Tsallis HDE-based reconstruction via correspondence scheme in a generalized torsion scalar theory

Abstract

The present study is devoted to calculate the form of generic function present in the Lagrangian of the most generalized teleparallel gravity based on higher-order derivative terms of torsion scalar using reconstruction scheme. Our primary objective is to obtain the form of Lagrangian coming from the Tsallis holographic dark energy model with Hubble horizon as IR cutoff through correspondence scheme of reconstruction. For this purpose, the flat FRW geometry filled with perfect fluid as ordinary matter contents is taken into account along with the power law form of scale factor. The cosmological significance of the reconstructed models is then explored graphically by using some cosmic measures like the EoS parameter for dark energy and the null energy condition. Also, we discuss the stability of the reconstructed models through speed of sound graphically. Further we discuss the reconstruction of this Lagrangian for power law and logarithmic corrected Tsallis HDE models. Lastly, we reconstruct the generic function of Lagrangian by taking new agegraphic Tsallis holographic dark energy into account. It is found that majority of the new reconstructed models are stable and cosmologically interesting (consistent with the cosmic observations) for the selected choices of involved free parameters.

Appendix

For the case $F(T, X_1)=f(T)+g(X_1)$, the speed of sound can be evaluated by using chain rule and is given by

$$\begin{aligned} v_{s}^2=&\frac{\dot{P}_m}{\dot{\rho _m}}=-4\left( 24 \dot{H}^3 \left( (H+3) \dot{g}_{\text {X1}}+24 H g_{\text {X1}}\right) +96 H \ddot{H}^2 g_{\text {X1}}\right) \nonumber \\& \quad -4\left( \ddot{H} \left( 72 H^2 \left( 2 H \dot{g}_{X1}+\ddot{g}_{X1}\right) +f_T-\kappa ^2\right) +H \left( \ddot{f}_T+72 H \dddot{H} \dot{g}_{\text {X1}}+24 H \ddddot{H} g_{\text {X1}}\right) \right) \nonumber \\& \quad -4\left( \dot{H} \left( 24 H \left( 14 \ddot{H}\dot{g}_{X1}+H \left( 3 H \ddot{g}_{X1}+\ddot{g}_{X1}\right) +\left( 9 H \ddot{H}+(3 H+2) \dddot{H}\right) g_{\text {X1}}\right) +2 \dot{f}_T-3 H \kappa ^2\right) \right) \nonumber \\& \quad -4\left( +24 \dot{H}^2 \left( H \left( 3 H (4 H+3) \dot{g}_{X1}+5 \ddot{g}_{X1}\right) +\left( 5\ddot{H} +4\dddot{H}\right) \left( f+g_{\text {X1}}\right) \right) \right) \big / (\dot{f}+\dot{g})\nonumber \\& \quad +12H\left( H \left( 24 H \left( 2 \ddot{H} \dot{g}_{X1}+\left( 3 H \ddot{H}+\dddot{H}\right) g_{\text {X1}}\right) +\dot{f}_{T}\right) +72 H \dot{H}^2 \left( \dot{g}_{X1}+4 H g_{\text {X1}}\right) \right) \nonumber \\& \quad +\dot{H}\left( 24 H \left( H \left( 3 H \dot{g}_{X1}+\ddot{g}_{X1}\right) +3 \ddot{H}g_{\text {X1}}\right) +2 f_T+1\right) . \end{aligned}$$

(49)

For the case $F(T,X_1)=f(T)g(X_1)$, time rate of energy density and pressure for DE are given by

