Exact Solutions

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In a row of the problems below [following the paper Marco A. Reyes, On exact solutions to the scalar field equations in standard cosmology, arXiv: 0806.2292] we present a simple algebraic method to find exact solutions for a wide variety of scalar field potentials. Let us consider the function $V_a(\varphi)$ defined as \begin{equation}\label{es_1} V_a(\varphi)\equiv V(\varphi)+\frac12\dot\varphi^2. \end{equation} Derivative of this function reads \[\frac{dV_a}{d\varphi}=\frac{dV}{d\varphi}+\ddot\varphi.\] Hence, equations \[H^2=\frac12\left(\frac12\dot\varphi^2+V(\varphi)\right),\] \[\ddot\varphi+3H\dot\varphi+\frac{dV}{d\varphi}=0\] can now be rewritten as \begin{align} \label{es_4}3H^2 & =V_a,\\ \label{es_5}3H\dot\varphi & =-\frac{dV_a}{d\varphi}. \end{align} To solve them, note that eq.(\ref{es_4}) defines $H$ as a function of $\varphi$, which when inserted into eq.(\ref{es_5}), gives the scalar field $\varphi(t)$ as a function of $t$, at least in quadratures \[-3H(\varphi)\left(\frac{dV_a}{d\varphi}\right)^{-1}d\varphi=dt.\] Finally, inserting $\varphi(t)$ into eqs.(\ref{es_1}) and (\ref{es_4}) gives $V(\varphi)$ and $a(t)$, respectively, and the solution is completed.

One could also use $H(t)$ to determine $\varphi(t)$, since \[\dot H=-\frac12\dot\varphi^2.\] implies that \[\Delta\varphi(t)=\pm\int\sqrt{-2\dot H}dt.\] Since $V_a(t)=3H^2(t)$, a complete knowledge of $H(t)$ fully determines the solution to the problem.

Problem 1

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For $H(t)=\alpha/t$ find $V(\varphi)$ and $\Delta\varphi(t)$.

Problem 2

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Reconstruct $V(\varphi)$ and find $\varphi(t)$ and $a(t)$ for $V_a=\lambda\varphi^2$.

Problem 3

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Reconstruct $V(\varphi)$ and find $\varphi(t)$ and $a(t)$ for $V_a=\lambda\varphi^4$.

Problem 4

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Reconstruct $V(\varphi)$ and find $\varphi(t)$ and $a(t)$ for $V_a=\lambda\varphi^n$, $n>2$.