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The study of accelerating DE models in Saez–Ballester theory of gravitation

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Abstract

The investigation in the present work is carried out on the spatially homogeneous and anisotropic axially symmetric space-time in the presence of two fluids, one being the pressureless matter and other being the different kinds of holographic dark energy (HDE). Eventually, in the present work, cosmological models with Tsallis HDE, Renyi HDE, Sharma–Mittal HDE by taking Hubble radius as infrared (IR) cutoff (\(L=H^{-1}\)) are obtained. The geometrical and matter parts of space-time are solved within the Saez–Ballester scalar-tensor theory of gravitation. Interestingly, in this study a time varying deceleration parameter (q) which exhibits a transition from deceleration to acceleration phase is obtained without assuming any scale factor. In the present work, the study of cosmic expansion is done through the scalar field (\(\phi\)) and various cosmological parameters like EoS, deceleration, statefinder, etc. The EoS parameter exhibits quintom-like behavior for Tsallis HDE model, the transition from matter-dominated phase to phantom phase for Renyi HDE model, whereas it shows quintessence behavior for Sharma–Mittal HDE. The stability analysis for three models is studied through the squared speed of sound (\(v_s^2\)). For all redshift (z) values \(v_s^2>0\), the Sharma–Mittal HDE model is stable throughout the universe’s expansion. In this work, the obtained results match with recent observational data.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

The authors are thankful to Dr. K. Sri Kavya, MVGR College of Engineering, Vizianagaram for her valuable suggestions rendered during this work. Also, the author Vinutha Tummala would like to thank the authorities of the Inter-University Centre for Astronomy and Astrophysics, Pune, India, for providing the research facilities. The authors are much delighted to thank the honorable editor and anonymous reviewer for their valuable suggestions and useful comments which helped us a lot to improve this paper in terms of quality as well as presentation.

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  1. Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India

    T. Vinutha & K. Venkata Vasavi

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  1. T. Vinutha
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Appendix

Appendix

$$\begin{aligned}{} & {} \iota _3=\frac{216\pi (-1+l)exp(-3\gamma _1t)\left( c_1exp(\gamma _1t)+c_2\gamma _1\right) \bigg (c_1^2exp(2\gamma _1t)(\delta \pi +\frac{\gamma _1^2}{9})+2\gamma _1c_2\pi \delta \bigg (c_1exp(\gamma _1t)+ \frac{c_2\gamma _1}{2}\bigg )\bigg )}{c^2\gamma _1^2c_1^3(2l+1)^2} \\{} & {} \iota _4=\frac{4\gamma _1c_2exp(-\gamma _1t) \bigg (c_1^2exp(2\gamma _1t)\bigg (\delta \pi +\frac{\gamma _1^2}{18}\bigg )+2\gamma _1c_2\pi \delta \bigg (c_1exp( \gamma _1t)+\frac{c_2\gamma _1}{2}\bigg )\bigg )}{c_1\bigg (c_1^2exp(2\gamma _1t)\bigg (\delta \pi +\frac{\gamma _1^2}{9}\bigg )+2\gamma _1c_2\pi \delta \bigg (c_1exp(\gamma _1t)+\frac{c_2\gamma _1}{2}\bigg )\bigg )} \\{} & {} \iota _5=-\frac{216\pi R(-1+l)exp(-3\gamma _1t)\bigg (c_1exp(\gamma _1t)+c_2\gamma _1\bigg )^3}{c^2\gamma _1^2c_1^3(2l+1)^2(\iota _7-1)} \\{} & {} \iota _6=-\frac{1}{(\iota _7-1)c_1\bigg (c_1^2exp(2\gamma _1t)\bigg (\delta \pi +\frac{\gamma _1^2}{9}\bigg )+2\gamma _1c_2 \pi \delta \bigg (c_1exp(\gamma _1t)+\frac{c_2\gamma _1}{2}\bigg )\bigg )}\\{} & {} \qquad 2\gamma _1c_2\bigg (\bigg (\pi c_2^2\gamma _1^2(R-2\delta )exp(-\gamma _1t)+c_1\bigg (c_1exp(\gamma _1t)\bigg ( \frac{-2\gamma _1^2}{9}+\pi (R-2\delta )\bigg )+2\pi \gamma _1c_2(R-2\delta )\bigg )\bigg )\iota _7\\{} & {} \qquad +2\pi c_2^2\delta \gamma _1^2exp(-\gamma _1t)+2c_1\bigg (c_1exp(\gamma _1t)\bigg (\delta \pi +\frac{\gamma _1^2}{9}\bigg )+2\gamma _1c_2\pi \delta \bigg )\bigg ) \\{} & {} \iota _7=\bigg (\frac{\big (9\pi exp(2\gamma _1t)c_1^2\delta +18\pi exp(\gamma _1t)c_1c_2\delta \gamma _1+9\pi c_2^2\delta \gamma _1^2+c_1^2exp(2\gamma _1t)\gamma _1^2\big )exp(-2\gamma _1t)}{c_1^2\gamma _1^2}\bigg )^{R/\delta } \end{aligned}$$

