** Next:** Perturbation Representation
** Up:** Metric and Stress-Energy Perturbations
** Previous:** Metric and Stress-Energy Perturbations

In linear theory, each eigenmode of the Laplacian for the perturbation
evolves independently, and so it is useful to decompose the perturbations
via the eigentensor , where
| |
(5) |

with ``|'' representing covariant differentiation with respect to
the three metric .Note that the eigentensor has |*m*| indices (suppressed
in the above). Vector and tensor modes also satisfy the auxiliary conditions
which represent the divergenceless and transverse-traceless conditions
respectively, as appropriate for vorticity and gravity waves.
In flat space, these modes are particularly simple and may be expressed as
| |
(6) |

where the presence of , which forms
a local orthonormal basis with , ensures the
divergenceless and transverse-traceless conditions.
It is also useful to construct (auxiliary) vector and tensor objects out
of the fundamental scalar and vector modes through covariant differentiation

| |
(7) |

| |
(8) |

The completeness properties of these eigenmodes are
discussed in detail in
[6], where it is shown that in terms of the generalized wavenumber

| |
(9) |

the spectrum is complete for
| |
(10) |

A deceptive aspect of this labelling is that for an open universe
the characteristic scale of
the structure in a mode is and *not* , so all
functions have structure only out to the curvature scale even as
.
We often go between the variable sets , and
for convenience.

** Next:** Perturbation Representation
** Up:** Metric and Stress-Energy Perturbations
** Previous:** Metric and Stress-Energy Perturbations
*Wayne Hu*

*9/9/1997*