$$\begin{aligned} \dot{P}_D & = -6H\dot{H}-2\ddot{H}+2\left[ 2\dot{F}_T\dot{H}+F_T\ddot{H}+H\ddot{F}_T+24\dot{H}(2H\ddot{H}+3\dot{H}^2+3H^2\dot{H})\dot{F}_{X_1}+ 24H\ddot{F}_{X_1}(2H\ddot{H}+3\dot{H}^2+3H^2\dot{H})\right. \nonumber \\& \quad +\left. 24H\dot{F}_{X_1}(8\dot{H}\ddot{H}+2H\dddot{H}+6H\dot{H}^2+3H^2\ddot{H}) +48H\dot{H}^2\ddot{F}_{X_1}+24H^2\ddot{H}\ddot{F}_{X_1}+24H^2\dot{H}\dddot{F}_{X_1}+48H\dot{H}\dddot{H}F_{X_1}\right. \nonumber \\& \quad +\left. 24H^2\ddddot{H}F_{X_1} +24H^2\dddot{H}\dot{F}_{X_1}+24\dot{F}_{X_1}\dot{H}^2(12H^2+\dot{H})+48F_{X_1}\dot{H}\ddot{H}(12H^2+\dot{H})+24F_{X_1}\dot{H}^2(24H\dot{H}+\ddot{H})\right. \nonumber \\& \quad +\left. 24\dot{H}F_{X_1}\ddot{H}(4\dot{H}+3H^2)+24H\dot{F}_{X_1}\ddot{H}(4\dot{H}+3H^2)+24HF_{X_1}\dddot{H}(4\dot{H}+3H^2) +24HF_{X_1}\ddot{H}(4\ddot{H}+6H\dot{H})\right] , \end{aligned}$$

(50)

$$\begin{aligned} \dot{\rho }_D & = -\frac{\dot{F}}{2}-6H^2\dot{F}_T-12H\dot{H}F_T+6H\dot{H}-6\dot{H}(24H^2F_{X_1})(3H\dot{H}+\ddot{H})-6H(48H\dot{H}F_{X_1} +24H^2\dot{F}_{X_1})(3H\dot{H}+\ddot{H})\nonumber \\& \quad -6H(24H^2F_{X_1})(3\dot{H}^2+3H\ddot{H}+\dddot{H})-(144)(3)H^2\dot{H}^2\dot{F}_{X_1}-144H^3\ddot{H}\dot{F}_{X_1}-144H^3\dot{H}\ddot{F}_{X_1}. \end{aligned}$$

(51)

Consequently, we calculate $v_s^2$ for this case as follows

$$\begin{aligned} v_{s}^2=&-4\left( 24 \dot{H}^3 \left( (H+3) \left( f \dot{g}_{X1}+\dot{f} g_{\text {X1}}\right) +24 f H g_{\text {X1}}\right) +96 f H \ddot{H}^2 g_{\text {X1}}\right) \nonumber \\& \quad -4\left( H \left( 72 H \dddot{H} \left( f \dot{g}_{X1}+\dot{f} g_{\text {X1}}\right) +2\dot{g}\dot{f}_T+g \ddot{f}_T+24 f H \ddddot{H} g_{\text {X1}}+f_T \ddot{g}\right) \right) \nonumber \\& \quad -4\left( \ddot{H}\left( 72 H^2\left( 2H\left( f\dot{g}_{X1}+\dot{f} g_{\text {X1}}\right) +2\dot{f}\dot{g}_{X1} +f \ddot{g}_{X1}+\ddot{f} g_{\text {X1}}\right) +g f_T-\kappa ^2\right) \right) \nonumber \\& \quad -4\left( 24\dot{H}^2\left( f \left( 5 \ddot{H}+4 \dddot{H}\right) g_{\text {X1}}+H\left( 3 H (4 H+3) \left( f \dot{g}_{X1}+\dot{f} g_{\text {X1}}\right) \right) \right) \right) \nonumber \\& \quad -4\left( 24\dot{H}^2\left( H \left( 3 H (4 H+3) \left( f \dot{g}_{X1}+\dot{f} g_{\text {X1}}\right) \right) +f \left( 5 \ddot{H}+4 \dddot{H}\right) g_{\text {X1}}\right) \right) \nonumber \\& \quad -20\left( 24\left( 2 \dot{f} \dot{g}_{X1}+f \ddot{g}_{X1}+\ddot{f} g_{\text {X1}}\right) \right) \nonumber \\& \quad -4\left( \dot{H} \left( 24 H \left( 14 \ddot{H}\left( f \dot{g}_{X1}+f_t g_{\text {X1}}\right) +f \left( 9 H \ddot{H}+(3 H+2) \dddot{H}\right) g_{\text {X1}}\right) +2 \left( g \dot{f}_T+f_T \dot{g}\right) -3 H \kappa ^2\right) \right) \nonumber \\& \quad -4\left( H \left( 3 H \left( 2 \dot{f} \dot{g}_{X1}+f \ddot{g}_{X1}+\ddot{f}g_{\text {X1}}\right) +3\ddot{f}\dot{g}_{X1}+3\dot{f}\ddot{g}_{X1}+f\dddot{g}_{X1}+\dddot{f} g_{\text {X1}}\right) \right) \big /f\dot{g}+f \dot{g}_{X1}\nonumber \\& \quad +12H\left( 72 H \dot{H}^2 \left( f \dot{g}_{X1}+4 f H g_{X1}+\dot{f}g_{X1}\right) +H(g\dot{f}_{T}+24H\left( f\left( 3H \ddot{H} +\dddot{H}\right) g_{\text {X1}}\right) +f_T \dot{g}\right) .\nonumber \\& \quad + 12H\left( \dot{H} \left( 24 H \left( H \left( 3 H \left( f \dot{g}_{X1}+\dot{f} g_{\text {X1}}\right) +2 \dot{f} \dot{g}_{X1}+f \ddot{g}_{X1}+\ddot{f} g_{\text {X1}}\right) +3 f \ddot{H} g_{\text {X1}}\right) +2 g f_T +2 \ddot{H} \left( f \dot{g}_{X1}+\dot{f}_T g_{\text {X1}}\right) \right) \right) . \end{aligned}$$

