 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2008

Paper 62

Vary the <scalar field> with compactly supported variation, leaving the <metric tensor> fixed. The <action> variation is
$$
\delta S_M=\int d^4x\sqrt{-g}
\left[-g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\delta\phi-V_{,\phi}\delta\phi\right].
$$
<Integration by parts> moves the derivative from $\delta\phi$ and gives the <Euler-Lagrange equation>
$$
\Box_g\phi-V_{,\phi}=0,\qquad
\Box_g\phi=\frac1{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi\right).
$$
For the <FRW metric>, $\sqrt{-g}=a^3$, so the <covariant wave operator> becomes
$$
\Box_g\phi=-a^{-3}\partial_t(a^3\dot\phi)+a^{-2}\nabla^2\phi
=-\ddot\phi-3H\dot\phi+a^{-2}\nabla^2\phi.
$$
Rearranging gives the requested <scalar field> evolution:
$$
\boxed{\ddot\phi+3\frac{\dot a}{a}\dot\phi-a^{-2}\nabla^2\phi=-V_{,\phi}.}
$$
The damping term is <Hubble friction>; it comes from the expanding volume factor $a^3$, not from a microscopic dissipative force.

Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 2 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 2 / c / Solution

 Ancestors (11)

 Discussion (0) New discussion

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 2 / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion