9.8 Determining cosmic parameters 387
in the current best-fit model. However, for some choices of parameters the destruc-
tive interference can annihilate this peak altogether.The consideration above refers to a spatially flat universe with (^) tot=1. Let
us now consider how the peak locations depend on the values of the fundamental
cosmological parameters. If the universe were curved, the angular size of the sound
horizon would change and the peaks would shift compared with the flat case. For
instance, as follows from (2.73), in a universe without the cosmological constant
l 1 ∝ −tot^1 /^2 .Could we then accurately determine the spatial curvature simply by
measuring the location of the first peak? The answer to this question is not as
straightforward as it seems at first glance. According to (9.94), the value of!
also depends on (^) m,h 75 and (^) b(throughξ), and so it is clear from (9.111) and
(9.112) that the peak positions depend on these parameters. Over the range of
realistic values, while the sensitivity to these parameters is not as strong as to the
spatial curvature, it is nevertheless significant. As an example, if we take a flat
universe with the current best-fit values of the cosmological parameters, and then
double the baryon density (ξ 0. 6 →ξ 1 .2), the first peak moves to the right
byl 1 ∼+ 20 ,the second byl 2 ∼+ 40 ,and the third byl 3 ∼+ 60 .We note
that the locations of the peaks depend onξ∝ (^) bh^275 ,whereas the dependence
on the cold matter density enters through!as (^) mh^375.^1 .An increase in (^) mhas
the opposite effect on peak locations: if we were to double the cold dark matter
density ( (^) mh^375.^1 0. 3 → (^) mh^375.^1 0 .6), the first peak would move to the left
byl 1 ∼−20 and the second and third peaks byl 2 ∼−40 andl 3 ∼− 60
respectively. Thus, even keeping the spatial curvature fixed, the first peak can be
shifted significantly (l 1 ∼40) by doubling the baryon density and simultaneously
halving the cold matter density. This limits our ability to determine the spatial
curvature precisely based on the first peak location only. Fortunately, the parameter
degeneracy can be resolved by combining measurements of peak locations with
peak heights, as described below.
Acoustic peak heights, the baryon and cold matter densities, and flatnessSubstitut-
ingln,given by (9.111) and (9.112), into (9.99) and using (9.92) for!, we see that
the factorIis canceled in the expression forP. Hence, the peak heights predicted
by (9.109) depend on the combinations (^) mh^275 and (^) bh^275 (orξ). For fixed (^) mh^275 ,
an increase in the baryon density increases the height of the the first acoustic peak
H 1. For instance, beginning from the current best-fit model, doubling the baryon
density increasesH 1 by a factor of 1.5, due principally toN 1 (proportional toξ^2 )
andO(sinceA 1 ∝ξ). An increase of the cold matter density for fixedξsuppresses
H 1 sincePdecreases as (^) mh^275 increases. For the cold matter density, the sensitiv-
ity comes from theN 2 andN 3 terms. Therefore, playing the various terms off one
another, the height of the first peak can be held fixed for certain combinations of