(52)

The second and third-order time rates of H,  T and $X_2$ can be shifted in terms of $X_2$ as follows

$$\begin{aligned}&\ddot{H}=-2b\left( \frac{X_2}{-36b^2(1-b)}\right) ^{3/4}, \quad \dddot{H}=\frac{X_2}{6(1-b)}, \quad \ddot{T}=-\frac{X_2}{1-b}, \quad \dddot{T}=-144b^2\left( \frac{X_2}{-36b^2(1-b)}\right) ^{5/4},\nonumber \\&\ddot{X}_2=-720b^2(1-b)\left( \frac{X_2}{-36b^2(1-b)}\right) ^{3/2}, \quad \dddot{X}_2=-4320b^2(1-b)\left( \frac{X_2}{-36b^2(1-b)}\right) ^{7/4}. \end{aligned}$$

(53)

Other higher-order time rates will be shifted in a similar fashion. For function $F(T,X_2)=f(T)+g(X_2)$, the squared speed of sound can be computed as follows

$$\begin{aligned} v_s^2 & = -4\left( \dot{H}\left( 2\dot{f}_{T}+6 H \ddot{g}_{X2}+2\dddot{g}_{X2}+12H\dddot{g}_{X2}-3H\kappa ^2\right) +12\dot{H}^2\dot{g}_{X2}\right. \nonumber \\& \quad +\left. \ddot{H} \left( (12 H+1) \ddot{g}_{X2}+f_T-\kappa ^2\right) +H \left( \ddot{f}_T+(3 H+1)\dddot{g}_{X2}\right) \right) \big /\nonumber \\& \quad +\dot{f}+\dot{g}+12\left( \dot{H}^2 \left( -\dot{g}_{X2}+(3 H+1) g_{\text {X2}}+H\right) +2 H \dot{H} f_T\right. \nonumber \\& \quad + \left. H \left( H \dot{f}_T+2\ddot{H} \dot{g}_{X2}+H \dddot{g}_{X2}+\dddot{H} g_{\text {X2}}\right) +2 \dot{H} \left( H\left( 6 H \dot{g}_{X2}+\ddot{g}_{X2}\right) +\ddot{H} g_{\text {X2}}\right) \right) . \end{aligned}$$

(54)

For the function $F(T,X_2)=f(T)g(X_2)$, the time rates of energy density and pressure for DE can be written as

$$\begin{aligned} \dot{P}_D & = -6H\dot{H}-2\ddot{H}+2\left( 2\dot{F}_T\dot{H}+F_T\ddot{H}+H\ddot{F}_T+12\dot{H}^2\dot{F}_{X_2}+12H\ddot{H}\dot{F}_{X_2}+ 12H\dot{H}\ddot{F}_{X_2}+(\ddot{H}+6H\dot{H})\ddot{F}_{X_2}\right. \nonumber \\& \qquad +\left. (\dot{H}+3H^2)\dddot{F}_{X_2}+\dot{H}\dddot{F}_{X_2}+H\ddddot{F}_{X_2}\right) , \end{aligned}$$

