# FixMe: A GitHub Bot for Detecting and Monitoring On-Hold Self-Admitted
Technical Debt

Saranphon Phaithoon1, Supakarn Wongnil1, Patiphol Pussawong1,
Morakot Choetkiertikul1, Chaiyong Ragkhitwetsagul1, Thanwadee Sunetnanta1,
Rungroj Maipradit2, Hideaki Hata3, and Kenichi Matsumoto2  1Faculty of
Information and Communication Technology (ICT), Mahidol University, Nakhon
Pathom, Thailand
2Nara Institute of Science and Technology (NAIST), Nara, Japan
3Shinshu University, Nagano, Japan  Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
{morakot.cho, chaiyong.rag<EMAIL_ADDRESS>
{maipradit.rungroj.mm6<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Self-Admitted Technical Debt (SATD) is a special form of technical debt in
which developers intentionally record their hacks in the code by adding
comments for attention. Here, we focus on issue-related “On-hold SATD”, where
developers suspend proper implementation due to issues reported inside or
outside the project. When the referenced issues are resolved, the On-hold SATD
also need to be addressed, but since monitoring these issue reports takes a
lot of time and effort, developers may not be aware of the resolved issues and
leave the On-hold SATD in the code. In this paper, we propose FixMe, a GitHub
bot that helps developers detecting and monitoring On-hold SATD in their
repositories and notify them whenever the On-hold SATDs are ready to be fixed
(i.e. the referenced issues are resolved). The bot can automatically detect
On-hold SATD comments from source code using machine learning techniques and
discover referenced issues. When the referenced issues are resolved,
developers will be notified by FixMe bot. The evaluation conducted with 11
participants shows that our FixMe bot can support them in dealing with On-hold
SATD. FixMe is available at https://www.fixmebot.app/ and FixMe’s VDO is at
https://youtu.be/YSz9kFxN_YQ.

##  I Introduction

On-hold Self-admitted Technical Debt (On-hold SATD) implies the debt that is
created when the implementation need to wait (i.e., be held) for a resolution
of a referenced issues in source code comments. These dependencies can refer
to same project’s or other project’s issues. Those On-hold SATDs are supposed
to be fixed after their referenced issues are resolved, so called Ready-to-be-
fixed On-hold SATD [1]. However, the study shows that a number of Ready-to-be-
fixed On-hold SATDs have actually never been fixed [1, 2, 3].

A software module usually relies on numerous dependencies such as calling
functions from within a project or across projects. When those issue
referencing in source code comments have been resolved, developers are not
aware of such issue resolving. Those On-hold SATDs thus do not usually be
maintained properly. For example, the source code
comments111https://bit.ly/3p6hGuZ in the Mono project222https://www.mono-
project.com/ shows that the problem of passing arrays can be fixed after the
issue ID 18245333https://bit.ly/3v2FToi was resolved. However, the issue was
resolved but the code have not been fixed (see Figure 1). Therefore, this
manual monitoring and maintaining of referenced issues are tedious and they
require tremendous effort.

In order to fill this gap and help code reviewers and developers overcome
these challenges, we present “FixMe bot”, a GitHub bot that help code
reviewers and developers to detect and monitor On-hold SATDs from source code
comments and inform them whenever an On-hold SATD is Ready-to-be-fixed by
adopting the existing On-hold SATD detection approach from Maipradit et al.
[1].

Figure 1: Example of Ready-to-be-fixed On-hold SATD in the Mono project

##  II Related work

Recently, there is an existing On-hold SATD detection model from Maipradit et
al. [2] who applied regular expressions to detect issue references in source
code comments and proposed a machine learning model to identify source code
comments as either “On-hold SATD” (the referenced issue is used to indicate
the need to wait for issue resolution) or “Cross reference” (the referenced
issue is used to document the code) [1]. Additionally, Huang et al. used text
mining to identify SATD in eight open source projects [4]. Potdar and Shihab
[5] reported 62 common patterns that can be used to identify SATD. Maldonado
et al. [6] present an approach to automatically identify design and
requirement SATD using Natural Language Processing (NLP) on code comments. Ren
et al. [7] proposed a Convolutional Neural Network (CNN) to classify code
comments into SATD or non-SATD. Yu et al. [8] introduce a framework that
reduce human effort in identifying SATD by using 2 steps. First, the framework
uses patterns to identify easy-to-find SATDs. After that, it uses machine
learning to assist human expert in manually identifying hard-to-find SATDs.
SonarQube [9] is a platform integrated with external tools like GitHub that
provides the capability to view and analyze technical debt in term of number
of working days required to solve all technical debt. Just-In-Time Bot
(JITBot) [10] is a GitHub application that automatically provide feedback
about riskiness of each commit and suggest risk mitigation plans to
developers. However, there is no On-hold SATD detection tool that is fully
integrated into software development process such as a code review in
practice.

##  III An overview of our FixMe bot

Figure 2: The scenario of using FixMe

We explain how FixMe bot works in Figure 2. FixMe bot analyzes source code in
GitHub repositories to find code comments. Using the existing On-hold SATD
detection model [1]. FixMe bot can automatically determine whether a code
comment is an On-hold SATD or an issue referencing comment (e.g., to note
related issues in the code). GitHub issue reports that are mentioned in On-
hold SATD comments are then discovered using regular expressions. After
extracting a list of issue reports and On-hold SATD code comment locations,
the bot monitors the issue status and notify software practitioners when those
issue are resolved. There are three form of outputs that FixMe bot informs the
developers of the Ready-to-be-fixed SATDs: 1) posting a comment containing a
list of Ready-to-be-fixed On-hold SATDs on a pull request with the confidence
level that shows the estimated probability obtained from the On-hold SATD
classification model (Figure 3), 2) posting a comment on a commit that shows
Ready-to-be-fixed On-hold SATDs found on the commit (i.e., code changes), and
3) creating a new issue report to list all the Ready-to-be-fixed On-hold SATDs
in a project (Figure 4). In addition, FixMe can be also configured via
https://www.fixmebot.app. For example, the developers can select the output
methods, select branches to monitor, and add new patterns of regular
expressions.

Figure 3: Example of pull request and commit comments posted by FixMe bot
Figure 4: Example of issue comment posted by FixMe bot

##  IV System Architecture

Figure 5: The overview architecture of FixMe

Figure 5 shows the FixMe’s architecture which consists of three components:
FixMe GitHub plug-in, FixMe backend, and FixMe frontend (configuration page).
First, a project owner (e.g., repository maintainer) needs to install the
FixMe plug-in into their GitHub repository. Whenever project contributors push
a commit or create a new pull request, GitHub sends a webhook to trigger the
FixMe backend module. The FixMe bot then make a request to GitHub’s API for
posting outputs based on the user’s configuration.

###  IV-A FixMe GitHub Plug-in

GitHub provides the GitHub API to allow integrations from third party
applications to perform their functionalities. The FixMe GitHub plug-in is
responsible for creating webhooks from GitHub to establish the connection
between GitHub and the FixMe bot’s back-end system.

###  IV-B FixMe Backend

The backend of FixMe bot contains three components: Comment extraction, On-
hold SATD detection, and GitHub issue monitoring (Figure 5).

####  IV-B1 Comment Extraction

The FixMe bot can extract source code comments from a specific commit in a
repository when the bot receives a webhook from the FixMe plug-in. The bot
then collects all files referenced in those requests and filters only source
code files based on file extensions. The bot then uses the comment parser
introduced in Maipradit et al. [1] which technically uses regular expressions
to extract source code comments through all source code files. Currently,
FixMe supports only Java language.

####  IV-B2 On-hold SATD Detection

The extracted code comments are then fed into the On-hold SATD detection model
adopted from Maipradit et al. [1] to determine whether the extracted code
comments are On-hold SATDs or not. The model is trained from N-gram features
of the code comments from 10 Java open source projects using Auto-sklearn
version 0.9.0 and Python version 3.7.10. It shows that the Extra Tree
algorithm was the best performer to classify On-hold SATDs. The model achieves
over 0.8 F-measure and 0.9 AUC scores [1].

####  IV-B3 GitHub Issue Monitoring

The referencing issues contained in those identified On-hold SATD comments are
extracted and recorded into FixMe’s database. We used a _cron job_ , a built-
in Linux utility, to periodically run the issue monitoring script. Thus,
whenever referenced issues are resolved, FixMe invokes the GitHub API to
notify software practitioners.

###  IV-C FixMe Configuration

There are three parts that users can configure FixMe: selecting On-hold SATD
detection techniques, specifying target branch(es), and choosing output
posting methods. In On-hold detection methods, FixMe allows users to define
their own regular expression patterns to detect On-hold SATD comments. Thus,
they can modify the detection methods to match with their own development
practices. For example, a team can define a common comment patterns that
reflect the On-hold SATD comment (e.g., `(after|once)\s+(issue \d+) is
resolved`). FixMe also supports multiple-branch monitoring and the users can
select a number of target branches to be monitored by FixMe. Lastly, the users
can select three output methods: pull request comment, commit comment, and
issue creation.

