solutions. The radius (scale factor) of such a universe is given by the expression [@SS] $$a^2 = \frac{\kappa}{\Lambda_{\rm RS}} \left( \frac32 + \frac{\sigma m^2}{M^6}
\right) \, .$$ The spatially flat case requires $\Lambda_{\rm RS} = 0$.

### Physical restrictions on the constants of the theory 

Observations indicate that, at the present cosmological epoch, our universe expands with acceleration, so that the simplest asymptotic condition for such a universe is that the energy density will decrease monotonically to zero. Thus, we can consider the low-energy behavior of the theory as $\rho\rightarrow 0$: $$\label{low-energy}
H^2+\frac{\kappa}{a^2}=C_1 + C_2\rho\, , \quad \qquad C_1
> 0\, , \quad C_2 > 0\, ,$$ which is valid under the condition $$\label{regime}
\frac{\ell^2 \rho}{3 m^2} \ll 1+\ell^2 \left( \frac{\sigma}{3m^2} -
\frac{\Lambda_{\rm b}}{6} \right) \, ,$$ and where we have assumed that the dark-radiation term (with the constant $C$ in (\[first\])) has negligible contribution. Assuming regime (\[low-energy\]) is realized today, we can write the following minimal physical restrictions on the constants of the theory:

[**BRANE1:**]{} The inequalities $$C_1 \equiv \frac{\sigma}{3m^2} + \frac2{\ell^2} \left[ 1  - \sqrt{1+\ell^2
\left( \frac{\sigma}{3m^2} - \frac{\Lambda_{\rm b}}{6} \right)} \right]
> 0$$ and $$C_2 \equiv \frac{1}{3m^2} \left[ 1 - \frac{1}{\sqrt{1+\ell^2
\left( \frac{\sigma}{3m^2} - \frac{\Lambda_{\rm b}}{6} \right)}}
