#  A note on inflation in dRGT massive gravity

B. Afshar1<EMAIL_ADDRESS>N. Riazi1<EMAIL_ADDRESS>H.
Moradpour2<EMAIL_ADDRESS>1 Department of Physics, Shahid
Beheshti University, Tehran 19839, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
University of Maragheh, P.O. Box 55136-553, Maragheh, Iran

###### Abstract

Although the dRGT massive gravity successfully explains the late-time cosmic
acceleration, it cannot justify inflation. On the other hand, and in the
frameworks of General Relativity and modified gravity, the interests and
attempts to describe dark energy and inflation by using Lagranginas, which may
have pole, have recently been enhanced. Subsequently, we are going to show
that this kind of Lagrangian may justify inflation in the framework of dRGT
massive gravity. The study is done focusing on the power and exponential
potentials, and the results show a plausible consistency with the Planck 2018
data and its combination with BK18 and BAO.

††preprint: APS/123-QED

##  I Introduction

Despite its enormous successes, General Relativity (GR) cannot be considered
as the final theory of gravity, and needs to be modified [1]. Two of the most
important motivations for modifying GR are the explanation of inflation, and
the late-time cosmic acceleration [2]. Massive gravity theory is an approach
to modify GR which includes a graviton of small mass [3]. In 1939, Fierz and
Pauli (FP) [4] proposed massive gravity theory for the first time. By adding
the fine-tuned mass term to the linearized Einstein-Hilbert action, they
developed a theory that correctly leads to 5 degrees of freedom for the
massive spin-2 particle [1, 5]. In 1970, van Dam, Veltman [6], and Zakharov
(vDVZ) [7] discovered that, in the massless limit, the FP theory does not
render GR [5]. In other words, the principle of conformity is violated leading
to a deviation in the gravitational lensing prediction of theory about 25%
compared to that of GR [8, 9]. This discontinuity is known as the vDVZ
discontinuity [8].

Focusing on the nonlinear extensions of the FP theory, Vainshtein [10] found
out a distance scale, called the Vainshtein radius
${{r}_{V}}={{(\frac{M}{M_{Pl}^{4}m_{g}^{4}})}^{\frac{1}{5}}}$, where $M_{Pl}$
is the Planck mass, and $m_{g}$ denotes the graviton mass) [11, 12, 13]. For
$r<{{r}_{V}}$, the nonlinear effects become highly important [11, 12, 13]
meaning that the outcomes differ from those predicted by FP. As the graviton
mass becomes less, the ${r}_{V}$ is boosted and at the massless limit,
${r}_{V}$ tends to infinity. Consequently, the results of the FP theory are no
longer valid [11, 12, 13]. So, it seems possible to rectify the vDVZ
discontinuity by considering the nonlinear effects [11, 12, 13], an approach
called the Vainshtein mechanism [13].

According to the Vainshtein’s idea, Boulware and Deser (BD) [14] find that all
nonlinear extensions of the FP theory have an additional degree of freedom (a
ghost-like scalar mode), known as the BD ghost [15, 16]. The discovery of the
late-time cosmic acceleration in 1998, and the unknown nature of cosmological
constant led scientists to investigate modified gravity theories, such as
massive gravity theory, more precisely. It is also realized that a non-
renormalizable theory, such as the FP theory and even one with apparent
instabilities, can be understood as an effective field theory, valid only at
energies below an ultraviolet cutoff scale [17].

In 2003, Arkani-Hamed, Georgi, and Schwartz [18] could return the gauge
invariance to massive gravity theory by employing the
St$\ddot{\text{u}}$ckelberg trick leading to develop an effective field theory
for massive gravity. The St$\ddot{\text{u}}$ckelberg trick also helps us solve
vDVZ discontinuity. In the massless limit, the effects of the strong
decoupling neutralize the ghost-like scalar mode, and consequently the results
of GR are reproduced. Moreover, in certain nonlinear extensions of FP, the
cutoff scale ${\Lambda_{5}}={{({{M}_{Pl}}m_{g}^{4})}^{{}^{1}/{}_{5}}}$ of FP
can also be raised to ${{\text{}\\!\\!\Lambda\\!\\!\text{
}}_{3}}={{({{M}_{Pl}}m_{g}^{2})}^{{}^{1}/{}_{3}}}$. De Rham, and Gabadadze
[19] determine the coefficients of higher-order interaction terms in a way
that the obtained action, as a generalization of FP, does not include the BD
ghost in the decoupling limit. One year later, in 2011, De Rham, Gabadadze,
and Tolley [20] could propose a covariant nonlinear theory of massive gravity
that correctly describes the massive spin-2 field; the birth of dRGT theory.
The dRGT theory is indeed ghost-free in the decoupling limit to all orders.

Although dRGT theory does not have flat, and closed FRW solutions, the open
FRW solution yields an effective cosmological constant proportional to the
graviton mass (${{m}_{g}}$) [21]. In the report of Ligo-Virgo Collaboration
about the discovery of the first gravitational wave, the constraint
${{m}_{g}}<1.2\times{{10}^{-22}}eV$ is determined [22]. If
${{m}_{g}}\sim{{H}_{0}}\sim{{10}^{-33}}eV$, then the effective cosmological
constant can explain the late-time cosmic acceleration [23]. Since the
graviton mass cannot be of the order of the Hubble constant during the
inflation, dRGT theory is not capable to produce enough number of e-foldings
[24].

Recently, a new type of Lagrangian, which may include pole and thus called
pole Lagrangian, has attracted attempts to itself in order to model and
describe the current and primary inflationary eras [25, 26, 27, 28]. This kind
of Lagrangian even supports evolving traversable wormholes satisfying energy
conditions in the framework of GR [29]. Motivated by these attampets and also
Ref. [24], here, we use the pole Lagrangian to build some models for the
primary inflationary era in the framework of dRGT theory.

The action of the scalar field $\sigma$ is written as

$S_{\sigma}=\int{{d^{4}}x}\sqrt{-g}[-\frac{1}{2}f(\sigma){g^{\mu\nu}}{{\partial}_{\mu}}{{\sigma}}{{\partial}_{\nu}}\sigma-V(\sigma)],$
(1)

where $V(\sigma)$ denotes the potential and $f(\sigma)$ is a function of the
scalar field $\sigma$ [30]. The kinetic term can also be brought into
canonical form

$S_{\varphi}=\int{{d^{4}}x}\sqrt{-g}[-\frac{1}{2}{{g}^{\mu\nu}}{{\partial}_{\mu}}{{\varphi}}{{\partial}_{\nu}}\varphi-V(\varphi)],$
(2)

using the transformation

$d\varphi=\sqrt{f(\sigma)}d\sigma,$ (3)

leading to [31]

$\varphi(\sigma)=\int_{\sigma_{0}}^{\sigma}{d{\sigma}}\sqrt{f({\sigma})},$ (4)

in which $\sigma_{0}$ denotes the desired point, which is considered as
${\sigma_{0}}=0$, here [31]. If $f(\sigma)$ is singular at some points, i.e.,
then it possesses pole, it is impossible to integrate from $f(\sigma)$ [31].
Assuming pole is located in $\sigma=\sigma_{p}$, if $\sigma_{0}<\sigma_{p}$,
then $\sigma$ lies within the branch $(-\infty,{\sigma_{p}})$. Also, when
$\sigma_{0}>\sigma_{p}$, it lies within the branch $({\sigma_{p}},\infty)$
[31]. To provide a general outlook on the possibility of describing inflation
in the dRGT theory, we first derive the corresponding formulation for the
general potential $V(\sigma)$ and the general function $f(\sigma)$. Then, we
concentrate on the power and exponential potentials, as well as the particular
function $f(\sigma)={A}{{|\sigma|}^{-p}}$ which attended in Refs. [25, 26, 27,
28, 29, 31].

The paper is organized as follows. The dRGT theory and corresponding open FRW
solutions are briefly reviewed in Sec. (II) and (III), respectively. In the
continuation of the third section, the number of e-foldings $N_{e}$, scalar
spectral $n_{s}$, and tensor-to-scalar ratio $r$ are also derived. One of our
main goals in Sec. (IV) is to show the power of models in producing the
appropriate number of e-foldings to solve the flatness problem, as well as the
consistency of models with the Planck 2018 data and its combination with BK18
and BAO is investigated through employing $r(n_{s})$. Special solutions that
lead to the invalidation of models are also discussed. A summary is finally
presented in the last section. It is assumed that ${M_{Pl}}=1$.

##  II The dRGT Theory

The dRGT theory is described by the physical metric ${{g}_{\mu\nu}}$, and the
non-dynamical (or fiducial) metric ${{f}_{\mu\nu}}$ so that

${{g}_{\mu\nu}}={{f}_{\mu\nu}}+{{H}_{\mu\nu}}.$ (5)

Here, ${{H}_{\mu\nu}}$ denotes the covariantization of the metric perturbation
(${{h}_{\mu\nu}}\equiv{{g}_{\mu\nu}}-{{\eta}_{\mu\nu}}$), and the fiducial
metric is defined as

${{f}_{\mu\nu}}={{\tilde{f}}_{ab}}{{\partial}_{\mu}}{{\phi}^{a}}{{\partial}_{\nu}}{{\phi}^{b}},$
(6)

where ${{\tilde{f}}_{\mu\nu}}$ is the reference metric, and
${{\phi}^{a}}(a=0,1,2,3)$ are the four fields introduced to restore general
covariance. For a Minkowskian reference metric, ${{\phi}^{a}}$ form a
4-vector, implying that the dRGT theory carries the global Poincare symmetry
[32].
The gravitational action consists of the Einstein-Hilbert term, and the mass
term

$\displaystyle{{S}_{g}}={{S}_{EH}}+{{S}_{mass}},$ (7)
$\displaystyle{{S}_{g}}=\int{{{d}^{4}}x}\sqrt{-g}[\frac{R}{2}+m_{g}^{2}\mathcal{U}\left({{g}_{\mu\nu}},{{H}_{\mu\nu}}\right)],$

in which $R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $\mathcal{U}$ is
the potential without derivatives of terms originated from the interaction
between ${{H}_{\mu\nu}}$ and ${{g}_{\mu\nu}}$ [23]. To prevent the appearance
of the BD ghost, $\mathcal{U}$ is constructed as

$\mathcal{U}\left({{g}_{\mu\nu}},{{H}_{\mu\nu}}\right)=\sum\limits_{n=0}^{4}{{{\alpha}_{n}}}{{\mathcal{L}}_{\text{n}}}.$
(8)

In this formula, ${{\alpha}_{n}}$ are the dimensionless parameters, and

$\displaystyle{{\mathcal{L}}_{0}}=1,~{}$ (9)
$\displaystyle{{\mathcal{L}}_{1}}=\left[\mathcal{K}\right],$
$\displaystyle{{\mathcal{L}}_{2}}=\frac{1}{2}~{}\left({{\left[\mathcal{K}\right]}^{2}}-\left[{{\mathcal{K}}^{2}}\right]\right),$
$\displaystyle{{\mathcal{L}}_{3}}=~{}\frac{1}{6}\left({{\left[\mathcal{K}\right]}^{3}}-3\left[\mathcal{K}\right]\left[{{\mathcal{K}}^{2}}\left]+2\right[{{\mathcal{K}}^{3}}\right]\right),$
$\displaystyle{{\mathcal{L}}_{4}}=\frac{1}{24}~{}\left({{\left[\mathcal{K}\right]}^{4}}-6{{\left[\mathcal{K}\right]}^{2}}\left[{{\mathcal{K}}^{2}}\right]+3{{\left[{{\mathcal{K}}^{2}}\right]}^{2}}+8\left[\mathcal{K}\right]\left[{{\mathcal{K}}^{3}}\left]-6\right[{{\mathcal{K}}^{4}}\right]\right),$

where $\mathcal{K}_{~{}\nu}^{\mu}$ is defined as

$\mathcal{K}_{~{}\nu}^{\mu}=\delta_{\nu}^{\mu}-\sqrt{{{g}^{\mu\lambda}}{{f}_{\lambda\nu}}},$
(10)

and the square brackets denote trace operation i.e.,

$\displaystyle\left[\mathcal{K}\right]=\mathcal{K}_{~{}\mu}^{\mu},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left[{{\mathcal{K}}^{2}}\right]=\mathcal{K}_{~{}\nu}^{\mu}\mathcal{K}_{~{}\mu}^{\nu},~{}~{}$
(11)
$\displaystyle\left[{{\mathcal{K}}^{3}}\right]=\mathcal{K}_{~{}\nu}^{\mu}\mathcal{K}_{~{}\lambda}^{\nu}\mathcal{K}_{~{}\mu}^{\lambda},\text{
}\left[{{\mathcal{K}}^{4}}\right]=\mathcal{K}_{~{}\nu}^{\mu}\mathcal{K}_{~{}\lambda}^{\nu}\mathcal{K}_{~{}\rho}^{\lambda}\mathcal{K}_{~{}\mu}^{\rho}.~{}$

${{g}_{\mu\nu}}$ raises, and lowers the indices, and moreover,
${\mathcal{L}}_{0}$ corresponds to a cosmological constant,
${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{2}$ to a tadpole, and the mass term,
respectively. ${\mathcal{L}}_{3,4}$ also include higher-order interaction
terms [16]. Setting ${{\alpha}_{0}}={{\alpha}_{1}}=0$ Minkowskian spacetime
becomes a vacuum solution [16], and in addition, if ${{\alpha}_{2}}=1$, then
the FP theory, at a linearized level, is recovered [33]. ${{\alpha}_{3}}$, and
${{\alpha}_{4}}$ are the free parameters of the dRGT theory. Accordingly, the
gravitational action (7) is finally rewritten as

$\\!{{S}_{g}}=\int{{{d}^{4}}x}\sqrt{-g}[\frac{R}{2}+m_{g}^{2}({{\mathcal{L}}_{2}}\text{
}\\!\\!~{}\\!\\!\text{ }+{{\text{ }\\!\\!\alpha\\!\\!\text{
}}_{3}}{{\mathcal{L}}_{3}}\text{ }\\!\\!~{}\\!\\!\text{ }+{{\text{
}\\!\\!\alpha\\!\\!\text{ }}_{4}}\text{ }\\!\\!~{}\\!\\!\text{
}{{\mathcal{L}}_{4}}\text{ }\\!\\!~{}\\!\\!\text{ })].$ (12)

##  III BACKGROUND COSMOLOGY

The total action consists of three parts including $i$) the gravitational
action (12), $ii$) the Lagrangian (1), and $iii$) the matter action $S_{m}$ so
that

${{S}_{tot}}={{S}_{g}}+{{S}_{\sigma}}+{{S}_{m}},$ (13)

where $S_{m}$ corresponds to a perfect fluid, described by the energy-momentum
tensor

$T_{~{}\nu}^{\mu}=diag(-{{\rho}_{m}},{{p}_{m}},{{p}_{m}},{{p}_{m}}),$ (14)

in which ${{\rho}_{m}},$ and ${{p}_{m}}$ denote the energy density, and the
pressure of matter, respectively.

Now, assuming the Universe is homogeneous and isotropic, the physical metric
${{g}_{\mu\nu}}$ is deemed as the Friedmann- Robertson-Walker (FRW) metric

${{g}_{\mu\nu}}d{{x}^{\mu}}d{{x}^{\nu}}=-N{{(t)}^{2}}{{(dt)}^{2}}+a{{(t)}^{2}}{{\Omega}_{ij}}d{{x}^{i}}d{{x}^{j}}.$
(15)

Here, $N$ is the cosmological time, $a$ is the scale factor, and
${{\Omega}_{ij}}$ denotes the metric of unit $3$-sphere

${{\Omega}_{ij}}={{\delta}_{ij}}+\frac{k}{1-k{{r}^{2}}}{{x}^{i}}{{x}^{j}},$
(16)

where $k$ is the spatial curvature, and
${{r}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}$. In this paper, the Minkowskian form
of the reference metric is chosen as

${{\tilde{f}}_{\mu\nu}}={{\eta}_{\mu\nu}}=diag(-1,1,1,1).$ (17)

As demonstrated in Ref [16], the flat FRW solution of the dRGT theory is equal
to $a=$const, inconsistent with a dynamic Cosmos. Furthermore, the dRGT theory
lacks closed FRW solutions, since the fiducial Minkowski metric cannot be
foliated by closed slices [21]. Therefore, we only focus on open FRW of
negative $k$ very close to zero, in accordance with the Planck collaboration
prescription [34, 35]. In order to carry the symmetries of an open FRW metric
by the fiducial metric, the St$\ddot{\textmd{u}}$ckelberg fields should be
chosen as

${{\phi}^{0}}=f(t)\sqrt{1-k{{r}^{2}}},\quad{{\phi}^{i}}=\sqrt{-k}f(t){{x}^{i}},$
(18)

leading to

${{f}_{\mu\nu}}d{{x}^{\mu}}d{{x}^{\nu}}=-{{\dot{f}(t)}^{2}}{{(dt)}^{2}}-kf{{(t)}^{2}}{{\Omega}_{ij}}d{{x}^{i}}d{{x}^{j}},$
(19)

where $\dot{f}=\frac{df(t)}{dt}$. Without loss of generality, we also assume
$a>0,~{}N>0,~{}f\geq 0,~{}\dot{f}\geq 0$.

###  III.1 Equations of motion

Inserting the physical metric (15), and the fiducial metric (19) in Eq. (13),
the total action is obtained as

$\displaystyle S_{tot}$ $\displaystyle=\int
d^{4}x\bigg{[}3\big{[}-\frac{a\dot{a}^{2}}{N}+kNa+m_{g}^{2}\big{(}NF(a,f)$
$\displaystyle-\dot{f}G(a,f)\big{)}\big{]}+a^{3}\big{(}\frac{1}{2N}f(\sigma)\dot{\sigma}^{2}-{N}V(\sigma)\big{)}\bigg{]}+S_{m},$
(20)

in which

$\displaystyle F(a,f)=a(a-\sqrt{-k}f)(2a-\sqrt{-k}f)+~{}~{}~{}~{}$ (21)
$\displaystyle\frac{{{\alpha}_{3}}}{3}{{(a-\sqrt{-k}f)}^{2}}(4a-\sqrt{-k}f)+\frac{{{\alpha}_{4}}}{3}{{(a-\sqrt{-k}f)}^{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}$

and

$\displaystyle
G(a,f)={{a}^{2}}(a-\sqrt{-k}f)+\alpha_{3}a{{(a-\sqrt{-k}f)}^{2}}+$ (22)
$\displaystyle\frac{{{\alpha}_{4}}}{3}{{(a-\sqrt{-k}f)}^{3}}.$

Now, variation of the total action (20) with respect to the field variables
$f,~{}N,~{}a$, and$~{}\sigma$ renders

${{\delta}_{f}}{{S}_{tot}}{{|}_{N=1}}:\mathop{{}}(\dot{a}-\sqrt{-k})\frac{\partial
G(a,f)}{\partial
a}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (23)
$\\!\,\,\,{{\delta}_{N}}{{S}_{tot}}{{|}_{N=1}}:\mathop{{}}3({{H}^{2}}+\frac{k}{{{a}^{2}}})={{\rho}_{MG}}+{{\rho}_{\sigma}}+{{\rho}_{m}},$
(24)

${{\delta}_{a}}{{S}_{tot}}{{|}_{N=1}}:\mathop{{}}-(2\dot{H}+3{{H}^{2}}+\frac{k}{{{a}^{2}}})={{p}_{MG}}+{{p}_{\sigma}}+{{p}_{m}},$
(25)

${{\delta}_{\sigma}}{{S}_{tot}}{{|}_{N=1}}:\mathop{{}}\frac{1}{2}{f}^{\prime}(\sigma){{\dot{\sigma}}^{2}}+f(\sigma)(\ddot{\sigma}+3H\dot{\sigma})+{V}^{\prime}(\sigma)=0,$
(26)

where $H=\frac{{\dot{a}}}{a}$ is the Hubble parameter, and
${V}^{\prime}(\sigma)=\frac{dV(\sigma)}{d\sigma}$.
In Eqs. (23)-(26), the effective energy density, and pressure of dRGT theory
are given by

${{\rho}_{MG}}=-3m_{g}^{2}\frac{F(a,f)}{{{a}^{3}}},$ (27)

and

${{p}_{MG}}=m{{{}_{g}^{2}}}\big{[}\frac{F(a,f)}{{{a}^{3}}}+(\dot{f}-\beta)\frac{\partial
G(a,f)}{\partial a}\big{]}.$ (28)

respectively. Also, we have

${{\rho}_{\sigma}}=\frac{1}{2}f(\sigma){{\dot{\sigma}}^{2}}+V(\sigma),$ (29)
${{p}_{\sigma}}=\frac{1}{2}f(\sigma){{\dot{\sigma}}^{2}}-V(\sigma),$ (30)

for the energy density and pressure of $\sigma$. For action (2), the equation
of motion yields

${{\delta}_{\varphi}}{{S}_{tot}}{{|}_{N=1}}:\mathop{{}}\ddot{\varphi}+3H\dot{\varphi}+\frac{dV(\varphi)}{d\varphi}=0.$
(31)

In this case, the energy density and pressure of $\varphi$ are also given by

${{\rho}_{\varphi}}=\frac{1}{2}{{\dot{\varphi}}^{2}}+V(\varphi),$ (32)

and

${{p}_{\varphi}}=\frac{1}{2}{{\dot{\varphi}}^{2}}-V(\varphi),$ (33)

respectively.