##  V Evaluation and Result

Figure 6: Evaluation setting

###  V-A Evaluation Setup

To evaluate the FixMe bot, we focus on the usability of the tool. Figure 6
illustrates the evaluation process. We conducted the study with 11
participants. Three participants are experienced developers from software
companies who have experience in managing technical debt in their projects,
and eight participants are computer science students who regularly use GitHub
and have experience in software development projects more than six months. A
mock-up Java project implementing TF-IDF searcher and Jaccard searcher is
stimulated on GitHub for our user study. There are 8 Java source code files
containing both regular code comments and 9 On-hold SATD comments in total
(e.g., // TODO: Use this for now then modify this once
https://github.com/mockito/mockito/issues/769 is fixed), and 4 unresolved
issue reports. The participants need to work on identifying On-hold SATD
comments and check the status of the related issue reports to identify Ready-
to-be-fixed On-hold SATDs. The participants must perform the task manually for
one hour before we introduce the FixMe bot for them to use. They then need to
validate the results from FixMe whether they agree with the On-hold SATD
comments identified by FixMe. After that, they fill the questionnaire for the
study. The evaluation focuses on five criteria using a five point Likert’s
scale (from strongly disagree to strongly agree). We would like to evaluate 1)
the participants’ agreement with the output from FixMe, 2) understandability
of the output, 3) ease of use, 4) helping on saving time, and 5) its potential
to be adopted in real software projects. In addition, the participants give
the overall satisfaction score scaled from 1 to 10 (from very poor to
excellent).

###  V-B Result

We separately report the evaluation results from three professional software
developers and eight computer science students as shown in Table I. Overall,
FixMe achieves over 9 out of 10 in the satisfaction score — 9.2 on average
across two groups.

Professional software developers: The professional software developers found
that the first assigned task in manual identification of Ready-to-be-fixed On-
hold SATD code comments are moderately difficult (4/5). After using FixMe, the
results indicate that they are satisfied with the supports provided by FixMe
with an overall satisfaction score of 9.3. The FixMe bot also receive high
scores in all evaluation aspects by achieving over 4 in all evaluation
criteria.

Computer science students: The results indicate that the output provided by
FixMe in forms of pull request comment, commit comment, and issue notification
are understandable (4.7/5). The overall satisfaction score is 9 out of 10.

##  VI Discussion

From the questionnaires, it shows that software developers are aware of
addressing SATD but the way to tackle SATD in practice consumes tremendous
effort from software developers. All participants indicate that finding On-
hold SATD comments without using any supporting tool is a non-trivial job.
They must go through all comments and check the related issue report’s
statuses. From the evaluation results, it can also infer that our FixMe bot
can provide advantages in terms of time-saving and ease of use that help them
realized Ready-to-be-fixed On-hold SATDs easier and faster. Moreover, the
evaluation results show that the tool can potentially be useful and adopted in
the real projects.

TABLE I: Evaluation results Criteria (1 to 5) |  Software Developers |  CS Students | AVG.
---|---|---|---
FixMe provides an accurate result |  4.7 |  4.1 | 4.4
A result is understandable |  4.4 |  4.7 | 4.6
FixMe is easy to use |  4.3 |  4.5 | 4.4
Using FixMe is time-saving |  5.0 |  5.0 | 5.0
FixMe can be used in real projects |  4.7 |  4.5 | 4.6
Satisfaction score (1 to 10) |  9.3 |  9.0 | 9.2

##  VII Conclusion

We introduce the FixMe bot which is an On-hold SATD detection tool seamlessly
integrated with GitHub to assist developers in monitoring issue status and
notify whenever On-hold SATD becomes ready-to-be-fixed. We adopt the existing
state-of-the-art On-hold SATD classification model using N-gram technique and
text filtering techniques complemented with automated machine learning
approach (Auto-sklearn). The FixMe bot allows developers to perform bot
configuration according to their project environment and practice including
detection techniques, target branch selection, and notification method
selection through the bot’s web application. Our tool can notify the
developers in forms of a pull request comment, commit comment, and issue
creation. We performed a usability testing to evaluate the performance of our
tool with 11 participants including experienced software developers and
computer science students who actively use GitHub. The evaluation results of
the FixMe bot shows promising adoption of the tool. We found that FixMe bot
can support developers in managing and detecting On-hold SATD resulting in
less effort required. Additionally, the bot helps saving time and gives more
information to handle On-hold SATDs in their projects. Our future work involve
adding a capability of FixMe to work with other programming languages and also
identify issue reports from other issue tracking platforms such as JIRA.

Acknowledgement. We thank anonymous reviewers for their constructive feedback.
This research is partially supported by the Japan Societyfor the Promotion of
Science [Grants-in-Aid for ScientificResearch (S) (No.20H05706)].

## References

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#
Modified scaling in $k$-essence model in interacting dark energy - dark matter
scenario

Anirban Chatterjee Corresponding Author<EMAIL_ADDRESS>Indian Institute
of Technology Kanpur, Kanpur 208016, India  Biswajit Jana
<EMAIL_ADDRESS>Ramakrishna Mission Vivekananda Educational and
Research Institute, Belur Math, Howrah 711202  Abhijit Bandyopadhyay
<EMAIL_ADDRESS>Ramakrishna Mission Vivekananda Educational and
Research Institute, Belur Math, Howrah 711202

(August 28, 2024)

###### Abstract

It has been shown by Scherrer and Putter et.al. in [1, 2] that, when dynamics
of dark energy is driven by a homogeneous $k-$essence scalar field $\phi$,
with a Lagrangian of the form $L=V_{0}F(X)$ with a constant potential $V_{0}$
and $X=\frac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi=\frac{1}{2}\dot{\phi}^{2}$,
one obtains a scaling relation $X(dF/dX)^{2}=Ca^{-6}$ , where $C$ is a
constant and $a$ is the FRW scale factor of the universe. The separate energy
conservation in the dark energy sector and the constancy of $k-$essence
potential are instrumental in obtaining such a scaling. In this paper we have
shown that, even when considering time-dependent interactions between dark
energy and dark matter, the constancy of $k-$essence potential may lead to a
modified form of scaling. We have obtained such a scaling relation for a
particular class of parametrisation of the source term occurring in the
continuity equation of dark energy and dark matter in the interacting
scenario. We used inputs from the JLA analysis of luminosity distance and
redshift data from Supernova Ia observations, to obtain the modified form of
the scaling.

##  1 Introduction

Several cosmological observations and surveys revealed that our universe is
presently undergoing an accelerated phase of expansion and a transition from
decelerated phase to this accelerated phase happened during the late time
phase of cosmic evolution. The luminosity distance and redshift measurements
of type Ia Supernova [3, 4, 5] are instrumental in establishing this fact.
Independent observations like Baryon Acoustic Oscillation [6, 7], Cosmic
Microwave Background Radiation [8] and Observed Hubble Data (OHD) [9] also
reinforce this conclusion. The cause of acceleration of the present-day cosmic
expansion still remains a mystery and has been presented in literature [3, 4,
5] as an effect due to presence of an unclustered form of energy with negative
pressure - dubbed ‘Dark Energy’ (DE). On the other hand, observed rotation
curves of spiral galaxies [10], observation of gravitational lensing [11],
bullet and other colliding clusters provide evidences for existence of non-
luminous matter in the universe, called ‘Dark Matter’ (DM), which reveals its
presence only through gravitational effects detected in above mentioned
observations. Results from satellite borne experiments like WMAP [12] and
Planck [13] have estimated that DE and DM together contribute $\sim 96\%$ of
total energy density of the present-day universe, with $\sim$69$\%$ and
$\sim$27$\%$ as their respective shares. The rest 4$\%$ contribution comes
from radiation and baryonic matter. Though a physical theory of dark energy is
still lacking, there exist diverse theoretical approaches aiming construction
of models for DE leading to present-day cosmic acceleration. These include the
$\Lambda$-CDM model, where $\Lambda$ refers to cosmological constant and ‘CDM’
corresponds to cold (non-relativistic) dark matter. This model, though fits
well with the present-day cosmological observations, is associated with the
problems of cosmological coincidence [14] and fine tuning [15] which motivate
construction of alternative DE models. One approach of constructing such
models, called modified gravity models [16], involves modification of Einstein
tensor in the geometric part of the Einstein field equations. Another class of
models treats DE to be driven by (scalar) fields with suitably chosen field
theoretic Lagrangians contributing to energy-momentum tensor in Einstein
equations. The second kind of models, as widely discussed in literature,
includes the Quintessence [17, 18, 19, 20, 21, 22, 23, 24, 25] and $k$-essence
models [26, 27, 28, 29, 30, 31, 32, 33]. A sub-class of such models involves
consideration of interaction between DE and DM [34, 35, 36, 37, 38, 39, 40,
41, 42, 43] to explain the present-day observed features of cosmic expansion.
We have considered such a model in the context of this paper.

To investigate the interacting scenario of DE and DM, we neglect the radiation
and baryonic matter contribution to the total energy density of universe
during its late time phase of cosmic evolution due to their small share ($\sim
4\%$) in the present-day energy-content as estimated from WMAP [12] and Planck
[13] observations. In this paper, we consider DE to be represented by a
homogeneous scalar field $\phi=\phi(t)$ whose dynamics is driven by a purely
kinetic ($k$-essence) Lagrangian with a constant potential. Purely kinetic
scalar field models have been widely discussed in [1, 27, 28, 31, 44, 45, 46,
47] and references therein. The DM component of the universe, on the other
hand, is chosen to be a non-relativistic pressure-less fluid (dust). We
consider the DE-DM interaction to be time-dependent and introduce it through a
source term $Q(t)$ in the non-conserving continuity equations for DM and DE
(see Eqs. (17) and (18) in Sec. 4). The function $Q(t)$ provides a measure of
instantaneous rate of energy transfer between DE and DM components of the
universe.

