A Kundu,1, S D Pathak,1, V K Ojha,21Department of Physics, Lovely Professional University, Phagwara, Punjab, 144 411, India 2School of Sciences, SAGE University Bhopal, Madhya Pradesh, 462 022, India
First author contact:*Author to whom any correspondence should be addressed Received:2020-06-24Revised:2020-11-12Accepted:2020-11-13Online:2021-01-25
Abstract We discuss the coupling between dark energy and matter by considering a homogeneous tachyonic scalar field as a candidate for dark energy. We obtained the functional form of scale factor by assuming that the coupling strength depends linearly on the Hubble parameter and energy density. We also estimated the cosmic age of the Universe for different values of coupling constant. Keywords：dark energy;dark matter;scalar field;tachyonic scalar field
In the year 1998, type Ia Supernova observation [1, 2] revealed the accelerated expansion of the Universe. To explain the observed accelerated expansion of the Universe, the idea of dark energy has been introduced. One of the remarkable features of dark energy is its negative pressure which results in the negative equation of state (ωde). The observed cosmic acceleration is possible only for ωde<−1/3. The cosmological constant is one of the potential candidates of the dark energy bearing equation of state ωλ =−1. The cosmological constant in the constant dark energy model suffers two serious issues namely the coincidence problem and cosmological constant problem. To resolve these two open problems interacting dark energy model has been proposed by a number of authors [3-24] by considering the dynamical behavior of dark energy.
In the interacting dark energy models, transfer of energy between the dominant components of the Universe (matter, dark energy, radiation) is considered. The interaction term Q in the interacting dark energy model concern the transfer of energy between the dominant components of the Universe. One of the major challenges in the interacting dark energy model is to fix the exact functional form of coupling term Q. Different functional forms of coupling term have been used over the years [3-10, 12].
In the absence of a fundamental theory of dark sector physics, the choice of coupling strength in interacting dark energy model might be purely phenomenological. One of the motivations to fix the form of coupling term comes from the dimensional argumentativeness applied to the continuity equation of energy for the components of the Universe. The form of coupling strength can be either the linear function of energy density and the Hubble parameter of the field or the function of the first time derivative of energy density.
In this article, we have discussed the scaling solution of energy density for the two dominant interacting components of the Universe. One component is the scalar field φ which we take as a candidate of dynamical dark energy and the other is dust matter. These two components are interacting via the transfer of their energy into each other. Further, we have estimated the age of the Universe and its variation with the varying coupling constant.
2. Background theory and motivation
In this section, we present a brief introduction to the working background theory of interacting dark energy model. The FLWR metric for flat (K=0) Universe given as$\begin{eqnarray}{\rm{d}}{s}^{2}=-{\rm{d}}{t}^{2}+{a}^{2}\left(t\right){\left({\rm{d}}{x}^{i}\right)}^{2},\end{eqnarray}$where a(t) is the expansion scale factor and i=1, 2, 3 represents the spatial component of spacetime. The Friedman equation can be obtained by solving the Einstein field equation for the above metric$\begin{eqnarray}{H}^{2}=\displaystyle \frac{8\pi G}{3}({\rho }_{m}+{\rho }_{{\rm{de}}}),\end{eqnarray}$where ρm, ρde are energy density of dust matter and dark energy respectively. We are considering a dynamical dark energy model. Although cosmological constant is a potential candidate of constant dark energy, we assume the scalar field as a candidate for dynamical dark energy. A number of scalar fields (quintessence, phantom, tachyonic, etc) have been introduced in physics in different contexts. We scrutinize the dynamical behavior of the tachyonic scalar field (TSF) in the interacting dark energy model by assuming it as one of the candidates for dynamical dark energy.
