Jaime Rudas said:
.. 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.
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