We consider spacetime geometry of the expanding universe, at large scales, to
be isotropic and homogeneous and flat which is described by the Friedmann -
Robertson - Walker (FRW) metric characterised by a time dependent scale factor
$a(t)$ and zero curvature constant. The evolution of such a universe with its
content modelled as an ideal fluid (characterised by its energy density
$\rho(t)$ and pressure $p(t)$) is governed by the Friedmann equations which
connect $a(t)$ and its derivatives with $\rho(t)$ and $p(t)$. Using the
luminosity distance vs. redshift measurements in Supernova Ia (SNe Ia)
observations, the temporal profile of the scale factor, the Hubble parameter
$H=\dot{a}/a$ may be extracted. From this knowledge, the time profile of total
energy density and pressure of the universe may as well be computed,
exploiting the Friedmann equations.

When we consider DM and DE to be non-interacting, both the components of total
dark fluid separately satisfy respective continuity equations (Eqs. (17) and
(18) with $Q(t)=0$) implying separate energy conservation in each individual
sector. In this non-interacting DE-DM scenario, if the dynamics of DE is
considered to be driven by a scalar field $\phi$ governed by a $k$-essence
Lagrangian of form $L=V_{0}F(X)$ with a constant potential $V_{0}$ and
$X=\frac{1}{2}\nabla^{\mu}\phi\nabla_{\mu}\phi=\frac{1}{2}\dot{\phi}^{2}$, one
obtains a scaling of the form: $X(dF/dX)^{2}=Ca^{-6}$ ($C$ is a constant) [1,
2]. Separate energy conservation in the DE sector and the constancy of
$k-$essence potential are instrumental in obtaining such a scaling. This
scaling relation connects the scale factor $a$ and $k$-essence scalar field
$\phi$ through its time derivatives appearing in $X$ and $dF/dX$.

In this paper we have shown that, even in presence of time-dependent
interactions between DE and DM ($Q(t)\neq 0$) implying continual exchange of
energy between the two sectors, the constancy of $k-$essence potential may
lead to a modified form of scaling. We have obtained such a scaling, by
parametrising the dependence of source term $Q(t)$ on $a(t)$ as a power law:
$Q(t)=Q_{0}[a(t)]^{k}$, where $k$ is a constant and $Q_{0}$ is the value of
$Q(t)$ at present epoch ($a(t)$ is normalised to 1 at present epoch). In
obtaining the form of scaling relation with above parametrisation of DE-DM
interactions, we have taken into consideration the observed feature of
temporal behaviour of the FRW scale factor $a(t)$, probed in the measurement
of luminosity distance and redshift of SNe Ia events. The observational
ingredient enters into the explicit form of scaling relation at different
levels of its derivation, through various constants which encode in them
features of the observational data. The obtained scaling depends on $k$ and
$Q_{0}$, which are the two parameters of the model. For computational
convenience, in stead of $Q_{0}$, we used a dimensionless parameter
$\beta_{0}\equiv Q_{0}/(\rho_{\rm de}^{0}+\rho_{\rm dm}^{0})$ all through the
work. We also find the region of the corresponding $k-\beta_{0}$ parameter
space that is allowed from SNe Ia data, in the context of interacting DE-DM
model considered here.

The paper is organised as follows. In Sec. 2, we briefly discussed Joint Light
curve Analysis (JLA) of Supernova Ia data using which we obtain the temporal
behaviour of scale factor and its derivatives. In Sec. 3, we gave a brief
outline of the framework of $k$-essence model of dark energy and discussed its
aspects for a constant potential scenario, which is relevant in the context of
the paper. In Sec. 4, we presented the theoretical framework of interacting
DE-DM scenario in FRW universe. We obtained temporal behaviour of equation of
state parameter ($\omega=p/\rho$) of the total dark fluid and the total energy
density $\rho$ of the universe using obtained time profile of the scale factor
and its derivatives obtained in Sec. 2. In the context of this interacting DE-
DM scenario, temporal behaviour of energy density of individual DE and DM
components ($\rho_{\rm de}$ and $\rho_{\rm dm}$) have also been obtained in
terms of model-parameters $k$ and $\beta_{0}$. In Sec. 5 we have shown how we
derived the corresponding modified scaling relation for $k-$essence model of
DE with a constant potential. The sensitivity of the modified scaling relation
on the parameters $k$ and $\beta_{0}$ are also graphically represented for
different chosen benchmark values of the parameters within their allowed
region. We summarised the results in the concluding Sec. 6.

##  2 Cosmological parameters from analysis of SNe Ia data

As discussed in Sec. 1 the luminosity distance and redshift measurements of
type Ia Supernova is the key observational ingredient in establishing the
transition from decelerated to an accelerated phase of cosmic expansion during
its late time phase of evolution. We have used the observational data as
input, in estimating the behaviour of modified scaling, on the parameters $k$,
$\beta_{0}$ which parametrises the time dependent interaction $Q(t)$ between
DE and DM. In this section we describe how we extract the relevant
cosmological parameters from the SNe Ia data to use them as direct inputs into
this estimation. There exist different compilations of the SNe Ia data
corresponding to supernova surveys in different redshift region using diverse
probes and measurements. Small redshift $(z>0.1)$ projects comprise Harvard-
Smithsonian Center for Astrophysics survey (cFa) [48], the Carnegie Supernova
Project (CSP)([49],[50],[51]), the Lick Observatory Supernova Search (LOSS)
[52] and the Nearby Supernova Factory (SNF) [53]. SDSS-II supernova surveys
([54],[55],[56],[57],[58]) are mainly focused on the redshift region of
($0.05<z<0.4$). Programmes like Supernova Legacy Survey (SNLS) ([59],[60]) the
ESSENCE project [61], the Pan-STARRS survey ([62],[63],[64]) correspond to
high redshift regime. More than one thousand SNe Ia events have been
discovered through all surveys. However, in the range between $z\sim 0.01$ and
$z\sim 0.7$, luminosity distances have been measured with a very high
statistical precision. ‘Joint Light-curve Analysis (JLA) data’
([63],[65],[66]) contains a total of 740 SNe Ia events from full three years
of SDSS survey, first three seasons of the five-year SNLS survey and 14 very
high redshift $0.7<z<1.4$ SNe Ia from space-based observations with the HST
[67]. For the present work, we consider this data set for analysis where the
different systematic uncertainties are taken care of by compiling the data
with flux-averaging technique whose technical details have been
comprehensively described in [68, 69, 70]. We briefly outline the methodology
of analysis here. The $\chi^{2}$ function corresponding to JLA data is defined
as

$\displaystyle\chi^{2}=\sum_{i,j=1}^{740}(\mu_{\rm obs}^{(i)}-\mu_{\rm
th}^{(i)})(\sigma^{-1})_{ij}(\mu_{\rm obs}^{(j)}-\mu_{\rm th}^{(j)})\,,$ (1)

where $\mu_{\rm obs}^{(i)}$ and $\mu_{\rm th}^{(i)}$ respectively denote the
observed value and theoretical expression for distance modulus at red-shift
$z_{i}$. In a flat FRW spacetime, $\mu_{\rm th}^{(i)}$ is given by

$\displaystyle\mu_{\rm th}^{(i)}$ $\displaystyle=$ $\displaystyle
5\log_{10}[d_{L}(z_{\rm hel},z_{\rm CMB})/{\rm Mpc})]+25$ (2)

where $d_{L}$ is the luminosity distance it’s given by

$\displaystyle d_{L}(z_{\rm hel},z_{\rm CMB})$ $\displaystyle=$
$\displaystyle(1+z_{\rm hel})r(z_{CMB})\quad\mbox{with}\quad
r(z)=cH_{0}^{-1}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\,,$ (3)

and $z_{\rm CMB}$ and $z_{\rm hel}$ respectively refer to CMB rest frame and
heliocentric frame value of SNe IA redshifts. The value of Hubble parameter at
present epoch is denoted by $H_{0}$. The observed distance modulus $\mu_{\rm
obs}^{(i)}$ is expressed in terms of the observed peak magnitude
$m_{B}^{\star}$, the time stretching parameter of the light-curve $X_{1}$ and
supernova color at maximum brightness, $C$ as

$\displaystyle\mu_{\rm obs}^{(i)}$ $\displaystyle=$ $\displaystyle
m_{B}^{\star}(z_{i})-M_{B}+\alpha X_{1}(z_{i})-\beta C(z_{i})$ (4)

$\alpha$, $\beta$ being the nuisance parameters and the absolute magnitude
$M_{B}$ is kept fixed at the value -19. $\sigma_{ij}$ in Eq. (1) is the
covariant matrix as given in Eq. (2.16) of [69]. Wang in [69] proposed a flux
averaging technique to reduce the effect of systematic uncertainties involved
in the covariant matrix owing to weak lensing of SNe Ia data. We take the
result of redshift($z$)-dependence of the function $E(z)=H(z)/H_{0}$
(corresponding to a zero red-shift cut-off $z=0$) obtained in [69] from the
$\chi^{2}$-marginalisation with respect to $(M_{B},\alpha,\beta)$. The
1$\sigma$ range of $E(z)$ resulting from the analysis is shown in Fig. 1. The
mean of $E(z)$ values for every $z$ in this $1\sigma$ range is shown by the
dashed line in Fig . 1. The temporal behaviour of relevant cosmological
quantities are obtained using this mean $E(z)$ vs $z$ curve as benchmark.

Figure 1: Plots of $E$ vs $z$ (Dashed line correspond to the average value of
$E(z)$ over the 1$\sigma$ range of $E(z)$) depicted by shaded region.

