# The scale of an extra dimension

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• Safinaz
In summary, the authors of this paper calculated the size of the extra dimension to be approximately 1 meter using the scale factor defined in Equation (3.3). They also compared this size to the observed 3D scale of our universe, which is much larger. The extra dimensions in this model are assumed to be infinite and not compactified at the Planck length, highlighting the validity of the physics at this scale.
Safinaz
TL;DR Summary
A paper about (n+1)-d Generalized Randall-Sundrum Model, where the authors calculated the size or the scale of the extra dimension in units of length using the scale factors. My question how they made these calculations.
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 ?

The authors of this paper calculated the size of the extra dimension by using the scale factor defined in Equation (3.3). This scale factor is related to the brane formation and is of the order of ##10^{35}## in Planck units, which is approximately equal to 1 meter. This means that the size of the extra dimension is on the same scale as the size of a proton.

The authors are referring to the observed 3D scale of our observable universe, which has a volume of approximately ##4 \times 10^{80} m^3##. This is much larger than the scale of the extra dimension, which is why it is not directly comparable. However, the authors are showing that even though the extra dimension is much smaller than the observable universe, it still has a significant impact on the physics of our universe.

In this paper, the extra dimensions are assumed to be infinite. This means that they extend infinitely in both directions and are not compactified, or curled up into a small size. This is different from other theories where the extra dimensions are assumed to be compactified at the Planck length.

The authors are highlighting that in this model, the scale of the extra dimensions is much larger than the Planck length. This means that the physics in this model are still valid and do not break down at this scale. This is important because in some theories, the extra dimensions are assumed to be compactified at the Planck length, which could potentially lead to issues with the validity of the physics at that scale.

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