Download A quintessence model Consider a real scalar φ, spatially

Introduction to Cosmology - a.y. 2013-14
Exercise 3 - A quintessence model
Consider a real scalar φ, spatially homogeneous, minimally couple to gravity
in a flat FLRW Universe, and with potential V of the form:
V (φ) =
M 4+α
,
φα
(1)
where M is a parameter of dimension of a mass and α > 0.
i) Assume a phase of either radiation or matter domination (namely, neglect
the energy density term associated to the field, ρφ , when computing the
Universe dynamics); use the ansatz:
φ(t) ∝ tβ
(2)
to find a solution of the equation of motion.
ii) Show that solution just found is a ”tracker” solution by introducing the
shift:
φ → φ + δφ ,
(3)
with |δφ/φ| 1 and imposing that φ+δφ is as well a solution of the equation
of motion, and by showing the |δφ/φ| decreases as t increases.
iii) Show that the tracking solution gives an energy density ρφ decreasing in
time slower than the dominant background component. Find the equation
of state parameter for the quintessence component wφ as a function of the
equation of state parameter for the background component wB ; how do the
two compare in the limit of large α?