Eq. (23) has three solutions. For one of these solutions, one obtains
$\dot{a}=\sqrt{-k}$ which leads to $a=\sqrt{-k}t+\text{const}$ that signals us
to an open FRW universe as the physical metric ${{g}_{\mu\nu}}$. Similar to
flat FRW solution, this solution is unacceptable. The other two solutions are
obtained from $\frac{\partial G(a,f)}{\partial a}=0$ which leads to

${{\beta}_{\pm}}=\frac{1+2{{\alpha}_{3}}+{{\alpha}_{4}}\pm\sqrt{1+{{\alpha}_{3}}+\alpha_{3}^{2}-{{\alpha}_{4}}}}{{{\alpha}_{3}}+{{\alpha}_{4}}},$
(34)

and

$a=\frac{\sqrt{-k}f}{{{\beta}_{\pm}}}.$ (35)

Now, assuming $a$ is positive, one finds that, according to Eq. (35),
${{\beta}_{\pm}}$ should also be positive. For the negative scale factor, Eq.
(35) becomes $a=-\frac{\sqrt{-k}f}{{{\beta}_{\pm}}}$. So, ${{\beta}_{\pm}}$ is
always positive. If $f(t)\propto a(t)$, then Eqs. (27)-(30) give us

$\displaystyle{{\rho}_{MG}}=-{{p}_{MG}}=\Lambda_{\pm}=$ (36) $\displaystyle
m_{g}^{2}({{\beta}_{\pm}}-1)[(3-{{\beta}_{\pm}})+{{\alpha}_{3}}(1-{{\beta}_{\pm}})],$

where $\Lambda_{\pm}$ denotes the effective cosmological constant of the
theory. Although ${{\alpha}_{3}}$ and ${{\alpha}_{4}}$ are free parameters,
Eq. (34) renders some constraints. Moreover, if
${{\alpha}_{3}}=-{{\alpha}_{4}}(\pm(1+{{\alpha}_{3}})>0)$, then the
cosmological constant $\Lambda_{\pm}$ becomes infinite. On the other hand,
${{\alpha}_{4}}={{\frac{3+2{{\alpha}_{3}}+3\alpha_{3}^{2}}{4}}}(\pm(1+{{\alpha}_{3}})>0)$
leads to $\Lambda_{\pm}=0$. Finally, the values of free parameters that lead
to negative radicand in Eq. (34) are not acceptable (see Ref. [21]).

###  III.2 Inflation

This subsection examines inflation in the slow-roll regime for which the scale
factor has an exponential behavior, ($a\sim{{e}^{Ht}}(H\sim\text{const})$),
and grows rapidly. Therefore, after a few number of e-foldings, the term
$\frac{k}{{{a}^{2}}}$ becomes ignorable. Hence, Eq. (24) can be rewritten as

$3{{H}^{2}}=\frac{1}{2}{f(\sigma)}{{\dot{\sigma}}^{2}}+V(\sigma)+\Lambda_{\pm}.$
(37)

Moreover, combining Eq. (24) with Eq. (25), one reaches at

$\dot{H}=-\frac{1}{2}f(\sigma){{\dot{\sigma}}^{2}}.$ (38)

The Hubble slow-roll parameters are also defined as

${\varepsilon}_{H}=-\frac{{\dot{H}}}{{{H}^{2}}},$ (39)

and

${{\eta}_{H}}=-\frac{{\ddot{\varphi}}}{H\dot{\varphi}}.$ (40)

In this regard, as the slow-roll conditions, we have
${{\varepsilon}_{H}},|{\eta}_{H}|<<1$, while the first condition leads to

$f(\sigma){{\dot{\sigma}}^{2}}<<V(\sigma)+\Lambda_{\pm},$ (41)

meaning that Eq. (37) can be recast as

$3{{H}^{2}}\approx V(\sigma)+\Lambda_{\pm}.$ (42)

Additionally, for the relation between the first time-derivatives of $\sigma$
and $\varphi$, we have

$\dot{\varphi}=\sqrt{f(\sigma)}{\dot{\sigma}}.$ (43)

On the other hand, the second time-derivatives of $\sigma$ and $\varphi$
address us to

$\ddot{\varphi}=\sqrt{f(\sigma)}{\ddot{\sigma}}+\frac{1}{2}{{f(\sigma)}^{\frac{-1}{2}}}{f^{\prime}(\sigma)}{{\dot{\sigma}}^{2}},$
(44)

where $f^{\prime}(\sigma)=\frac{df}{d\sigma}$. Now, inserting $\dot{\varphi}$
and $\ddot{\varphi}$ in Eq. (40), ${{\eta}_{H}}$ is obtained as

$\eta_{H}=-\frac{\ddot{\sigma}+\frac{1}{2}f(\sigma)^{-1}f^{\prime}(\sigma)\dot{\sigma}^{2}}{H\dot{\sigma}}.$
(45)

Using $|{{\eta}_{H}}|<<1,$ Eq. (26) is approximately equal to

$3H\dot{\sigma}\approx-\frac{1}{f(\sigma)}{V}^{\prime}(\sigma).$ (46)

Combining Eqs. (38), (39), and (42) with each other, we get

${{\varepsilon}_{H}}=\frac{1}{2}{f(\sigma)}{{(\frac{{\dot{\sigma}}}{H})}^{2}}\approx\frac{1}{2f(\sigma)}{{\big{(}\frac{{V}^{\prime}(\sigma)}{V(\sigma)+\Lambda_{\pm}}\big{)}}^{2}}.$
(47)

and thus

$3\dot{H}\dot{\sigma}+3H\ddot{\sigma}\approx-\frac{1}{f(\sigma)}\big{[}{V}^{\prime\prime}(\sigma)\dot{\sigma}-{\frac{{f}^{\prime}(\sigma)}{f(\sigma)}}\dot{\sigma}{V}^{\prime}(\sigma)\big{]},$
(48)

in which
${V}^{\prime\prime}(\sigma)=\frac{{{d}^{2}}V(\sigma)}{d{{\sigma}^{2}}}$.
Finally, it is straightforward to achieve

${{\eta}_{H}}\approx\frac{1}{f(\sigma)}\big{(}\frac{{V}^{\prime\prime}(\sigma)-{\frac{{f}^{\prime}(\sigma)}{2f(\sigma)}}{V}^{\prime}{{(\sigma)}}}{V(\sigma)+\Lambda_{\pm}}\big{)}+\frac{{\dot{H}}}{{{H}^{2}}}.$
(49)

The best method for investigating a specific potential is the calculation of
parameters including ${{\varepsilon}_{V}}$, and ${{\eta}_{V}}$ combined with
${{\varepsilon}_{V}}\approx{{\varepsilon}_{H}}$, and
${{\eta}_{V}}\approx{{\eta}_{H}}+{{\varepsilon}_{H}}$ to reach [36]

${{\varepsilon}_{V}}=\frac{1}{2f(\sigma)}{{\big{(}\frac{{V}^{\prime}(\sigma)}{V(\sigma)+\Lambda_{\pm}}\big{)}}^{2}},$
(50)

and

${{\eta}_{V}}=\frac{1}{f(\sigma)}\big{(}\frac{{V}^{\prime\prime}(\sigma)-{\frac{{f}^{\prime}(\sigma)}{2f(\sigma)}}{V}^{\prime}{{(\sigma)}}}{V(\sigma)+\Lambda_{\pm}}\big{)}.$
(51)

${\varepsilon}_{V},{|{\eta}_{V}|}<<1$ are the neccessary conditions for the
slow-roll inflation [36, 37]. The amount of inflation that occurs is measured
by the number of e-foldings [38],

$N_{e}=\int_{a}^{a_{f}}d\ln a=\int_{t}^{t_{f}}{Hdt},$ (52)

Thus, using $Hdt=\pm\frac{d\sigma}{\sqrt{2{\varepsilon_{V}}}}$, $N_{e}$ is
rewritten as

${N_{e}}(\sigma)=\left|\int_{\varphi}^{\varphi_{f}}d\varphi\frac{1}{\sqrt{2{\varepsilon_{V}}}}\right|=\left|\int_{\sigma}^{\sigma_{f}}{d\sigma}f(\sigma)\big{(}\frac{V(\sigma)+\Lambda_{\pm}}{{V}^{\prime}(\sigma)}\big{)}\right|.$
(53)

In Eqs. (52) and (53), the subscript $f$ point to the end of inflation [38].
The scalar spectral index $n_{s}$, and tensor-to-scalar ratio $r$ are thus
calculated as

$n_{s}=1-6{{\varepsilon}_{V}}+2{{\eta}_{V}},$ (54)

and

$r=16{{\varepsilon}_{V}},$ (55)

respectively. Here, $r$ is a significant observational quantity, and it is
used to distinguish different inflationary models.

##  IV NUMERICAL DYNAMICS

This section focuses on the specific function
$f(\sigma)={A}{{|\sigma|}^{-p}}(p\not=0)$ which has pole at $\sigma=0$ for
$p>0$ [25, 26, 27, 28, 29]. Because the field does not cross zero due to the
pole, $\sigma$ can be lie within one of the branches $\sigma>0$ or $\sigma<0$.
The case $p=2$, and $V(\sigma)=0$ corresponds to minimal $k$-essence model
[29].
Using Eq. (4), the relationships between $\sigma$ and $\varphi$ are easily
achieved as

$\varphi=\left\\{\begin{aligned}
&\frac{2\sqrt{A}}{2-p}{{|\sigma|}^{\frac{2-p}{2}}}~{}~{}~{}\text{for}~{}~{}p\neq
2\underset{\underset{{}}{\mathop{}}\,}{\mathop{}}\,\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},\\\
&\sqrt{A}\ln(|\sigma|)~{}~{}~{}~{}~{}\text{for}~{}~{}p=2\\\
\end{aligned}\right.$ (56)

and

$\sigma=\left\\{\begin{aligned}
&\pm\big{(}{\frac{(2-p)\varphi}{2\sqrt{A}}}\big{)}^{\frac{2}{2-p}}~{}~{}~{}\text{for}~{}~{}p\neq
2\underset{\underset{{}}{\mathop{}}\,}{\mathop{}}\,\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},\\\
&\pm
e^{\frac{\varphi}{\sqrt{A}}}~{}~{}~{}~{}\,\qquad\qquad\text{for}~{}~{}p=2\\\
\end{aligned}\right.$ (57)

respectively, up to an integration constant.

