3.4. Equations of motion and solutions

Let us assume there is only a single dynamical scalar field, and that any extra contribution to the energy density from a second field is included in its potential V(). The equations for an expanding Universe containing a homogeneous scalar field are easily obtained by substituting Eqs. (21) and (22) into the Friedmann and fluid equations, giving

(23)

(24)

where prime indicates d / d. Here I have ignored the curvature term k, as it will quickly become negligible once inflation starts. This is done for simplicity only; there is no obstacle to including that term if one wished.

Since

(25)

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 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

(26)

(27)

If we define slow-roll parameters [6]

(28)

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 ⁽³⁾

(29)

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 [7], which is sufficient as well as necessary [8].

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 [7, 8] that the solution to the full equations possesses an attractor property, eliminating the dependence on the extra parameter.

³ Note that is positive by definition, whilst can have either sign. Back.