(55)

$$\begin{aligned} \dot{\rho }_{D} & = -\frac{\dot{F}}{2}-6H^2\dot{F}_T-12H\dot{H}F_T+6H\dot{H}-6\dot{F}_{X_2}\dot{H}^2-12F_{X_2}\dot{H}\ddot{H}-6\dot{H}F_{X_2}(3H\dot{H}+\ddot{H}) -6\dot{F}_{X_2}H(3H\dot{H}+\ddot{H})\nonumber \\& \quad -6HF_{X_2}(3\dot{H}^2+\dddot{H})-6\dot{H}\dot{F}_{X_2}(3H^2-\dot{H})-6H\ddot{F}_{X_2}(3H^2-\dot{H}) -6H\dot{F}_{X_2}(6H\dot{H}-\ddot{H})-12H\dot{H}\ddot{F}_{X_2}-6H^2\dddot{F}_{X_2}. \end{aligned}$$

(56)

Using above relations, we can check the stability of the solution by the following speed of sound parameter

$$\begin{aligned} v_s^2 & = -4 \left( \ddot{H} \left( (12 H+1) \left( 2 \dot{f} \dot{g}_{X2}+f \ddot{g}_{X2}+\ddot{f} g_{\text {X2}}\right) +gf_T-\kappa ^2\right) +12 \dot{H}^2 \left( f \dot{g}_{X2}+\dot{f} g_{\text {X2}}\right) \right) \nonumber \\& \quad -4\dot{H}\left( 6 H \left( 2 \dot{f} \dot{g}_{X2}+f \ddot{g}_{X2}+\ddot{f} g_{\text {X2}}\right) +2 \left( g \dot{f}_T+f_T \dot{g}\right) +2 \left( 3 \ddot{f}\dot{g}_{X2}+3\dot{f}\ddot{g}_{X2}+f \dddot{g}_{X2}+\dddot{f} g_{\text {X2}}\right) -3 H \kappa ^2\right) \nonumber \\& \quad -4\left( 12 H \dot{H} \left( 3\ddot{f} \dot{g}_{X2}+3\dot{f}\ddot{g}_{X2}+f \dddot{g}_{X2}+\dddot{f} g_{\text {X2}}\right) \right) \nonumber \\& \quad -4H\left( (3H+1)\left( 3\ddot{f}\dot{g}_{X2}+3\dot{f}\ddot{g}_{X2}+f\dddot{g}_{X2}+\dddot{f} g_{\text {X2}}\right) +2\dot{g}\dot{f}_{T}+g\ddot{f}_{T}+f_T \ddot{g}_{X2}\right) \big /\nonumber \\& \quad \dot{f}+\dot{g}+12\left( 2 g H \dot{H} f_T+\dot{H}^2 \left( -f \dot{g}_{X2}-\dot{f} g_{\text {X2}}+(3 H+1) g_{\text {X2}}+H\right) \right) \nonumber \\& \quad +24 \dot{H} \left( H \left( 6 H \left( f \dot{g}_{X2}+\dot{f} g_{\text {X2}}\right) +2 \dot{f} \dot{g}_{X2}+f \ddot{g}_{X2}+\ddot{f} g_{\text {X2}}\right) +\ddot{H} g_{\text {X2}}\right) \nonumber \\& \quad +12 H \left( H \left( g \dot{f}_T +\dot{f}_T \dot{g}\right) +H\left( 3 \ddot{f} \dot{g}_{X2} +3 \dot{f} \ddot{g}_{X2}+f \dddot{g}_{X2}+\dddot{f} g_{\text {X2}}\right) +2 \ddot{H} \left( f \dot{g}_{X2}+f_t g_{\text {X2}}\right) +\dddot{H} g_{\text {X2}}\right) . \end{aligned}$$

(57)