###  IV.1 The Power Potential

First, let us consider $V(\sigma)={V_{0}}{{\sigma}^{n}}$, where $V_{0}$ and
$n$ are non-zero constants. The general form of canonicalized potential is
obtained using Eq. (57) as

$V(\varphi)=\left\\{\begin{aligned}
&\pm{V_{0}}\big{(}{\frac{(2-p)\varphi}{2\sqrt{A}}}\big{)}^{\frac{2n}{2-p}}~{}~{}~{}\text{for}~{}~{}p\neq
2\underset{\underset{{}}{\mathop{}}\,}{\mathop{}}\,\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}.\\\
&\pm{V_{0}}e^{\frac{n\varphi}{\sqrt{A}}}~{}~{}~{}\,\,\qquad\quad\,\quad\text{for}~{}~{}p=2\\\
\end{aligned}\right.$ (58)

Inserting ${V_{0}}{{\sigma}^{n}}$ in Eq. (50), and using Eq. (51),
$\varepsilon_{V}$ and $\eta_{V}$ are calculated as

$\varepsilon_{V}=\frac{{|\sigma|}^{p}}{2A}{{\big{(}\frac{n{{\sigma}^{n-1}}}{{{\sigma}^{n}}+c}\big{)}}^{2}},$
(59)

and

$\eta_{V}=\frac{{|\sigma|}^{p}}{2A}\big{[}\frac{n(2n+p-2){{\sigma}^{n-2}}}{{{\sigma}^{n}}+c}\big{]},$
(60)

respectively, where $c$ is defined as $c=\frac{{\Lambda}_{\pm}}{V_{0}}$. As
inflation ends when ${\varepsilon_{V}}=|{\eta_{V}}|=1$ [39, 40], we have

${\tilde{\sigma}}_{f}={\big{(}\frac{2n+p-2}{-3n-p+2}\big{)}}^{\frac{1}{n}},$
(61)

and

$A=\frac{1}{2}{{\big{(}c\frac{2n+p-2}{-3n-p+2}\big{)}}^{\frac{p-2}{n}}}{{(2n+p-2)}^{2}},$
(62)

(a)

(b)

(c)

(d)

Figure 1: Plots of $r(n_{s})$ and ${N_{e}}(n_{s})$ for $V_{0}\sigma$ ($n=1$)
with ${V_{0}}>0$ (top plots) and ${V_{0}}<0$ (bottom plots). In left plots,
green and blue regions are related to the Planck 2018 data [34] and its
combination with BK18 and BAO [35], respectively. In addition, the 68% conf-
idence regions are distinguished from the 95% confidence regions by
highlighting the corresponding regions. Depending on the values of
$p$, the curves of $r(n_{s})$ for ${V_{0}}>0$ lie within the 68% or 95%
confidence regions of the Planck 2018 TT,TE,EE+lowE+lensing data, wher-
eas the curves of $r(n_{s})$ for ${V_{0}}<0$ and $2<p<200$ lie within the 68%
confidence region of the Planck 2018 TT,TE,EE+lowE+lensing+
BK18+BAO data. More details on the text.

where $\tilde{\sigma}\equiv{\sigma}{c^{\frac{-1}{n}}}$. Only $n=1$ is
acceptable for $\sigma<0$ because other values of $n$ cause $\sigma_{f}$ or
$V(\sigma)$ to become imaginary numbers, thus the branch $\sigma>0$ is
considered here. In table. (i@), three ranges of $p$ for each $n$ are obtained
using condition $\sigma_{f}>0$, whereas ${V_{0}}>0$ in case 1 and ${V_{0}}<0$
in cases 2 and 3. It should be noted that $p=-3n+2$ and $p=-2n+2$ are
unacceptable because they lead to $A=0$ or $A=\infty$, causing undifinable
expressions for $\varphi$ (56) and the canonicalized potential (58). Moreover,
in this manner, the kinetic term in the Lagrangian (1) becomes zero or
infinite. Consequently, we do not pay attention to these improper solutions.
Case 3 leads to $V(\sigma)+\Lambda_{\pm}<0$, indicating that the slow-roll
condition (41) has been violated [41]. Hence, this case is not also studied in
the following.

Case |  Range of $p$ for $n>0$ |  Range of $p$ for $n<0$
---|---|---
1 |  $-3n+2<p<-2n+2$ |  $-2n+2<p<-3n+2$
2 |  $-2n+2<p$ |  $-2n+2>p$
3 |  $-3n+2>p$ |  $-3n+2<p$

Table 1: Allowed ranges of $p$ depending value of $n$ whereas ${V_{0}}>0$ in
case 1 and ${V_{0}}<0$ in cases 2 and 3.

Focusing on cases 1 (${V_{0}}>0$) and 2 (${V_{0}}<0$), two ranges of $n$
(named $i$ and $j$) for which the Lagrangian (1) may include pole gathered in
Table.(ii@). Indeed, while we have always $p>0$ in the range $i$, both of the
positive and negative values of $p$ are allowed in range $j$.

Case |  Range $i$ |  Range $j$
---|---|---
1 |  $0<n\leq\frac{2}{3}$ |  $\frac{2}{3}<n<1$
2 |  $1\geq n$ |  $1<n$
Table 2: The ranges of $n$ which lead to $p>0$ and thus pole.

Inserting $V(\sigma)=V_{0}{{\sigma}^{n}}$ in Eq. (53), the number of
e-foldings $N_{e}$ is obtained as

${N_{e}}(\sigma)=\left|\frac{{A}^{\prime}}{n}\int_{{\tilde{\sigma}}}^{{\tilde{\sigma}}_{f}}{d\tilde{\sigma}}({{\tilde{\sigma}}^{-p+1}}+{{\tilde{\sigma}}^{-p-n+1}})\right|,$
(63)

in which

$\int_{{\tilde{\sigma}}}^{{\tilde{\sigma}}_{f}}{d\tilde{\sigma}}{{\tilde{\sigma}}^{-p+1}}=\left\\{\begin{aligned}
&\frac{1}{-p+2}({{\tilde{\sigma}}_{f}}^{-p+2}-{{\tilde{\sigma}}}^{-p+2})~{}~{}\,\text{for}~{}~{}p\neq
2\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},\\\
&\ln(\frac{{\tilde{\sigma}}_{f}}{{\tilde{\sigma}}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}~{}p=2\\\
\end{aligned}\right.$ (64)

(a)

(b)

(c)

(d)

Figure 2: Plots of $r(n_{s})$ and ${N_{e}}(n_{s})$ for the exponential
potential $V_{0}e^{-\tilde{\sigma}}$, $p>0$ and $c=0$ (top plots) and $c=0.5$
(bottom plots). In left plots, green and blue regions are related to the
Planck 2018 data [34] and its combination with BK18 and BAO [35],
respectively. In ad-
dition, the 68% confidence regions are highlighted compared to the 95%
confidence regions. Depending on the values of $p$, the curves of $r(n_{s})$
for $c=0$ lie within the 68% or 95% confidence regions of the Planck 2018
TT,TE,EE+lowE+lensing+BK18+BAO data. In addit-
ion, all curves of $r(n_{s})$ for $c=0.5$ lie within the 68% confidence region
of the Planck 2018 TT,TE,EE+lowE+lensing+BK18+BAO data. More details on the
text.

and

$\int_{\tilde{\sigma}}^{\tilde{\sigma}_{f}}{d\tilde{\sigma}}{{\tilde{\sigma}}^{-p-n+1}}=$
$\left\\{\begin{aligned}
&\frac{1}{-p-n+2}({{\tilde{\sigma}}_{f}}^{-p-n+2}-{{\tilde{\sigma}}}^{-p-n+2})~{}~{}~{}~{}\text{for}~{}~{}p\neq
2\underset{\underset{{}}{\mathop{}}\,}{\mathop{}}\,\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},\\\
&\ln(\frac{{\tilde{\sigma}}_{f}}{\tilde{\sigma}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad\qquad\,\,\text{for}~{}~{}p=2\\\
\end{aligned}\right.$ (65)

where $A^{\prime}\equiv c^{\frac{2-p}{n}}{A}$. Now, equipped with Eqs. (54),
(55), (59), and (60), the scalar spectral index $n_{s}$, and tensor-to-scalar
ratio $r$ are obtained as

$n_{s}=1+n\frac{{\tilde{\sigma}}^{p-2}}{{A}^{\prime}}\big{[}\frac{(-n+p-2)+(2n+p-2){{\tilde{\sigma}}^{-n}}}{{{(1+{{\tilde{\sigma}}^{-n}})}^{2}}}\big{]},$
(66)

and

$r=8\frac{{\tilde{\sigma}}^{p-2}}{{A}^{\prime}}\big{(}{\frac{n}{1+{{\tilde{\sigma}}^{-n}}}}\big{)}^{2},$
(67)

respectively. From the given equations, two striking conclusions can be drawn.
First, $r(n_{s})$ and ${N_{e}}(n_{s})$ are independent of $c\,(c\not=0)$
because $\sigma_{f}$ and $\sigma$ are proportional to $c^{\frac{1}{n}}$ for
each specific $n$ and $p$. Secondly, the curves of $r(n_{s})$ and
${N_{e}}(n_{s})$ are repeatable for any values of $n$, depending on the values
of $p$. In this regard, three instances for each of cases 1 and 2 are given in
table. (ii@). Note that for $c=0$, $p=-3n+2$, $A$ becomes a free parameter,
and therefore the second conclusion is rendered invalid.

Using Eq. (66), $\sigma$ is obtained for $0.942<n_{s}<0.988$. The values of
$r(n_{s})$ and ${N_{e}}(n_{s})$ are then calculated by substituting them into
Eqs. (63) and (67). In Fig. (1), $r(n_{s})$ and ${N_{e}}(n_{s})$ are shown for
$V(\sigma)=V_{0}\sigma$, with top and bottom plots corresponding to cases 1
and 2, respectively. Depending on the values of $p$, the curves of $r(n_{s})$
for case 1 lie within the 68% or 95% confidence regions of the Planck 2018
TT,TE,EE+lowE+lensing data, whereas the curves of $r(n_{s})$ for case 2 and
$2<p<200$ lie within the 68% or 95% confidence regions of the Planck 2018
TT,TE,EE+lowE+lensing+BK18+ BAO data. For $p>200$, the curves of $r(n_{s})$
increase very gradually, get away from the Planck 2018
TT,TE,EE+lowE+lensing+BK18+BAO data, and fall into the 68% confidence region
of the Planck 2018 TT,TE,EE+lowE+lensing data. Furthermore, ${N_{e}}(n_{s})$
is highly similar to $p=200$ for $p>200$. As previously stated, only
$\sigma>0$ is admissible and correspondingly $p<2$, leading to $\sigma<0$ is
ignored.

(a)

(b)

(c)

(d)

Figure 3: Plots of $r(n_{s})$ and ${N_{e}}(n_{s})$ for the exponential
potential $V_{0}e^{-\tilde{\sigma}}$. Here, $p<0$ and $c=0$ (top plots) and
$c=10$ (bottom plots). In left plots, green and blue regions are related to
the Planck 2018 data [34] and its combination with BK18 and BAO [35],
respectively. In addition, the 68%confidence regions are highlighted compared
to the 95% confidence regions. Depending on the values of $p$ and $c$, the
cur-
ves of $r(n_{s})$ lie within the 68% or 95%confidence regions of the Planck 2018 TT,TE,EE+lowE+lensing data, in such a way that $c=0$ is obviously more consistent with observational data than $c=10$. More details on the text. Case |  Power $n$ |  Range of $p$
---|---|---
1 |  1 |  $-1<p<0$
|  -1 |  $4<p<5$
|  2 |  $-4<p<-2$
2 |  1 |  $0<p$
|  -1 |  $4>p$
|  2 |  $2<p$
Table 3: Different values of $n$ and their corresponding intervals of $p$
generating the same results as those of Fig. (1).

