Escursionedelcampo:

(Homework: derive this expression). For � = 0, this is the dilution equation for non-relativistic matter. � represents the inflation decay rate. Indeed, in this period of timethe inflation is supposed to decay into other particles. These thermalize and, once the in-flation has decayed enough, start dominating the universe. This is the start of the standardbig-bang universe.

1.5 Simplest Models of Inflation

1.5.1 Large Field Inflation

The simplest versions of inflation are based on scalar fields slowly rolling down their potential.These typically fall into two categories: large fields and small fields. Large field models arethose characterized by a potential of the form

V (�) =�↵

M↵�4

. (57)

� �� ��obs.

�reheat�begin

V (�)

�

Figure 6: A ‘large-field’ inflationary model.

For any M and ↵, if we put the scalar field high enough, we can have an inflationarysolution. Let us see how this happens by imposing the slow roll conditions.

✏ ⇠M2

Pl

✓V,�

V

◆2

⇠ ↵2

M2

Pl

�2

(58)

For ↵ ⇠ 1, we have✏⌧ 1 ) ��M

Pl

. (59)

The field vev has to be super planckian. Further, notice that the field travels an amount oforder

�� =

Z �end

�in

d� =

Z tend

tin

�̇dt ' �̇

H

Z Htend

Htin

d(Ht) =�̇

HNe ⇠ ✏1/2Ne M

Pl

(60)

16

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