The scale factor $a$, which we have taken to be normalised to $a=1$ at present
epoch, is related to redshift $z$ by the relation $1/a=1+z$. Using
$H=\dot{a}/a$ we can write

$\displaystyle dt$ $\displaystyle=$
$\displaystyle-\frac{dz}{(1+z)H(z)}=-\frac{dz}{(1+z)H_{0}E(z)}$ (5)

which on integration gives

$\displaystyle\frac{t(z)}{t_{0}}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{H_{0}t_{0}}\int_{z}^{0}\frac{dz^{\prime}}{(1+z^{\prime})E(z^{\prime})}$
(6)

where $t_{0}$ denotes present epoch. Using $E(z)$ vs $z$ profile as shown in
Fig. 1 we perform the above integration numerically to obtain $z$ dependence
of $t(z)$. Using Eq. (6) and the relation $1/a=1+z$, simultaneous values of
$a$ and $t$ at any given redshist $z$ are computed. This amounts to
elimination of $z$ from them to obtain the time profile of $a(t)$ from the
analysis of observational data. The obtained profile is shown in left panel of
Fig. 2 and this corresponds to the redshift range $0\leqslant z\leqslant 1$ or
to the equivalent $t$-range: $0.44\leqslant t(z)\leqslant 1$ as obtained from
Eq. (6). $t$ is also normalised to unity ($t_{0}=1$) at present epoch $z=0$.

Figure 2: Plot of $a$ (left panel), $\dot{a}$ (middle panel) and $\ddot{a}$
(right panel) vs $t$ corresponding to the central line of Fig. 1.

##  3 $k-$essence scalar field model of dark energy with field-independent
potential

The field theoretic models of dark energy involve suitably chosen Lagrangians
contributing to energy-momentum tensor in the Einstein’s field equations. In
this paper, we consider dark energy described by a homogeneous scalar field
$\phi$ whose dynamics is governed by a $k$-essence Lagrangian with a field-
independent (constant) potential. Such models involve actions with non-
canonical kinetic terms and have phenomenologically rich properties. Theories
with a non-canonical kinetic term was first proposed by Born and Infeld in
[71] to get rid of the infinite self-energy of the electron and were also
investigated by Heisenberg in [72] in the context of physics of cosmic rays
and meson productions and further developed by Dirac in [73]. Such non-
canonical Lagrangians have also been investigated in low energy effective
string theory and in D-branes models [74, 75, 76]. In cosmology, the kinetic
energy driven acceleration was first proposed in the context of cosmic
inflation [27] and also applied to explain the late time cosmic acceleration
i.e. dark energy, in [28, 29, 30, 31, 32, 33]. In this section, we present a
brief outline of the general framework and features of the $k-$essence model
that would be relevant in the context of this paper.

The dynamics of the $k-$essence scalar field $\phi$, which is minimally
coupled to the gravitational field $g_{\mu\nu}$ is driven by the action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{4}x\sqrt{-g}{\cal
L}(X,\phi)$ (7)

where $X\equiv\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, $g$ is
determinant of the metric $g_{\mu\nu}$ and $\nabla_{\mu}$ denotes covariant
derivative. Variation of the action with respect to the field $g_{\mu\nu}$
gives the energy momentum tensor for the $k-$essence field as

$\displaystyle T_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta
g^{\mu\nu}}=\frac{\partial{\cal L}}{\partial
X}\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}{\cal L}$ (8)

which when compared with the form of that of perfect fluid (characterised by
energy density $\tilde{\rho}$ and pressure $\tilde{p}$):
$T_{\mu\nu}=(\tilde{\rho}+\tilde{p})u_{\mu}u_{\nu}-\tilde{p}g_{\mu\nu}$ gives

$\displaystyle\tilde{p}={\cal
L}(\phi,X)\,\quad,\quad\tilde{\rho}=2X\frac{\partial\tilde{p}}{\partial
X}-\tilde{p}$ (9)

where $u_{\mu}$ is the effective four-velocity given by $u_{\mu}={\rm
sgn}(\partial_{0}\phi)\frac{\nabla_{\mu}\phi}{2X}$. We take the form of the
non-canonical Lagrangian in $k$-essence model as

$\displaystyle{\cal L}(\phi,X)=V(\phi)F(X)$ (10)

Motivations for consideration of such restricted form of the $k$-essence
Lagrangian density also comes from low energy effective string theory with a
suitable redefinition of the field discussed comprehensively in [77, 78, 79].
For such $k-$essence Lagrangians in Eq. (10), the pressure and energy density
as given in Eq. (9) may be written as

$\displaystyle\tilde{p}=V(\phi)F(X)\,\quad,\quad\tilde{\rho}=V(\phi)(2XF_{X}-F)$
(11)

where $F_{X}\equiv dF/dX$. In an expanding FRW universe, described by the FRW
scale factor $a(t)$, when the dark energy component does not interact with any
other component of the universe, the energy density and pressure due to this
component satisfies the continuity equation
$\dot{\tilde{\rho}}+3H(\tilde{\rho}+\tilde{p})=0$, which using with Eq. (11)
one obtains

$\displaystyle\Big{[}\frac{d}{dt}V(\phi)\Big{]}(2XF_{X}-F)+V(\phi)\Big{[}\frac{d}{dt}(2XF_{X}-F)\Big{]}+6V(\phi)HXF_{X}$
$\displaystyle=$ $\displaystyle 0$ (12)

Taking the $k$-essence field in the FRW spacetime background to be
homogeneous: $\phi(t,\vec{x})\equiv\phi(t)$, we have $X=(1/2)\dot{\phi}^{2}$,
$\Big{[}\frac{d}{dt}V(\phi)\Big{]}=\Big{[}\frac{d}{d\phi}V(\phi)\Big{]}\dot{\phi}$
and
$\Big{[}\frac{d}{dt}(2XF_{X}-F)\Big{]}=\Big{[}\frac{d}{dX}(2XF_{X}-F)\Big{]}\dot{\phi}\ddot{\phi}$.
From Eq. (12), we then have the equation of motion for the $k-$essence field
in the evolving in FRW background spacetime as

$\displaystyle(F_{X}+2XF_{XX})\ddot{\phi}+3HF_{X}\dot{\phi}+(2XF_{X}-F)\frac{V_{\phi}}{V}$
$\displaystyle=$ $\displaystyle 0$ (13)

where $F_{XX}=d^{2}F/dX^{2}$ and $V_{\phi}=dV/d\phi$.

A simple class of such $k-$essence models, called ‘purely’ kinetic $k$-essence
models was investigated in [1], where the Lagrangian involves only the kinetic
factor - a function of $X$, without having any explicit dependence on the
field $\phi$. This corresponds to $k-$essence models with Lagrangians given by
Eq. (10) with a constant potential term: $V(\phi)\equiv V_{0}$, a constant.
Such models have been shown to generate an exponential cosmic inflation but
remains unable to explain the graceful exit from the phase of inflation.
However, in the context of dark energy, these purely kinetic $k-$essence
models have been shown to generate the observed transition from the phase of
decelerated expansion to a phase of accelerated expansion, during late time
evolution of the universe.

For such models with constant $V(\phi)$, $V_{\phi}=0$ and using
$X=\dot{\phi}^{2}/2$, $\dot{X}=\dot{\phi}\ddot{\phi}$, the equation of motion
(13) for the $k-$essence field can be written as

$\displaystyle(F_{X}+2XF_{XX})\dot{X}+6HXF_{X}$ $\displaystyle=$
$\displaystyle 0$ (14)

Writing $H=\dot{a}/a$ and changing the independent variable from $t$ to the
scale factor $a$, the above equation can be expressed in a differential form
as

$\displaystyle d\ln(XF_{X}^{2})+d\ln a^{6}$ $\displaystyle=$ $\displaystyle 0$
(15)

which on integration gives a scaling relation:

$\displaystyle XF_{X}^{2}=Ca^{-6}$ (16)

where $C$ is the constant of integration. In the context of a field
independent potential, such a form of scaling relation holds only when the
dark energy component does not interact with dark matter and continuity
equation in each of these sectors are satisfied without any source term. In
the context of interacting DE-DM scenario, the Eq. (16) is no longer valid. In
the following sections, we obtained the modified scaling relation, in the
interacting DE-DM scenario for $k-$ essence model of Dark energy with a
constant potential.

