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The universal traveling wave solution to the Balitsky-Kovchegov equation withrunning coupling (and other equations in the same universality class) isextended to subleading orders at large rapidity and small dipole size $r$. Thelarge rapidity expansion of the logarithm of the saturation scale $Q_s(Y)$ isderived from that traveling wave solution. In addition to the two already knownleading terms in $Y^{1/2}$ and $Y^{1/6}$, which are determined only by the LLBFKL kernel, the three following terms in $Y^0$, $Y^{-1/6}$ and $Y^{-1/3}$ areobtained. They are universal and sensitive to NLL BFKL effects. Initialcondition dependence and NNLL BFKL effects would both start to appear only atthe next order, which is in $Y^{-1/2}$. In the light of these results, theimpact of the precise implementation of the running coupling in gluonsaturation is discussed, \emph{i.e.} parent dipole size prescription \emph{vs.}Balitsky's prescription, and also the effect of the full NLL corrections, notyet implemented in numerical simulations.

Universal behavior of the gluon saturation scale at high energy including full NLL BFKL effects