We consider the cosmology of a ``3-brane universe'' in a five dimensional(bulk) space-time with a cosmological constant. We show that Einstein'sequations admit a first integral, analogous to the first Friedmann equation,which governs the evolution of the metric in the brane, whatever the timeevolution of the metric along the fifth dimension. We thus obtain thecosmological evolution in the brane for any equation of state describing thematter in the brane, without needing the dependence of the metric on the fifthdimension. In the particular case $p = w \rho$, $(w = constant)$, we giveexplicit expressions for the time evolution of the brane scale factor, whichshow that standard cosmological evolution can be obtained (after an early nonconventional phase) in a scenario \`a la Randall and Sundrum, where a branetension compensates the bulk cosmological constant. We also show that a tinydeviation from exact compensation leads to an effective cosmological constantat late time. Moreover, when the metric along the fifth dimension is static, weare able to extend the solution found on the brane to the whole spacetime.
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机译：我们考虑具有宇宙学常数的五维（本体）时空中的``三脑宇宙''的宇宙学。我们证明，爱因斯坦方程组接受第一个积分，类似于第一个弗里德曼方程，该方程控制度量在米糠中的演化，而不考虑度量沿第五维的时间演化。因此，对于描述它们在麸皮中的任何状态方程，我们都可以得到麸皮的宇宙学演化，而无需度量依赖于第五维。在特定情况下$ p = w \ rho $，$（w =常数）$，我们给出了麸皮比例因子时间演化的明确表达式，表明可以在早期非常规阶段之后获得标准的宇宙学演化。拉兰德尔（Randall）和桑德拉姆（Sundrum）场景，其中低屈曲度补偿了整体宇宙学常数。我们还表明，与精确补偿的微小偏差会导致后期的有效宇宙常数。而且，当沿第五维的度量是静态的时，穿戴者能够将在麸上找到的解扩展到整个时空。
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