The Lagrangian of TSF appears in string theory in the formulation of tachyon condensate [25-27] given as$\begin{eqnarray*}{ \mathcal L }=-V\left(\phi \right)\sqrt{1-{\partial }^{\mu }\phi {\partial }_{\mu }\phi },\end{eqnarray*}$where V(φ) is the potential of the field. The equation of motion for the spatially homogeneous TSF can be written as$\begin{eqnarray*}\displaystyle \frac{\ddot{\phi }}{1-\dot{{\phi }^{2}}}+3H\dot{\phi }+\displaystyle \frac{V^{\prime} (\phi )}{V(\phi )}=0,\end{eqnarray*}$and the energy-momentum tensor is$\begin{eqnarray}{T}^{\mu \nu }=\displaystyle \frac{\partial { \mathcal L }}{\partial \left({\partial }_{\mu }\phi \right)}{\partial }^{\nu }\phi -{g}^{\mu \nu }{ \mathcal L }.\end{eqnarray}$From the energy-momentum tensor equation (3), the T00 component gives the energy density while the T11 component leads to pressure for the TSF, i.e.$\begin{eqnarray*}\rho =\displaystyle \frac{V\left(\phi \right)}{\sqrt{1-{\partial }^{\mu }\phi {\partial }_{\mu }\phi }},\end{eqnarray*}$and$\begin{eqnarray*}p=-V\left(\phi \right)\sqrt{1-{\partial }^{\mu }\phi {\partial }_{\mu }\phi }.\end{eqnarray*}$For the spatially homogeneous TSF, the energy density and the pressure reduces to the following form$\begin{eqnarray*}\rho =\displaystyle \frac{V\left(\phi \right)}{\sqrt{1-\dot{{\phi }^{2}}}},\ \ \ p=-V\left(\phi \right)\sqrt{1-\dot{{\phi }^{2}}}.\end{eqnarray*}$
3. Interaction between the components
We consider the two dominant components of the Universe are spatially homogeneous TSF (as a candidate of dynamical dark energy) and the dust matter. These two components interact via the transfer of energy. The continuity equation leads to conservation of energy for the two-components. For non-interacting case as, we have$\begin{eqnarray*}{\dot{\rho }}_{\phi }+3H\left(1+{\omega }_{\phi }\right){\rho }_{\phi }=0,\end{eqnarray*}$and$\begin{eqnarray*}{\dot{\rho }}_{m}+3H\left(1+{\omega }_{m}\right){\rho }_{m}=0,\end{eqnarray*}$where wφ=pφ/ρφ. Here wφ, wm is the equation of state parameter for the scalar field φ and the dust matter respectively. During the interaction, individual components can violate the energy conservation but the total energy is conserved. Thus, the energy conservation with the interaction term Q can be represented by the following equations$\begin{eqnarray}\begin{array}{rcl} & & {\dot{\rho }}_{\phi }+3H\left(1+{\omega }_{\phi }\right){\rho }_{\phi }=-Q,\\ & & {\dot{\rho }}_{m}+3H\left(1+{\omega }_{m}\right){\rho }_{m}=Q.\end{array}\end{eqnarray}$In the absence of the fundamental theory of the dark sector, there are some possible linear and nonlinear functional forms of interaction term Q. One can fix the functional form of Q purely based upon the phenomenological argument which is it should be dimensionally compatible with the left-hand side of equation (4). Thus, it is natural that the coupling parameter Q should be a function of energy density and Hubble parameter or the rate of change of energy density of components. Based on phenomenological motivation, several authors [28-32] proposed different forms of interaction term in the interacting dark energy model.
