Averaging generalized scalar-field cosmologies III: Kantowski–Sachs and closed Friedmann–Lemaître–Robertson–Walker models

Abstract

<jats:p>Scalar-field cosmologies with a generalized harmonic potential and matter with energy density <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\rho _m$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msub>\n <mml:mi>ρ</mml:mi>\n <mml:mi>m</mml:mi>\n </mml:msub>\n </mml:math></jats:alternatives></jats:inline-formula>, pressure <jats:inline-formula><jats:alternatives><jats:tex-math>$$p_m$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mi>m</mml:mi>\n </mml:msub>\n </mml:math></jats:alternatives></jats:inline-formula>, and barotropic equation of state (EoS) <jats:inline-formula><jats:alternatives><jats:tex-math>$$p_m=(\\gamma -1)\\rho _m, \\; \\gamma \\in [0,2]$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mi>m</mml:mi>\n </mml:msub>\n <mml:mo>=</mml:mo>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>γ</mml:mi>\n <mml:mo>-</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>ρ</mml:mi>\n <mml:mi>m</mml:mi>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:mspace />\n <mml:mi>γ</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mrow>\n <mml:mo>[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>]</mml:mo>\n </mml:mrow>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> in Kantowski–Sachs (KS) and closed Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function <jats:italic>D</jats:italic>. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for <jats:inline-formula><jats:alternatives><jats:tex-math>$$0\\le \\gamma \\le 2$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>≤</mml:mo>\n <mml:mi>γ</mml:mi>\n <mml:mo>≤</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>, and flat FLRW matter-dominated universe if <jats:inline-formula><jats:alternatives><jats:tex-math>$$0\\le \\gamma \\le \\frac{2}{3}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>≤</mml:mo>\n <mml:mi>γ</mml:mi>\n <mml:mo>≤</mml:mo>\n <mml:mfrac>\n <mml:mn>2</mml:mn>\n <mml:mn>3</mml:mn>\n </mml:mfrac>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for <jats:inline-formula><jats:alternatives><jats:tex-math>$$0\\le \\gamma \\le \\frac{2}{3}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>≤</mml:mo>\n <mml:mi>γ</mml:mi>\n <mml:mo>≤</mml:mo>\n <mml:mfrac>\n <mml:mn>2</mml:mn>\n <mml:mn>3</mml:mn>\n </mml:mfrac>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> as in KS and Einstein–de Sitter solution for <jats:inline-formula><jats:alternatives><jats:tex-math>$$0\\le \\gamma <1$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>≤</mml:mo>\n <mml:mi>γ</mml:mi>\n <mml:mo><</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein–Gordon (KG) equation can be controlled and smoothed out when <jats:italic>D</jats:italic> goes monotonically to zero, and incidentally for the whole <jats:italic>D</jats:italic>-range for KS and closed FLRW (if <jats:inline-formula><jats:alternatives><jats:tex-math>$$0\\le \\gamma < 1$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>≤</mml:mo>\n <mml:mi>γ</mml:mi>\n <mml:mo><</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>) too. However, for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\gamma \\ge 1$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>γ</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> closed FLRW solutions of the full system depart from the solutions of the averaged system as <jats:italic>D</jats:italic> is large. Our results are supported by numerical simulations.</jats:p>