##  4 Framework of interactive Dark energy - Dark matter model

In this section we describe the theoretical framework of interacting DE-DM
scenario in FRW flat spacetime background filled with an ideal fluid with its
components as DE and DM. We focus only on the late time era of cosmic
evolution where we neglect the contribution to energy density due to baryonic
matter and radiation, as supported by small estimated value of their combined
share $(\sim 4\%)$ in present universe based on measurements from WMAP [12]
and Planck [13] experiments. The conservation of energy momentum tensor for
the total dark fluid with interactions between its components may be expressed
through the non-conserving continuity equations,

$\displaystyle\Big{[}\dot{\rho}_{\rm dm}+3H{\rho}_{\rm dm}\Big{]}$
$\displaystyle=$ $\displaystyle Q(t)$ (17)
$\displaystyle\Big{[}\dot{\rho}_{\rm de}+3H(\rho_{\rm de}+p_{\rm de})\Big{]}$
$\displaystyle=$ $\displaystyle-Q(t)$ (18)

where, $\rho_{\rm de}$ and $\rho_{\rm dm}$ denote the instantaneous energy
densities of DE and DM components respectively and $p_{\rm de}$ is the
pressure of DE fluid. Dark matter is considered to be (non-relativistic)
pressure-less dust. The function $Q(t)$ in the source term of the above
equation gives a measure of instantaneous rate of energy transfer between DE
and DM components. Eqs. (17) and (18) imply

$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$ $\displaystyle 0$ (19)

which implies conservation of energy momentum tensor of the total dark fluid
with energy density $\rho=\rho_{\rm de}+\rho_{\rm dm}$ and pressure $p=p_{\rm
de}$. The dot ($\cdot$) represents derivatives with respect to the
dimensionless time parameter $t$ which is normalised to $t=1$ at present
epoch. As discussed in Sec. 2, the time profile of $a(t)$ has been extracted
from analysis of observational data over the late time domain
$0.44\raisebox{-3.69899pt}{\leavevmode\nobreak\ \shortstack{$<$ \\\ [-0.07cm]
$\sim$}}\leavevmode\nobreak\ t\leqslant 1$ accessible in SNe Ia observations.
We express the time dependence of the source term $Q(t)$ through the scale
factor $a(t)$ and a constant parameter $k$ as

$\displaystyle Q(t)=Q_{0}\big{[}a(t)\big{]}^{k}$ (20)

Where $Q_{0}$ is the value of $Q(t)$ at present epoch since we used the
normalisation $a(t)=1$ at present epoch. For convenience, we express time
dependences of various quantities in terms of a time parameter $\eta$ defined
as $\eta\equiv\ln a(t)$ ($\eta=0$ then corresponds to the present epoch and
$a=e^{\eta}$). Eqs. (17) and (18) then take the form

$\displaystyle\rho_{\rm dm}^{\prime}+3\rho_{\rm dm}$ $\displaystyle=$
$\displaystyle Q_{0}\frac{e^{k\eta}}{H}$ (21) $\displaystyle\rho_{\rm
de}^{\prime}+3(\rho_{\rm de}+p_{\rm de})$ $\displaystyle=$ $\displaystyle-
Q_{0}\frac{e^{k\eta}}{H}$ (22)

where all the time-dependent quantities involved, are regarded as functions of
$\eta$ and ′ denotes derivative with respect to $\eta$. Multiplying both sides
of Eq. (21) by $e^{3\eta}$ and writing the right hand side as a total
derivative we get

$\displaystyle\frac{d}{d\eta}\Big{[}e^{3\eta}\rho_{\rm dm}\Big{]}$
$\displaystyle=$ $\displaystyle Q_{0}\frac{e^{(k+3)\eta}}{H}$ (23)

which on integration between limits $\eta=0$ and $\eta=\eta$ gives

$\displaystyle\rho_{\rm dm}(\eta)$ $\displaystyle=$ $\displaystyle
e^{-3\eta}\left[\rho_{\rm dm}^{0}+Q_{0}\int_{0}^{\eta}\frac{d\eta_{1}\
e^{(k+3)\eta_{1}}}{H(\eta_{1})}\right]\,$ (24)

where, $\rho_{\rm dm}^{0}$ denotes the DM density at present epoch. Note that,
in absence of any interaction between DE and DM ($Q_{0}=0$), we get $\rho_{\rm
dm}(\eta)=e^{-3\eta}\rho_{\rm dm}^{0}=a^{-3}\rho_{\rm dm}^{0}$, which is as
expected from $\Lambda$-CDM model. Dividing both sides by the total dark fluid
density at present epoch, $(\rho_{\rm de}^{0}+\rho_{\rm dm}^{0})$, we have

$\displaystyle\frac{\rho_{\rm
dm}(\eta;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{de}^{0})}$ $\displaystyle=$
$\displaystyle e^{-3\eta}\left[\Omega_{\rm
dm}^{0}+\beta_{0}\int_{0}^{\eta}\frac{d\eta_{1}\
e^{(k+3)\eta_{1}}}{H(\eta_{1})}\right]$ (25)

Here, $\beta_{0}\equiv Q_{0}/(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})$ and
$\Omega_{\rm dm}^{0}(\equiv\rho_{\rm dm}^{0}/(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0}))$ denotes fractional content of DM in the present universe whose
value is $\sim 0.268$ from WMAP and PLANCK observations [12, 13]. $k$ and
$\beta_{0}$ are also put in the argument of $\rho_{\rm dm}$ to emphasize the
point that temporal behaviour of the DM energy density, in this scenario,
depends on values of these parameters.

The time profile of the Hubble parameter $H(t)=\dot{a}/a$ may be obtained from
the temporal profile of $a$ and $\dot{a}$ as shown in Fig. 2. Numerically
eliminating $t$ from the $a(t)$ and $H(t)$ we may express the Hubble parameter
as a function of $a$ or subsequently as a function of $\eta$ as $H(\eta)$
which appear in the right hand side of Eq. (25). We find that the quantity
$1/H(\eta)$, thus obtained, corresponding to the central line in Fig. 1, may
be fitted with a polynomial of order 3, which we express as

$\displaystyle\frac{1}{H(\eta)}=\sum_{m=0}^{3}A_{m}\eta^{m}\quad\mbox{where,
}A_{0}=1,A_{1}=0.45,A_{2}=-0.32,A_{3}=-0.15.$ (26)

Using Eq. (26) in the integral appearing in right hand side of Eq. (25) we
have

$\displaystyle\frac{\rho_{\rm
dm}(\eta;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{de}^{0})}$ $\displaystyle=$
$\displaystyle e^{-3\eta}\left[\Omega_{\rm
dm}^{0}+\beta_{0}\sum_{m=0}^{3}A_{m}I_{k,m}(\eta)\right]$ (27)

where

$\displaystyle I_{k,m}(\eta)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\eta}d\eta_{1}\ e^{(k+3)\eta_{1}}{(\eta_{1})}^{m}$
(28)

For $k\neq-3$, the function $I_{k,m}(\eta)$ satisfies the recursion relation

$\displaystyle I_{k,m}(\eta)$ $\displaystyle=$
$\displaystyle\frac{\eta^{m}e^{(k+3)\eta}}{k+3}-\frac{m}{k+3}I_{k,m-1}(\eta)$
$\displaystyle\mbox{with}\quad
I_{k,0}(\eta)=\int_{0}^{\eta}d\eta_{1}e^{(k+3)\eta_{1}}=\frac{e^{(k+3)\eta}}{k+3}$

However, for $k=-3$, the exponential term in Eq. (28) becomes unity and we can
easily compute the integral as

$\displaystyle I_{-3,m}(\eta)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\eta}d\eta_{1}\
{(\eta_{1})}^{m}=\frac{\eta^{m+1}}{m+1}\,$ (30)

where $m$ can take 4 values viz. 0, 1, 2, 3 as evident from Eq. (26). Using
Eq. (4) or (30), according as $k\neq-3$ or $k=-3$, we can write the term
$\displaystyle\sum_{m=0}^{3}A_{m}I_{k,m}(\eta)$ appearing in right hand side
of Eq. (27) as

$\displaystyle\sum_{m=0}^{3}A_{m}I_{k,m}(\eta)$ $\displaystyle=$
$\displaystyle\begin{cases}(B_{k0}+B_{k1}\eta+B_{k2}\eta^{2}+B_{k3}\eta^{3})e^{(k+3)\eta}\quad\mbox{for
}k\neq-3\\\
A_{0}\eta+\frac{A_{1}\eta^{2}}{2}+\frac{A_{2}\eta^{3}}{3}+\frac{A_{3}\eta^{4}}{4}\quad\mbox{for
}k=-3\end{cases}$ (31)

where, the constants $B_{ki}$’s can be expressed as

$\displaystyle B_{k0}$ $\displaystyle=$
$\displaystyle\left(\frac{A_{0}}{k+3}-\frac{A_{1}}{(k+3)^{2}}+\frac{2A_{2}}{(k+3)^{3}}-\frac{6A_{3}}{(k+3)^{4}}\right)$
$\displaystyle B_{k1}$ $\displaystyle=$
$\displaystyle\left(\frac{A_{1}}{k+3}-\frac{2A_{2}}{(k+3)^{2}}+\frac{6A_{3}}{(k+3)^{3}}\right)$
$\displaystyle B_{k2}$ $\displaystyle=$
$\displaystyle\left(\frac{A_{2}}{k+3}-\frac{3A_{3}}{(k+3)^{2}}\right)$
$\displaystyle B_{k3}$ $\displaystyle=$
$\displaystyle\left(\frac{A_{3}}{k+3}\right)$

With the aid of Eq. (31) the $\eta$-dependence appearing in right hand side of
Eq. (27) can be expressed in an algebraic form in terms of parameters $k$ and
$\beta_{0}$, with known values of all other factors involved in the
expression, e.g $A_{i}$’s and $\Omega_{\rm dm}^{0}\sim 0.268$. This allows us
to numerically compute the time($\eta$)-profile of DM energy density term for
any chosen benchmark values of the parameters: $k$ and $\beta_{0}$. Note that,
determination of values of constants $A_{i}$’s (and so also $B_{ki}$’s for any
given $k$) uses time profile of scale factor as extracted from analysis of SNe
Ia data. The obtained $\eta$-dependence of DM energy density term in DE-DM
interacting scenario is, therefore, consistent with the Sne Ia data. The
features of the data are encoded in the corresponding expression through the
constants $A_{i}$’s. We may now also use the substitution $\eta=\ln a$ in Eqs.
(27) and (31) to express the time dependence in terms of scale factor $a$
itself as

$\displaystyle\frac{\rho_{\rm
dm}(a;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{de}^{0})}$ $\displaystyle=$
$\displaystyle a^{-3}\left[\Omega_{\rm
dm}^{0}+\beta_{0}\sum_{m=0}^{3}A_{m}I_{k,m}(\ln a)\right]$ (32)

