# Wrong scale factor on wikipedia

From wikipedia we have for Scale-Factor $$a(t)$$:

".. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time "t" to their distance at some reference time t0". The formula for this is:" $$d(t) = a(t) * d_0\tag {F1}$$ "..if at present time we receive light from a distant object with a redshift of z, then the scale factor at the time the object originally emitted that light is" $$a(t) = 1/(1+z)\tag{F2}$$

and after:

"..For a dark-energy-dominated universe, the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric is easily obtained solving the Friedmann equations:"

$$a(t) = \text{constant} * \exp(H_0 t) \tag{F3}$$

"...{ Here, the coefficient H0 in the exponential is the Hubble constant.)"

https://en.wikipedia.org/wiki/Scale_factor_(cosmology)

Well, I think the formula [F2] is wrong, the correct should be :

$$a(t) = (1+z)$$

because the greater the speed, the greater the distance and also the greater the redshift. Or am I mistaken?

• Wiki has its own discussion page. I believe our question should be posted there: en.wikipedia.org/wiki/Talk:Scale_factor_(cosmology) Commented Oct 20, 2022 at 18:18
• Please use latex/mathjax, you just need to add dollar signs \$ between equations. Here is a guide math.meta.stackexchange.com/questions/5020/… Commented Oct 20, 2022 at 18:30
• Needs to be performed elsewhere Commented Oct 20, 2022 at 20:51

Formula F2 is correct: the universe was smaller in the past, $$a(t)$$ for $$t$$ before the current time ($$t_0$$) is smaller than $$a(t_0)$$.
The argument that "the greater the speed, the greater the distance and also the greater the redshift" is not without any merit, but we must be careful about the specific distances we are discussing. It is true that high-redshift objects are more distant than low-redshift objects, but that distance is not $$a(t)$$!
As you quote, $$a(t)$$ is "the scale factor at the time the object originally emitted that light", and it decreases with redshift. The luminosity distance from us to that object, on the other hand, does increase with redshift.