By using Eqs. (32), (37), (42), (45), (46), (47) and its derivatives in (50), we obtain the expressions for \(v_s^2\). The following equations (62), (63), (64) represent \(v_s^2\) for Tsallis, Renyi, Sharma–Mittal HDE models, respectively. These are as follows

$$\begin{gathered} v_{s}^{2} = \frac{{2\exp ( - \gamma _{1} t)}}{{\alpha c_{1}^{3} \gamma _{1}^{2} ( - 2 + \delta )\left( {l + \frac{1}{2}} \right)^{2} }}\left( {\frac{{ - 819^{{ - \delta }} ( - 1 + l)\exp ( - 2\gamma _{1} t)(c_{1} \exp (\gamma _{1} t) + c_{2} \gamma _{1} )^{3} (3H)^{{2\delta }} }}{{16}}} \right. \hfill \\ \left. {\quad \quad \, + \alpha c_{1}^{2} \gamma _{1}^{3} c_{2} ( - 2 + \delta )\left( {\delta - 32)\left( {l + \frac{1}{2}} \right)^{2} } \right)} \right) \hfill \\ \end{gathered}$$
(62)
$$\begin{aligned}{} & {} v_s^2=\frac{\iota _8}{\iota _9} \end{aligned}$$
(63)
$$\begin{aligned}{} & {} v_s^2= \frac{1}{\iota _{10}}\iota _{11} \end{aligned}$$
(64)

