Abstract
We show that the standard theory of thermal production and chemical decoupling of WIMPs is incomplete. The hypothesis that WIMPs are produced and decouple from a thermal bath implies that the rate equation the bath particles interacting with the WIMPs is an algebraic equation that constraints the actual WIMPs abundance to have a precise analytical form down to the temperature \(x_*=m_\chi /T_*\). The point \(x_*\), which coincides with the stationary point of the equation for the quantity \(\varDelta = Y-Y_0\), is where the maximum departure of the WIMPs abundance \(Y\) from the thermal value \(Y_0\) is reached. For each mass \(m_\chi \) and total annihilation cross section \(\langle \sigma _\text {ann}v_\text {r}\rangle \), the temperature \(x_*\) and the actual WIMPs abundance \(Y(x_*)\) are exactly known. This value provides the true initial condition for the usual differential equation that have to be integrated in the interval \(x\ge x_*\). The matching of the two abundances at \(x_*\) is continuous and differentiable. The dependence of the present relic abundance on the abundance at an intermediate temperature is an exact result. The exact theory suggests a new analytical approximation that furnishes the relic abundance accurate at the level of 1–2 % in the case of \(S\)-wave and \(P\)-wave scattering cross sections. We conclude the paper studying the evolution of the WIMPs chemical potential and the entropy production using methods of non-equilibrium thermodynamics.
Highlights
Freeze-out [1,2] refers to the idea that stable particles, once in thermal and chemical equilibrium in the hot and dense early Universe, left a relic abundance because at a certain stage of the evolution, the expansion and cooling made their density so small that the annihilation reaction rates became frozen
Where a is the Friedmann–Robertson–Walker scale factor, a3 the comoving volume, H = (1/a)da/dt the Hubble parameter, σannvr the total thermally averaged annihilation unitary rate, and n0 the number density at zero chemical potential determined by the equilibrium statistics obeyed by WIMPs
It is clear that these codes, leaving aside the uncertainties intrinsic in the numerical methods, for a given WIMP mass and total annihilation cross section inevitably will differ in the value of the relic abundance, even if within a few percent in normal situations
Summary
Freeze-out [1,2] refers to the idea that stable particles, once in thermal and chemical equilibrium in the hot and dense early Universe, left a relic abundance because at a certain stage of the evolution, the expansion and cooling made their density so small that the annihilation reaction rates became frozen. The problem of the initial condition is important for the relic abundance calculation because it determines the precision with which the asymptotic value is obtained. It is clear that these codes, leaving aside the uncertainties intrinsic in the numerical methods, for a given WIMP mass and total annihilation cross section inevitably will differ in the value of the relic abundance, even if within a few percent in normal situations. A conceptual problem arises: it would tell us that the relic abundance really depends on an intermediate temperature, in a way similar to the freeze-out approximation, and not on the asymptotic value at x = 0 as it should be if the first order differential equation described the evolution over the whole range.
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