Qualitative stability analysis of cosmological parameters in f(T, B) gravity

Abstract

Abstract We analyze the cosmological solutions of f ( T , B ) gravity using dynamical system analysis where T is the torsion scalar and B be the boundary term scalar. In our work, we assume three specific cosmological models. For first model, we consider $$ f(T,B)=f_{0}(B^{k}+T^{m})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where k and m are constants. For second model, we consider $$f(T,B)=f_{0}T B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>T</mml:mi> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> , for third model, we consider $$f(T,B)=\alpha T^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . We generate an autonomous system of differential equations for each models by introducing new dimensionless variables. To solve this system of equations, we use dynamical system analysis. We also investigate the critical points and their natures, stability conditions and their behaviors of Universe expansion. For first and second models, we get two stable critical points, while for third model we get one stable critical point. The phase plots of this system are analyzed in detail and study their geometrical interpretations also. For these three models, we evaluated density parameters such as $$\Omega _{r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> , $$\Omega _{m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> , $$\Omega _{\Lambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>Λ</mml:mi> </mml:msub> </mml:math> and $$\omega _{eff}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> </mml:msub> </mml:math> and deceleration parameter ( q ) and find their suitable range of the parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> for stability. For first model, we get $$\omega _{eff}=-0.833,-0.166$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>0.833</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>0.166</mml:mn> </mml:mrow> </mml:math> and for second model, we get $$\omega _{eff}=-\frac{1}{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> . This shows that both the models are in quintessence phase. For third model we get accelerated expansion of the Universe. Further, we compare the values of EoS parameter and deceleration parameter with the observational values.

Topics & Concepts