The normal argument for flatness from the CMB starts with the comoving horizon size at last scattering

(147) |

and divides it by the present-day horizon size for a zero- universe,

(148) |

to yield a main characteristic angle that scales as
_{m}^{1/2}.
Large curvature (i.e. low
_{m}) is
ruled out because the
main peak in the CMB power spectrum is not seen at very small
angles. However, introducing vacuum energy changes the conclusion.
If we take a family of models with fixed initial perturbation spectra,
fixed physical densities
_{m}
_{m}
*h*^{2},
_{b}
_{b}
*h*^{2}, it is possible to vary both
_{v} and the
curvature to keep a fixed value of the angular size distance to last
scattering, so that the resulting CMB power spectra are identical.
This degeneracy was analyzed comprehensively by
Efstathiou & Bond
(1999),
and we now summarize the main results.

The usual expression for the comoving angular-diameter distance is

(149) |

where =
_{m} +
_{v}.
Defining _{i}
_{i}
*h*^{2}, this can
be rewritten in a way that has no explicit *h* dependence:

(150) |

where _{k}
(1 -
_{m} -
_{v})*h*^{2}. This parameter
describes the curvature of the universe, treating it effectively as
a physical density that scales as
*a*^{-2}.
This is convenient for the present formalism, but
it is important to appreciate that curvature differs fundamentally
from a hypothetical fluid with such an equation of state.

The sound horizon distance at last scattering is
governed by the relevant physical densities,
_{m} and
_{b};
if _{m} and _{b} are given, the
shape of the spatial power spectrum is determined. The translation of this
into an angular spectrum depends on the angular-diameter distance,
which is a function of these parameters, plus
_{k} and
_{v}. Models in
which
_{m}^{1/2}
*R*_{0} *S*_{k}(*r*) is
a constant have the same angular horizon size.
For fixed _{m}
and _{b}, there
is therefore a degeneracy between
curvature (_{k})
and vacuum (_{v}):
these two parameters
can be varied simultaneously to keep the same apparent distance,
as illustrated in figure 19.

In short, this degeneracy occurs because the physical densities
control the structure of the perturbations in physical Mpc at last
scattering, while curvature,
_{v} and
_{m}
govern the proportionality between length at last scattering and
observed angle. The degeneracy is not exact, and is weakly broken by the
Integrated Sachs-Wolfe effect from evolving potentials at very low
multipoles, and second-order effects at high
.
However, strong breaking of the degeneracy requires additional information.
This could be in the form of external data on the Hubble
constant, which obeys the relation

(151) |

so specifying *h* in addition to the physical matter density
fixes _{v} +
_{k} and removes
the degeneracy.
A more elegant approach is to add results from large-scale structure,
so that conclusions are based only on the shapes of power spectra.
Efstathiou et al. (2002)
show that doing this yields a total density
(| - 1| < 0.05) at
95% confidence.