The total energy density $\rho=\rho_{\rm dm}+\rho_{\rm de}$ and pressure
$p=p_{\rm de}$ of the dark fluid, on the other hand, are independent of the
parameters $\beta_{0}$ and $k$, as the continuity equation of the total dark
fluid involves no source term. We can obtain their temporal behaviour,
directly from the time profile of the scale factor obtained from the analysis
discussed in Sec. 2. To see this, we write the Friedmann equations governing
late-time cosmic evolution, which in a flat FRW spacetime background with DE
and DM as its primary contents take the forms

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\kappa^{2}}{3}(\rho_{\rm
dm}+\rho_{\rm de})$ (33) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}}{6}\Big{[}(\rho_{\rm dm}+\rho_{\rm
de})+3p_{\rm de}\Big{]}$ (34)

where $\kappa^{2}\equiv 8\pi G$ ($G$ is the Newton’s Gravitational constant).
Using Eqs. (33) and (34), we may express the equation of state $w$ of the
total dark fluid in terms of the scale factor and its time derivatives as

$\displaystyle w$ $\displaystyle\equiv$ $\displaystyle\frac{p_{\rm
de}}{\rho_{\rm de}+\rho_{\rm
dm}}=-\frac{2}{3}\frac{a\ddot{a}}{\dot{a}^{2}}-\frac{1}{3}$ (35)

From the obtained time profile of scale factor $a(t)$ and its derivatives as
shown in Fig. 2 we can obtain $t-$dependence of $w(t)$. Numerically
eliminating $t$ from the $a(t)$ and $w(t)$ we may express the equation of
state parameter $w$ as a function of $a$ or $\eta$, making use of the
substitution $\eta=\ln a$. We find that, $w(\eta)$ thus obtained,
corresponding to the central line in Fig. 1, may be fitted with a polynomial
of order 5. This we express as

$\displaystyle w(\eta)$ $\displaystyle=$
$\displaystyle-1+\sum_{i=0}W_{i}\eta^{i}$ (36)

with values of the coefficients $W_{i}$ at best-fit are given by

$\displaystyle W_{0}=-0.70\,,W_{1}=-0.61\,,W_{2}=-0.49\,,W_{3}=-2.29\,,$
$\displaystyle W_{4}=-2.81\,,W_{5}=-0.92\,,\mbox{ and }W_{i}=0\,\mbox{ for
}i>5$ (37)

In terms of the parameter $\eta$ the continuity equation (19) for the total
dark fluid takes the form

$\displaystyle\frac{d}{d\eta}\ln\Big{(}\rho_{\rm dm}+\rho_{\rm de}\Big{)}$
$\displaystyle=$ $\displaystyle-3\Big{(}1+w(\eta)\Big{)}$ (38)

which on integration between the limits $\eta=0$ and $\eta=\eta$ gives

$\displaystyle\frac{(\rho_{\rm de}+\rho_{\rm dm})_{\eta}}{(\rho_{\rm
de}^{0}+\rho_{\rm dm})_{0}}$ $\displaystyle=$
$\displaystyle\exp\left[-3\int_{0}^{\eta}\big{(}1+w(\eta_{1})\big{)}d\eta_{1}\right]$
(39)

The integral within exponent can be performed by using Eq. (36), with
$W_{i}$’s as given in Eq. (37). This gives

$\displaystyle\frac{\rho_{\rm de}(\eta)+\rho_{\rm dm}(\eta)}{(\rho_{\rm
de}^{0}+\rho_{\rm dm}^{0})}$ $\displaystyle=$
$\displaystyle\exp\left[-3\int_{0}^{\eta}\left(\sum_{i=0}^{5}W_{i}\eta_{1}^{i}\right)d\eta_{1}\right]=\exp\left[-3\left(\sum_{i=0}^{5}\frac{W_{i}\eta^{i+1}}{i+1}\right)\right]$
(40)

We may again use the substitution $\eta=\ln a$ in the above to express the
energy density of the total dark fluid $\rho=\rho_{\rm de}+\rho_{\rm dm}$ as a
function of scale factor $a$ as,

$\displaystyle\frac{\rho(a)}{\rho_{\rm de}^{0}+\rho_{\rm dm}^{0}}$
$\displaystyle=$ $\displaystyle\exp\left[-3\left(\sum_{i=0}^{5}\frac{W_{i}(\ln
a)^{i+1}}{i+1}\right)\right]$ (41)

We find that this scale factor dependence of the total energy density, thus
obtained, can be fitted best with a fourth order polynomial expressed as,

$\displaystyle\frac{\rho(a)}{\rho_{\rm de}^{0}+\rho_{\rm dm}^{0}}$
$\displaystyle=$ $\displaystyle\sum_{m=0}^{4}R_{m}a^{m}$ (42)

with the best-fit values of coefficients as

$\displaystyle R_{0}=29.3,\quad R_{1}=-120.3,\quad R_{2}=199.96,\quad
R_{3}=-151.75,\quad R_{4}=43.8$ (43)

Note that, the constants $W_{i}$’s and $R_{i}$’s encode the inputs from the
observational data used here.

Similarly we may also obtain temporal behaviour of the pressure ($p_{\rm de}$)
of DE. Since dark matter dust has zero pressure, the equation of state
parameter of the dark fluid can be written as

$\displaystyle w(\eta)$ $\displaystyle=$ $\displaystyle\frac{p_{\rm
de}(\eta)}{\rho_{\rm de}(\eta)+\rho_{\rm dm}(\eta)}\,,$ (44)

from which we may write

$\displaystyle\frac{p_{\rm de}(\eta)}{\rho_{\rm de}^{0}+\rho_{\rm dm}^{0}}$
$\displaystyle=$ $\displaystyle w(\eta)\frac{\rho_{\rm de}(\eta)+\rho_{\rm
dm}(\eta)}{\rho_{\rm de}^{0}+\rho_{\rm dm}^{0}}$ (45)

Using Eqs. (36) and (40) in Eq. (45) and by making the substitution $\eta=\ln
a$ we may express this functional behaviour of the $p_{\rm de}$ in terms of
$a$ as

$\displaystyle\frac{p_{\rm de}(a)}{\rho_{\rm de}^{0}+\rho_{\rm dm}^{0}}$
$\displaystyle=$ $\displaystyle\Big{[}-1+\sum_{i=0}^{5}W_{i}(\ln
a)^{i}\Big{]}\cdot\exp\left[-3\left(\sum_{i=0}^{5}\frac{W_{i}(\ln
a)^{i+1}}{i+1}\right)\right]$ (46)

We find that, this scale factor dependence of $p_{\rm de}$ can be fitted best
with a polynomial of order 6 expressed as,

$\displaystyle\frac{p_{\rm de}(a)}{\rho_{\rm de}^{0}+\rho_{\rm dm}^{0}}$
$\displaystyle=$ $\displaystyle\sum_{m=0}^{6}P_{m}a^{m}$ (47)

with the best-fit values of coefficients as

$\displaystyle\quad\quad\quad\quad P_{0}=-11.84,\quad P_{1}=90.04,\quad
P_{2}=-297.92,$ $\displaystyle P_{3}=515.97,\quad P_{4}=-495.22,\quad
P_{5}=250.83,\quad P_{6}=-52.57$ (48)

The scale factor dependence of $\rho(a)$ and $p_{\rm de}(a)$, thus obtained
using the temporal profile of $a(t)$ from SNe Ia data, are shown in left panel
and right panel of Fig. 3 respectively.

Figure 3: Temporal behaviour of cosmological parameters obtained from SNe Ia
observation (Left panel: Total energy density as a function of scale factor,
Right panel: Pressure of dark sector as a function of scale factor).

Note that, the density of DE as a function of the scale factor $a$ may now be
expressed as

$\displaystyle\frac{\rho_{\rm
de}(a;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{de}^{0})}$ $\displaystyle=$
$\displaystyle\frac{\rho(a)}{(\rho_{dm}^{0}+\rho_{de}^{0})}-\frac{\rho_{\rm
dm}(a;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{de}^{0})}\,,$ (49)

which involves the parameters $\beta_{0}$ and $k$, due to their appearance in
$\rho_{\rm dm}(a;k,\beta_{0})$. Since energy density is always a positive
quantity and the scale factor or time dependence of energy density of total
dark fluid has already been obtained directly from observation (left panel of
Fig. 3), the estimated profile of the dark matter density $\rho_{\rm
dm}(a,k,\beta_{0})$ computed from Eq. (25) for given values of $k$ and
$\beta_{0}$ are subject to the constraint

$\displaystyle 0<\frac{\rho_{\rm dm}(a;k,\beta_{0})}{(\rho_{\rm
de}^{0}+\rho_{\rm dm}^{0})}<\frac{\rho(a)}{(\rho_{\rm de}^{0}+\rho_{\rm
dm}^{0})}$ (50)

for the accessible domain of $a$ ($0.5\lesssim a<1$) in SNe Ia observations.
Using Eqs. (32) and (42), we may express the above condition as

$\displaystyle 0<a^{-3}\left[\Omega_{\rm
dm}^{0}+\beta_{0}\sum_{m=0}^{3}A_{m}I_{k,m}(\ln
a)\right]<\displaystyle\sum_{m=0}^{4}R_{m}a^{m}$ (51)

where the function $I_{k,m}(\ln a)$ contains the parameter $k$ as seen from
its explicit form in Eq. (31) with $\eta=\ln a$. Using the form of
$I_{k,m}(\ln a)$ for $k\neq-3$, we find the range in the parameter space
spanned by $k$ and $\beta_{0}$, every point ($k,\beta_{0}$) of which satisfies
the condition in Eq. (51). This allowed region in parameter space has been
shown by the shaded region in Fig. 4. However, for $k=-3$ (when
$Q(t)=Q_{0}a^{-3})$, we find the range of $\beta_{0}$ for which the condition
in Eq. (51) is satisfied, is $-0.2\leqslant\beta_{0}\leqslant 0.42$.

