Distance at redshift z=0.666

[Jaime Rudas]

TL;DR Summary

At what distance today is a galaxy that in the future receives with a redshift of $z=0.666$ a signal that we emit today in its direction?

According to the ΛCDM model, the light we receive today from a galaxy located at a distance of 8.10 Gyr took 6.31 Gyr to reach us and has a redshift of $z=0.666$ (cosmological calculator).

At what distance today is a galaxy that in the future receives with a redshift of $z=0.666$ a signal that we emit today in its direction? How long will it take for that signal to arrive? At what distance will that galaxy be when it receives our signal?

 

[Halc]

According to the ΛCDM model, the light we receive today from a galaxy located at a distance of 8.10 Gyr took 6.31 Gyr to reach us

For a redshift of z=0.66, I get a current proper distance along a line of constant cosmological time of about 6.3 GLy, not 8.1, so perhaps the calculator is returning a different value than current proper distance along a line of constant cosmological time. Coincidence that 6.3 current distance is nearly the same as the time it took to get here.
The emission event 6.3 Gyr ago (I agree with that) was a proper distance of about 4.5 GLy away.

At what distance today is a galaxy that in the future receives with a redshift of $z=0.666$ a signal that we emit today in its direction? How long will it take for that signal to arrive? At what distance will that galaxy be when it receives our signal?

That's a hard one. Expansion is accelerating so the galaxy would be a closer one, currently perhaps 5.6 GLy away, and it will take perhaps 10 Gyr of cosmological time to get there. That's a serious rough guess. I don't have a calculator.

 

[Jaime Rudas]

For a redshift of z=0.66, I get a current proper distance along a line of constant cosmological time of about 6.3 GLy, not 8.1

How do you get that distance?
I ask because, for $H_0=67.4$ km/s/Mpc, $Ω_m=0.315$ and $z=0.666$, Ned Wright's cosmological calculator gives us a comoving distance of 8.10 Gly and a light travel time of 6.31 Gyr.

 

[Halc]

for $H_0=67.4$ km/s/Mpc

An amazingly precise value for a constant known to 1.5 digits at best. Irrelevant comment, I know.

I used a picture, not a calculator. There are several, but one is marked with redshifts, and I used that. It being decades old, perhaps the models/constants have been since updated, so I'm not saying your values are wrong.

z=0.66 (labeled along the red light cone line) falls just left of that v=c/2 dotted line.

Notice that for high redshifts like 13 (some of the most distant things that Webb has spotted), light was emitted from only 2.5 GLy away, much closer than the light from this z=0.66 object.

It's harder to find a similar picture extrapolating into the next 25 Gyr or so, but they're out there.

 

[Ibix]

Jorrie's calculator agrees Jaime's figure.

I think the calculation is simple enough if you can find a source for $a(t)$. From memory, the Insights linked from Jorrie's calculator describe the necessary maths to get it, but I think you have to do a numerical integral for any universe that isn't pure radiation/matter/dark energy.

 

[Jaime Rudas]

An amazingly precise value for a constant known to 1.5 digits at best. Irrelevant comment, I know.

I took that value from the Planck 2018 results which reports $(H_0=67.4±0.5)$ km/s/Mpc

Notice that for high redshifts like 13 (some of the most distant things that Webb has spotted), light was emitted from only 2.5 GLy away, much closer than the light from this z=0.66 object.

It's harder to find a similar picture extrapolating into the next 25 Gyr or so, but they're out there.

Yes, that's the distance it was at when the signal was emitted, but I'm referring to the distance the galaxy is currently at, which is established by the green line in your graph and corresponds to about 7.8 Gly.

 

[PeterDonis]

for $H_0=67.4$ km/s/Mpc

You can't use a constant $H_0$ for this calculation; $H$ will change significantly over the periods of time you are considering. You have to do the appropriate integral.

 

[PeterDonis]

There are several

A better one for this purpose would be a conformal diagram, in which light signals move on 45 degree lines.

 

[Halc]

Jorrie's calculator agrees Jaime's figure.

Well, the modern calculators are going to be more up to date than an image that is 'decades old'. Why the big discrepancy though? Could be as simple as where those redshift number lie on that red light cone line? Were they so far off only a short time ago? Have no more up to date pictures like that been made? You'd think they'd be everywhere, but the searches keep finding that one and the yellow-grey ones that are so common, dating similarly far back, showing only worldlines for z= 1,3,10,1000 respectively.

I'm referring to the distance the galaxy is currently at, which is established by the green line in your graph and corresponds to about 7.8 Gly.

