158 Cosmic Inflation
Thus the Universe at that time must have been flat to within 54 decimal places, a
totally incredible situation. If this were not so the Universe would either have reached
its maximum size within one Planck time (10−^43 s), and thereafter collapsed into a
singularity, or it would have dispersed into a vanishingly small energy density. The
only natural values for훺are therefore 0, 1 or infinity, whereas to generate a universe
surviving for several Gyr without a훺value of exactly unity requires an incredible
fine-tuning. It is the task of the next sections to try to explain this.
7.2 Consensus Inflation
Let us assume that the푟P-sized universe then was pervaded by a homogeneous scalar
classical field휙,theinflatonfield, and that all points in this universe were causally
connected. The idea with inflation is to provide a mechanism which blows up the Uni-
verse so rapidly, and to such an enormous scale, that the causal connection between
its different parts is lost, yet they are similar due to their common origin. This should
solve the horizon problem and dilute the monopole density to acceptable values, as
well as flatten the local fluctuations to near homogeneity.
We already have tools to achieve this. In Section 5.2 on the de Sitter cosmology we
saw that the solution to the FLRW Equation (5.57) for a constant expansion [Equa-
tion (5.58)] leads to an exponentially expanding universe [Equation (5.59)]. Inflation-
ary models assume that there is a moment when the inflaton domination starts and
subsequently drives the Universe into a de Sitter-like exponential expansion in which
the temperature푇≃0.
Slow-roll Inflation Slow-roll inflation is a very simple idea which could be an effec-
tive representation of a variety more complicated underlying theories. It consists of
one spatially homogeneous classical scalar field휙with a minimal kinetic term and a
potential푉(휙). It does not really matter what this field represents, here it is just an
order parameter for a phase transition.
The total inflaton energy is of the form
1
2
휙̇^2 +^1
2
(∇휙)^2 +푉(휙). (7.22)
The equation of motion for the classical field휙is given by the gravity model we met
in Equation (5.85), now simplified with푓(푅,휙)=1,푀=0, and with the Lagrangian
휙=1
2
푔휇휈휕휇휙휕휈휙−푉(휙). (7.23)
In addition we need theKlein–Gordonequation for the scalar field
휙̈+ 3 퐻휙̇+푉′(휙)= 0. (7.24)Here the prime refers to derivation with respect to휙. The only unknown function is
the potential푉(휙)which contains all the important physics. Friedmann’s equation
becomes
퐻^2 +푘
푎^2