Figure 4: Range of parameter space of model parameters $k$ vs. $\beta_{0}$ for
which the condition in Eq. (51) is satisfied. A black line is drawn at
$\beta_{0}=0$ to reflect the fact that for $\beta_{0}=0$, the parameter $k$
loses its relevance in the context (described in text). Figure 5: Variation of
energy density of dark energy and dark matter with scale factor for chosen
benchmark values of the parameters ($k$ and $\beta_{0}$).

For some benchmark values of $k$ and $\beta_{0}$ chosen within this allowed
region we have also depicted the variation of the dark energy and dark matter
densities with the scale factor in Fig. 5. Note from Eq. (20) that for
$\beta_{0}=0=Q_{0}$, the parameter $k$ becomes irrelevant and corresponding
continuity equation for DM (Eq. (17)) has no source term ($Q(t)=0$). This
implies $\rho_{\rm dm}(a)=\rho_{\rm dm}^{0}a^{-3}$ with $\rho_{\rm
de}(a)=\rho(a)-\rho_{\rm dm}^{0}a^{-3}$, where $\rho(a)$ as directly obtained
from observation has been expressed thorough a fitted polynomial in Eq. (42).
These profiles for $\beta_{0}=0$ (which corresponds to ‘non interacting DE-DM’
scenario) are presented in left panel of Fig. 5, and we find that DM energy
density falling as $\sim 1/a^{3}$ falls below dark energy density at an epoch
corresponding to $a\sim 0.71$. For $\beta_{0}=0.5$ and $k=1$ (middle panel of
Fig. 5) the source term $Q(t)$ grows with time having the same profile as that
$a(t)$ itself, and the corresponding DM and DE density profiles evaluated
using Eqs. (32) and (49) show that DM energy density falls below that of DE at
an epoch when $a\sim 0.75$. For $\beta_{0}=0.1$ and $k=-3$ the source term
$Q(t)$ falls as $\sim 1/a(t)^{3}$ as the universe expands, and the
corresponding plots presented in right panel of Fig. 5 show beginning of
dominance of DE density over that of DM at a relatively earlier epoch marked
by $a\sim 0.63$.

##  5 $k$-essence model with constant potential and modification of scaling
relation in presence of DE-DM interaction

We assume dynamics of dark energy of universe to be driven by a homogeneous
scalar field $\phi=\phi(t)$ governed by non-canonical $k-$essence Lagrangian
of the form $L=V(\phi)F(X)$ where the potential $V(\phi)=V_{0}$ is considered
to be constant and the dynamical term $F(X)$ is a function of
$X\equiv(1/2)g_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi=(1/2)\dot{\phi}^{2}$.
The energy density and pressure of DE are identified with the corresponding
quantities in the context of the $k$-essence model which may be expressed as

$\displaystyle\rho_{\rm de}$ $\displaystyle=$ $\displaystyle V_{0}(2XF_{X}-F)$
(52) $\displaystyle p_{\rm de}$ $\displaystyle=$ $\displaystyle V_{0}F(X)$
(53)

where, $F_{X}\equiv dF/dX$. In the context of this constant potential
$k-$essence model, it has been shown in [1, 2] that, in the non-interacting
DE-DM scenario when the energy conservation is separately conserved in the DE
sector (and also in DM sector) implying corresponding continuity equation (18)
being satisfied with $Q(t)=0$, we have a scaling relation of the form
$XF_{X}^{2}=Ca^{-6}$ where $C$ is a constant. In this section we show that,
even in presence of time-dependent interactions implying continual exchange of
energy between the two sectors ($Q(t)\neq 0$), the constancy of $k$-essence
potential may lead to a modified form of the scaling. Below we discuss how we
obtain the modified form of scaling taking into consideration observational
inputs from SNe Ia data.

To obtain this we use Eqs. (52) and (53) in Eq. (18) with $Q(t)$ parametrised
as $Q(t)=Q_{0}[a(t)]^{k}$ (Eq. (20)) to get

$\displaystyle\frac{d}{dt}\Big{[}V_{0}(2XF_{X}-F)\Big{]}+3H(2V_{0}XF_{X})$
$\displaystyle=$ $\displaystyle-Q_{0}a^{k}\,.$ (54)

Writing $\displaystyle\frac{d}{dt}=\dot{a}\frac{d}{da}$ and $H=\dot{a}/a$,
after some rearrangements the above equation takes the form

$\displaystyle d\Big{[}\ln\\{a^{6}(XF_{X}^{2})\\}\Big{]}$ $\displaystyle=$
$\displaystyle-\frac{Q_{0}}{V_{0}}\frac{a^{k-1}}{H}\frac{da}{(XF_{X})}$ (55)

where $F_{XX}=d^{2}F/dX^{2}$ and the quantities $H$, $X$, $F$ and its
derivatives are regarded as functions of scale factor $a$. Integrating both
sides of Eq. (55) between the limits $a=1$ (present epoch) and $a=a$ we have,

$\displaystyle\Big{[}\ln\\{a^{6}(XF_{X}^{2})\\}\Big{]}-\Big{[}\ln\\{(XF_{X}^{2})\\}\Big{]}_{a=1}=-\frac{Q_{0}}{V_{0}}\int_{1}^{a}\frac{f(a^{\prime})da^{\prime}}{(XF_{X})}$
or, $\displaystyle\quad\quad
XF_{X}^{2}=Ca^{-6}\exp\left(-\frac{Q_{0}}{V_{0}}\int_{1}^{a}\frac{f(a^{\prime};k)da^{\prime}}{(XF_{X})}\right)$
(56)

where, $C\equiv XF_{X}^{2}\big{|}_{a=1}$ is a constant and
$f(a;k)\equiv\frac{a^{k-1}}{H(a)}$. In Eq. (26) we have shown how $H^{-1}$
depends on $\eta=\ln a$, with the help of a polynomial of $\ln a$, with given
of values of coefficients $A_{i}$’s encoding the observational inputs
extracted from the analysis of JLA data. We may, equivalently, fit this scale
factor dependence of $H^{-1}$ by a polynomial in $a$ (instead of $\ln a$) and
found that, the corresponding best-fit curve may be given by a second order
polynomial of scale factor $a$ as

$\displaystyle\frac{1}{H(a)}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{2}D_{n}a^{n}\quad\mbox{with
}D_{0}=-0.272\,,D_{1}=2.174\,,D_{2}=-0.9.$ (57)

To perform the integration involved in Eq. (56), we can then express the
function $f(a,k)$ as a polynomial

$\displaystyle f(a;k)$ $\displaystyle=$ $\displaystyle
a^{k-1}(D_{0}+D_{1}a+D_{2}a^{2})\,,$

and we also need to express $(XF_{X})$ as function of $a$. To do so, we
eliminate $F(X)$ from Eqs. (52) and (53) and write

$\displaystyle XF_{X}$ $\displaystyle=$
$\displaystyle\frac{1}{2V_{0}}\Big{[}\rho_{\rm de}+p_{\rm de}\Big{]}$ (58)

Using Eq. (49) and writing all variables and parameters explicitly, we may
write the above equation as

$\displaystyle XF_{X}$ $\displaystyle=$ $\displaystyle\frac{(\rho_{\rm
dm}^{0}+\rho_{\rm de}^{0})}{2V_{0}}\Bigg{[}\frac{\rho(a)}{(\rho_{\rm
dm}^{0}+\rho_{\rm de}^{0})}-\frac{\rho_{\rm
dm}(a;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{\rm de}^{0})}+\frac{p_{\rm
de}(a)}{(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})}\Bigg{]}$ (59) $\displaystyle=$
$\displaystyle\frac{(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})}{2V_{0}}g(a;k,\beta_{0})$

where

$\displaystyle g(a;k,\beta_{0})$ $\displaystyle\equiv$
$\displaystyle\Bigg{[}\frac{\rho(a)}{(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})}-\frac{\rho_{\rm dm}(a;k,\beta_{0})}{(\rho_{dm}^{0}+\rho_{\rm
de}^{0})}+\frac{p_{\rm de}(a)}{(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})}\Bigg{]}$
(60)

Finally, using Eq. (59) in Eq. (56) we obtain

$\displaystyle\frac{XF_{X}^{2}}{Ca^{-6}}$ $\displaystyle=$
$\displaystyle\exp\left(-2\beta_{0}\int_{1}^{a}\frac{f(a_{1};k)da_{1}}{g(a_{1};k,\beta_{0})}\right)$
(61)

All the three terms in the right hand side of Eq. (60) have been expressed in
algebraic form in Eqs. (32), (42) and (47), using which we can numerically
compute the function $g(a;k,\beta_{0})$ for any input values of
$a,k,\beta_{0}$. With this and using the form of the function $f(a;k)$ in Eq.
(5), we may numerically evaluate the integral within the exponent appearing in
Eq. (61). This is the modified scaling relation arising out of the constancy
of the potential $V=V_{0}$ of $k$-essence model of dark energy, in presence of
interaction between DE and DM parametrised in terms of $\beta_{0}$ and $k$.
The inputs from observational data are encoded in the form of the functions
$f(a;k)$ and $g(a;k,\beta_{0})$ through the various coefficients $D_{i}$’s,
$A_{i}$’s, $P_{i}$’s, $R_{i}$’s etc. introduced in Secs. 4 and 5, while
establishing connections with the temporal profile of the scale factor
obtained from the SNe Ia data in Sec. 2. In the context of this constant
potential $k-$essence model of dark energy in interacting DE-DM scenario, the
modified scaling relation (61) establishes a connection between the dynamical
terms $X$, $F(X)$ involved in the $k-$essence Lagrangian and the scale factor
$a(t)$ of FRW universe along with the parameters $k$ and $\beta_{0}$. The
constancy of the $k$-essence potential is instrumental in establishing the
relation.

