There are two other major programmes which result in constraints on combinations of H0, ,
(now considered as a general density in dark energy rather than specifically a cosmological constant energy
density) and
. The first is the study of type Ia supernovae, which as we have seen function as standard
candles, or at least easily calibratable candles. Studies of supernovae at cosmological redshifts by
two different collaborations [116, 115, 123] have shown that distant supernovae are fainter
than expected if the simplest possible spatially flat model (the Einstein–de Sitter model, for
which
= 1,
= 0) is correct. The resulting determination of luminosity distance has
given constraints in the
–
plane which are more or less orthogonal to the WMAP
constraints.
The second important programme is the measurement of structure at more recent epochs than the epoch
of recombination. This is interesting because fluctuations prior to recombination can propagate at the
relativistic () sound speed which predominates at that time. After recombination, the sound speed
drops, effectively freezing in a characteristic length scale to the structure of matter which corresponds to the
propagation length of acoustic waves by the time of recombination. This is manifested in the real Universe
by an expected preferred correlation length of
100 Mpc between observed baryon structures, otherwise
known as galaxies. The largest sample available for such studies comes from luminous red galaxies
(LRGs) in the Sloan Digital Sky Survey [177]. The expected signal has been found [35
] in the
form of an increased power in the cross-correlation between galaxies at separations of about
100 Mpc. It corresponds to an effective measurement of angular diameter distance to a redshift
z
0.35.
As well as supernova and acoustic oscillations, several other slightly less tightly-constraining measurements should be mentioned:
Tegmark et al. [154] have considered the effect of applying the SDSS acoustic oscillation detection together
with WMAP data. As usual in these investigations, the tightness of the constraints depends on what is
assumed about other cosmological parameters. The maximum set of assumptions (called the “vanilla”
model by Tegmark et al.) includes the assumption that the spatial geometry of the Universe is exactly flat,
that the dark energy contribution results from a pure w = –1 cosmological constant, and that tensor
modes and neutrinos make neglible contributions. Unsurprisingly, this gives them a very tight
constraint on H0 of 73
1.9 km s–1 Mpc–1. However, even if we now allow the Universe to be
not exactly flat, the use of the detection of baryon acoustic oscillations in [35] together with
the WMAP data yields a 5%-error measurement of H0 =
km s–1 Mpc–1. This is
entirely consistent with the Hubble Key Project measurement from Cepheid variables, but
only just consistent with the version in [134]. The improvement comes from the extra distance
measurement, which provides a second joint constraint on the variable set (H0,
,
,
w).
Even this value, however, makes the assumption that w = –1. If we relax this assumption as well,
Tegmark et al. [154] find that the constraints broaden considerably, to the point where the 2
bounds on
H0 range lie between 61 and 84 km s–1 Mpc–1 (see Figure 5
and [154
]), even if the HST Key Project
results [45
] are added. It has to be said that both w = –1 and
= 0 are highly plausible
assumptions11,
and if only one of them is correct, H0 is known to high accuracy. To put it another way, however, an
independent measurement of H0 would be extremely useful in constraining all of the other cosmological
parameters provided that its errors were at the 5% level or better. In fact [59], “The single most important
complement to the CMB for measuring the dark energy equation of state at z
0.5 is a determination of
the Hubble constant to better than a few percent”. Olling [105] quantifies this statement by modelling
the effect of improved H0 estimates on the determination of w. He finds that, although a 10%
error on H0 is not a significant contribution to the current error budget on w, but that once
improved CMB measurements such as those to be provided by the Planck satellite are obtained,
decreasing the errors on H0 by a factor of five to ten could have equal power to much of the
(potentially more expensive, but in any case usefully confirmatory) direct measurements of w
planned in the next decade. In the next Section 4 explore various ways by which this might be
achieved.
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It is of course possible to put in extra information, at the cost of introducing more data sets and hence
more potential for systematic error. Inclusion of the supernova data, as well as Ly- forest data [95],
SDSS and 2dF galaxy clustering [155, 24] and other CMB experiments (CBI, [120]; VSA, [31]; Boomerang,
[93], Acbar, [85]), together with a vanilla model, unsurprisingly gives a very tight constraint
on H0 of 70.5
1.3 km s–1 Mpc–1. Including non-vanilla parameters one at a time also
gives extremely tight constraints on the spatial flatness (
= –0.003
0.006) and w
(–1.04
0.06), but the constraints are again likely to loosen if both w and
are allowed to depart
from vanilla values. A vast literature is quickly assembling on the consequences of shoehorning
together all possible combinations of different datasets with different parameter assumptions (see
e.g. [26, 1, 67, 52, 175, 139]).
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