MODERN COSMOLOGY

Model-independent cosmological constraints 253

length at the time of last scattering subtends an angle on the sky of around two
degrees. For a low-density universe with
= 0 .3, this angle becomes smaller by
half, roughly.

A change in angular scale of this magnitude will change the apparent scale
of all physical scales in the microwave background. A model-independent
determination of
thus requires a physical scale of known size to be imprinted on
the primordial plasma at last scattering; this physical scale can then be compared
with its apparent observed scale to obtain a measurement of
.Themicrowave
background fluctuations actually depend on two basic physical scales. The first is
the sound horizon at last scattering,rs(cf equation (7.29)). If coherent acoustic
oscillations are visible, this scale sets their characteristic wavelengths. Even if
coherent acoustic oscillations are not present, the sound horizon represents the
largest scale on which any causal physical process can influence the primordial
plasma. Roughly, if primordial perturbations appear on all scales, the resulting
microwave background fluctuations appear as a featureless power law at large
scales, while the scale at which they begin to depart from this assumed primordial
behaviour corresponds to the sound horizon. This is precisely the behaviour
observed by current measurements, which show a prominent power spectrum
peak at an angular scale of a degree (l=200), arguing strongly for a flat universe.
Of course, it is logically possible that the primordial power spectrum has power
on scales only significantly smaller than the horizon at last scattering. In this case,
the largest scale perturbations would appear at smaller angular scales for a given
geometry. But then the observed power-law perturbations at large angular scales
must be reproduced by the Integrated Sachs–Wolfe effect, and resulting models
are contrived. If the microwave background power spectrum exhibits acoustic
oscillations, then the spacing of the acoustic peaks depends only on the sound
horizon independent of the phase of the oscillations; this provides a more general
and precise probe of flatness than the first peak position.

The second physical scale provides another test: the Silk damping scale is
determined solely by the thickness of the surface of last scattering, which in turn
depends only on the baryon density
bh^2 , the expansion rate of the universe and
standard thermodynamics. Observation of an exponential suppression of power at
small scales gives an estimate of the angular scale corresponding to the damping
scale. Note that the effects of reionization and gravitational lensing must both be
accounted for in the small-scale dependence of the fluctuations. If the reionization
redshift can be accurately estimated from microwave background polarization
(see later) and the baryon density is known from primordial nucleosynthesis
or from the alternating peak heights signature (section 7.4.4), only a radical
modification of the standard cosmology altering the time dependence of the
scale factor or modifying thermodynamic recombination can change the physical
damping scale. If the estimates of
based on the sound horizon and damping
scales are consistent, this is a strong indication that the inferred geometry of the
universe is correct.