A late epoch cosmic acceleration may be naturally entangled with cosmic coincidence – the observation that at the onset of acceleration the vacuum energy density fraction nearly coincides with the matter density fraction .
In this Letter we show that this is indeed the case with the cosmology of a Friedmann-Lamaître-Robertson-Walker ( FLRW ) 3-brane in a five-dimensional anti-de Sitter spacetime .
We derive the four-dimensional effective action on a FLRW 3-brane , from which we obtain a mass-reduction formula , namely , M _ { \lower 2.0 pt \hbox { $ \scriptstyle P$ } } ^ { 2 } = \rho _ { \lower 2.0 pt \hbox { $% \scriptstyle b$ } } / | \Lambda _ { \lower 2.0 pt \hbox { $ \scriptstyle 5 $ } } | , where M _ { \lower 2.0 pt \hbox { $ \scriptstyle P$ } } is the effective ( normalized ) Planck mass , \Lambda _ { \lower 2.0 pt \hbox { $ \scriptstyle 5 $ } } is the five-dimensional cosmological constant , and \rho _ { \lower 2.0 pt \hbox { $ \scriptstyle b$ } } is the sum of the 3-brane tension V and the matter density \rho .
Although the range of variation in \rho _ { \lower 2.0 pt \hbox { $ \scriptstyle b$ } } is strongly constrained , the big bang nucleosynthesis bound on the time variation of the effective Newton constant G _ { \lower 2.0 pt \hbox { $ \scriptstyle N$ } } = ( 8 \pi M _ { \lower 2.0 pt \hbox { $% \scriptstyle P$ } } ^ { 2 } ) ^ { -1 } is satisfied when the ratio V / \rho \gtrsim { O } ( 10 ^ { 2 } ) on cosmological scales .
The same bound leads to an effective equation of state close to -1 at late epochs in accordance with astrophysical and cosmological observations .