In our model we considered a specific functional form of coupling term which is linearly proportional to Hubble parameter H as well as the energy density ρφ of the TSF. This form of coupling term has been used in the literature [33]. Thus we have the following form for interaction term$\begin{eqnarray}Q=\alpha H{\rho }_{\phi },\end{eqnarray}$where α is the proportionality constant. One can solve equation (4) for given interaction strength equation (5) to obtain the following scaling solution of the energy density of components [34]$\begin{eqnarray*}\displaystyle \frac{{\rho }_{\phi }}{{\rho }_{\phi }^{0}}={\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-\beta },\end{eqnarray*}$and$\begin{eqnarray*}\begin{array}{l}\displaystyle \frac{{\rho }_{m}}{{\rho }_{m}^{0}}={\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-3\left(1+{\omega }_{m}\right)}+\displaystyle \frac{\beta {\rho }_{\phi }^{0}}{{\rho }_{m}^{0}\left[\beta -3\left(1+{\omega }_{m}\right)\right]}\\ \quad \times \ \left[{\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-3\left(1+{\omega }_{m}\right)}-{\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-\beta }\right],\end{array}\end{eqnarray*}$where β=α+3(1+ωφ), ${\rho }_{\phi }^{0}$ (${\rho }_{m}^{0}$) is the present value of scalar field (dust matter), and a0 is the present value of scale factor. If $\dot{\phi }\to 0$, i.e. for constant φ we get ρφ→ρλ (someconstant). In this approximation limit TSF mimics the cosmological constant, and we have β=α. Hence, the scaling solution reduces to the following form$\begin{eqnarray}\displaystyle \frac{{\rho }_{\lambda }}{{\rho }_{\lambda }^{0}}={\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-\alpha },\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rho }_{m}}{{\rho }_{m}^{0}}={\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-3\left(1+{\omega }_{m}\right)}+\displaystyle \frac{\alpha {\rho }_{\lambda }^{0}}{{\rho }_{m}^{0}\left[\alpha -3\left(1+{\omega }_{m}\right)\right]}\\ \quad \times \ \left[{\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-3\left(1+{\omega }_{m}\right)}-{\left(\displaystyle \frac{a}{{a}_{0}}\right)}^{-\alpha }\right].\end{array}\end{eqnarray}$
4. Functional form of scale factor
In this section, we investigate the evolution of the expansion scale factor (a) for the two important cases. The interaction proportionality constant term (α) can be either zero or non-zero. So, we are considering the following two case
4.1. For α=0
From the Friedmann equations (equation (2)) along with the scaling solutions equations (6) and (7), we have$\begin{eqnarray}H=\sqrt{\displaystyle \frac{8\pi G}{3}}\sqrt{{\rho }_{\lambda }^{0}+{\rho }_{m}^{0}{x}^{-3(1+{\omega }_{m})}},\end{eqnarray}$where $x=\displaystyle \frac{a}{{a}_{0}}$, a0 is the present value scale factor. Equation (8) gives$\begin{eqnarray}{\rm{d}}t=\displaystyle \frac{A{\rm{d}}x}{x\sqrt{1+{B}^{2}{x}^{-3(1+{\omega }_{m})}}},\end{eqnarray}$where$\begin{eqnarray*}A=\sqrt{\displaystyle \frac{3}{k{\rho }_{\lambda }^{0}}},\,\,B=\sqrt{\displaystyle \frac{{\rho }_{m}^{0}}{{\rho }_{\lambda }^{0}}},\ \ \,\mathrm{and}\,\,k=8\pi G.\end{eqnarray*}$Solving equation (9), we get the analytical expression for x as a function of cosmic time t$\begin{eqnarray}x={\left[{B}^{-2}{\left(\tanh \left(\displaystyle \frac{3(1+{\omega }_{m})t}{2A}\right)\right)}^{2}-{B}^{-2}\right]}^{-\tfrac{1}{3(1+{\omega }_{m})}}.\end{eqnarray}$
