the existence of SF
clumps, with the latest predictions being that they are $10^{7}-10^{8}$
$M_{\odot}$ in mass, SFR $\sim$1% of galaxy-integrated SFR, and size $\sim
50-100$ pc (e.g., Mandelker et al., 2017; Tamburello et al., 2017; Faure et
al., 2021; Fensch & Bournaud, 2021). So far, the predictions are at odds with
the handful of typical SFGs at $z\sim 2-3$ with high-resolution, high-S/N ALMA
observations: the central SF regions of UDF2 and other HUDF galaxies are
smooth and the strongly-lensed ‘Eyelash’ is also clumpless down to 1% SFR at
$\simeq$80 pc resolution (Rujopakarn et al., 2019; Ivison et al., 2020). It is
hence natural to ask if we are finally seeing off-center submm SF clumps here.
Perhaps not directly: the northern ridge is significantly larger in stellar
mass, SFR, and especially the size than the predicted clumps, though we cannot
exclude a scenario that the northern ridge is a structure comprising multiple
clumps of SF, which can be tested using higher resolution ALMA observations at
higher S/N.

This Letter highlights the importance of the combination of ALMA and JWST to
probe dusty galactic structure at $z\sim 3$. Apparent voids at shorter
wavelengths due to dust obscuration/attenuation are being filled by JWST. In
the case of UDF2, the location of the bulk of the stellar mass buildup is
revealed to be co-located with the starburst core, affording a clearer picture
of the galactic bulge during its assembly, and also galactic-disk
substructures actively forming stars. Building a statistical sample of disk
substructures in typical SFGs with optical/submm counterparts will be crucial
to help bridge the gaps between optical/near-IR and far-IR/submm morphologies,
as well as between the compact core SF and the rest of the galaxy.

## Acknowledgments

We thank the anonymous referee for their helpful suggestions and comments.
This Letter makes use of observations from JWST Cycle 1 GO program #1963.
Support for program JWST-GO-1963 was provided in part by NASA through a grant
from the Space Telescope Science Institute, which is operated by the
Associations of Universities for Research in Astronomy, Incorporated, under
NASA contract NAS 5-26555. The research of CCW is supported by NOIRLab, which
is managed by the Association of Universities for Research in Astronomy (AURA)
under a cooperative agreement with the National Science Foundation. This
Letter makes use of the ALMA data from ADS/JAO.ALMA#2017.1.00001.S. ALMA is a
partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. W.R. acknowledges support
from Chulalongkorn University’s CUniverse and the Ratchadapiseksompot
Endowment Fund.

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