There is no green line in my pic, and nothing drawn at 7.8 Gly, although the c=1 worldline is crosses that proper distance about 7.5 Gyr ago.

A better one for this purpose would be a conformal diagram, in which light signals move on 45 degree lines.

That works, but doesn't give an easy answer (proper distance, cosmic time) to this distant galaxy event when it receives our signal emitted today, redshifted by z=0.66. Given a transformation from conformal time to proper time, the numbers can be had, yes.

 

[Ibix]

Well, the modern calculators are going to be more up to date than an image that is 'decades old'. Why the big discrepancy though?

You can change data sources in Jorrie's calculator and see some history of our estimates. You can actually get your 6.3 Gly by using Jorrie's calculator with the WMAP 2013 data and turning off dark energy (click the WMAP button, then set $\Omega_L=0.001$, which is as low as it'll go, set the redshift range to a small range around 0.66, then Calculate). That may be a coincidence - with the various parameters you can play with there are doubtless many ways to get that particular result - but that was literally the first thing I tried. Do you have a source for your graph?

 

[Jaime Rudas]

You can't use a constant $H_0$ for this calculation;

To perform the calculation, not only can $H_0$ be considered, but it must be taken into account. If it isn't taken into account, it isn't possible to calculate how $H$ evolves.

 

[Jaime Rudas]

There is no green line in my pic,

I drew the green line in your drawing, as seen in post #6.

 

[Ibix]

To perform the calculation, not only can $H_0$ be considered, but it must be taken into account. If it isn't taken into account, it isn't possible to calculate how $H$ evolves.

To note, Jorrie's calculator agrees with your number derived from Ned Wright's and I know that Jorrie's calculator does account for the evolution of $H$. Hence you are allowing correctly for the evolution (or we're all doing it wrong... ).

I think that in principle you can get the numbers you want from Jorrie's calculator by messing around with the $z_{eq}$ parameter to adjust the Present Time figure. But the allowed range doesn't seem to be sufficient to change the present time very much.

 

[PeterDonis]

To perform the calculation, not only can $H_0$ be considered, but it must be taken into account. If it isn't taken into account, it isn't possible to calculate how $H$ evolves.

None of this is responsive to what I actually said. @Ibix's post #13, on the other hand, is.

 

[Jaime Rudas]

None of this is responsive to what I actually said. @Ibix's post #13, on the other hand, is.

In post #7 you said that I can't use a constant $H_0$ for this calculation. My point is that the Hubble constant $H_0$ should be used for this calculation. Is this incorrect?

 

[Halc]

I drew the green line in your drawing, as seen in post #6.

Was no drawing in post 6. I refreshed the topic, and it appeared. Tech problem.
OK, you drew a line of constant time at just under 6 Gyr ago (close enough to the 6.3 quoted). It hits the z=0.66 mark on the red light cone, about 4.6 GLy away. The line doesn't go out to 7.8 GLy, where there isn't anything special.
Perhaps you mean that 6 GYr ago, the universe was of age 7.8 GYr, to which I'll agree. But that's not a distance.

Do you have a source for your graph?

Possibly here, but maybe even this reference was copied from some prior work, but it says "figures made for this website", an educational page
http://lifeng.lamost.org/courses/astr553/Topic16/t16_light_cones.html

 

[pbuk]

You can't use a constant $H_0$ for this calculation;

I think there is some confusion here: $H_0$ is by definition a constant: it is the value of $H$ at the current time.

$H$ will change significantly over the periods of time you are considering. You have to do the appropriate integral.

I can confrim that Jorrie's calculator does integrate the Friedmann equation appropriately, and results from Ned's calculator have been cross-checked for various time intervals.

 

[Jaime Rudas]

OK, you drew a line of constant time at just under 6 Gyr ago (close enough to the 6.3 quoted). It hits the z=0.66 mark on the red light cone, about 4.6 GLy away. The line doesn't go out to 7.8 GLy, where there isn't anything special.
Perhaps you mean that 6 GYr ago, the universe was of age 7.8 GYr, to which I'll agree. But that's not a distance.

The 4.6 Gly are the distance at which the galaxy was when it emitted the light that reaches us today with z=0.666. According to your picture, that galaxy is now 7.8 Gly away, as can be seen in the left branch of the red line, which corresponds to the comoving distance, that is, the distance at which the galaxy is TODAY.

 

[PeterDonis]

In post #7 you said that I can't use a constant $H_0$ for this calculation.