###  IV.2 The Exponential Potential

In this subsection, we study the inflationary era generated by the exponential
potential $V(\sigma)=V_{0}e^{-\gamma\sigma}$ where $\gamma$ is a constant.
Using Eq. (57), the general form of canonicalized potential is achieved as

$V(\varphi)=\left\\{\begin{aligned}
&{V_{0}}e^{\pm\gamma{\big{(}\frac{(2-p)\varphi}{2\sqrt{A}}\big{)}}^{\frac{2}{2-p}}}~{}~{}~{}\text{for}~{}~{}p\neq
2\underset{\underset{{}}{\mathop{}}\,}{\mathop{}}\,\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}.\\\
&{V_{0}}e^{\pm\gamma
e^{\frac{\varphi}{\sqrt{A}}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\,\,\,\text{for}~{}~{}p=2\\\
\end{aligned}\right.$ (68)

Inserting $V_{0}e^{-\gamma\sigma}$ in Eq. (50), and Eq. (51), the slow-roll
parameters are obtained as

${\varepsilon}_{V}=\frac{|\sigma|^{p}}{2A}\big{(}\frac{\gamma}{1+ce^{\gamma\sigma}}\big{)}^{2},$
(69)

and

${\eta}_{V}=\frac{|\sigma|^{p}}{A}\big{(}\frac{\gamma(\gamma-\frac{p}{2}\sigma^{-1})}{1+ce^{\gamma\sigma}}\big{)}.$
(70)

At the point of ${\varepsilon}_{V}=|{\eta}_{V}|=1$, inflation is ended, a fact
helps us find out

$c(e^{{\tilde{\sigma}}_{f}})=\frac{-3{{\tilde{\sigma}}_{f}}+p}{2{{\tilde{\sigma}}_{f}}-p},$
(71)

and

$A=\gamma^{2-p}\big{[}\frac{1}{2}{{|{\tilde{\sigma}}_{f}|}^{p-2}}(2{{\tilde{\sigma}}_{f}}-p)^{2}\big{]},$
(72)

where $\tilde{\sigma}$ is defined as $\tilde{\sigma}=\gamma\sigma$. Inserting
$V_{0}e^{-\tilde{\sigma}}$ in Eq. (53), the number of e-foldings $N_{e}$ are
obtained as

${N_{e}}(\sigma)=\left|A^{\prime}\int_{\tilde{\sigma}}^{{\tilde{\sigma}}_{f}}d\tilde{\sigma}\big{[}{{|{\tilde{\sigma}}|}^{-p}}(1+ce^{\tilde{\sigma}})\big{]}\right|,$
(73)

in which

$\int_{{\tilde{\sigma}}}^{{\tilde{\sigma}}_{f}}{d\tilde{\sigma}}{|{\tilde{\sigma}|}^{-p}}=$
$\left\\{\begin{aligned}
&\frac{1}{-p+1}({|{{\tilde{\sigma}}_{f}}|}^{-p+1}-{|{\tilde{\sigma}}|}^{-p+1})~{}~{}\,~{}\text{for}~{}~{}p\neq
1\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,,\\\
&\ln(\left|\frac{\tilde{\sigma}_{f}}{\tilde{\sigma}}\right|)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad\,\,\quad\text{for}~{}~{}p=1\\\
\end{aligned}\right.$ (74)

and

$\displaystyle\int_{\tilde{\sigma}}^{{\tilde{\sigma}}_{f}}d\tilde{\sigma}{{|{\tilde{\sigma}}|}^{-p}}e^{{\tilde{\sigma}}}=$
$\displaystyle({|{{\tilde{\sigma}}_{f}}|}^{-p}e^{{\tilde{\sigma}}_{f}}-{|{\tilde{\sigma}}|}^{-p}e^{\tilde{\sigma}})+p\int_{\tilde{\sigma}}^{{\tilde{\sigma}}_{f}}d\tilde{\sigma}{{|{\tilde{\sigma}}|}^{-p-1}}e^{\tilde{\sigma}},$
(75)

where $A^{\prime}\equiv\gamma^{p-2}{A}$. Now, combining Eqs. (54), (55) with
(69) and (70), the scalar spectral index $n_{s}$, and tensor-to-scalar ratio
$r$ are calculated as

$n_{s}=1+\frac{{|{\tilde{\sigma}}|}^{p}}{A^{\prime}}\big{[}\frac{-1-p{{\tilde{\sigma}}^{-1}}+c(2-p{{\tilde{\sigma}}^{-1}})e^{\tilde{\sigma}}}{(1+ce^{\tilde{\sigma}})^{2}}\big{]},$
(76)

and

$r=8\frac{{|{\tilde{\sigma}}|}^{p}}{A^{\prime}}\big{(}\frac{1}{1+ce^{\tilde{\sigma}}}\big{)}^{2},$
(77)

respectively. According to our investigations, inflation occurs in the
branches $\tilde{\sigma}>0$ ($\sigma$ and $\gamma$ have same signs) and
$\tilde{\sigma}<0$ ($\sigma$ and $\gamma$ have opposite signs) for $p>0$ and
$p<0$, respectively. Note that $\sigma_{f}$ and $\sigma$ are obtained
inversely proportional to $\gamma$, for any specific $c$ and $p$, implying
that $r(n_{s})$ and ${N_{e}}(n_{s})$ are independent of $\gamma$.

In Fig. (2), $r(n_{s})$ and ${N_{e}}(n_{s})$ are shown for
$V_{0}e^{-\tilde{\sigma}}$, $p>0$, and two values of $c$. Depending on the
values of $p$ and $c$, the curves of $r(n_{s})$ lie within the 68% or 95%
confidence regions of the Planck 2018 TT,TE,EE+lowE+lensing+BK18+BAO data.
When $p>8$ and $c=0$, $r(n_{s})$ increases, but $c\not=0$ leads to
$\tilde{\sigma}_{f}=\frac{p}{2}$ and ${A}=0$, undifinable expressions for
$\varphi$ (56) and canonicalized potential (68), and zero for the kinetic term
in the Lagrangian (1). Furthermore, $r(n_{s})$ approaches zero when $p$ tends
to zero. Similarly, $r(n_{s})$ and ${N_{e}}(n_{s})$ have been plotted in Fig.
(3) for $V_{0}e^{-\tilde{\sigma}}$, $p<0$, and two values of $c$. The curves
of $r(n_{s})$ lie within the 68% or 95% confidence regions of the Planck 2018
TT,TE,EE+lowE+lensing data, again depending on the values of $p$ and $c$. When
$p<-80$, $r(n_{s})$ and ${N_{e}}(n_{s})$ decrease gradually, and as $p$
approaches zero, $r(n_{s})$ gets away from observations, and ${N_{e}}(n_{s})$
grows rapidly. As shown in Figs. (2) and (3), for $p>0$, $r(n_{s})$ decreases
by increasing in $c$, and thus, the curves of $r(n_{s})$ are more consistent
with the 68% confidence region of the Planck 2018
TT,TE,EE+lowE+lensing+BK18+BAO data. For $p<0$, $r(n_{s})$ increases as a
function of $c$, and consequently, the consistency between $r(n_{s})$ and
observations is reduced. It is worth noting that the evaluation $r(n_{s})$
versus $c$ becomes more clear as the values of $p$ increase.

##  V Summary and discussions

dRGT theory cannot provide a unique source for both dark energy and inflation.
Motivated by this shortcoming of dRGT theory, and also, the successes of
Lagrangian (1) in describing the primary and current inflationary eras of the
Universe [25, 26, 27, 28, 29, 30, 31], we addressed some inflationary
scenarios in the framework of dRGT theory. Throughout our analysis, we only
focused on the power and exponential potentials, both of which can
satisfactorily produce 50-70 number of e-foldings, required to solve the
flatness problem. Also, comparing the curves of $r(n_{s})$ with the Planck
2018 data [34] and its combination with BK18 and BAO [35], it has been
obtained that the established models provide acceptable outcomes.

## References

  * Kürekçi [2015] Ş. Kürekçi,  _Basics of massive spin-2 theories_ , Master’s thesis, Middle East Technical University (2015).
  * Akbarieh  _et al._ [2021] A. R. Akbarieh, S. Kazempour, and L. Shao, Cosmological perturbations in gauss-bonnet quasi-dilaton massive gravity, Physical Review D 103, 123518 (2021) [arXiv:2105.03744v2 [gr-qc]].
  * Clifton  _et al._ [2012] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Modified gravity and cosmology, Physics reports 513, 1 (2012) [arXiv:1106.2476v3 [astro-ph.CO]].
  * Fierz and Pauli [1939] M. Fierz and W. E. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 173, 211 (1939).
  * Crisostomi [2010] M. Crisostomi, Modification of gravity at large distances, PhD Dissertation, University of L’Aquila (2010).
  * van Dam and Veltman [1970] H. van Dam and M. Veltman, Massive and mass-less yang-mills and gravitational fields, Nuclear Physics B 22, 397 (1970).
  * Zakharov [1970] V. I. Zakharov, Linearized graviton theory and the graviton mass, JETP Lett 12, 312 (1970).
  * Gambuti and Maggiore [2020] G. Gambuti and N. Maggiore, A note on harmonic gauge (s) in massive gravity, Physics Letters B 807, 135530 (2020) [arXiv:2006.04360v2 [gr-qc]].
  * Moynihan and Murugan [2018] N. Moynihan and J. Murugan, Comments on scattering in massive gravity, vdvz and bcfw, Classical and Quantum Gravity 35, 155005 (2018) [arXiv:1711.03956v2 [hep-th]].
  * Vainshtein [1972] A. I. Vainshtein, To the problem of nonvanishing gravitation mass, Physics Letters B 39, 393 (1972).
  * Chiang  _et al._ [2012] C.-I. Chiang, K. Izumi, and P. Chen, Spherically symmetric analysis on open flrw solution in non-linear massive gravity, Journal of Cosmology and Astroparticle Physics 12, 025 (2012) [arXiv:1208.1222v2 [hep-th]].
  * Alberte [2013] L. Alberte, Non-linear massive gravity, PhD Dissertation, Munich University (2013).
  * Babichev and Deffayet [2013] E. Babichev and C. Deffayet, An introduction to the vainshtein mechanism, Classical and Quantum Gravity 30, 184001 (2013) [arXiv:1304.7240v2 [gr-qc]].
  * Boulware and Deser [1972] D. G. Boulware and S. Deser, Can gravitation have a finite range?, Physical Review D 6, 3368 (1972).
  * Schmidt-May and Von Strauss [2016] A. Schmidt-May and M. Von Strauss, Recent developmen
ts in bimetric theory, Journal of Physics A: Mathematical and Theoretical 49,
183001 (2016) [arXiv:1512.00021v2 [hep-th]].

