Evolution of a domain wall in expanding universe: inflation and after it

We study the evolution of thick domain walls in the expanding universe. We have found that the domain wall evolution crucially depends on the time-dependent parameter C (t ) = 1/(H (t )δ0 )2 , where H (t ) is the Hubble parameter and δ0 is the width of the wall in flat space-time. For C (t ) > 2 the physical width of the wall, a (t )δ (t ), tends with time to constant value δ 0 , which is microscopically small. Otherwise, when C (t ) ≤ 2, the wall steadily expands and can grow up to a cosmologically large size.

It is well known that a symmetry, which is broken in vacuum, at high temperatures tends to be restored. But in general, the situation is not that simple and straightforward. It is also possible that a symmetry is broken only in a particular range of temperatures, i.e. it is restored at the highest as well as at the lowest temperatures. This is just what is needed for a matter-antimatter domain generation without domain wall problem. A.D. Dolgov, S.I. Godunov, A.S. Rudenko, I.I. Tkachev, JCAP 1510, 027 (2015 The model with spontaneous CP violation is suggested.
CP violation appears due to interaction of additional scalar field with inflaton.
BAU is generated just after inflation due to interaction of introduced scalar field with quarks and leptons.
This scenario leads to the generation of matter-antimatter domains in the Early Universe.
To avoid annihilation at the domain boundary, the distance between the domains should grow exponentially fast during inflation.
How fast the domain wall width can grow in the Early Universe?

Domain wall during inflation
Let us consider a simple model of inflation with quadratic inflaton potential U = m 2 Φ 2 /2, then the Hubble parameter is and the equation of motion of the inflaton in the slow-roll regime is the following: where m pl is the Planck mass, m is the inflaton mass.
where Φi is the initial value of inflaton field.

S.I. Godunov
Evolution of a domain wall in expanding universe: inflation and after it May 30, 2018 8 / 19 Inflation: equations of motion The Hubble parameter and the scale factor can also be easily found: These formulas are valid only till the end of inflation, t < te ≡ √ 12πΦ i m pl m −1 . it is convenient to use 1/m units in equation of motion: In numerical calculations the following parameters were used: Φi = 2 m pl , ti = 0, a0 = 1.

S.I. Godunov
Evolution of a domain wall in expanding universe: inflation and after it May 30, 2018 9 / 19 Inflation: C(t) dependence Time tC at which C(tC ) = 2: .

S.I. Godunov
Evolution of a domain wall in expanding universe: inflation and after it May 30, 2018 15 / 19 Universe with p = wρ: C(t) dependence The parameter C(t) increases as The time tC at which C(tC ) = 2: tC δ0 = √ 2α.
We obtain that tC > ti for

Conclusions
The evolution of the domain walls was considered in the following cases: de Sitter universe For C = λη 2 /H 2 = 1/(Hδ 0 ) 2 > 2 the solutions tend to the stationary ones.
For C = λη 2 /H 2 = 1/(Hδ 0 ) 2 < 2 the wall width grows rapidly. For C 0.1 the growth is practically exponential, so the wall expands with the universe.
during inflation: For m · δ 0 0.173 the deviation of the wall width from δ 0 is small.
For 0.173 m · δ 0 1 the wall width can reach cosmologically large values, but then it quickly diminishes and reaches δ 0 .
For m · δ 0 1 the wall width grows with the scale factor and by the end of inflation it reaches cosmologically large size. p = wρ universe: Domain walls with cosmologically large width can exist only in the beginning of this phase.
For t/δ 0 √ 2α the wall width is close to δ 0 .