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According to Hubble's Law, the farther a galaxy is, the farther it is moving away. But do we take into account the fact that we are actually looking in the past?

For example, there are two galaxies A and B at distance of 5 and 10 billion years respectively. Now, when we observe A we are looking at how it was moving 5 billion years ago. The same applies for B. So, now we conclude that 5 billion years ago space was expanding at a slower rate while it was expanding comparatively faster 10 billion years ago. What's wrong with this conclusion?

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The light we see now is not a direct indication of how the galaxy was moving at the time the light was emitted. In the cosmological frame, the galaxies aren't moving (on average, at least); rather, space between them is expanding. The rate at which it is expanding is the Hubble parameter.

If cosmic expansion is homogeneous and isotropic, then all distances on the cosmological scale should be affected in the same manner, proportionally, unless other forces are involved. Thus the distance between some galaxy and us is affected in the same proportion as the wavelength of the light: $$\frac{D_\text{now}}{D_\text{then}} = \frac{\lambda_\text{now}}{\lambda_\text{then}}\text{.}$$ The light we see now is a direct indication of the scale factor, i.e. by what factor the distance has grown since the time of emission. We can also consider distance $D$ as a function of cosmological time $t$ and define a recession velocity as the rate at which it is changing: $$v_\text{r} \equiv \frac{\mathrm{d}D}{\mathrm{d}t} = \underbrace{\left[{\dot{D}}/{D}\right]}_{H(t)}D\text{.}$$ Therefore, the recession velocity now is given by the Hubble parameter $H(t)$ now.

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But as I mentioned in the question, can it not be (mis)interpreted as the expansion being faster 10 billion years ago and slower 5 billion years ago? – Yashbhatt Jun 11 '14 at 6:48
@Yashbhatt: What you observe is the redshift, which is (related to) the ratio of distances, present vs past. You're talking about changing recessional velocities. A changing rate of expansion is possibly (hence $H(t)$, not a constant), but the conclusion in your question is logically disconnected from the premises. – Stan Liou Jun 11 '14 at 7:09
I think I have misinterpreted something. In $v = Hd$, the $d$ is the present distance to the galaxy or the distance to the galaxy when the light was emitted? – Yashbhatt Jun 11 '14 at 7:39
@Yashbhatt: It's the present distance to the galaxy. Above, $\dot{D}$ is the rate of change of the distance to the galaxy. Note the that the Hubble parameter has $D$ in the denominator. (Assuming the galaxy has no "peculiar velocity", i.e. zero velocity through space in the cosmological frame--only recession due to expansion.) – Stan Liou Jun 11 '14 at 7:53
But how do we find the present distance to the galaxy, it may be well beyond the cosmic horizon. – Yashbhatt Jun 11 '14 at 7:55

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