  * de Rham [2014] C. de Rham, Massive gravity, Living reviews in relativity 17, 7 (2014) [arXiv:1401.4173v2 [hep-th]].
  * Hinterbichler [2012] K. Hinterbichler, Theoretical aspects of massive gravity, Reviews of Modern Physics 84, 671 (2012) [arXiv: 1105.3735v2 [hep-th]].
  * Arkani-Hamed  _et al._ [2003] N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Effective field theory for massive gravitons and gra, Annals of Physics 305, 96 (2003) [arXiv:hep-th/021084v2].
  * De Rham and Gabadadze [2010] C. De Rham and G. Gabadadze, Generalization of the fierz-pauli action, Physical Review D 82, 044020 (2010) [arXiv:1007.0443v2 [hep-th]].
  * De Rham  _et al._ [2011] C. De Rham, G. Gabadadze, and A. J. Tolley, Resummation of massive gravity, Physical Review Letters 106, 231101 (2011) [arXiv:1011.1232v2 [hep-th]].
  * Gümrükçüoğlu  _et al._ [2012] A. E. Gümrükçüoğlu, C. Lin, and S. Mukohyama, Cosmological perturbations of self-accelerating universe in, Journal of Cosmology and Astroparticle Physics 03, 006 (2012) [arXiv:1111.4107v2 [hep-th]].
  * Zakharov  _et al._ [2018] A. F. Zakharov, P. Jovanović, D. Borka, and V. B., Different ways to estimate graviton mass, in  _International Journal of Modern Physics: Conference Series_ , Vol. 47, p. 1860096. World Scientific (2018) [arXiv:1712.08339v1 [gr-qc]].
  * Pereira  _et al._ [2016] S. Pereira, E. Mendonça, A. Pinho SS, and J. Jesus, Cosmological bounds on open FLRW solutions of massive gravity, Revista mexicana de astronomía y astrofísica 52, 125 (2016) [arXiv:1504.02295v2 [gr-qc]].
  * Hinterbichler  _et al._ [2013] K. Hinterbichler, J. Stokes, and M. Trodden, Cosmologies of extended massive gravity, Physics Letters B 725, 1 (2013) [arXiv:1301.4993v3 [astro-ph.CO]].
  * Broy  _et al._ [2015] B. J. Broy, M. Galante, D. Roest, and A. , Westphal, Pole inflation—shift symmetry and universal corrections, Journal of High Energy Physics 12, 149 (2015) [arXiv:1507.02277 [hep-th]].
  * Terada [2016] T. Terada, Generalized pole inflation: Hilltop, natural, and chaotic inflationary attractors, Physics Letters B 760, 674-680 (2016) [arXiv:1602.07867 [hep-th]].
  * Kobayashi  _et al._ [2017] T. Kobayashi, O. Seto, and T. H. Tatsuishi, Toward pole inflation and attractors in supergravity: Chiral matter field inflation, Progress of Theoretical and Experimental Physics, 2017, no. 12, 123B04 (2017) [arXiv:1703.09960 [hep-th]].
  * Linder [2020] E. V. Linder, Pole dark energy, Physical Review D 101, 023506 (2020) [arXiv:1911.01606v1 [astro-ph.Co]].
  * Zangeneh  _et al._ [2021] M. K. Zangeneh, F. S. Lobo, and H. Moradpour, Evolving traversable wormholes satisfying the energy conditions in the presence of pole dark energy, Physics of the Dark Universe 31, 100779 (2021) [arXiv:2008.04013v3 [gr-qc]].
  * Karamitsos [2019] S. Karamitsos, Beyond the poles in attractor models of inflation, Journal of Cosmology and Astroparticle Physics 09, 022 (2019) [arXiv:1903.03707v3 [hep-th]].
  * Feng  _et al._ [2020] C.-J. Feng, X.-H. Zhai, and X.-Z. Li, Multi-pole dark energy, Chinese Physics C 44, 105103 (2020) [arXiv: 1912.10830v1 [gr-qc]].
  * Huang  _et al._ [2012] Q.-G. Huang, Y.-S. Piao, and S.-Y. Zhou, Mass-varying massive gravity, Physical Review D 86, 124014 (2012) [arXiv:1206.5678v4 [hep-th]].
  * Kenna-Allison  _et al._ [2020] M. Kenna-Allison, A. E. Gümrükçüoğlu, and K. Koyama, Cosmic acceleration and growth of structure in massive gravity, Physical Review D 102, 103524 (2020) [arXiv:2009.05405v1 [gr-qc]].
  * Aghanim  _et al._ [2020] N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. Banday, R. Barreiro, N. Bartolo, S. Basak, _et al._ , Planck 2018 results-VI. cosmological parameters, Astronomy & Astrophysics 641, A6 (2020) [arXiv:1807.06209v4 [astro-ph.Co]].
  * Ade  _et al._ [2016] P. A. R. Ade, Z. Ahmed, M. Amiri, D. Barkats, R. Basu Thakur, C. A. Bischoff, D. Beck, J. J. Bock, H. Boenish, E. Bullock, V. Buza, _et al._ , Improved Constraints on Pri- mordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Ob- serving Season, Physics Review Letters 127, 151301 (2021) [arxiv: 2110.0048v1 [astro-ph.Co]].
  * Remmen and Tomberg [2019] S. Räsänen and E. Tomberg, Planck scale black hole dark matter from Higgs inflation, Journal of Cosmology and
Astroparticle Physics 1901, 038 (2019) [arXiv:1810.1260
8v2 [astro-ph.Co]].

  * Felder  _et al._ [2002] C. Pattison, V. Vennin, H. Assadullahi, and D. Wands,T-
he attractive behaviour of Ultra-slow-roll inflation, 1808, 048 (2018)
[arXiv:1806.09553v2 [astro-ph.Co]].

  * Remmen and Carroll [2014] G. N. Remmen and S. M. Carroll, How many e-folds should we expect from high-scale inflation?, Physical Review D 90, 063517 (2014) [arXiv:1405.5538v2 [hep-th]].
  * Salah [2016] M. Amin, S. Khalil, and M. Salah, A viable logarithmic f(R) model for inflation, 1608, 043 (2016) [arXiv:1512.09324v2 [hep-th]].
  * Oda [2014] Y. Hamada, H. Kawai, and K. -y. Oda, Minimal Higgs inflation, 2014, 023B02 (2014) [arXiv:1308.6651v3 [hep-th]].
  * Felder  _et al._ [2002] G. Felder, A. Frolov, L. Kofman, and A. Linde, Cosmology with negative potentials, Physical Review D 66, 023507 (2002) [arXiv:hep-th/0202017v2].


# Velocity fluctuations for bubbly flows at small Re

Mithun Ravisankar  Roberto Zenit  School of Engineering, Brown University, 184
Hope St, Providence, RI 02912, USA

###### Abstract

We experimentally investigate the effect of Reynolds number (Re) on the
turbulence induced by the motion of bubbles in a quiescent Newtonian fluid at
small Re. The energy spectra, $E(k)$, is determined from the decaying
turbulence behind the bubble swarm obtained using particle image velocimetry
(PIV). As the bubble Reynolds number decreases, the signature $k^{-3}$ of the
energy spectra for the bubble induced turbulence is replaced by a scaling
similar to the classical $k^{-5/3}$ Kolmogorov’s scaling, where $k$ is the
wavenumber. We find that for Re $<$ $O$(100) the slope of the normalized
energy spectra is no longer independent on the gas volume fraction. This is
further corroborated using spatial correlation of the velocity vector behind
the bubble swarm. Thus, we provide experimental evidence to the claim by
Mazzitelli et al. (2003) that at low Reynolds numbers the normalized energy
spectra of the bubble induced turbulence will no longer show the $k^{-3}$
scaling because of the absence of bubble wake and that the energy spectra will
depend on the number of bubbles, thus non-universal.

###### keywords:

Viscous bubbly flows; pseudoturbulence;

##  1 Introduction

When a swarm of bubbles rises in an otherwise stagnant fluid due to the motion
of bubbles disturbances are created in the surrounding fluid giving rise to
velocity fluctuations. At a moderate to high Reynolds number, Re, these
velocity fluctuations in the wake behind the bubbles interacts with one
another giving rise to the emergence of $k^{-3}$ scaling of the energy
spectra, $E(k)$, of velocity fluctuations. This signature $k^{-3}$ scaling of
the energy spectra instead of the classical $k^{-5/3}$ Kolmogorov’s scaling is
regarded as the bubble induced turbulence, also referred to as
pseudoturbulence (Risso, 2018).

Lance & Bataille (1991) were the first to observe the emergence of $k^{-3}$
scaling of energy spectra, $E(k)$, with respect to the wavenumber, $k$ for the
bubble induced turbulence in Newtonian fluids. They provided an argument that
in a spectral space, the balance between the energy produced by the motion of
bubbles and the viscous dissipation in a statistically steady state gives rise
to the emergence of $k^{-3}$ scaling. Following this pioneering work, a number
of numerical (Esmaeeli & Tryggvason, 1996; Bunner & Tryggvason, 2002b;
Balachandar & Eaton, 2010) and experimental (Zenit et al., 2001; Martínez-
Mercado et al., 2007; Riboux et al., 2013; Prakash et al., 2016; Alméras et
al., 2017) works supported the emergence of $k^{-3}$ scaling for both the
spatial and temporal velocity fluctuations generated by the bubble motion
(Risso, 2018). Amoura et al. (2017) showed experimentally that irrespective of
the dispersed phase, be it bubbles or random fixed solid spheres, the velocity
fluctuations generated in the continuous phase gives rise to the $k^{-3}$
scaling. Recently, Pandey et al. (2023) identified the coexistence of
Kolmogorov’s turbulence with the bubble induced turbulence for a wide range of
Reynolds number and Galilei number (ratio of buoyancy to viscous forces).

Determining the liquid velocity fluctuations in two-phase gas-liquid flows is
challenging. Due to the dispersed nature of the bubbly flow, laser-based
techniques within the bubble swarm can only be used for very dilute flows. On
the other hand, the hot-wire-based techniques (Martínez-Mercado et al., 2007;
Mendez-Diaz et al., 2013; Alméras et al., 2017), are far from perfect as their
use implies elaborate signal processing. To overcome these difficulties,
Riboux et al. (2010) measured the liquid velocity fluctuations by abruptly
stopping the bubble formation using a solenoid valve thus leaving the wake
behind the bubble swarm free of bubbles to be analyzed using particle image
velocimetry (PIV). They showed that the $k^{-3}$ scaling , independent of the
gas volume fraction and bubble diameter, is observed between the Eulerian
length scale ($\Lambda=D/C_{d}$) and the bubble diameter ($D$), where $C_{d}$
is the drag coefficient of a single rising bubble. Later Risso (2018) modified
the Eulerian length scale, $\Lambda={D}/{\sqrt{C_{d}\text{Re}}}$ to include
the Reynolds number.

