content_image/402207.jpg content_image/402210.jpg content_image/402208.jpg content_image/402209.jpg content_image/402211.jpg content_image/402212.jpg content_image/402222.jpg content_image/402218.jpg content_image/402217.jpg content_image/402216.jpg content_image/402215.jpg content_image/402214.jpg content_image/402213.jpg content_image/402221.jpg content_image/402220.jpg content_image/402219.jpg
and replace the upper limit in (24) by infinity. Thus we obtain 

<img src='content_image/402207.jpg'>

This obviously dominates the second term 2 ρ ( τ ) on the right-hand side of (10). 

On the left-hand side of (10) both terms are compara- ble. The integral 

is gathered in the region η = O (1). Thus we can replace E ( η, τ ) by e − η and integrate up to infinity. Using 

we conclude that the left-hand side of (21) is asymp- totically equal to 1 − 1 / 2 = 1 / 2. Equating this to the asymptotic expression (26) for the right-hand side we ar- rive at the announced results (10) for the final density of domain walls. 

## IV. GENERAL COOLING PROCEDURE 

For a cooling scheme characterized by an arbitrary function γ ( t ), the solution is found using the same pro- cedure as in Sect. III A. Instead of the integral equation (19) we get 

<img src='content_image/402210.jpg'>

with η = 2 ∫ t t ′ dt ′′ γ ( t $^{′′}$). The final density of domain walls satisfies 

Note that t ′ that appears in ρ ( t $^{′}$) and e 2( t $^{′}$− τ ) should be expressed via η by inverting 

Further, the upper limit in integrals in Eq. (29) is given by ζ = η ( t ′ = 0) = 2 τ 0 dt ′′ γ ( t $^{′′}$). 

∫ The dominant contributions to both integrals in (29) are gathered in the region where 1 − t $^{′}$/τ ≪ 1. In this region the relation between t ′ and η simplifies. We now perform the calculations for two families of cooling pro- cedures introduced in Sect. II. 

<img src='content_image/402208.jpg'>

<img src='content_image/402209.jpg'>

<img src='content_image/402211.jpg'>

<img src='content_image/402212.jpg'>

For the family of cooling procedures (11) we have 

<img src='content_image/402222.jpg'>

or equivalently 

which leads to 

Using (31) and (25) we find that the integral on the right- hand side of Eq. (29) is asymptotically 

Computing the integral in (32) we find that the asymp- totic of the right-hand side of Eq. (29) is given by 

<img src='content_image/402218.jpg'>

The left-hand side of Eq. (29) is asymptotically 

<img src='content_image/402217.jpg'>

where in the last step we have used (27). Equating the RHS to the LHS we arrive at (12)–(13). 

For the family of cooling procedures (14) we have 

<img src='content_image/402216.jpg'>

Computing the integral and inverting we find (keeping two leading terms) 

<img src='content_image/402215.jpg'>

Hence the exponential factor in the integrands in (29) is 

<img src='content_image/402214.jpg'>

and therefore the RHS of (29) is asymptotically 

<img src='content_image/402213.jpg'>

## A. Algebraic Cooling 

<img src='content_image/402221.jpg'>

<img src='content_image/402220.jpg'>

<img src='content_image/402219.jpg'>

## B. Exponential Cooling 

