Physical Foundations of Cosmology

178 The very early universe

Taking different values for the Higgs mass (10 GeV,30 GeV,100 GeV), find out
when the potentialVeffbecomes negative. In this way, assuming that the standard
electroweak theory is valid up to the scale ̄χmand requiring the absence of the
dangerous second minimum inVeffat very large ̄χ>χ 0 ,one can obtain a lower
bound on the Higgs mass. However, this bound is not as robust as the Linde–
Weinberg bound.
The high-temperature expansion of the potential (4.134) is derived using the
same methods as for (4.122). The result is

Veff(χ, ̄ T)

λT

4

χ ̄^4 −

3

Tχ ̄^3 +

Υ(T^2 −T 02 )

2

χ ̄^2 +R, (4.135)

where the temperature-dependent coupling constant

λT=

m^2 H

2 χ 02

+

3

16 π^2 χ 04

(

M^4 Zln

bT^2

M^2 Z

+ 2 M^4 Wln

bT^2

MW^2

− 4 Mt^4 ln

bFT^2

Mt^2

)

(4.136)

is expressed through the masses of the gauge bosons andtquark in the broken
symmetry phase,e.g.MZ≡mZ(χ 0 ). The constantb is defined in (4.123) and
lnbF=2lnπ− 2 C 1. 14.
The dimensionless constantsandΥare

=

3

(

MZ^3 + 2 MW^3

)

4 πχ 03

 2. 7 × 10 −^2 ,Υ=

M^2 Z+ 2 M^2 W+ 2 Mt^2

4 χ 02

 0. 3 ,

and the temperature

T 02 =

1

2 Υ

(

m^2 H−

3

(

M^4 Z+ 2 M^4 W− 4 Mt^4

)

8 π^2 χ 02

)

 1. 7

(

m^2 H+(44 GeV)^2

)

(4.137)

depends explicitly on the unknown Higgs mass.
The terms due to the vector bosons are of the same type as in (4.122). This
becomes clear upon rewriting (4.122)–(4.124) using MG=eχ 0 instead of the
coupling constante.To find the contribution of the fermions we have used the high-
temperature expansion (3.44) forJ(+^1 ).Note that this expansion does not contain a
nonanalytic term which would contribute to the numerical coefficientin front of
the ̄χ^3 term.
Now we turn to the temperature behavior of potential (4.134) and study the
symmetry breaking in the early universe. To get an idea of the expected character
of the transition it is enough to consider the high-temperature expansion (4.135).
For a given temperatureT, the contribution from different fields to ( 4.135)
can be trusted only for those ̄χfor which the induced masses of the corresponding
fields are smaller than the temperature. For instance, thetquark terms should be