where

$$\begin{aligned} \iota _{8} & = - 3exp( - 3\gamma _{1} t)\left( {\gamma _{1} c_{1}^{6} \left( {\gamma _{1}^{6} \left( {\left( {\pi \left( {\frac{1}{{162}} - \frac{l}{{162}}} \right) + \frac{{c^{2} (l + \frac{1}{2})^{2} }}{{486}}} \right) + \frac{{\delta \pi \gamma _{1}^{4} \left( {\pi ( - 9l + 9) + c^{2} (l + \frac{1}{2})^{2} } \right)}}{{18}}} \right.} \right.} \right. \hfill \\ & \left. {\qquad + \delta ^{2} \pi ^{2} \gamma _{1}^{2} \left( {\left( {\frac{{15}}{2} - \frac{{15l}}{2}} \right)\pi + c^{2} \left( {l + \frac{1}{2}^{2} } \right)} \right) - \frac{{63\pi ^{4} \delta ^{3} ( - 1 + l)}}{2}} \right)c_{2} exp(6\gamma _{1} t) + \left( {6\gamma _{1}^{3} c_{1}^{4} \left( {\left( {\pi \left( { - \frac{l}{{36}} + \frac{1}{{36}}} \right)} \right.} \right.} \right. \hfill \\ & \left. {\left. {\qquad + \frac{{c^{2} \left( {l + \frac{1}{2}} \right)^{2} }}{{108}}} \right)\gamma _{1}^{4} + \delta \pi \gamma _{1}^{2} \left( {\pi \left( {\frac{{ - 5l}}{2} + \frac{5}{2}} \right) + c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right) - \frac{{105\pi ^{3} \delta ^{2} ( - 1 + l)}}{4}} \right)\delta c_{2}^{3} exp(4\gamma _{1} t) \hfill \\ & \left. {\qquad + 4\gamma _{1}^{2} \left( {\left( {\pi \left( { - \frac{l}{8} + \frac{1}{8}} \right) + \frac{{c^{2} \left( {l + \frac{1}{2}} \right)^{2} }}{{36}}} \right)\gamma _{1}^{4} + \delta \pi \gamma _{1}^{2} \left( {\pi \left( { - \frac{{15l}}{4} + \frac{{15}}{4}} \right) + c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right) - \frac{{189\pi ^{3} \delta ^{2} ( - 1 + l)}}{8}} \right)} \right) \hfill \\ & \qquad _{1}^{5} \delta c_{2}^{2} exp(5\gamma _{1} t) + \gamma _{1}^{5} c_{1}^{2} \left( {\left( {\pi \left( {\frac{3}{2} - \frac{{3l}}{2}} \right) + c^{2} (l + 1/2)^{2} } \right)\gamma _{1}^{2} - \frac{{189\delta \pi ^{2} ( - 1 + l)}}{2}} \right)\delta ^{2} \pi c_{2}^{5} exp(2\gamma _{1} t) \hfill \\ & \qquad - \frac{{9\left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right)^{3} c_{1}^{7} ( - 1 + l)exp(7\gamma _{1} t)}}{2} + 4\gamma _{1}^{4} \delta ^{2} \pi c_{2}^{4} \left( {c_{1}^{3} \left( {\left( {\pi \left( {\frac{{ - 15l}}{8} + \frac{{15}}{8}} \right) + c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right)\gamma _{1}^{2} } \right.} \right. \hfill \\ & \left. {\left. {\left. {\left. {\qquad - \frac{{315\delta \pi ^{2} ( - 1 + l)}}{8}} \right)exp(3\gamma _{1} t) - \frac{{63\gamma _{1}^{2} \left( {c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{7}} \right)( - 1 + l)\delta \pi ^{2} c_{2}^{2} }}{8}} \right)} \right)\pi } \right) \hfill \\ \iota _{9} & = \gamma _{1}^{2} c^{2} c_{1}^{3} \left( {l + \frac{1}{2}} \right)^{2} \left( {c_{1}^{2} exp(2\gamma _{1} t)\left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right) + 2\gamma _{1} \delta \pi c_{2} \left( {c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{2}} \right)} \right)\left( {c_{1}^{2} \left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{{18}}} \right)exp(2\gamma _{1} t)} \right. \hfill \\ & \left. {\qquad + 2\gamma _{1} \left( {c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{2}} \right)\delta \pi c_{2} } \right) \hfill \\ \iota _{{10}} & = 2c^{2} \left( {\frac{{c_{1}^{2} \left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right)exp(2\gamma _{1} t)}}{2} + \gamma _{1} (c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{2})\delta \pi c_{2} } \right)\left( {l + \frac{1}{2}} \right)^{2} \gamma _{1}^{2} \left( {\frac{{c_{1}^{2} \left( {\frac{{ - 2\gamma _{1}^{2} }}{9} + \pi (R - 2\delta )} \right)exp(2\gamma _{1} t)}}{2}} \right. \hfill \\ & \left. {\left. {\qquad + \pi \gamma _{1} \left( {c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{2}} \right)(R - 2\delta )c_{2} } \right)\iota _{7} + c_{1}^{2} \left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right)exp(2\gamma _{1} t) + 2\gamma _{1} \left( {c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{2}} \right)\delta \pi c_{2} } \right) \hfill \\ \iota _{{11}} & = - \left( {\left( { - \left( {c_{1}^{4} exp(6\gamma _{1} t)\left( {\frac{{\gamma _{1}^{4} }}{{27}} - \frac{{5\pi \left( {R - \frac{{12\delta }}{5}} \right)\gamma _{1}^{2} }}{{18}} + \pi ^{2} (R - 2\delta )\left( {R - \frac{{3\delta }}{2}} \right)} \right) + \pi \left( {6\left( {\gamma _{1}^{2} \left( {\frac{{ - 5R}}{{108}} + \frac{\delta }{9}} \right)} \right.} \right.} \right.} \right.} \right. \hfill \\ & \left. {\qquad + \pi \left( {R - \frac{{3\delta }}{2}} \right)(R - 2\delta )} \right)\gamma _{1} c_{2} c_{1}^{2} exp(4\gamma _{1} t) + 4\left( {\gamma _{1}^{2} \left( {\frac{{ - 5R}}{{36}} + \frac{\delta }{3}} \right) + \pi (R - 2\delta )\left( {R - \frac{{3\delta }}{2}} \right)} \right)c_{1}^{3} exp(5\gamma _{1} t) \hfill \\ & \left. {\left. {\qquad + \left( {R - \frac{{3\delta }}{2}} \right)\pi \left( {exp(2\gamma _{1} t)c_{2} \gamma _{1} + 4c_{1} exp(3\gamma _{1} t)} \right)\gamma _{1}^{2} (R - 2\delta )c_{2}^{2} } \right)\gamma _{1} c_{2} } \right)c^{2} \left( {l + \frac{1}{2}} \right)^{2} \gamma _{1}^{3} c_{2} c_{1}^{2} \iota _{7} \hfill \\ & \qquad + \left( {\gamma _{1} c_{2} c_{1}^{6} exp(6\gamma _{1} t)\left( {\frac{{2c^{2} \gamma _{1}^{6} \left( {l + \frac{1}{2}} \right)^{2} }}{{27}} - \frac{{5\pi \gamma _{1}^{4} \left( {\frac{{ - 24\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{5} + R\left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{9l}}{5} + \frac{9}{5}} \right)} \right)}}{{18}}} \right.} \right. \hfill \\ & \left. {\qquad + \pi ^{2} \left( {6c^{2} \left( {l + \frac{1}{2}} \right)^{2} \delta ^{2} - \frac{{7R\left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{30l}}{7} + \frac{{30}}{7}} \right)\delta }}{2} + R^{2} c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right)\gamma _{1}^{2} + \frac{{189\pi ^{3} R\delta ^{2} ( - 1 + l)}}{2}} \right) \hfill \\ & \qquad + \left( {6\gamma _{1}^{3} c_{2}^{3} \left( {\gamma _{1}^{4} \left( {\frac{{2\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{9} - \frac{{5R\left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{3l}}{5} + \frac{3}{5}} \right)}}{{108}}} \right) + \left( {6c^{2} \left( {l + \frac{1}{2}} \right)^{2} \delta ^{2} } \right.} \right.} \right. \hfill \\ & \left. {\left. {\qquad - \frac{{7R\delta \left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{10l}}{7} + \frac{{10}}{7}} \right)}}{2} + R^{2} c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right)\pi \gamma _{1}^{2} + \frac{{315\pi ^{2} R\delta ^{2} ( - 1 + l)}}{4}} \right)c_{1}^{4} exp(4\gamma _{1} t) \hfill \\ &\qquad + 4\left( {\gamma _{1}^{4} \left( {\frac{{2\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{3} - \frac{{5R\left( {c^{2} (l + 1/2)^{2} - \frac{{9l}}{{10}} + \frac{9}{{10}}} \right)}}{{36}}} \right) + \pi \left( {6c^{2} \left( {l + \frac{1}{2}} \right)^{2} \delta ^{2} } \right.