An intriguing question is to explore what happens to the energy spectra of the
bubbly flows at low Reynolds number (Bunner & Tryggvason, 2002a, b). When Re
$<$ 20, the bubbles will not have a significant wakes behind them (Blanco &
Magnaudet, 1995; Mougin & Magnaudet, 2001). A clue on what to expect for
smaller Reynolds numbers was summarized by Mazzitelli et al. (2003), who
conducted numerical simulations for bubbly Newtonian fluids considering
bubbles as point particles. Specifically, they did not observe the $k^{-3}$
scaling. Thus concluding the essentiality of the wakes. They further argued
that the energy spectrum slope for bubbly flows with little to no wakes behind
them will depend on the number of bubbles and will therefore be non-universal
(Mazzitelli et al., 2003; Mazzitelli & Lohse, 2009). Most surprisingly, to our
knowledge, there are only a few experimental studies in this regime
(Cartellier & Rivière, 2001; Martínez-Mercado et al., 2007; Mendez-Diaz et
al., 2013). Cartellier & Rivière (2001) studied bubble induced agitations at
Re $\sim$ $O$(1), however to delay the onset of large scale instabilities
(transition into heterogeneous bubbly regime) an inner tube within the bubble
column was used to create the liquid flow rate due to the gas lift. Further,
they did not address the nature of energy spectra of liquid velocity
fluctuations.

In the present study we will focus only on the homogeneous bubbly regime to
address the effect of Reynolds number on the bubble induced turbulence. Using
the particle image velocimetry technique proposed by Riboux et al. (2010), the
liquid velocity fluctuations are determined to calculate the energy spectra.
We find that as the Reynolds number decreases the slope of the energy spectra
emerges as $k^{-5/3}$ instead of the $k^{-3}$ scaling observed for the bubble
induced turbulence. Further, we show that this slope of the energy spectra
depends on the number of bubbles in the column. Understanding the spectral
structure fluid velocity fluctuations in two-phase flows at low Re could be
justified by their relevance in modern applications such as microfluidics
(Anna, 2016) and hydrogen production (Avci & Toklu, 2022).

##  2 Experimental setup and methods

###  2.1 Experimental setup

Figure 1(a) shows the schematic of the experimental setup used in the present
study. It consists of a transparent acrylic tank of height 1000 mm with a
cross-section of 100 mm $\times$ 50 mm. Monodispersed air bubbles are injected
at the bottom of the tank through a removable capillary bank. The capillary
bank is custom made with identical capillaries of inner diameter 0.6 mm in
tandem with a secondary capillary of inner diameter 0.25 mm, arranged in a
hexagonal array to increase the number of identical capillaries per unit area
(Martínez-Mercado et al., 2007). The tandem arrangement provides the
sufficient hydraulic resistance through the capillaries such that individual
bubbles are formed in a quasi-steady manner, thereby avoiding the generation
of gas jets (Oguz & Prosperetti, 1993). By adjusting the gas flow rate using a
needle valve, the mean gas volume fraction is measured from the increase in
the liquid level after the injection of bubbles, $\alpha=(H_{0}/\Delta
H+1)^{-1}$, where $H_{0}$ is the initial liquid level and $\Delta H$ is the
liquid level increase.

Figure 1: (a) Schematic of the experimental setup used in the current study.
Using a solenoid valve, the formation of bubbles are abruptly stopped and the
wake behind the bubble swarm is visualized using PIV in the measurement
section. A sample snapshot of a bubbly flow with gas volume fraction of
$\alpha\approx 0.025$ in Newtonian fluid with viscosity (b) Re = 626 and We =
3.2, and (c) Re = 6 and We = 0.5, respectively. The scale bar is 10 mm.

The velocity fluctuations in the liquid phase is measured from the wake behind
the bubble swarm using particle image velocimetry (PIV), following the work by
Riboux et al. (2010). The airflow is abruptly stopped by using a rapid
solenoid valve. Then, the region behind the bubble swarm is studied using
high-speed particle image velocimetry (Photron FASTCAM SA5). The recorded
images were then analyzed using PIVLab in MATLAB. From the PIV data, the
liquid velocity fluctuations in the bubble swarm wake are obtained as follows
$\mathbf{u^{\prime}}=\mathbf{u}-\langle\mathbf{u}\rangle$. Here
$\langle\rangle$ represent the average in space. The energy spectra, $E(k)$,
of the velocity fluctuations are determined from the decaying turbulence
behind the bubble swarm using particle image velocimetry following Riboux et
al. (2010). The energy spectra of horizontal and vertical velocity
fluctuations are then obtained using Welch method by averaging the energy
spectra of each rows and columns, respectively.

###  2.2 Test fluids

To prepare viscous Newtonian fluids, mixture of glycerin water solutions were
used. The properties of the fluids used in the study, shear viscosity, density
and surface tension were measured using ARES-G2 Rheometer (TA Instruments),
Density meter (Anton Paar) and Bubble pressure tensiometer (KRUSS Scientific
Instruments), are listed in Table 1. To reduce bubble coalescence, a small
amount of magnesium sulfate salt (0.05 mol/l) was added to all the fluids used
in the study, following (Lessard & Zieminski, 1971). The relevant
dimensionless numbers used in the current study are (i) Reynolds number, Re =
$\rho UD/\eta$, where $U$ is the average bubble velocity, $D$ is the average
bubble diameter, $\rho$ is the fluid density, and $\mu$ is the fluid
viscosity, and (ii) Weber number, We = $\rho U^{2}D/\sigma$, where $\sigma$ is
the surface tension. The average bubble diameter and velocity were obtained
from the probability distribution obtained from the image processing. The
Kolmogorov’s length scale, $\eta=\left(\frac{\nu^{3}}{\epsilon}\right)^{1/4}$,
where $\nu=\frac{\mu}{\rho}$ is the kinematic viscosity and
$\epsilon\approx\frac{U^{3}}{D}$ is the dissipation rate. The Eulerian length
scale, $\Lambda={D}/{\sqrt{C_{d}\text{Re}}}$, where $C_{d}=\frac{4gD}{3U^{2}}$
is the drag coefficient.

Fluids |  $\rho$ |  $\sigma$ |  $\mu$ |  $D$ |  $U$ |  $\Lambda$ |  $\eta$ |  Re |  We
---|---|---|---|---|---|---|---|---|---
(in water) |  (kg/m3) |  (mN/m) |  (Pa.s) |  (mm) |  (mm/s) |  (mm) |  (mm) |  |
10% Glycerin |  1003.0 |  73.23 |  0.001 |  2.8 $\pm$ 0.2 |  290 |  0.169 |  0.022 |  626 |  3.2
50% Glycerin |  1140.6 |  72.38 |  0.005 |  2.6 $\pm$ 0.1 |  243 |  0.285 |  0.062 |  144 |  2.1
60% Glycerin |  1142.1 |  71.97 |  0.007 |  2.6 $\pm$ 0.1 |  206 |  0.311 |  0.090 |  87 |  1.8
75% Glycerin |  1195.3 |  70.07 |  0.029 |  2.5 $\pm$ 0.2 |  114 |  0.475 |  0.394 |  11 |  0.6
85% Glycerin |  1224.0 |  70.04 |  0.048 |  2.3 $\pm$ 0.1 |  109 |  0.590 |  0.572 |  6 |  0.5
Table 1: Physical properties of the fluids: $\rho$ \- density; $\sigma$ \-
surface tension; $\mu$ \- viscosity;
$D$ \- average bubble diameter; $U$ \- average bubble velocity; $\Lambda$ \-
Eulerian length scale;
$\eta$ \- Kolmogorov’s length scale; Re - Reynolds number; We - Weber number.

##  3 Results and Discussions

###  3.1 Energy spectra of the velocity fluctuations

We first investigate the energy spectra of velocity fluctuations for different
bubble Reynolds number at a fixed gas volume fraction. Figure 2 shows the
horizontal and vertical energy spectra of velocity fluctuations normalized by
the corresponding variance for a family of bubble Reynolds number at a
constant gas volume fraction of $\alpha\approx 0.025$. Here, to be able to
compare different Reynolds numbers, the abscissa and the ordinate are
normalized with the modified Eulerian length scale
($\Lambda={D}/{\sqrt{C_{d}\text{Re}}}$) proposed by Risso (2018). As seen in
the Fig. 2, for Reynolds numbers, Re $\gg$ $O$(10), the signature $k^{-3}$
scaling of the energy spectra is obtained similar to (Martínez-Mercado et al.,
2007; Riboux et al., 2010), which also serves as a validation of our
experimental method. On further decreasing the Reynolds number, for Re = 11
and 6, the signature $k^{-3}$ scaling for the pseudoturbulence does not
emerge. The numerical results obtained for the bubble induced turbulence by
Bunner & Tryggvason (2002b) for Re between 12 and 30 found energy spectra
decays with a $k^{-3}$ scaling, which agrees with the present findings. To our
knowledge no other data for Re $<$ $O$(10) exists.

It has been argued that the observed $k^{-3}$ scaling is the result of the
contribution from the bubble wakes (Risso, 2018). In the current study,
however, for Re $\sim$ $O$(1-10), no significant wake behind the bubbles are
expected to appear (Blanco & Magnaudet, 1995). Thus, instead of the signature
$k^{-3}$ scaling for the pseudoturbulence, a scaling similar to the classical
$k^{-5/3}$ Kolmogorov’s scaling is observed for the energy spectra of velocity
fluctuations induced by the bubbles in our experiments. It is surprising,
however, to find such scaling, typically found in turbulent flows, for such a
low Reynolds number flow. We argue that $k^{-5/3}$ scaling emerges from the
small scale disturbances generated by the bubble motion, which are strong
enough to generate a large scale flow with the same characteristics as the
Kolmogorov’s turbulence (Mazzitelli & Lohse, 2009). Thus, we provide the first
experimental evidence to the claim by Mazzitelli & Lohse (2009), that the
$k^{-3}$ scaling is the signature of the bubble wake interactions and at very
low Reynolds numbers the classical $k^{-5/3}$ scaling is recovered. Further,
we note that the steeper drop in the energy spectra beyond the normalized
wavenumber of 0.5 may be due to the dominance of viscosity beyond the
Kolmogorov’s microscale, $\eta$, as listed in the Table 1. In other words, the
length scale at which the energy is being injected into the system is close to
the length scale at which the energy is dissipated by the viscosity. This
confirms that the turbulence induced due to the bubble motion is
instantaneously diffused by viscosity.

Figure 2: (a) Horizontal, and (b) vertical spectra of the liquid velocity
fluctuations normalized by the Eulerian length scale and the variances for a
family of Reynolds number at a constant gas volume fraction of $\alpha\approx$
0.025.

Figure 3: Horizontal spectra of the liquid velocity fluctuations normalized by
the Eulerian length scale and the variances for a family of gas volume
fraction at (a) Re = 114 and (b) Re = 11.