The term $\Theta _1$ present in Eq. (34) is given by the following expression:

$$\begin{aligned} \Theta _1 & = 91\times 2^{(\alpha +2\delta )/3}3^{(3+\alpha +2\delta )/3}BX_1^{2/3}-49(-1)^{2\delta }2^{(2+\alpha )/3}3^{(11+\alpha )/3}X_1^{(1+\delta )/3}\nonumber \\& \quad -455\times 2^{(3+\alpha +2\delta )/3}\times 3^{(\alpha +2\delta )/3}BX_1^{2/3}\alpha +245(-1)^{2\delta }2^{(5+\alpha )/3}\times 3^{(8-\alpha )/3} X_1^{(1+\delta )/3}\alpha +91\times 6^{(\alpha +2\delta )/3}BX_2^{2/3}\alpha ^2\nonumber \\& \quad - 49(-1)^{2\delta }2^{(2+\alpha )/3}\times 3^{(8+\alpha )/3}X_1^{(1+\delta )/3}\alpha ^2+11466(-1)^{\alpha +2\delta }6^{1/3}X_1^{(\alpha +2\delta )/6}\beta _1 +7(-1)^{2\delta }6^{(8+\alpha )/3}X_1^{(1+\delta )/3}\delta \nonumber \\& \quad - 35(-1)^{-2\delta }2^{(11+\alpha )/3}\times 3^{(5+\alpha )/3}X_1^{(1+\delta )/3}\alpha \delta +7(-1)^{2\delta }2^{(8+\alpha )/3} \times 3^{(5+\alpha )/3}X_1^{(1+\delta )/3}\delta \alpha ^2\nonumber \\& \quad - 2184(-1)^{\alpha +2\delta }6^{1/3}\beta _1\delta X_1^{(\alpha +2\delta )/6}+7(-1)^{2\delta }6^{(8+\alpha )/3}X_1^{(1+\alpha )/3}\delta ^2 -35(-1)^{2\delta }2^{(11+\alpha )/3}\times 3^{(5+\alpha )/3}X_1^{(1+\delta )/3}\alpha \delta ^2\nonumber \\& \quad + 7(-1)^{2\delta }2^{(8+\alpha )/3}\times 3^{(5+\alpha )/3}X_1^{(1+\alpha )/3}\delta ^2\alpha ^2-2184(-1)^{\alpha +2\delta }6^{1/3}\beta _1\delta ^2X_1^{(\alpha +2\delta )/6}\nonumber \\& \quad +455(-1)^{2\delta }2^{(4+\alpha )/3}\times 3^{(10+\alpha )/3}X_1^{(3+\delta )/3}\lambda -2275(-1)^{2\delta }6^{(7+\alpha )/3}X_1^{(3+\delta )/3}\alpha \lambda \nonumber \\& \quad +455(-1)^{2\delta }2^{(4+\alpha )/3}\times 3^{(7+\alpha )/3}X_1^{(3+\delta )/3}\alpha ^2\lambda -65(-1)^{2\delta }2^{(10+\alpha )/3}\times 3^{(7+\alpha )/3} X_1^{(3+\delta )/3}\delta \lambda \nonumber \\& \quad +325(-1)^{2\delta }2^{(13+\alpha )/3}\times 3^{(4+\alpha )/3}X_1^{(3+\delta )/3}\alpha \delta \lambda + 65(-1)^{2\delta }2^{(10+\alpha )/3}\times 3^{(4+\alpha )/3}X_1^{(3+\delta )/3}\alpha ^2\delta \lambda \nonumber \\& \quad -65(-1)^{2\delta }2^{(10+\alpha )/3}\times 3^{(7+\alpha )/3}X_1^{(3+\delta )/3}\delta ^2\lambda +325(-1)^{2\delta }2^{(13+\alpha )/3}\times 3^{(4+\alpha )/3}X_1^{(3+\delta )/3}\alpha \delta ^2\lambda \nonumber \\& \quad -65(-1)^{2\delta }2^{(10+\alpha )/3}\times 3^{(4+\alpha )/3}X_1^{(3+\delta )/3}\alpha ^2\delta ^2\lambda \end{aligned}$$

(58)

