Anisotropic separate universe and Weinberg's adiabatic mode
  In the separate universe approach, an inhomogeneous universe is rephrased as
a set of glued numerous homogeneous local patches. This is the essence of the
gradient expansion and the $\delta N$ formalism, which have been widely used in
solving a long wavelength evolution of the universe. In this paper, we show
that the separate universe approach can be generically used, as long as a
theory under consideration is local and preserves the spatial diffeomorphism
invariance. Focusing on these two conditions, we also clarify the condition for
the existence of the so-called Weinberg's adiabatic mode. Remarkably, the
separate universe approach and the $\delta N$ formalism turn out to be
applicable also to models with shear on large scales and also to modified
theories of gravity, accepting violation of four-dimensional diffeomorphism
invariance. The generalized $\delta N$ formalism enables us to calculate all
the large scale fluctuations, including gravitational waves. We also argue
several implications on anisotropic inflation and ultra slow-roll inflation.
