If the scale factor as a function of time represents or is equal to the average distance between the galaxies, then it increases altogether with the CMB redshift $z$, but it also must be equal to the inverse $1/(z+1)$ of the redshift at each point on its curve. This is contradiction, because the redshift cannot increase with the scale factor and decrease as its inverse at the same time. What are we missing here?
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3$\begingroup$ $z$ is not the CMB redshift. It's the redshift factor of light received today that was emitted at redshift $z$ or scale factor $a=1/(1+z)$. $\endgroup$– StenCommented Jul 21 at 3:44
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1$\begingroup$ You are saying that $z$ is not the redshift, but the it's a redshift of light that was emitted at redshift $z$. Illogical. What is the present CMB redshift? $\endgroup$– user57748Commented Jul 21 at 3:51
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1$\begingroup$ Between last scattering and redshift $z$, the CMB redshifts by a factor of about $1091/(1+z)$. $\endgroup$– StenCommented Jul 21 at 3:57
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1$\begingroup$ $1091/(1+z)$ What is the value of $z$? $\endgroup$– user57748Commented Jul 21 at 3:58
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1$\begingroup$ $z=1/a-1$, and you can relate that to time by integrating $\mathrm{d}t=\mathrm{d}a/(aH)$. $\endgroup$– StenCommented Jul 21 at 4:00
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1 Answer
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This is contradiction, because the redshift cannot increase with the scale factor and decrease as its inverse at the same time
This is your problem. If we were to look at the CMB in the future, the scale factor of the universe would be larger and the redshift of the measured CMB would be larger, and it is not a contradiction at all.
The relationship is $$\frac{a}{a_0} = \frac{1}{1+z}\ ,$$ where $a$ is the scale factor of the universe when the light was emitted, $z$ is the redshift measured for that light and $a_0$ is the scale factor of the universe when the redshift is measured (usually set to 1 for the present day).
If you were to observe the CMB in the future then $a$ is the same, because the absolute scale factor for when the CMB was emitted cannot change; $a_0$ however is larger, because the CMB light is received at a larger scale factor as the universe has expanded. Thus $a/a_0$ becomes smaller than 1 and this is entirely consistent with $z$ becoming larger.
Addition
To try and come at it from a different angle. The relationship between scale factor and redshift is actually the relationship between a scale factor ratio (the ratio of scale factors at emission and reception of light) and redshift. Since it is a ratio, one can ask about what happens to the redshift if either the numerator or denominator is allowed to change as the universe expands according to the Friedmann equation. The former corresponds to asking: "what is the observed redshift I see at a fixed epoch when the light was emitted at various times in the past, at scale factors given by the Friedmann equation?" The latter corresponds to asking: "for light emitted at a given scale factor, what will its redshift be at some later time, at a scale factor given by the Friedmann equation?"
You have to be clear about which of these questions you are trying to answer and what the definition of a scale factor ratio is on any plot you consider or construct.