The term $\Theta _2$ present in Eq. (35) can be defined as follows

$$\begin{aligned} \Theta _2 & = 637\times 2^{2\delta /3}\times 3^{(6+2\delta )/3}(-1)^{-2\delta }BX_1^{4/3}-27783\times 6^{2/3}X_1^{(3+\delta )/3}+1092\times X_1^{(4+\delta )/3}\beta _2+5733\times X_1^{(4+\delta )/3}\beta _3\nonumber \\& \quad +5292\times 6^{2/3}X_1^{(3+\delta )/3}\delta -208X_1^{(4+\delta )/3}\beta _2\delta -1092\times X_1^{(4+\delta )/3}\times \beta _3\delta +5292\times 6^{2/3}X_1^{(3+\delta )/3}\delta ^2-208X_1^{(4+\delta )/3}\beta _2\delta ^2\nonumber \\& \quad -1092X_1^{(4+\delta )/3}\beta _3\delta ^2+515970\times 6^{1/3}X_1^{(5+\delta )/3}\lambda -98280\times 6^{1/3}X_1^{(5+\delta )/3}\delta \lambda -98280\times 6^{1/3}X_1^{(5+\delta )/3}\delta ^2\lambda \nonumber \\& \quad +5733X_1^{(4+\delta )/3}\beta _2\ln \left(\frac{6^{2/3}}{X_1^{1/3}}\right)-1092X_1^{(4+\delta )/3}\beta _2\delta \ln \left(\frac{6^{2/3}}{X_1^{1/3}}\right) -1092X_1^{(4+\delta )/3}\beta _2\delta ^2\ln \left(\frac{6^{2/3}}{X_1^{1/3}}\right). \end{aligned}$$

(59)

The terms $\Psi _1$ and $\Psi _2$ present in Eq. (36) are defined by

$$\begin{aligned} \Psi _1 & = -1175\times 2^{2\delta /3}\times 3^{(3+2\delta )/3}X_1^{2/3}+6075\times 6^{2/3}X_1^{(1+\delta )/3} +47\times 2^{2\delta /3}\times 3^{(9+2\delta )/3}X_1^{(\alpha +2\delta )/6}\nonumber \\& \qquad +47\times 2^{(12+2\delta )/3}\times 3^{2\delta /3}X_1^{2/3}\alpha -1296\times 6^{2/3}X_1^{(1+\delta )/3}\alpha -47\times 6^{2\delta /3}X_1^{2/3}\alpha ^2+81\times 6^{2/3}X_1^{(1+\delta )/3}\alpha ^2\nonumber \\& \qquad +3600\times 6^{2/3}X_1^{(1+\delta )/3}\delta +47\times 2^{(12+2\delta )/3}\times 3^{2\delta /3}X_1^{(\alpha +2\delta )/6}\delta -768\times 6^{2/3}X_1^{(1+\delta )/3}\alpha \delta +48\times 6^{2/3}X_1^{(1+\alpha )/3}\alpha ^2\delta \nonumber \\& \qquad+900\times 6^{2/3}X_1^{(1+\delta )/3}\delta ^2 47\times 2^{(6+2\delta )/3}\times 3^{2\delta /3}X_1^{(\alpha +2\delta )/6}\delta ^2 -192\times 6^{2/3}X_1^{(1+\delta )/3}\alpha \delta ^2+12\times 6^{2/3}X_1^{(1+\delta )/3}\alpha ^2\delta ^2, \end{aligned}$$

(60)

$$\begin{aligned} \Psi _2 & = -1175\times 2^{2\delta /3}\times 3^{(3+2\delta )/3}X_1^{2/3}-6075\times 6^{2/3}X_1^{(1+\delta )/3} +47\times 2^{2\delta /3}\times 3^{(9+2\delta )/3}X_1^{(\alpha +2\delta )/6}\nonumber \\& \qquad +47\times 2^{(12+2\delta )/3}\times 3^{2\delta /3}X_1^{2/3}\alpha -1296\times 6^{2/3}X_1^{(1+\delta )/3}\alpha -47\times 6^{2\delta /3}X_1^{2/3}\alpha ^2+81\times 6^{2/3}X_1^{(1+\delta )/3}\alpha ^2\nonumber \\& \qquad +3600\times 6^{2/3}X_1^{(1+\delta )/3}\delta +47\times 2^{(12+2\delta )/3}\times 3^{2\delta /3}X_1^{(\alpha +2\delta )/6}\delta - 768\times 6^{2/3}X_1^{(1+\delta )/3}\alpha \delta +48\times 6^{2/3}X_1^{(1+\delta )/3}\alpha ^2\delta \nonumber \\& \qquad+900\times 6^{2/3}X_1^{(1+\delta )/3}\delta ^2+47\times 2^{(6+2\delta )/3}\times 3^{2\delta /3}X_1^{(\alpha +2\delta )/6}\delta ^2 -192\times 6^{2/3}X_1^{(1+\delta )/3}\alpha \delta ^2+12\times 6^{2/3}X_1^{(1+\delta )/3}\alpha ^2\delta ^2]. \end{aligned}$$

