Logolinear series expansions with applications to primordial cosmology

We develop a method for computing series expansions for solutions to ordinary differential equations when the asymptotic form contains both linear and logarithmic terms. Such situations are common in primordial cosmology when considering series expansions out of a singularity in the equations arising from a preinflationary phase of the universe. We develop mathematical techniques for generating these series expansions, and we apply them to polynomial and Starobinsky inflationary potentials with kinetic initial conditions. Code for analytic and numerical computation of logolinear series is provided on GitHub.

Figure 1

Inflation using $V=\frac{1}{2}{m}^2{\varphi }^2$. We see generic features of kinetic initial conditions; the universe initially expands out of a singularity in kinetic dominance with ${\dot{\varphi }}^2\gg V$, $a\propto t^{1/3}$. At later times, the potential comes to dominate ${\dot{\varphi }}^2\ll V$, and a period of inflation begins at $t\sim m^{-1}$. Inflation then exits as the field $\varphi$ oscillates about the minimum of the potential with the scale factor consequently expanding as some power law. For this example, we choose $m={10}^{-5}{m}_{\mathrm{p}}$, consistent with current observations. As initial conditions at $t=1$ we set ${\varphi }_{\mathrm{p}}=-23m_{\mathrm{p}}$, $N_{\mathrm{p}}=0$, in order to give 50–60 $e$-folds of inflation, and choose a slightly closed universe, $b=2\times {10}^{-6}$. Using only the first term in the kinetic dominance solution in order to start the numerical integration at $t=m_{\mathrm{p}}^{-1}$ gives the dotted line, while using the correction for curvature provided by the next term proves sufficient to match the true solution, which is indistinguishable from the solid curve.

Figure 2

Absolute error in the expansion of $\varphi$ for differing numbers of terms. Before $t\lesssim 17000m_{\mathrm{p}}$ the expansion becomes more accurate as the number of terms is increased, whereas for $t\gtrsim 17000m_{\mathrm{p}}$ the reverse is true. We use $V(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$ with $m={10}^{-5}{m}_{\mathrm{p}}$, $b=0$, ${\varphi }_{\mathrm{p}}=-23m_{\mathrm{p}}$ on the positive branch and compare to a numerical solution defined by using the highest order approximation at $t=0$ as initial conditions. Plots for $H$ and $N$ are similar in quality.

Figure 3

The Starobinsky potential for $R+R^2$ inflation in the Einstein frame. Its shape is typified by a plateau region at positive $\varphi$, with an exponential slope at negative $\varphi$.

Figure 4

Evolution of the field $\varphi$ in a Starobinsky potential. The two branches of the kinetically dominated solutions eventually come to rest in a slow-roll inflating phase at $\varphi \sim 5m_{\mathrm{p}}$, before exiting inflation and dropping into tight oscillations about $\varphi =0$. Here we choose ${\mathrm{\Lambda }}^2={10}^{-5}{m}_{\mathrm{p}}$ for consistency with current observations, and ${\varphi }_{\mathrm{p}}^{+}=-4.2m_{\mathrm{p}}$, ${\varphi }_{\mathrm{p}}^{-}=14.9m_{\mathrm{p}}$, $b=0$, to give 50–60 $e$-folds of inflation in both cases.