I came across two different formulas for the transverse comoving distance in cosmology from GR based on Friedmann's solution in the FLRW metric for an expanding space homogenoeus and isotropic:
a) ..the first one directly derived from GR (as shown in the scientific lieterature)
$$ d=\frac{c}{H_0} \frac{1}{\sqrt{Ω_k}}\mathrm{sinn}\left[{\sqrt{Ω_k}}\int \frac{dz}{\sqrt{Ω_r(1+z)^4+Ω_m(1+z)^3+Ω_k(1+z)^2+Ω_Λ}}\right] $$
where the operator sinn varies depending if we have an open, close or flat Universe. Obviously, $Ω_r$, $Ω_m$, $Ω_k$, $Ω_Λ$ are the omega density parameters whose sum is 1.
b) ..the second one derived by Carroll in his "The Cosmological Constant" Annu. Rev. Astron. Astrophy. 1992 30 499-542 (Eq. 25 page 511) which is slightly different
$$ d=\frac{c}{H_0} \frac{1}{\sqrt{Ω_k}}\mathrm{sinn}\left[{\sqrt{Ω_k}}\int \frac{dz}{\sqrt{(1+z)^2(1+Ω_mz)-z(2+z)Ω_Λ}}\right] $$
Indeed, if I multiply all factors in the square root at the denominator in this second integral b), I do not obtain the correspective part of formula a). I infer that the conceptual meaning is different. b) is defined in Carroll's document as proper motion which should actually be the transverse comoving distance for definition. Did I miss any important conceptual and mathematical steps bewteen the two formulas? Do you know when we have to apply Carroll's formula b) instead of general formula a)?
