Yes, cosmological time dilation is indeed taken into account when measuring the pulsation rate of Cepheid variables at large redshifts. Cosmological time dilation arises due to the expansion of the universe, which causes time to pass at different rates in different regions of space.
In the context of Cepheid variables, these are pulsating stars whose brightness oscillates in a regular pattern. The period of their pulsation is directly related to their intrinsic luminosity. By observing the pulsation period of Cepheids, astronomers can determine their intrinsic brightness and use it as a standard candle to measure distances to galaxies.
However, when observing Cepheids in distant galaxies with large redshifts (indicating the universe's expansion), the cosmological time dilation needs to be considered. As light travels through an expanding universe, its wavelength is stretched, leading to an increase in observed redshift. This stretching of light also affects the apparent pulsation period of Cepheids.
To account for cosmological time dilation, astronomers use appropriate corrections in their calculations. They consider the time dilation factor associated with the redshift of the galaxy where the Cepheid is located. By applying this correction, they can estimate the intrinsic pulsation period of the Cepheid, enabling accurate distance measurements even for galaxies at large redshifts.
Taking cosmological time dilation into account is crucial to ensure accurate and meaningful measurements when studying distant objects and understanding the evolution and properties of the universe on cosmological scales.