4.2. For α≠0
In this case, equation (8) leads to the following equation$\begin{eqnarray}{\rm{d}}t=\displaystyle \frac{A^{\prime} {\rm{d}}x}{{x}^{\left(1-\tfrac{\alpha }{2}\right)}\sqrt{1+B{{\prime} }^{2}{x}^{\alpha -3(1+{\omega }_{m})}}},\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}A^{\prime} & = & \sqrt{\displaystyle \frac{3(1+{\omega }_{m})-\alpha }{k{\rho }_{\lambda }^{0}(1+{\omega }_{m})}},\\ {\rm{and}}\,\,B^{\prime} & = & \sqrt{\displaystyle \frac{3(1+{\omega }_{m}){\rho }_{m}^{0}-\alpha ({\rho }_{\lambda }^{0}+{\rho }_{m}^{0})}{3(1+{\omega }_{m}){\rho }_{\lambda }^{0}}}.\end{array}\end{eqnarray*}$Solving equation (11) to obtain an expression of t in terms of x, we have$\begin{eqnarray}\begin{array}{rcl}t & = & \displaystyle \frac{2A^{\prime} }{{\left(-1\right)}^{n}B{{\prime} }^{\alpha ^{\prime} }[\alpha -3(1+{\omega }_{m})]}\left[{\sqrt{1+B{{\prime} }^{2}{x}^{\alpha -3(1+{\omega }_{m})}}}_{2}{{F}}_{1}\right.\\ & & \left.\times \left(\displaystyle \frac{1}{2},n,\displaystyle \frac{3}{2},{\left(\sqrt{1+B{{\prime} }^{2}{x}^{\alpha -3(1+{\omega }_{m})}}\right)}^{2}\right)\right],\end{array}\end{eqnarray}$where$\begin{eqnarray}\alpha ^{\prime} =\displaystyle \frac{\alpha }{\alpha -3(1+{\omega }_{m})},\ \ n=\displaystyle \frac{\tfrac{\alpha }{2}-3(1+{\omega }_{m})}{\alpha -3(1+{\omega }_{m})}.\end{eqnarray}$The function 2F1 hypergeometric functional series defined by [35]$\begin{eqnarray*}{}_{2}{{F}}_{1}(a,b,c,z)=\displaystyle \sum _{m=0}^{\infty }\displaystyle \frac{{\left(a\right)}_{m}{\left(b\right)}_{m}}{{\left(c\right)}_{m}}\displaystyle \frac{{z}^{m}}{m!},\end{eqnarray*}$where$\begin{eqnarray*}{\left(a\right)}_{m}=\left\{\begin{array}{ll}1 & \,m=0,\\ a(a+1)..(a+m-1) & \,m\gt 0.\end{array}\right.\end{eqnarray*}$By using the definition of 2F1, equation (12) can be rewritten as$\begin{eqnarray}\begin{array}{rcl}t & = & \displaystyle \frac{2A^{\prime} }{{\left(-1\right)}^{n}B{{\prime} }^{\alpha ^{\prime} }\left[\alpha -3(1+{\omega }_{m})\right]}\left(\Space{0ex}{3.6ex}{0ex}\sqrt{1+B{{\prime} }^{2}{x}^{\alpha -3(1+{\omega }_{m})}}\right.\\ & & +\displaystyle \frac{1}{3}n\displaystyle \frac{{\left(\sqrt{1+B{{\prime} }^{2}{x}^{\alpha -3(1+{\omega }_{m})}}\right)}^{3}}{1!}\\ & & +\left.\displaystyle \frac{1}{5}n(n+1)\displaystyle \frac{{\left(\sqrt{1+B{{\prime} }^{2}{x}^{\alpha -3(1+{\omega }_{m})}}\right)}^{5}}{2!}+...\right).\end{array}\end{eqnarray}$When α=0 (from equations (13)), we have n=1, $\alpha ^{\prime} =0$, $A^{\prime} =A$ and, $B^{\prime} =B$. Hence, the term inside the bracket of equations (14) can be written as$\begin{eqnarray}\begin{array}{l}{\tanh }^{-1}\sqrt{1\,+\,{B}^{2}{x}^{-3(1+{\omega }_{m})}}=\sqrt{1\,+\,{B}^{2}{x}^{-3(1+{\omega }_{m})}}\\ \quad +\,\displaystyle \frac{1}{3}{\left(\sqrt{1+{B}^{2}{x}^{-3(1+{\omega }_{m})}}\right)}^{3},\\ \quad +\,\displaystyle \frac{1}{5}{\left(\sqrt{1+{B}^{2}{x}^{-3(1+{\omega }_{m})}}\right)}^{5}+....\end{array}\end{eqnarray}$Using equation (15) and equation (14) we get$\begin{eqnarray}t=\displaystyle \frac{2A}{3(1+{\omega }_{m})}{\tanh }^{-1}\sqrt{1+{B}^{2}{x}^{-3(1+{\omega }_{m})}}.\end{eqnarray}$In the limit α→0, equation (14) reduces to the same form which we will obtain by integrating the equation (9).
