[Physics] Distance between two galaxies of different redshift

cosmologydistancegeneral-relativityredshiftspace-expansion

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?

Best Answer

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.

Related Solutions

[Physics] Can the distance of a quasar be determined accurately

You may gain some intuition on cosmological distances by exploring cosmocalc which gives a light travel time of 12.9 Gyrs for the redshift of 7. This site is nice because it shows you the parameters for the "concordant" cosmology that describe our Universe. These parameters describe how the universe is expanding and how it expanded in the past. One cool thing to try out is varying the redshift and seeing what ages for the universe you get at that redshift. This gives you a feeling for how a big change in redshift at high $z$ gives you little change in the age of the universe. For example a change from $z=7$ to $z=8$. But this represents a pretty big shift in the spectrum (challenging detection). A decade ago astronomers did not know whether the universe was flat or what $\Omega_M$ was (density in units of a flat universe). But now there are a number of different astronomical measurements that have all agreed on (or given small error bars around) these parameters. Hence the term "concordant."

To answer your question, a light travel time is a proxy for distance, it is an integrated quantity and so a meaningful metric for distance. However you can't really send a message back as the universe has been expanding since the quasar emitted the light that we are detecting. The second part of your question concerned accuracy. Probably the most meaningful error is the error in the estimated redshift. As you mention the emission lines can be broad and this would contribute to the error. The emission lines could be misidentified though since there is a Lyman alpha forest giving absorption lines from material in between us and the quasar this is unlikely for this object. The quasar could be moving w.r.t to the `Hubble flow' or mean expansion of the universe. For example the quasar could be a black hole that ejected from a galaxy. This type of error is likely to be less than the speed of light and so would at most contribute something like an error of 1 to the redshift (and that would be a really big error and so it's probably less than this). Likewise the emission lines could be broad but at most giving an error of about this size. At most. You can turn an error in redshift into an error in the frequency or wavelength shift of the spectrum. Even for low resolution spectra an error of 0.1 in $z$ would be just enormous. Errors in the concordant cosmology parameters would also affect the estimated light travel time. These days these parameters have errors of a few percent or so, -- using the cosmocalc you can figure out what affect variations in these ($\Omega_M$, $\Lambda$, and the hubble constant) would have on the light travel time.

[Physics] Why is the Baryon Acoustic Oscillations (BAO) calculated like this

The BAOs became imprinted in the background density of the Universe at the time of decoupling, which happened at the same time as recombination, i.e. when the Universe was a factor $z \sim 1100$ smaller. At this time, they had a characteristing wavelength given by the speed of sound in the plasma at that time. This happened to be $\lambda_{\mathrm{BAO},\,z=1100} \sim 0.14 \,\mathrm{Mpc}$ in physical length, i.e. what you would measure if you were there. At this point, the waves got "frozen" in the gas so that it was compressed and rarefied on these scales, but as the Universe expanded, these compressions grew proportionally.

Now when we, today, measure the BAOs, we observe them at various redshifts and with correspondingly different wavelengths. But multiplying the physical length by $(1+z)$, where $z$ is the redshift at which we observe them, we can convert to comoving lengths so that the expansion is factored out, yielding the same result at all $z$, namely ~150 Mpc. Because by definition comoving and physical coordinates coincide today, this is the wavelength that the BAOs have today.

The angular diameter distance $d_\mathrm{A}$ is defined as the distance that relates the physical, not comoving, extend $D$ of an object and the angle $\theta$ covered on the sky by that object: $\theta = D/d_\mathrm{A}$.

For instance, the angle covered by a BAO at a redshift of, say, $z=2$ can be calculated in Python as:

from numpy import pi
from astropy.cosmology import Planck15        # Use a Planck 2016 cosmology
from astropy import units as u
z     = 2
dA    = Planck15.angular_diameter_distance(z) # Angular diameter distance
d_com = 150 * u.Mpc                           # Comoving size of the BAOs
d_phy = d_com / (1+z)                         # Physical size of BAO at z
theta = d_phy / dA * u.rad                    # Angle covered by BAO in radians
print('BAO size in degrees at z = '+str(z)+': ', theta.to(u.deg))

giving $1.6^\circ$ (at $z=2$).

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[Physics] Can the distance of a quasar be determined accurately

[Physics] Why is the Baryon Acoustic Oscillations (BAO) calculated like this

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