(61)

The term $\Upsilon$ present in Eq. (40) is given by

$$\begin{aligned} \Upsilon & = -38\times 3^{9+\delta }X_2^{3/2}(-1)^{-2\delta }+9565938X_2^{(2+\delta )/2}\delta +134631720X_2^{(4+\delta )/2}\delta -612009X_2^{(3+\delta )/2}\beta _2\delta \nonumber \\&\qquad -1417176X_2^{(2+\delta )/2}\delta ^2(1+\delta )-19945440X_2^{(4+\delta )/2}\delta ^2(1+\delta )+90668\beta _2X_2^{(3+\delta )/2}\delta ^2(1+\delta ) -858762X_2^{(3+\delta )/2}\beta _2\delta \ln (\frac{3}{\sqrt{X_2}})\nonumber \\&\qquad+127224X_2^{(3+\delta )/2}\beta _2\delta ^2\ln (\frac{3}{\sqrt{X_2}})(1+\delta )-373977X_2^{(3+\delta )/2}\beta _2\delta \ln (\frac{3}{\sqrt{X_2}})^2 +55404X_2^{(3+\delta )/2}\beta _2\delta ^2\ln (\frac{3}{\sqrt{X_2}})^2(1+\delta ). \end{aligned}$$

(62)

The term $\Pi _1$ present in Eq. (44) is defined by the relation:

$$\begin{aligned} \Pi _1 & = 413\times 2^{\frac{1+8\delta }{3}}\times 3^{\frac{1+2\delta }{3}}X_1-3835\times 2^{1+\frac{8\delta }{3}}\times 3^{\frac{2\delta }{3}}X_1^{5/3} +17472\times 6^{1/3}(-1)^{-2\delta }{a_0}^{4-2\delta }B{X_1}^{\delta /3}\nonumber \\&\qquad -7\times 2^{\frac{7+8\delta }{3}}\times 3^{\frac{7+2\delta }{3}}\delta +65\times 2^{3+\frac{8\delta }{3}}\times 3^{2+\frac{2\delta }{3}} X_1^{5/3}\delta +7\times 2^{\frac{7+8\delta }{3}}\times 3^{\frac{1+2\delta }{3}}\delta ^2-65\times 2^{3+\frac{8\delta }{3}}\times 3^{2\delta /3}X_1^{5/3}\delta ^2. \end{aligned}$$

(63)

The term $\Xi$ present in Eq. (47) is given by

$$\begin{aligned} \Xi & = \left( 53\times 2^{2+2\delta }\times 3^{3+\delta }X_2^{3/2}+5035\times \times 2^{4+2\delta }\times 3^{\delta } X_2^{5/2}+73872(-1)^{-2\delta }a_0^{4-2\delta }BX_2^{\delta /2}\right. \nonumber \\&\qquad -\left. 197\times 2^{2\delta }\times 3^{3+\delta }X_2^{3/2}\delta -18715\times 2^{2+2\delta } \times 3^{\delta }X_2{5/2}\delta +13\times 2^{2+2\delta }3^{3+\delta }X_2^{3/2}\delta ^2+1235\times 2^{4+2\delta }3^{\delta }X_2^{5/2}\delta ^2\right. \nonumber \\& \qquad -\left. 2^{2+2\delta }\times 3^{3+\delta }X_2^{3/2}\delta ^3-95\times 2^{4+2\delta }\times 3^{\delta }X_2^{5/2}\delta ^3\right) \left( (-3+2\sqrt{7})(-1+2\sqrt{7})(1+2\sqrt{7})(3+2\sqrt{7})\right. \nonumber \\& \qquad\times \left. X_2(9+2\sqrt{7}-2\delta )(-4+\delta )(-9+2\sqrt{7}+2\delta )\right) ^{-1} \end{aligned}$$

(64)