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We discuss the cooling of hybrid stars by considering the neutrino emissionfrom quark matter. As a current topic the appearance of various inhomogeneouschiral phases have been studied near the chiral transition. Here we considerthe dual-chiral-density-wave (DCDW) specified by the spatially modulated quarkcondensates. Since the DCDW state can be represented as a chirally rotatedstate from the normal quark matter, the quark weak-current is accordinglytransformed to have an additional phase factor which modifies theenergy-momentum conservation at the vertex, and makes the quark direct Urcaprocess possible. The direct evaluation of the neutrino emissivity shows thatit is proportional to and the magnitude is comparable with the quark or pioncooling. Since the DCDW phase develops only in the limited density region, thisnovel mechanism may give an interesting scenario about cooling of hybrid starsthat lower-mass stars should be cooler than higher-mass ones.

Quark beta decay in an inhomogeneous chiral phase and cooling of hybrid stars