<p>We explore the dynamical behavior of two <i>f</i>(<i>T</i>,&#xa0;<i>B</i>) gravity models with a scalar field: 1. <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq1.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="188" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(T,B)=T-\gamma log\bigg [\frac{\psi B_{0}}{B}\bigg ]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>T</mi> <mo>-</mo> <mi>γ</mi> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">[</mo> </mrow> <mfrac> <mrow> <mi>ψ</mi> <msub> <mi>B</mi> <mn>0</mn> </msub> </mrow> <mi>B</mi> </mfrac> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and 2. <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(T,B)=\eta T+\frac{\zeta }{B^{n}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>η</mi> <mi>T</mi> <mo>+</mo> <mfrac> <mi>ζ</mi> <msup> <mi>B</mi> <mi>n</mi> </msup> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, using the potential <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">\(V(\phi )=V_{0}(\alpha +e^{-\beta \phi })^{-\delta }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>β</mi> <mi>ϕ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mi>δ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and an interaction term <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar{Q} = \epsilon H \dot{\phi }^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>Q</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>=</mo> <mi>ϵ</mi> <mi>H</mi> <msup> <mover accent="true"> <mi>ϕ</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. A phase space analysis reveals four fixed points in Model <InternalRef RefID="Sec4">3.1</InternalRef> (three stable, one saddle) and five in Model <InternalRef RefID="Sec5">3.2</InternalRef> (four stable), indicating transitions from matter to dark energy dominance. With interaction, Model <InternalRef RefID="Sec5">3.2</InternalRef> exhibits seven fixed points, including five stable, one unstable (stiff matter era) and one saddle point. Evolution of the deceleration parameter <i>q</i> and the total EoS parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega _{tot}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mrow> <mi mathvariant="italic">tot</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> confirms sustained cosmic acceleration, with present values <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(q_{0} = -1.005\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mn>1.005</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega _{0} = -0.556\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mn>0.556</mn> </mrow> </math></EquationSource> </InlineEquation> (Model <InternalRef RefID="Sec4">3.1</InternalRef>) and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(q_{0} = -1.245\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mn>1.245</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega _{0}=-1.0404\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mn>1.0404</mn> </mrow> </math></EquationSource> </InlineEquation> (Model <InternalRef RefID="Sec5">3.2</InternalRef>). Comparisons of our observationally constrained parameters with <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10701_2025_856_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>CDM show strong consistency, supporting the viability of these models in describing the late-time accelerated expansion of the Universe.</p>

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Cosmological Dynamics and Stability Analysis in f(TB) Gravity with Interacting Scalar Field

  • Amit Samaddar,
  • S. Surendra Singh

摘要

We explore the dynamical behavior of two f(TB) gravity models with a scalar field: 1. \(f(T,B)=T-\gamma log\bigg [\frac{\psi B_{0}}{B}\bigg ]\) f ( T , B ) = T - γ l o g [ ψ B 0 B ] and 2. \(f(T,B)=\eta T+\frac{\zeta }{B^{n}}\) f ( T , B ) = η T + ζ B n , using the potential \(V(\phi )=V_{0}(\alpha +e^{-\beta \phi })^{-\delta }\) V ( ϕ ) = V 0 ( α + e - β ϕ ) - δ and an interaction term \(\bar{Q} = \epsilon H \dot{\phi }^2\) Q ¯ = ϵ H ϕ ˙ 2 . A phase space analysis reveals four fixed points in Model 3.1 (three stable, one saddle) and five in Model 3.2 (four stable), indicating transitions from matter to dark energy dominance. With interaction, Model 3.2 exhibits seven fixed points, including five stable, one unstable (stiff matter era) and one saddle point. Evolution of the deceleration parameter q and the total EoS parameter \(\omega _{tot}\) ω tot confirms sustained cosmic acceleration, with present values \(q_{0} = -1.005\) q 0 = - 1.005 and \(\omega _{0} = -0.556\) ω 0 = - 0.556 (Model 3.1) and \(q_{0} = -1.245\) q 0 = - 1.245 and \(\omega _{0}=-1.0404\) ω 0 = - 1.0404 (Model 3.2). Comparisons of our observationally constrained parameters with \(\Lambda \) Λ CDM show strong consistency, supporting the viability of these models in describing the late-time accelerated expansion of the Universe.