# Calculating the size of extra dimensions from the scale factor

Any help to understand how the authors of this paper

Fine Tuning Problem of the Cosmological Constant in a Generalized Randall-Sundrum Model

calculted this size of the extra dimension Equ. (3.8) from the scale factor defined by Equ. (3.3) ? Specifically, this paragraph after Equ. (3.8)

The brane just formed is of order $$10^{35}$$ in planck unit ($$\sim 1$$ m), in order to form the presently observed 3D scale of order $$10^{61}$$, we obtain the scale of extra dimensions is of order $$10^9$$ with $$n_2= 3$$.

• How they calculated the extra dimensions’ scale factor to be $$\sim 10^{35}$$ in planck unit or ($$\sim 1$$ m)
• I suppose they mean by the (presently observed 3D scale of order $$10^{61}$$), our observable universe, but $$10^{61}$$ in planck unit equals $$10^{26}$$ meter, how to compare this value by the universe’s volume $$4 \times 10^{40} m^3$$ ?
• As they mentioned :

the scale of extra dimensions can be much larger than Planck length, which leads to the fact that physics are still valid in this model.

So are these extra dimensions compactified or infinite ? Usually the scale of the compactified extra dimensions is of order Planck length $$\sim 10^{-35} m$$. So what do they mean by this sentence ?

I’m familiar with the scale factor of $$\Lambda$$CDM as given by solving Friedmann equations as for instance this thread The scale factor of ΛCDM as a function of time

Can we similar to the paper get the size of the universe from this $$\Lambda$$CDM’s scale factor?