Redshift-space fluctuations in stochastic gravitational wave background

We study the redshift-space fluctuations induced by a stochastic gravitational wave background (SGWB) via the Sachs-Wolfe effect. The redshift-space fluctuations can be encapsulated in a line-of-sight integral that is useful for studying the imprint of short-wavelength gravitational waves on the cosmic microwave background (CMB) anisotropy. We thus derive constraints on the SGWB from small-scale CMB anisotropy measurements. Our results reproduce the constraint on the short-wavelength SGWB, previously derived from the Planck and BICEP/Keck array CMB data with a CMB Boltzmann numerical code. Furthermore, we improve the constraint and extend it to shorter wavelengths by using the CMB measurements made by the Atacama Cosmology Telescope and the South Pole Telescope. Also, the integral provides us with a precise redshift fluctuation correlation between a pair of pulsars in pulsar timing measurements, which conveniently incorporates the effect of the pulsar term into a small-angle correlation. We further discuss the observation of pulsar pairs in globular clusters to look for this small-angle correlation.