The Power-Law Cosmology

1

Problem 1

problem id: PWL_1

Show that for power law $a(t)\propto t^n$ expansion slow-roll inflation occurs when $n\gg1$.

solution

Slow-roll inflation corresponds to \[\varepsilon\equiv-\frac{\dot H}{H}\ll1.\] For power law expansion $H=n/t$ so that $\varepsilon=n^{-1}$. Consequently, slow-roll inflation occurs when $n\gg1$.

2

Problem 2

problem id: PWL_2

Show that in the power-law cosmology the scale factor evolution $a\propto\eta^q$ in conformal time transforms into $a\propto t^p$ in physical (cosmic) time with \[p=\frac{q}{1+q}.\]

3

Problem 3

problem id: PWL_3

Show that if $a\propto\eta^q$ then the state parameter $w$ is related to the index $q$ by the following \[w=\frac{2-q}{3q}=const.\]

solution

\[\bar H=\frac q t,\quad \bar H'=-\frac q{t^2}.\] Using \[\bar H'=-\frac{1+3w}{2}\bar H^2,\] one obtains \[w=\frac{2-q}{3q}.\]