**5.2. Equations of motion and solutions**

The equations for an expanding Universe containing a homogeneous scalar field are easily obtained by substituting Eqs. (33) and (34) into the Friedmann and fluid equations, giving

(38)
(39)

where prime indicates *d* /
*d*. Here I have ignored the
curvature term
*k*, since we know that by definition it will quickly become negligible
once inflation starts. This is done for simplicity only; there is no
obstacle to including that term.

Since

we will have inflation whenever the potential energy dominates. This should be possible provided the potential is flat enough, as the scalar field would then be expected to roll slowly. The potential should also have a minimum in which inflation can end.

The standard strategy for solving these equations is the **slow-roll
approximation** (SRA); this assumes that a term can be neglected in each of
the equations of motion to leave the simpler set

(41)
(42)

If we define **slow-roll parameters**
[3]

where the first measures the slope of the potential and the second the
curvature, then necessary conditions for the slow-roll approximation to hold
are ^{(4)}

Unfortunately, although these are necessary conditions for the slow-roll approximation to hold, they are not sufficient, since even if the potential is very flat it may be that the scalar field has a large velocity. A more elaborate version of the SRA exists, based on the Hamilton-Jacobi formulation of inflation, [4] which is sufficient as well as necessary. [5]

Note also that the SRA reduces the order of the system of equations by one, and so its general solution contains one less initial condition. It works only because one can prove [4, 5] that the solution to the full equations possesses an attractor property, eliminating the dependence on the extra parameter.

^{4} Note that
is positive by definition,
whilst can have
either sign.
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