5. Age of the Universe (AoU)
The age of the Universe has been discussed in the [14] by considering two-phase (interacting and non-interacting dark energy) evolution of the Universe. Recent data [36] provide the present value of Hubble parameter H0 to be approximately 67.66±0.42 km s−1 Mpc−1 and the present value of normalized energy density parameters are ${{\rm{\Omega }}}_{\lambda }^{0}=0.6889\pm 0.0056$, ${{\rm{\Omega }}}_{m}^{0}=0.3111\pm 0.0056$. In this section, we estimate the age of the Universe with and without coupling between the matter and the TSF (candidate of dynamical dark energy).
5.1. Without interaction (α=0)
One can find the cosmic age in the absence of interaction by integrating the equation (9) to get$\begin{eqnarray*}{t}_{a}={\int }_{0}^{1}\displaystyle \frac{A{\rm{d}}x}{x\sqrt{1+{B}^{2}{x}^{-3(1+{\omega }_{m})}}}.\end{eqnarray*}$For the Hubble constant H0=67.66 km s−1 Mpc−1, energy density parameter ${{\rm{\Omega }}}_{\lambda }^{0}=0.6889$, and ${{\rm{\Omega }}}_{m}^{0}=0.3111$, we found the age of the Universe to be ${t}_{a}\approx 0.9543{H}_{0}^{-1}\approx 13.52\,\mathrm{Gyr}$. The value of ${H}_{0}^{-1}$ is ≈14.167 Gyr.
5.2. With interaction (α≠0)
In presence of interaction the age of the Universe can be derived from equation (11) and it comes out be [37]$\begin{eqnarray*}{t}_{a}(x)={\int }_{0}^{x}\displaystyle \frac{{\rm{d}}x^{\prime} }{x^{\prime} H(x^{\prime} )},\end{eqnarray*}$which can be modified as$\begin{eqnarray}{t}_{a}(x){H}_{0}={\int }_{0}^{x}\displaystyle \frac{{\rm{d}}x^{\prime} }{x^{\prime} E(x^{\prime} )},\end{eqnarray}$where $E(x)=\displaystyle \frac{H(x)}{{H}_{0}}$ and, $x=\displaystyle \frac{a}{{a}_{0}}$. Using equations (2), (6), and equation (7) for ωm=0, we have$\begin{eqnarray*}E(x)=\sqrt{\left({{\rm{\Omega }}}_{m}^{0}+\displaystyle \frac{\alpha }{\alpha -3}{{\rm{\Omega }}}_{\lambda }^{0}\right){x}^{-3}-{{\rm{\Omega }}}_{\lambda }^{0}\left(\displaystyle \frac{3}{\alpha -3}\right){x}^{-\alpha }}.\end{eqnarray*}$Thus the equation (17) gives$\begin{eqnarray}{t}_{a}(x){H}_{0}={\int }_{0}^{1}\displaystyle \frac{{\rm{d}}x^{\prime} }{\sqrt{\left({{\rm{\Omega }}}_{m}^{0}+\tfrac{\alpha }{\alpha -3}{{\rm{\Omega }}}_{\lambda }^{0}\right){\left(x^{\prime} \right)}^{-1}-{{\rm{\Omega }}}_{\lambda }^{0}\left(\tfrac{3}{\alpha -3}\right){\left(x^{\prime} \right)}^{2-\alpha }}}.\end{eqnarray}$The cosmic age of Universe in the presence of coupling α for its various numerical values is (from α=0 to α=0.9) given in the table 1 for ${{\rm{\Omega }}}_{\lambda }^{0}=0.7$, ${{\rm{\Omega }}}_{m}^{0}=0.3$ and, ${{\rm{\Omega }}}_{\lambda }^{0}\,=0.6889$, ${{\rm{\Omega }}}_{m}^{0}=0.3111$.
