Erratum: Thermal decoupling of WIMPs from first principles

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Here we address three issues that we recently identified in our original article [1]. The first concerns the derivation of the Liouville operator and is purely conceptual; no result or equation derived in our paper is affected. The second concerns an overall factor of 2 in the definition of the collision term; including it slightly increases the kinetic decoupling temperature, in typical cases by a factor of 2 1/4 1.19. The third point in the list is not an actual correction but a brief update about more recent work that generalized our main results by lifting some of the specific assumptions that we made in the interest of a clear and pedagogic derivation.
1. Contrary to the impression given in appendix A, we are throughout using physical rather than co-moving momenta. The text after eq. (A.1), and until the end of that paragraph (". . . drop the bars over the p"), should thus for consistency be replaced by the following: "Here, the Liouville operatorL is the covariant generalization of the convective derivative familiar from hydrodynamics, or -in more technical terms -the variation with respect to an affine parameter λ along a geodesic: Article funded by SCOAP 3 . Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. where in the last step we have chosen λ = τ (i.e. the eigentime). Note that the sum is only over spatial momenta p i since we consider on-shell particles and hence p 0 is not an independent variable of f ; likewise, we only sum over spatial coordinates x i because we consider a Hamiltonian system where f cannot have an explicit time-dependence. In a (flat) Friedmann-Robertson-Walker (FRW) spacetime, the homogeneity of space implies f = f (|p| , t) and eq. (1) becomeŝ where we have introduced H ≡ȧ/a. In the second step, we have adopted the standard convention of treating f explicitly as a function of two (t, p) rather than only one (p) variable, using that to leading order the time-dependence of p derives exclusively from the scalefactor a, i.e. p(t) =p/a(t). Writing f as a function of the co-moving momentā p instead, f = f (t,p), the Liouville operator would simply becomeL = p 0 ∂ t ." 2. For the choice of λ = τ adopted for the Liouville operator, the correctly normalized collision term reads [2, 3] where M is the scattering amplitude, summed over final and initial spin states, and This is a factor of 2 smaller than the collision term stated in appendix B of [1] (note that we are now adopting a slightly different convention for the spin sum). The need for this additional factor is most easily seen by integrating the Boltzmann equation with gχ E , which results in a rate equation for the number density n χ ; only in the presence of this additional factor the right-hand side becomes symmetric in the sense that all four external particles are integrated over the full Lorentz-invariant phase-space. Alternatively, one can derive the Boltzmann equation from its quantum analogue, the Kadanoff-Baym equation, which directly leads to the above stated result in the appropriate limit (see, e.g., [4][5][6][7]). Note that this overall factor of 1/2 is independent of the presence of identical particles in initial or final states, in which case the usual symmetry factors have additionally to be taken into account.
3. Subsequently to the original publication [1], the limitation to relativistic scattering partners was overcome [8]. Including the above-mentioned factor of 1/2, the collision term stated in eq. (B.20) then becomes (initial and final spin states are again being summed over) -2 -

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The above expression is only valid, to lowest order in the expansion parameter T χ /M χ , if |M| 2 is Taylor expandable around t = 0 (in the sense that |M| 2 − |M| 2 t=0 T 2 χ /M 2 χ |M| 2 t=0 ). If this assumption is relaxed, the collision term still takes the above form, but with the replacement [10,11] which describes an effective average over t.
Finally, a simplification in the original treatment [1] was to assume a constant effective number of heat bath degrees of freedom, which was later generalized [8,9]. Assuming no non-standard entropy production, in particular, it is advantageous to consider the dimensionless parameters x ≡ M χ /T and y ≡ M χ T χ s −2/3 , because the main process equation -eqs. (3) and (A.12) in [1] -then takes a particularly simple form [9]: where g * is the effective number of entropy degrees of freedom of the heat bath. This equation, with the collision term stated above, is valid in full generality (as long as the co-moving DM number density does not change during or after kinetic decoupling, like for example in the presence of the Sommerfeld effect [9]). The kinetic decoupling temperature, in analogy with eq. (5), is then obtained as Note that y(x) typically converges very quickly to a constant value for x > x kd , such that this definition is both unique and rather independent of the assumed cosmological evolution (in contrast to what is indicated in [12]), cf. figure 1 in [8].