} \right. \hfill \\ & \left. {\left. {\qquad - \frac{{7R\left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{15l}}{7} + \frac{{15}}{7}} \right)\delta }}{2} + R^{2} c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right)\gamma _{1}^{2} + \frac{{567\pi ^{2} R\delta ^{2} ( - 1 + l)}}{8}} \right)\gamma _{1}^{2} c_{2}^{2} c_{1}^{5} exp(5\gamma _{1} t) \hfill \\ & \qquad + \left( {\left( {6c^{2} \left( {l + \frac{1}{2}} \right)^{2} \delta ^{2} - \frac{{7\left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{6l}}{7} + \frac{6}{7}} \right)R\delta }}{2} + R^{2} c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right)\gamma _{1}^{2} + \frac{{567\pi R\delta ^{2} ( - 1 + l)}}{2}} \right)\pi \gamma _{1}^{5} c_{2}^{5} c_{1}^{2} \hfill \\ & \qquad exp(2\gamma _{1} t) + 4\pi \gamma _{1}^{4} c_{2}^{4} c_{1}^{3} exp(3\gamma _{1} t)\gamma _{1}^{2} \left( {6c^{2} \left( {l + \frac{1}{2}} \right)^{2} \delta ^{2} - \frac{{7R\left( {c^{2} \left( {l + \frac{1}{2}} \right)^{2} - \frac{{15l}}{{14}} + \frac{{15}}{{14}}} \right)\delta }}{2} + R^{2} c^{2} \left( {l + \frac{1}{2}} \right)^{2} } \right) \hfill \\ & \qquad \left. {\left. {\left. { + \frac{{945\pi R\delta ^{2} ( - 1 + l)}}{8}} \right) + \frac{{189( - 1 + l)R\left( {\frac{{\left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right)^{2} c_{1}^{7} exp(7\gamma _{1} t)}}{7} + \pi ^{2} c_{2}^{6} \delta ^{2} \gamma _{1}^{6} \left( {c_{1} exp\gamma _{1} t + \frac{{c2\gamma _{1} }}{7}} \right)} \right)}}{2}} \right)\pi } \right)\iota _{7} \hfill \\ & \qquad - \frac{1}{2}\left( {189\gamma _{1} \left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right)c_{2} c_{1}^{6} exp(6\gamma _{1} t)\left( {\frac{{(2c^{2} (l + \frac{1}{2})^{2} \gamma _{1}^{4} )}}{{567}} + \frac{{\pi \gamma _{1}^{2} \left( {\frac{{2\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{3} + R( - 1 + l)} \right)}}{{21}} + \pi ^{2} R\delta ( - 1 + l)} \right)} \right) \hfill \\ & \qquad - \frac{1}{2}189\pi \left( {5\left( {\gamma _{1}^{4} \left( {\frac{{4\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{{2835}} + \frac{{R( - 1 + l)}}{{2835}}} \right) + \frac{{4\delta \left( {\frac{{3\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{5} + R( - 1 + l)} \right)\pi \gamma _{1}^{2} }}{{63}} + \pi ^{2} R\delta ^{2} } \right.} \right. \hfill \\ & \qquad \left. {( - 1 + l)} \right)\gamma _{1}^{3} c_{2}^{3} c_{1}^{4} exp(4\gamma _{1} t) + 3\gamma _{1}^{2} c_{2}^{2} c_{1}^{5} exp(5\gamma _{1} t)\left( {\gamma _{1}^{4} \left( {\frac{{8\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{{1701}} + \frac{{R( - 1 + l)}}{{567}}} \right)} \right. \hfill \\ & \left. {\qquad + \frac{{20\delta \pi \left( {\frac{{2\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{5} + R( - 1 + l)} \right)\gamma _{1}^{2} }}{{189}} + \pi ^{2} R\delta ^{2} ( - 1 + l)} \right) + 3\delta \pi \gamma _{1}^{5} c_{2}^{5} c_{1}^{2} exp(2\gamma _{1} t) \hfill \\ & \qquad \left( {\gamma _{1}^{2} \left( {\frac{{2\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{{189}} + \frac{{2R( - 1 + l)}}{{189}}} \right) + \pi R\delta ( - 1 + l)} \right) + 5\delta \pi \gamma _{1}^{4} c_{2}^{4} c_{1}^{3} \hfill \\ & \qquad \left( {\gamma _{1}^{2} \left( {\frac{{8\delta \left( {l + \frac{1}{2}} \right)^{2} c^{2} }}{{315}} + \frac{{2R( - 1 + l)}}{{63}}} \right) + \pi R\delta ( - 1 + l)} \right)exp(3\gamma _{1} t) + ( - 1 + l)R \hfill \\ & \qquad \left. {\left. {\left. {\left. {\frac{{\left( {\delta \pi + \frac{{\gamma _{1}^{2} }}{9}} \right)^{2} c_{1}^{7} exp(7\gamma _{1} t)}}{7} + \pi ^{2} c_{2}^{6} \delta ^{2} \gamma _{1}^{6} \left( {c_{1} exp(\gamma _{1} t) + \frac{{c_{2} \gamma _{1} }}{7}} \right)} \right)} \right)} \right)} \right)exp( - 3\gamma _{1} t) \end{aligned}$$

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Vinutha, T., Vasavi, K.V. The study of accelerating DE models in Saez–Ballester theory of gravitation. Eur. Phys. J. Plus 137, 1294 (2022). https://doi.org/10.1140/epjp/s13360-022-03477-x

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