6. Variation of Ωλ, Ωm with ${\rm{ln}}\,a$
Here we have observed the variation of energy density parameter Ωλ and Ωm as a function of $\eta =\mathrm{ln}\,a$ by following the method of [38]. We can define two new variable ${ \mathcal X }$ and ${ \mathcal Y }$ such that ${ \mathcal X }=\displaystyle \frac{8\pi G}{3{H}^{2}}{\rho }_{\lambda }={{\rm{\Omega }}}_{\lambda }$, and ${ \mathcal Y }=\displaystyle \frac{8\pi G}{3{H}^{2}}{\rho }_{m}={{\rm{\Omega }}}_{m}$. Differentiating ${ \mathcal X }$ and ${ \mathcal Y }$ with η, we have$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{ \mathcal X }}{{\rm{d}}\eta }=\displaystyle \frac{1}{H}\left[\displaystyle \frac{8\pi G}{3{H}^{2}}{\dot{\rho }}_{\lambda }-2\displaystyle \frac{8\pi G}{3{H}^{3}}{\rho }_{\lambda }\dot{H}\right],\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{ \mathcal Y }}{{\rm{d}}\eta }=\displaystyle \frac{1}{H}\left[\displaystyle \frac{8\pi G}{3{H}^{2}}{\dot{\rho }}_{m}-2\displaystyle \frac{8\pi G}{3{H}^{3}}{\rho }_{m}\dot{H}\right].\end{eqnarray}$From equations (4), we get$\begin{eqnarray}{\dot{\rho }}_{\lambda }=-\alpha H{\rho }_{\lambda },\ \ \ \ \ \ {\dot{\rho }}_{m}=\alpha H{\rho }_{\lambda }-3H{\rho }_{m},\end{eqnarray}$and from equations (2) we get$\begin{eqnarray}\displaystyle \frac{\dot{H}}{{H}^{2}}=-\displaystyle \frac{3}{2}\displaystyle \frac{8\pi G}{3{H}^{2}}{\rho }_{m},\ \ \ \ \ \ { \mathcal X }+{ \mathcal Y }=1.\end{eqnarray}$So using, equations (21), (22) in equations (19), equation (20), we get the following set of equations$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{ \mathcal X }}{{\rm{d}}\eta }+3{{ \mathcal X }}^{2}+(\alpha -3){ \mathcal X }=0,\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{ \mathcal Y }}{{\rm{d}}\eta }-3{{ \mathcal Y }}^{2}+(\alpha +3){ \mathcal Y }-\alpha =0.\end{eqnarray}$
We have plotted equations (23), (24) in figure 1. In figure 2 we have observed the variation of normalized cosmic time t with scale factor x (equations (12)), and in figure 3 we plotted the data of table 1 which shows the age of Universe for different value of coupling constant α.
Figure 1.
New window|Download| PPT slide Figure 1.Plots of Ωλ and Ωm as a function of $\eta (=\mathrm{ln}\,a)$ for different values coupling parameter α. The vertical axis represents the present cosmological time.
Figure 2.
