Physical Foundations of Cosmology

5.4 How to realize the equation of state p≈−ε 237

φ

attractor

φ

√m

12 π

√−m

12 π

Fig. 5.3.

is ultra-hard,p≈+ε. Neglectingmφcompared to ̇φin (5.27), we obtain

dφ ̇

dφ



√

12 πφ. ̇ (5.28)

The solution of this equation is

φ ̇=Cexp

(√

12 πφ

)

, (5.29)

whereC<0 is a constant of integration. In turn, solving (5.29) forφ(t)gives

φ=const−

1

√

12 π

lnt. (5.30)

Substituting this result into (5.25) and neglecting the potential term, we obtain

H^2 ≡

(

a ̇

a

) 2



1

9 t^2

. (5.31)

It immediately follows thata∝t^1 /^3 andε∝a−^6 in agreement with the ultra-hard
equation of state. Note that the solution obtained is exact for a massless scalar field.
According to (5.29) the derivative of the scalar field decays exponentially more
quickly than the value of the scalar field itself. Therefore, the large initial value
of|φ ̇|is damped within a short time interval before the fieldφitself has changed
significantly. The trajectory which begins at large|φ ̇|goes up very sharply and
meets the attractor. This substantially enlarges the set of initial conditions which
lead to an inflationary stage.

Inflationary solutionIf a trajectory joins the attractor where it is flat, at|φ|1,
then afterwards the solution describes a stage of accelerated expansion (recall that
we work in Planckian units). To determine the attractor solution we assume that