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I had read somewhere that light from very distant sources can be measured to be increasingly red shifted the further away the object is (due to cosmic inflation?).
Suppose you had an object emitting cosmic rays or neutrinos other physical matter (not photons), is there an equivalent effect for these objects? ex: a red-shifted helium nuceli in a cosmic ray or red-shifted neutrinos coming from a neutrino source?
Yes, there is a "retardation of the co-moving velocity" of particles. It is important to take it into account to understand the time history of peculiar velocities of galaxies and for determining the energy distribution of cosmic rays coming from other galaxies.
Peculiar velocities, the difference between an object's velocity and the local rest velocity with respect to the microwave background, decay as 1/a(t), if there are no forces. The global scale factor a(t) is the relative size of the universe compared to today. So there is a "redshift" that applies to particles similar to that of the frequency of light. The ratio of "observed" velocity to "emitted" velocity is $\frac{v_o}{v_e} = \frac{1}{1 + z}$. (Peebles, Principles of Physical Cosmology). The $z = \frac{1}{a_e}-1$ here is different from the $z$ of the galaxy because the time of the particle's emission is different from the emission time of the light that we see now.
@Eric's comment reworded as a preface), and would improve the answer by providing not only the extra information, but the clarification on how you've approached the question's intent. :-)
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Apr 26, 2021 at 23:07
The de Broglie relation suggests the "wavelength of a particle" is proportional to the reciprocal of its momentum.
Like all wavelengths, the de Broglie wavelength is effectively stretched by a factor $(1+z)$, where $z$ is the redshift.
The net effect is to increase the wavelength and hence reduce the momentum when measured in the co-moving rest frame.
This, for example, is why the cosmic neutrino background should be very cold $\sim 2$K, despite the neutrinos decoupling from the rest of the universe when they were ultra-relativistic. i.e. the temperature that characterises their momentum distribution is redshifted in a similar (but not identical) way to photons in the cosmic microwave background.
i.e. the redshifting does produce a (potentially) measurable effect on the momentum distribution.