By which I meant assuming that $H = H_0$ for the entire time the light is traveling.

My point is that the Hubble constant $H_0$ should be used for this calculation.

As a starting point for calculating how $H$ evolves in time, yes.

 

[Jaime Rudas]

By which I meant assuming that $H = H_0$ for the entire time the light is traveling.

But I didn't assume or hint that I had assumed that.

As a starting point for calculating how $H$ evolves in time, yes.

Yes, that was precisely how I presented it.

 

[PeterDonis]

I didn't assume or hint that I had assumed that.

I realize that now after reading @Ibix's post.

 

[Jaime Rudas]

Jorrie's calculator agrees Jaime's figure.

I think the calculation is simple enough if you can find a source for $a(t)$. From memory, the Insights linked from Jorrie's calculator describe the necessary maths to get it, but I think you have to do a numerical integral for any universe that isn't pure radiation/matter/dark energy.

Since Jorrie's calculator allows negative values for redshift, I would like to know if the following reasoning is correct:

For a galaxy to receive our current signal at a redshift $z=0.666$, the scale factor must grow from $a_0=1$ to $a_1=1.6$.

Now, this means that the current equivalent redshift of that galaxy must be $z=\frac{a_0}{a_1}-1=-0.375$.
Putting this redshift into Jorrie's calculator, we find that the galaxy is currently at 5.87 Gly.

The above would mean that if we were to now emit a signal towards a galaxy that is now at 5.87 Gly, that signal would be received in that galaxy at a redshift of z=0.666.

 

[Ibix]

For a galaxy to receive our current signal at a redshift $z=0.666$, the scale factor must grow from $a_0=1$ to $a_1=1.6$.

$a_1=1.666$, surely, if $z=a_1/a_0-1$? Or it's grown from 0.6 to 1.

The above would mean that if we were to now emit a signal towards a galaxy that is now at 5.87 Gly, that signal would be received in that galaxy at a redshift of z=0.666.

If I follow you, I think that reasoning implies that a (comoving) galaxy that was at z=0.666 is eternally at z=0.666. I'm not sure that's correct in general - an obvious counterexample is a closed universe where redshifted galaxies become blueshifted eventually.

 

[Jaime Rudas]

$a_1=1.666$, surely, if $z=a_1/a_0-1$? Or it's grown from 0.6 to 1.

You are right $a_1=1.666$

 

[Jaime Rudas]

You are right $a_1=1.666$

For a galaxy to receive our current signal at a redshift $z=0.666$, the scale factor must grow from $a_0=1$ to $a_1=1.666$.

Now, this means that the current equivalent redshift of that galaxy must be $z=\frac{a_0}{a_1}-1=-0.4$.
Putting this redshift into Jorrie's calculator, we find that the galaxy is currently at 6.29 Gly.

The above would mean that if we were to now emit a signal towards a galaxy that is now at 6.29 Gly, that signal would be received in that galaxy at a redshift of z=0.666.

If I follow you, I think that reasoning implies that a (comoving) galaxy that was at z=0.666 is eternally at z=0.666.

No, no. When the signal arrives, it will be at z=0.666, but now that galaxy is at 6.29 Gly, which corresponds to z=0.493.

 

[Jaime Rudas]

.. this means that the current equivalent redshift of that galaxy must be $z=\frac{a_0}{a_1}-1=-0.4$.
Putting this redshift into Jorrie's calculator, we find that the galaxy is currently at 6.29 Gly.

The above would mean that if we were to now emit a signal towards a galaxy that is now at 6.29 Gly, that signal would be received in that galaxy at a redshift of z=0.666.

On another forum, @Alriga performed the calculations without using the, from my point of view, unintuitive negative redshift:

A cosmological instant $t_0$ corresponds to a scale factor $a_0$.

Another cosmological instant $t_1$ corresponds to a scale factor $a_1$.

The relationships between the scale factors and the respective electromagnetic signals exchanged are always:

$\dfrac{a_0}{a_1}=\dfrac{\lambda_0}{\lambda_1}=\dfrac{f_1}{f_0}$

Since $t_0=$now, it is usual for us to study signals that reach us now $t_1<t_0$ and therefore it is usual (for the expanding universe) that $a_1< a_0$

The case of the galaxy in question is "not usual" since we are the ones who send the signal $t_1>t_0$ and then (for expanding universe) it is true that $a_1>a_0$. If we apply the above relationship to the case at hand:

$\dfrac{a_1}{a_0}=\dfrac{f_0}{f_1}=\dfrac{f_0}{0.6f_0}=1.666667$

The usual convention is to define that now the scale factor is $a_0=1 \rightarrow a_1=1.666667$

