# What is the co-moving distance to a point, at which the late-time acceleration of the universe began?

First of all sorry for my English - it is not my native language.

According to the following wonderful diagram made by Pulsar,

the late-time acceleration of the universe began at $$t_{acc}=$$ 7.7 billion years (indicated by the horizontal black dashed line). I want to know a co-moving distance ($$\mathbf R$$) of an object or region that is closest to us, among astronomical objects or regions being accelerated by the late-time acceleration of the universe.

I think that the value ($$\mathbf R$$) corresponds to the x-coordinate of the intersection of the horizontal black dashed line and the orange thick line, in the above diagram; that is, about 9 Gly.

Q1) Is this interpretation correct?

Meanwhile, in order to more precisely indicate $$t_{acc}=$$ 7.7 billion years, it seem that it is necessary to slightly move up the horizontal black dashed line, as depicted by the red dotted line in the following diagram. If so, it seem that $$\mathbf R$$ should be about 8 Gly, which is indicated by the blue dotted line.

Q2) Which of 8 or 9 Gly is the right value of $$\mathbf R$$?

In addition, the Pulsar's diagram was made based on the condition that the Hubble parameter is 67.3 (km/s)/Mpc. Does anyone have a diagram made under the condition of 70 (km/s)/Mpc? Or, does anyone know what the value of $$\mathbf R$$ under the condition of 70 (km/s)/Mpc is?

• Every galaxy is accelerating, in the sense that if you specify any two galaxies A and B, the proper distance $r_{AB}$ will have $\ddot{r}_{AB}\ne0$. On small scales, this will be dominated by the peculiar motion, which is basically Newtonian, and can have either sign. As you go to larger scales, the accelerations will tend to be systematically positive, due to dark energy. When you talk about a galaxy that's accelerating, do you mean one for which its proper distance from our own galaxy has $\ddot{r}>0$? The closest such will be one whose acceleration is mainly peculiar, not cosmological. – Ben Crowell Mar 25 at 16:48
• @BenCrowell Right! I found that I did not express the meaning of acceleration clearly. So, I revised the post. The acceleration mentioned in the above post means the late-time acceleration of the universe (so-called, caused by dark energy). – SOQEH Mar 26 at 1:22
• The turning point when the deceleration turns into acceleration occurs when d²a/dt²=0, see yukterez.net/lcdm/hubble.differential.2.png which would be 7.6 billion years after the big bang, using the cosmological parameters from the Planck mission Ωm=0.315, Ωλ=0.685 – Yukterez Mar 26 at 3:13
• @Yukterez Thank you for the image showing the informative mathematica code. However, what I want to know is the co-moving distance of the turning point. If you know the distance, I would appreciate it if you let me know the value. – SOQEH Mar 26 at 4:33
• That happened everywhere, not only at a special distance. However the distance associated with the light travel travel time of 7.6 Gyr is 3.5 Glyr at emission, which is a comoving distance of 29 Glyr, see redshift.yukterez.net – Yukterez Mar 26 at 12:28