Further, Riboux et al. (2010) suggested that the $k^{-3}$ scaling observed in
the pseudoturbulence does not depend on the bubble diameter and gas volume
fraction for a wide range of Reynolds numbers. We show that this is not the
case at low Reynolds number. Figure 3(a) shows the horizontal spectra of
liquid velocity fluctuations normalized by the Eulerian length scale and the
variance for a family of gas volume fraction at Re = 114. It is to be noted
that at very low gas volume fractions, though the bubbles have significant
wake, the energy spectra does not show the $k^{-3}$ scaling. However, as the
gas volume fraction increases, the $k^{-3}$ scaling emerges. This is because
as the number of bubbles (i.e the gas volume fraction) increases the average
inter distance between the bubbles decreases, thus leading to wake
interactions. This is further corroborated using spatial correlation in the
next section. Whereas, for Re = 11 as seen in the Fig. 3(b), the $k^{-3}$
scaling is not observed, instead the classical $k^{-5/3}$ Kolmogorov’s scaling
is obtained. This is in agreement with the results from Mazzitelli et al.
(2003) that the slope of the energy spectra no longer show the signature
$k^{-3}$ scaling for the pseudoturbulence and that the slope of the energy
spectra depend on the number of bubbles and therefore is non-universal.

###  3.2 Spatial correlation

To further investigate why the slope of the energy spectra depends on the gas
volume fraction for low to moderate Reynolds number, the spatial correlation
of the horizontal velocity fields are considered. Here, the horizontal
velocity field is preferred over the vertical velocity field because the
probability density functions (PDFs) of the horizontal velocity component is
symmetric, whereas the vertical velocity component is positively skewed
because of the rising bubbles (Risso, 2018).

Figure 4: Spatial correlation of the horizontal velocity observed in the wake
of the bubble swarm (a) at a constant gas volume fraction of $\alpha\approx
0.025$ for a family of Reynolds number. Spatial correlation of the horizontal
velocity for a family of gas volume fractions at (b) Re = 114 and (c) Re = 11.

The spatial correlation of the horizontal velocity, $\mathbf{u}_{x}$ defined
as,

${\overline{R}}_{xx}=\frac{\langle\mathbf{u}_{x}(\mathbf{x})\cdot\mathbf{u}_{x}(\mathbf{x+\Delta
x})\rangle}{\langle u^{2}_{x}\rangle}.$ (1)

Here $\langle\rangle$ represent the average in space. Figure 4(a) shows the
spatial correlation of the horizontal velocity behind the bubble swarm at a
constant gas volume fraction of $\alpha\approx 0.025$. It is evident that as
the Re decreases from 626 to 144 to 87, we observe a higher correlation over
longer distances. This is because as we increase the viscosity of the fluid to
decrease the Reynolds number, the size of the bubble decreases as listed in
the Table 1. Therefore, more number of bubbles are required to achieve a fixed
gas volume fraction (say $\alpha\approx 0.025$). This is in agreement with the
numerical results from Esmaeeli & Tryggvason (1996), who showed that there is
an emergence of flow structures many times larger than the bubble size, which
is also observed in particle suspensions at finite Reynolds numbers (Climent &
Maxey, 2003). However, on further decreasing the Reynolds number (Re = 11 and
6), the velocity is decorrleated quickly as seen in the Fig. 4(a). This
indicates that the presence of wake is essential for the signature $k^{-3}$
scaling to emerge as seen in the Fig. 2. Figure 4(b) shows the spatial
correlation of the horizontal velocity observed behind the bubble swarm for a
family of gas volume fractions at Re = 114. It is immediately evident that as
the gas volume fraction increases, the horizontal velocity is correlated over
a longer distance. Thus, agreeing with our results for the energy spectra, as
seen in Fig. 3(a), dependence on the number of bubbles. However, as seen in
the Fig. 4(c) when the Re = 11, even with higher gas volume fraction, the
horizontal velocity is quickly decorrelated beyond one bubble radius.

##  4 Conclusions

The bubbly flow properties in viscous Newtonian fluids have been studied
experimentally by varying the concentration of glycerin in water mixture.
Particle image velocimetry (PIV) technique is used to visualize the wake
behind the bubble swarm to determine the velocity fluctuations in the decaying
agitations. We demonstrated experimentally that the signature $k^{-3}$ scaling
of the pseudoturbulence is replaced by the $k^{-5/3}$ scaling for the energy
spectra of velocity fluctuations induced by low Reynolds number bubbles. To
compare a range of Reynolds numbers bubbly flows, the energy spectra and the
wavenumbers are normalized with the modified Eulerian length scale,
$\Lambda={D}/{\sqrt{C_{d}\text{Re}}}$ proposed by Risso (2018).

Further, we showed that as the Reynolds number decreases, the slope of the
energy spectra depends on the number of bubbles in the flow. These
experimental results coincides with the numerical results by Mazzitelli et al.
(2003) that the energy spectra is non-universal for bubbles with Re $\sim$
$O$(1-10). To understand why the slope of the energy spectra depends on the
gas volume fraction, the spatial correlation of the velocity field was
considered. For a constant gas volume fraction, as the Re $\sim$ $O$(10-100),
a higher correlation over longer distances was observed in agreement with the
results from Esmaeeli & Tryggvason (1996). As the bubble size decreases with
Reynolds number, more bubbles (i.e. higher gas volume fraction) are required
to maintain the same gas volume fraction. Thus, as the inter-bubble distance
among bubbles decreases the wake interactions are more pronounced. Whereas for
Re $\sim$ $O$(1-10) the velocity is quickly decorrelated beyond the bubble
radius, thus the signature $k^{-3}$ scaling for the bubble induced turbulence
does not emerge.

## References

  * Alméras et al. (2017) Alméras, E., Mathai, V., Lohse, D. & Sun, C. 2017 Experimental investigation of the turbulence induced by a bubble swarm rising within incident turbulence. Journal of Fluid Mechanics 825, 1091–1112.
  * Amoura et al. (2017) Amoura, Z., Besnaci, C., Risso, F. & Roig, V. 2017 Velocity fluctuations generated by the flow through a random array of spheres: a model of bubble-induced agitation. Journal of Fluid Mechanics 823, 592–616.
  * Anna (2016) Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annual Review of Fluid Mechanics 48, 285–309.
  * Avci & Toklu (2022) Avci, A. C. & Toklu, E. 2022 A new analysis of two phase flow on hydrogen production from water electrolysis. International Journal of Hydrogen Energy 47 (11), 6986–6995.
  * Balachandar & Eaton (2010) Balachandar, S. & Eaton, J.K. 2010 Turbulent dispersed multiphase flow. Annual Review of Fluid Mechanics 42, 111–133.
  * Blanco & Magnaudet (1995) Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetric high-reynolds number flow around an ellipsoidal bubble of fixed shape. Physics of Fluids 7 (6), 1265–1274.
  * Bunner & Tryggvason (2002a) Bunner, B. & Tryggvason, G. 2002a Dynamics of homogeneous bubbly flows part 1. rise velocity and microstructure of the bubbles. Journal of Fluid Mechanics 466, 17–52.
  * Bunner & Tryggvason (2002b) Bunner, B. & Tryggvason, G. 2002b Dynamics of homogeneous bubbly flows part 2. velocity fluctuations. Journal of Fluid Mechanics 466, 53–84.
  * Cartellier & Rivière (2001) Cartellier, A. & Rivière, N. 2001 Bubble-induced agitation and microstructure in uniform bubbly flows at small to moderate particle reynolds numbers. Physics of Fluids 13 (8), 2165–2181.
  * Climent & Maxey (2003) Climent, E. & Maxey, M.R. 2003 Numerical simulations of random suspensions at finite reynolds numbers. International Journal of Multiphase Flow 29 (4), 579–601.
  * Esmaeeli & Tryggvason (1996) Esmaeeli, A. & Tryggvason, G. 1996 An inverse energy cascade in two-dimensional low reynolds number bubbly flows. Journal of Fluid Mechanics 314, 315–330.
  * Lance & Bataille (1991) Lance, M. & Bataille, J. 1991 Turbulence in the liquid phase of a uniform bubbly air–water flow. Journal of Fluid Mechanics 222, 95–118.
  * Lessard & Zieminski (1971) Lessard, R.R. & Zieminski, S.A. 1971 Bubble coalescence and gas transfer in aqueous electrolytic solutions. Industrial & Engineering Chemistry Fundamentals 10 (2), 260–269.
  * Martínez-Mercado et al. (2007) Martínez-Mercado, J., Palacios-Morales, C.A. & Zenit, R. 2007 Measurement of pseudoturbulence intensity in monodispersed bubbly liquids for 10 $<$ Re $<$ 500\. Physics of Fluids 19 (10), 103302.
  * Mazzitelli & Lohse (2009) Mazzitelli, I.M. & Lohse, D. 2009 Evolution of energy in flow driven by rising bubbles. Physical Review E 79 (6), 066317.
  * Mazzitelli et al. (2003) Mazzitelli, I.M., Lohse, D. & Toschi, F. 2003 The effect of microbubbles on developed turbulence. Physics of Fluids 15 (1), L5–L8.
  * Mendez-Diaz et al. (2013) Mendez-Diaz, S., Serrano-Garcia, J.C., Zenit, R. & Hernandez-Cordero, J.A. 2013 Power spectral distributions of pseudo-turbulent bubbly flows. Physics of Fluids 25 (4).
  * Mougin & Magnaudet (2001) Mougin, G. & Magnaudet, J. 2001 Path instability of a rising bubble. Physical Review Letters 88 (1), 014502.
  * Oguz & Prosperetti (1993) Oguz, H.N. & Prosperetti, A. 1993 Dynamics of bubble growth and detachment from a needle. Journal of Fluid Mechanics 257, 111–145.
  * Pandey et al. (2023) Pandey, V., Mitra, D. & Perlekar, P. 2023 Kolmogorov turbulence coexists with pseudo-turbulence in buoyancy-driven bubbly flows. Physical Review Letters 131 (11), 114002.
  * Prakash et al. (2016) Prakash, V.N., Mercado, J.M., Van Wijngaarden, L., Mancilla, E., Tagawa, Y., Lohse, D. & Sun, C. 2016 Energy spectra in turbulent bubbly flows. Journal of Fluid Mechanics 791, 174–190.
  * Riboux et al. (2013) Riboux, G., Legendre, D. & Risso, F. 2013 A model of bubble-induced turbulence based on large-scale wake interactions. Journal of Fluid Mechanics 719, 362–387.
  * Riboux et al. (2010) Riboux, G., Risso, F. & Legendre, D. 2010 Experimental characterization of the agitation generated by bubbles rising at high reynolds number. Journal of Fluid Mechanics 643, 509–539.
  * Risso (2018) Risso, F. 2018 Agitation, mixing, and transfers induced by bubbles. Annual Review of Fluid Mechanics 50, 25–48.
  * Zenit et al. (2001) Zenit, R., Koch, D.L. & Sangani, A.S. 2001 Measurements of the average properties of a suspension of bubbles rising in a vertical channel. Journal of Fluid Mechanics 429, 307–342.