# Cosmological redshift vs doppler redshift

I'm reading Harrison's "Cosmology: Science of the universe" because Harrison focuses on the distinction between cosmological redshift (he calls it expansion redshift) and the Doppler redshift.

He states that "they [Doppler redshifts] are produced by peculiar and not by recession velocities" and "[expansion redshifts] are produced by recession and not peculiar velocities".

I understand the concepts of both kinds of redshifts but have a hard time understanding this strict separation. Please correct me, where/if I am wrong:

Suppose a galaxy has no peculiar movement. This means that its position will stay (approximately) the same in comoving coordinates. In fact though ( proper distance), it IS moving away from Earth with recession velocity V, caused by the expansion of the universe. So it should have a Doppler effect based on this recession velocity AND a cosmological (expansion) redshift should also take effect because the light gets stretched with the expansion of the universe. Even though the recession velocity is not due to a peculiar movement, it means that the source of light is moving away from the observer and hence the light should be redshifted and on top the light gets redshifted on its way through expanding space.

Please correct me or tell me if I am right or wrong, I have spent a lot of time reading but still don't fully understand this. Thanks.

• An idea that I have, is that my assumption, that recession velocities cause a Doppler redshift is wrong and that they cannot be compared to other "ordinary" or peculiar velocities. Maybe recession velocities don't cause Doppler redshifts because the galaxy moving away due to recession velocity is not moving compared to its environment. Objects in a Doppler scenario (moving light source on Earth or galaxy with peculiar motion), are moved against their environment (or their environment is moved, if in the objects rest frame). Maybe this solves my problem. I'd love to discuss/get feedback. – user120112 Sep 16 at 2:02

I think your confusion has to do with terms and semantics, rather than physics:

The cosmological redshift has nothing to do with the velocity of the emitter and the observer with respect to each other. That's why it's not a Doppler shift. The cosmological redshift is caused by the expansion of space. It is a direct measure of the relative size of the Universe at emission and observation.

In principle you could have a universe (non-capitalized, since it's not our universe, the Universe) that were static when a distant galaxy emitted a photon, then at some point expanded quickly by a factor of 2, and then again is static. In this hypothetical case, the observer would still measure the photon to have been redshifted by a factor of 2 (i.e. $$z=1$$).

That this is true can bee seen from considering the mathematical derivation of the cosmological redshift (see e.g. here) which involves an integral, the result of which only depends on the initial and the final state, not on the expansion history.

In contrast, if you and your friend stand still with respect to each other while your friend shines her flashlight at you, then run away from each other with a relative velocity of $$0.6c$$, then stand still before you receive the light (i.e. analogously to the hypothetical universe above), then you would measure no redshift; you wouldn't measure the special relativistic Doppler shift of $$z+1 = \sqrt{\frac{1+v/c}{1-v/c}} = 2$$ that you would if you were receding from each other while either emitting or observing.

In the real Universe, galaxies move through space (i.e. they change their comoving coordinate $$\chi$$), and space expands (i.e. the scale factor $$a$$ evolves). If the physical distance to a galaxy is $$d = a \chi,$$ then the change in this distance gives their total velocity wrt. us, and is obtained through differentiation: $$\begin{array}{rcl} v_\mathrm{tot} & = & \dot{a}\chi + a\dot{\chi} \\ & \equiv & v_\mathrm{rec} + v_\mathrm{pec}, \end{array}$$ where dots denote differentiation with respect to time, and the two terms have been identified as the cosmological recession velocity, and the peculiar, "normal" velocity. Each of these terms give rise to a redshift, but through two very different mechanisms. Only the latter term is called a Doppler shift.

• The last part, however, makes it seem like the cosmological redshift would be an instantaneous thing again and you could just plug v_rec in the standard formula for the doppler shift. Therefore giving rise to a redshift that only depends on the da/dt at the moment of emission. – SpaceBread Sep 16 at 14:57
• Up vote because the reference to semantics. For many cosmological red shift is a doppler shift. I think focusing on comoving and proper coordinates remove the confusion or whatever the need to call things a way or another. – Alchimista Sep 17 at 10:29
• @user120112 Yes, exactly! :) – pela Sep 17 at 15:08
• @SpaceBread Yes, maybe it's a bit confusing to write it like this, but it's just Hubble's law. Without (or with negligible) peculiar velocities, $v_\mathrm{rec} = \dot{a}\chi = \dot{a}d/a \equiv H_0 d$. – pela Sep 17 at 15:29
• @SpaceBread I think I misunderstood your comment. The "$\dot{a}$" doesn't, in general, refer to the change in $a$ at the time of emission, but to the general value at the time you wish to know the recession velocity. So in the hypothetical universe that is static except for a brief, intermediate expansion, you have $\dot{a}(t=0) = 0$ and $\dot{a}(t=t_\mathrm{em}) = 0$, meaning that $v_\mathrm{rec}=0$ at $t=0$ and at $t_\mathrm{em}$. In our Universe, however, $\dot{a}(t=0) = H_0 \ne 0$. – pela Sep 19 at 11:03

There are several sorts of redshifts and they are not always easy to separate from one another. First you have cosmological redshift. These redshifts tend to be so high they could not be caused by anything else. Then you have gravitational redshifts, which when they originate with distant quasars having a mass equivalent to trillions of suns might become entangled with their cosmological redshifts. We also have redshift (sometimes blueshift) which arises from the proper motion of the galaxy in question. These red or blue shifts tend to mask any cosmological redshift for nearby galaxies, which redshift is in any case trivial. The Andromeda galaxy, for example, is slightly blue shifted, as it is coming towards us and is 'only' 2.5 million light years away. The distant quasars I referred to are supermassive black holes many billions of light years away.

• This doesn’t address the question; also, quasars do not have masses of “trillions of suns”. – Peter Erwin Sep 16 at 0:20