New window|Download| PPT slide Figure 2.Graphical variation of equations (12) for ${{\rm{\Omega }}}_{\lambda }^{0}=0.7$ and ${{\rm{\Omega }}}_{m}^{0}=0.3$ where cosmic time (in the unit of ${H}_{0}^{-1}$) is plotted against normalized dimensionless scale factor x in presence and absence of coupling.
Figure 3.
New window|Download| PPT slide Figure 3.Graphical variation of table 1. Here we observe the evolution of age of the Universe with the coupling constant α.
Table 1. Table 1.Age of Universe (AOU) ta for different values of coupling constant α. First table (left one) is for ${{\rm{\Omega }}}_{\lambda }^{0}=0.7,\,{{\rm{\Omega }}}_{m}^{0}=0.3$ and second one (right one) is for ${{\rm{\Omega }}}_{\lambda }^{0}=0.6889$, ${{\rm{\Omega }}}_{m}^{0}=0.3111$.
In this article we studied the evolution of scale factor by considering the two interacting components (matter and dark energy) in the spatially flat Universe. We considered the spatially homogeneous TSF as a candidate of dark energy. In the interacting dark energy model, the two components are mutually coupled via coupling parameter Q, and a transfer of energy between the two components is possible. During the interaction, the individual components can violate the energy conservation, but overall energy is conserved. In the absence of a fundamental theory of dark sector the choice of coupling parameter is purely phenomenological. We obtained the age of the Universe in interacting dark energy model and presented in table 1 for different value of coupling constant. We found that with the increase in the value of coupling constant (α), the age of the Universe is also increasing. But as we further increased the value of coupling constant (beyond 1), the age of the Universe turns out to be imaginary which is a non-physical situation. This puts an upper bound to the coupling constant α, and it should be less than 1. We plotted the normalized energy density of dark energy (Ωλ) and matter (Ωm) as a function of $\mathrm{ln}\,a$ in figure 1. Figure 2 showed the relationship between the cosmic time and the normalized dimensionless scale factor x. And in figure 3, we plotted the data of table 1 which shows the variation of the age of the Universe with the coupling constant α.
Acknowledgments
The authors are thankful to reviewers for their useful comments and suggestions.
RiessA G et al. 1998 Observational evidence from supernovae for an accelerating universe and a cosmological constant Astron. J 116 10091038 DOI:10.1086/300499 [Cited within: 1]
PerlmutterS et al. 1999 Measurements of Ω and Λ from 42 high-redshift supernovae Astrophys. J. 517 565586 DOI:10.1086/307221 [Cited within: 1]
ChimentoL PForteMKremerG M2008 Cosmological model with interactions in the dark sector Gen. Relativ. Gravit. 41 11251137 DOI:10.1007/s10714-008-0694-5
BertolamiOCarrilhoPPáramosJ2012 Two-scalar-field model for the interaction of dark energy and dark matter Phys. Rev. D 86 103522 DOI:10.1103/PhysRevD.86.103522
WangBZangJLinC-YAbdallaEMichelettiS2007 Interacting dark energy and dark matter: observational constraints from cosmological parameters Nucl. Phys. B 778 6984 DOI:10.1016/j.nuclphysb.2007.04.037
LuJWuYJinYWangY2012 Investigate the interaction between dark matter and dark energy Results Phys. 2 1421 DOI:10.1016/j.rinp.2012.02.001
FarajollahiHRavanpakAFadakarG2012 Interacting agegraphic dark energy model in tachyon cosmology coupled to matter Phys. Lett. B 711 225231 DOI:10.1016/j.physletb.2012.04.001