Note that, in absence of any interaction ($\beta_{0}=Q_{0}/(\rho_{\rm
dm}^{0}+\rho_{\rm de}^{0})=0$), the exponential term in Eq. (61) becomes unity
and the modified scaling relation reduces to the usual form
$XF_{X}^{2}=Ca^{-6}$ as obtained in [1, 2]. Therefore, the deviation from
unity, of the quantity
$\exp\left(-2\beta_{0}\int_{1}^{a}\frac{f(a_{1};k)da_{1}}{g(a_{1};k,\beta_{0})}\right)$
in right hand side of (Eq. (61) evaluated for any parameter set
($k,\beta_{0}$) gives the extent of modification in the scaling relation due
to presence of (time-dependent) interaction between DE and DM parametrised in
terms of ($k,\beta_{0}$). The behaviour of modifications in the scaling
relation are shown in three panels Fig. 6 for three chosen benchmark values of
$k$ viz. $k=0$ (left panel), $k=1$ (middle panel) and $k=-3$ (right panel). In
each of the panels, corresponding to a given value of $k$, we have plotted the
quantity $XF_{X}^{2}/Ca^{-6}$ as a function of the scale factor, for few
different values of the parameter $\beta_{0}$ chosen from the corresponding
allowed domain of $k-\beta_{0}$ parameter space discussed (and also shown in
Fig. 4) in the end of Sec. 4. The benchmark cases: $k=0$, $k=1$ and $k=-3$
respectively correspond to $Q(t)$ = constant, $Q(t)=Q_{0}a(t)$ and
$Q(t)=Q_{0}/[a(t)]^{3}$ in the non-conserving continuity equations (17) and
(18) of DM and DE sectors. The plots shown in Fig. 6 show exponential
behaviour for $\beta_{0}\neq 0$ as evident from Eq. (61). Since we have
assigned the value $(XF_{X}^{2})\Big{|}_{a=1}$ to the constant $C$ the plots
of $XF_{X}^{2}/Ca^{-6}$ approach unity as the scale factor approaches to its
(normalised) value $a=1$ (at present epoch). The $\beta_{0}=0$ case,
corresponds to non-interacting DE-DM scenario and is represented by the
$XF_{X}^{2}/Ca^{-6}=1$ line in Fig. 6.

Figure 6: Plot of $XF_{X}^{2}/Ca^{-6}$ as a function of scale factor ($a$) for
some benchmark values of the model parameters $k$ and $\beta_{0}$. The plots
within the small boxes drawn on the top of each panel depict the same plots of
the panel in an appropriately zoomed domain $0.7<a<1$ to bring out a better
resolution of each of the distinct curves corresponding to different
$\beta_{0}$ values in that domain.

##  6 Conclusion

In this paper we considered a scenario of interacting dark matter and dark
energy during the late time evolution of cosmic evolution, neglecting the
contribution due to radiation and baryonic matter to the total energy density.
We describe the dynamics of DE to be driven by a homogeneous $k-$essence field
($\phi$) driven by a non-canonical Lagrangian of the form $L=V(\phi)F(X)$ with
a constant potential $V(\phi)=V_{0}$, where the dynamical term $F(X)$ is a
function of $X=\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi=\dot{\phi^{2}}$
(for homogeneous field). Under such considerations, we showed existence of a
scaling relation in the theory, which connects the dynamical quantities
$X,dF/dX$ (i.e. $\dot{\phi}$) to the FRW scale factor $a(t)$ of the universe
along with two relevant parameters ($\beta_{0}$ and $k$) of the model. The
time-dependent interaction between DM and DE has been incorporated through a
function $Q(t)$ in the continuity equations ((17) and (18)) of the two fluids.
The source term $Q(t)$ is parametrised in terms of the dimensionless
parameters $\beta_{0}$ and $k$ as $Q(t)=Q_{0}[a(t)]^{k}$ where
$Q_{0}\equiv\beta_{0}(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})$, $\rho_{\rm
dm}^{0}$ and $\rho_{\rm de}^{0}$ being the present-day observed energy
densities of the dark matter and dark energy respectively. The constancy of
the $k-$essence potential is instrumental in proving the scaling relation (Eq.
(61)) in interacting DE-DM scenario. For $\beta_{0}=0=Q(t)$, the obtained
scaling relation reduces to the usual scaling relation $XF_{X}^{2}=Ca^{-6}$
obtained in [1, 2] for non-interacting DE-DM scenario.

We have expressed the modification to the usual scaling relation due to the
effect of DE-DM interaction in terms of an exponential term of form:
$\exp\left(-2\beta_{0}\int_{1}^{a}\frac{f(a_{1};k)da_{1}}{g(a_{1};k,\beta_{0})}\right)$
(see Eq. (61)). In obtaining such a form we have taken into consideration the
observed feature of temporal behaviour of the FRW scale factor $a(t)$, probed
in the measurement of luminosity distance and redshift of SNe Ia events. This
key observational ingredient enters into the above exponential form at
different levels of its derivation, through various constants which finally
got twined together in the obtained expressions for the functions $f(a;k)$ and
$g(a_{1};k,\beta_{0})$. The modified scaling expressed in Eq. (61), in the
form of the exponential function, thus encodes in it the features of the SNe
Ia data. The values of parameters $\beta_{0}$ and $k$, involved in the
scaling, are also restricted from observed data. This has been imposed by the
condition $0<\rho_{\rm dm}(a;k,\beta_{0})<\rho(a)$, where $\rho_{\rm
dm}(a;k,\beta_{0})$ is the dark matter density at an epoch corresponding to
scale factor value $a$ in the interacting DE-DM scenario and is expressed by
Eq. (32). $\rho(a)$ is the profile of energy density of the total dark fluid
extracted from observation and expressed thorough a fitted polynomial in Eq.
(42). This constraint puts a bound in the $k-\beta_{0}$ parameter space as
shown in Fig. 4. We also observe that, the values of parameters $\beta_{0}$
and $k$ which determines the time dependence of the source term $Q(t)$
responsible for DE-DM interactions, decide the epoch in the past where density
of DE starts dominating over that of DM. This has been demonstrated in Fig. 5.

This modified form of scaling relation in Eq. (61), obtained in the context of
DE-DM interacting DE-DM scenario, may also be used to eliminate $F_{X}$ from
Eqs. (59), to obtain $X$ in terms of the functions $f(a;k)$ and
$g(a_{1};k,\beta_{0})$, along with the parameters $k$ and $\beta_{0}$. Since
for homogeneous field $\phi$, we have $X=\frac{1}{2}\dot{\phi}^{2}$, one may
thus obtain the scale factor dependence or the temporal profile of the
$k-$essence scalar field in the context of DE-DM interacting scenario using
the modified scaling.

Finally, we also note that, for a given set of values of the parameters $k$
and $\beta_{0}$, eliminating $F_{X}$ from Eq. (58) and the modified scaling
relation (61), we may obtain the scale factor dependence of the term $X$ as

$\displaystyle X$ $\displaystyle=$ $\displaystyle\left[\frac{(\rho_{\rm
dm}^{0}+\rho_{\rm
de}^{0})^{2}}{4CV_{0}^{2}}\right]a^{6}\Big{[}g(a;k,\beta_{0})\Big{]}^{2}\exp\left(2\beta_{0}\int_{1}^{a}\frac{f(a^{\prime};k)da^{\prime}}{g(a^{\prime};k,\beta_{0})}\right)\,.$
(62)

Up to the constant multiplicative factor $\frac{(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})^{2}}{4CV_{0}^{2}}$, the right hand side of the Eq. (62) is
numerically computable at different values of the scale factor $a$, with the
knowledge of the functions $g(a;k,\beta_{0})$ and $f(a;k)$ as extracted from
observation, for any given choice of values of the parameters $\beta_{0}$ and
$k$. Also computing $F(X)=p_{\rm de}(a)/V_{0}$ from Eq. (53), upto the factor
$1/V_{0}$, at different values of the scale factor $a$, we may eliminate $a$
from these two dependencies to obtain $V_{0}F(X)$ as a function of
$\frac{4CV_{0}^{2}}{(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})^{2}}X$. Such a
dependence, as extracted from observation, using the modified scaling in the
constant potential $k-$essence model contains the information in the form of
$F(X)$ for given choices of the parameters $\beta_{0}$ and $k$ in the context
of interacting DE-DM scenario. A given choice of the constant $C$ (appearing
in scaling relation), the value of the constant potential $V_{0}$ and the
total density of dark matter and dark energy $(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})$ would uniquely determine the form of the function $F(X)$.

### Acknowledgement

We thank the honourable referee for valuable suggestions. A.C. would like to
thank Indian Institute of Technology, Kanpur for supporting this work by means
of Institute Post-Doctoral Fellowship (Ref.No.DF/PDF197/2020-IITK/970).

### Data Availability Statement

No Data associated in the manuscript.

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