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Let $Q_1$ and $Q_2$ two different objects in the Universe (we can think to two galaxies or quasars), that we observe from the Earth at different angular position $(\alpha_1,\delta_1)$, $(\alpha_2,\delta_2)$ and at different redshift $z_1$,$z_2$.

I know how to find the distances of the two objects from the Earth at the present epoch ( the comoving distance): $$ d_C(Q_i)=\dfrac{c}{H_0}\int_0^{z_i}\dfrac {dz}{E(z)} $$

Now I want to find the comoving distance (at epoch $z=0$) between the two objects.

My first idea is to use the classical formulas for the transformation of spherical to cartesian coordinates and find the cartesian coordinates of the two objects, than calculate the Pythagorean distance. But this can work only in a flat space, so it seems not useful in general.

My final goal is to find the distances between the two objects at any epoch and the relative redshift of one of them observed by the other.

Searching on the books that I have and on Internet I don't find a general solution of this problem. Someone know the solution or has some reference?

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    $\begingroup$ Wow, I wish I could help, but all I can tell you is something that won't help in the slightest. I've seen the general solution for this before. I can't for the life of me recall what it is called or what exactly it is, but I have seen it, so it does exist. I also recall that it was very simple. Like I said, I wish I could help further, but I'm drawing a blank about this for some reason. $\endgroup$ – Jim Jul 23 '15 at 14:40
  • $\begingroup$ @Jimself: Thank you for your interest, I'm really interested to some reference to such problem. It seems strange to me that It's so difficult to find a general solution to a problem that is not so strange ! $\endgroup$ – Emilio Novati Jul 23 '15 at 19:51
  • $\begingroup$ Oh, I'm pretty sure the general solution is simple. I just can't quite recall what it looks like or what it's called. I know that's frustrating, but at least you know the solution exists, right? $\endgroup$ – Jim Jul 23 '15 at 20:18
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Like Jimself, I know I've seen this somewhere (and will try to dig it up), but in the meantime I'll give you the answer off the top of my head. Can't guarantee this is entirely correct until I do dig some things up, but some parts are true (confident in the flat Universe part!).

As long as you're interested in our Universe, your idea will actually work, since our Universe is flat (or close enough to flat, anyway).

Furthermore, for objects with zero peculiar velocity, the comoving distance is constant with redshift, so if you find the comoving distance at $z=0$, this is also the comoving distance at any $z$! If the peculiar velocities are small compared to the comoving velocities, you can safely ignore the peculiar velocities. This will in practice be true for any sufficiently high redshift objects. An object at 100 Mpc has a recession velocity of 7000 km/s, which will be substantially higher than any peculiar velocities for things like galaxies and quasars.

In a curved Universe things will be a bit trickier, but not too bad. I think you can still find the cartesian coordinates of the two objects in the same way, but instead of the Pythagorean distance you'll need to solve for the geodesic connecting those two points and find its length.

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  • $\begingroup$ Thank you for your answer. Clearly I'm not interesting on peculiar velocities. My problem can be formulated as "find the distances between any two points in an expanding universe for points comoving with the expansion". And finde the relative redshift at any epoch. It seems a standard problem in cosmography, but I have not found a standard solution in textbooks. And I'm interesting to know how the solution depends on $\Omega_M$, $\Omega_K$ and $\Omega_\Lambda$ in the standard model. $\endgroup$ – Emilio Novati Jul 23 '15 at 20:02
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You are correct that the flat equations will fail you when $\Omega_k \neq 0$. What you need is the correct version of the law of cosines for the geometry in question. Assuming that maximizing numerical accuracy when object separations are small is important, then you'll want the law of haversines for $\Omega_k < 0$ and the hyperbolic law of haversines for $\Omega_k > 0$. Thus the comoving distance between source at comoving distance $D_A$ and one at $D_B$ with angular separation on the sky $\theta$ are separated by comoving distance: $$D_{AB} = \left\{\begin{array}{lc} \left(\frac{2D_H}{\sqrt{\Omega_k}}\right)\operatorname{asinh}\sqrt{\sinh^2\frac{(D_A-D_B)\sqrt{\Omega_k}}{2D_H} + \sinh\frac{D_A\sqrt{\Omega_k}}{D_H} \sinh\frac{D_B\sqrt{\Omega_k}}{D_H} \sin^2 \frac{\theta}{2}} & \Omega_k > 0 \\ \sqrt{(D_A-D_B)^2 + 4D_A D_B \sin^2 \frac{\theta}{2}} & \Omega_k = 0 \\ \left(\frac{2D_H}{\sqrt{|\Omega_k|}}\right)\operatorname{asin}\sqrt{\sin^2\frac{(D_A-D_B)\sqrt{|\Omega_k|}}{2D_H} + \sin\frac{D_A\sqrt{|\Omega_k|}}{D_H} \sin\frac{D_B\sqrt{|\Omega_k|}}{D_H} \sin^2 \frac{\theta}{2}} & \Omega_k < 0. \end{array}\right.$$ If you plug in $D_A = D_B$ then do a Taylor expansion in small $\theta$, the linear term will reproduce Hogg 1999 Equation 16, as required.

Translating this comoving distance into a physical distance at any redshift is a simple matter of applying the right scale factor.

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