ZimdahlW2012 Models of interacting dark energy
YangWPanSValentinoE DNunesR CVagnozziSMotaD F2018 Tale of stable interacting dark energy, observational signatures, and the H0 tension J. Cosmol. Astropart. Phys. JCAP18(2018)019 DOI:10.1088/1475-7516/2018/09/019 [Cited within: 1]
CaoSLiangNZhuZ-H2011 Testing the phenomenological interacting dark energy with observational H(z) data Mon. Not. R. Astron. Soc. 416 10991104 DOI:10.1111/j.1365-2966.2011.19105.x
VermaM MPathakS D2012 A tachyonic scalar field with mutually interacting components Int. J. Theor. Phys. 51 23702379 DOI:10.1007/s10773-012-1116-8
VermaM MPathakS D2013 Shifted cosmological parameter and shifted dust matter in a two-phase tachyonic field universe Astrophys. Space Sci. 344 505512 DOI:10.1007/s10509-012-1348-3 [Cited within: 1]
VäliviitaJMaartensRMajerottoE2010 Observational constraints on an interacting dark energy model Mon. Not. R. Astron. Soc. 402 23552368 DOI:10.1111/j.1365-2966.2009.16115.x
PanSMukherjeeABanerjeeN2018 Astronomical bounds on a cosmological model allowing a general interaction in the dark sector Mon. Not. R. Astron. Soc. 477 11891205 DOI:10.1093/mnras/sty755
AmendolaLRubioJWetterichC2018 Primordial black holes from fifth forces Phys. Rev. D 97 081302 DOI:10.1103/PhysRevD.97.081302
BéguéDStahlCXueS-S2019 A model of interacting dark fluids tested with supernovae and baryon acoustic oscillations data Nucl. Phys. B 940 312320 DOI:10.1016/j.nuclphysb.2019.01.001
PanSYangWSinghaCSaridakisE N2019 Observational constraints on sign-changeable interaction models and alleviation of the H0 tension Phys. Rev. D 100 083539 DOI:10.1103/PhysRevD.100.083539
PapagiannopoulosGTsiapiPBasilakosSPaliathanasisA2020 Dynamics and cosmological evolution in λ Λ-varying cosmology Eur. Phys. J. C 80 55 DOI:10.1140/epjc/s10052-019-7600-z
SavastanoSAmendolaLRubioJWetterichC2019 Primordial dark matter halos from fifth forces Phys. Rev. D 100 083518 DOI:10.1103/PhysRevD.100.083518
von MarttensRCasariniLMotaDZimdahlW2019 Cosmological constraints on parametrized interacting dark energy Phys. Dark Universe 23 100248 DOI:10.1016/j.dark.2018.10.007
YangWBanerjeeNPaliathanasisAPanS2019 Reconstructing the dark matter and dark energy interaction scenarios from observations Phys. Dark Universe 26 100383 DOI:10.1016/j.dark.2019.100383
AsghariMJiménezJ BKhosraviSMotaD F2019 On structure formation from a small-scales-interacting dark sector J. Cosmol. Astropart. Phys. JCAP19(2019)042 DOI:10.1088/1475-7516/2019/04/042 [Cited within: 1]
PathakS DVermaM MLiS2016 Thermodynamics of interacting tachyonic scalar field National Conf. on Current Issues in Cosmology, Astrophysics and High Energy PhysicsDibrugarh, India2-5 November 201577arXiv:1612.00860 [Cited within: 1]
AmendolaLQuercelliniCTocchini-ValentiniDPasquiA2003 Cosmic microwave background as a gravity probe Astrophys. J. 583 L53L56 DOI:10.1086/368064
BergerM SShojaeiH2008 Possible equilibria of interacting dark energy models Phys. Rev. D 77 123504 DOI:10.1103/PhysRevD.77.123504
VermaM MPathakS D2014 The bicep2 data and a single higgs-like interacting scalar field Int. J. Mod. Phys. D 23 1450075 DOI:10.1142/S0218271814500758
BahamondeSBöhmerC GCarloniSCopelandE JFangWTamaniniN2018 Dynamical systems applied to cosmology: dark energy and modified gravity Phys. Rep. 775-777 1122 DOI:10.1016/j.physrep.2018.09.001 [Cited within: 1]
,1, S D Pathak
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