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We explore a plausible mechanism that the hemispherical power asymmetry inthe CMB is produced by the spatial variation of the primordial sound speedparameter. We suggest that in a generalized approach of the $\delta N$formalism the local e-folding number may depend on some other primordialparameters besides the initial values of inflaton. Here the $\delta N$formalism is extended by considering the effects of a spatially varying soundspeed parameter caused by a super-Hubble perturbation of a light field. Usingthis generalized $\delta N$ formalism, we systematically calculate theasymmetric primordial spectrum in the model of multi-speed inflation by takinginto account the constraints of primordial non-Gaussianities. We furtherdiscuss specific model constraints, and the corresponding asymmetry amplitudesare found to be scale-dependent, which can accommodate current observations ofthe power asymmetry at different length scales.

Scale-dependent CMB power asymmetry from primordial speed of sound and a generalized $\delta$N formalism