321: Astro Ination 2 Slow Roll Relations
Recall the equation of motion for the unperturbed scalar eld a 0 + 2 0 + a2 V = 0 , a the denitions of the slow-roll parameters = = V 1 , 16G V 1 V , 8G V
2
(1)
(2) (3)
where primes are derivatives with repect to the argument, for V (), and the formulae for the curvature and gravity wave power spectra 2 2 h where mpl = G1/2 . = = H mpl H mpl
2
1 , 4 .
(4) (5)
2
1. Chaotic Ination
Consider polynomial chaotic ination where V = m2 2 /2. Write down and . Ination will occur if the initial eld 0 (0) = i meets what conditions? 1) with H() ( 1) evaluated with
Write down the slow roll equation in coordinate time (d2 0 /dt2 = 0; the Friedmann equation. Solve for 0 (t). Solve for a(t) using the H() relation assume and a(t = 0) = ai . Take
= 1 to dene the end of ination. Show that the number of efoldings of ination can be written as N = ln(aend /ai ) = 2 2 1 i 2 2 mpl (6)
what is the condition on i such that sucient ination occurs (N > 70). Is it compatible with the slow roll conditions? Write down the curvature power spectrum 2 and gravity wave power spectrum 2 for this model in terms of h . Taking = i dened now as N = 70 above, what is ...