And therefore the distance now to that galaxy is:

$\displaystyle d=\dfrac c{H_0} \ \int_1^{1.666667} \dfrac{da}{\sqrt{\Omega_{R_0}+\Omega_{M_0} a + \Omega_{K_0} a^2 + \Omega_{\Lambda_0} a^4}}$

The value obtained using the Planck Collaboration cosmological parameters, which naturally takes into account the accelerated expansion of the universe since it uses $\Omega_{\Lambda_0}=0.6689$, is current distance d=6.256 billion light years. The signal will reach the galaxy in 8.036 billion years and when it reaches that galaxy the distance will be 10.426 billion light years.

 

[Vanadium 50]

On another forum,

Lets not argue with people in other forums by proxy.

 

[Jaime Rudas]

Lets not argue with people in other forums by proxy.

I'm sorry. I don't understand what you asked me for.

 

[PeterDonis]

I'm sorry. I don't understand what you asked me for.

He means that what is posted in other forums should stay in other forums. If there's a problem with it, there's no point in discussing it here since the person who posted it in the other forum is not here to discuss it with.

If you are confident that the calculation you posted in post #26 is correct, then you should post it as coming from you, not some random person on some other forum, and you should be prepared to defend it on your own.

 

[Jaime Rudas]

He means that what is posted in other forums should stay in other forums. If there's a problem with it, there's no point in discussing it here since the person who posted it in the other forum is not here to discuss it with.

If you are confident that the calculation you posted in post #26 is correct, then you should post it as coming from you, not some random person on some other forum, and you should be prepared to defend it on your own.

Understood. I will do so.

 

[PAllen]

He means that what is posted in other forums should stay in other forums. If there's a problem with it, there's no point in discussing it here since the person who posted it in the other forum is not here to discuss it with.

If you are confident that the calculation you posted in post #26 is correct, then you should post it as coming from you, not some random person on some other forum, and you should be prepared to defend it on your own.

But don't you want to give credit to an idea/analysis you are using? I certainly would want to give such credit.

 

[PeterDonis]

don't you want to give credit to an idea/analysis you are using?

If it's a valid reference, sure. But some random person's post on some other forum is not a valid reference.

 

[Jaime Rudas]

If you are confident that the calculation you posted in post #26 is correct, then you should post it as coming from you,

$1+z_1=\dfrac{a_1}{a_0}=a_1=1.666$

The radial component of the FLRW metric for a flat universe is:

$ds^2=-c^2 dt^2+a^2 dr^2$

For light $ds=0$ from which we obtain:

$dr=\dfrac{c \, dt}a$

$H=\dfrac{\dot a} a$

$\dfrac{da}{dt}=H \, a$

$dr=\dfrac{c \, da}{a^2 H}$

From the Friedman equation for a flat universe and $\Omega_R=0$ we obtain:

$H=H_0 \sqrt{\dfrac {\Omega_{M_0}}{a^3} + (1-\Omega_{M_0)}}$

$\displaystyle r=\dfrac c{H_0} \ \int \dfrac{da}{\sqrt{\Omega_{M_0} a + (1-\Omega_{M_0}) a^4}}$

For the case at hand:

$\displaystyle d_a=\dfrac c{H_0} \ \int_1^{a_r} \dfrac{da}{\sqrt{\Omega_{M_0} a + (1-\Omega_{M_0}) a^4}}=6.28$ Gly

The above would mean that if we were to now emit a signal towards a galaxy that is now at 6.28 Gly, that signal would be received in that galaxy at a redshift of z=0.666.

 

[Jaime Rudas]

For a redshift of z=0.66, I get a current proper distance along a line of constant cosmological time of about 6.3 GLy, [...]. Coincidence that 6.3 current distance is nearly the same as the time it took to get here.

For an expanding universe, it is impossible for the current proper distance to coincide with the time taken for the signal to arrive here. Precisely because of the expansion, the current proper distance is necessarily greater than the path traveled by the light.

 

[Halc]

Precisely because of the expansion, the current proper distance is necessarily greater than the path traveled by the light.

Exactly so. So my labeling it as coincidence was more "not enough time for the two values to differ much".

My chart is also misleading since it seems to be older than the discovery of dark energy. Look at the v=c/2 worldline in my post4. It curves left the whole way, but after about half the age of the universe, dark energy became dominant and that worldline should start curving right, putting the current proper distance to it far larger than what this old picture shows.