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3.2 Degeneracies and implications for H0

If the Universe is exactly flat and the dark energy is a cosmological constant, then the debate about H0 is over. The Wilkinson Anisotropy Probe (WMAP) has so far published two sets of cosmological measurements, based on one year and three years of operation [146145Jump To The Next Citation Point]. From the latest data, the assumption of a spatially flat Universe requires H0 = 73 ± 3 km s–1 Mpc–1, and also determines other cosmological parameters to two and in some cases three significant figures [145Jump To The Next Citation Point]. If we do not assume the Universe to be exactly flat, then we obtain a degeneracy with H0 in the sense that every decrease of 20 km s–1 Mpc–1 increases the total density of the Universe by 0.1 in units of the closure density (see Figure 5View Image). The WMAP data by themselves, without any further assumptions or extra data, do not supply a significant constraint on H0.

There are two other major programmes which result in constraints on combinations of H0, Ω m, Ω Λ (now considered as a general density in dark energy rather than specifically a cosmological constant energy density) and w. 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 [116115123] have shown that distant supernovae are fainter than expected if the simplest possible spatially flat model (the Einstein–de Sitter model, for which Ωm = 1, ΩΛ = 0) is correct. The resulting determination of luminosity distance has given constraints in the ΩmΩ Λ 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 (√ -- c∕ 3) 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 [35Jump To The Next Citation Point] 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:

3.2.1 Combined constraints

Tegmark et al. [154Jump To The Next Citation Point] 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 = +4.7 71.6−4.3 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, Ω m, Ω Λ, w).

Even this value, however, makes the assumption that w = –1. If we relax this assumption as well, Tegmark et al. [154Jump To The Next Citation Point] find that the constraints broaden considerably, to the point where the 2σ bounds on H0 range lie between 61 and 84 km s–1 Mpc–1 (see Figure 5View Image and [154Jump To The Next Citation Point]), even if the HST Key Project results [45Jump To The Next Citation Point] are added. It has to be said that both w = –1 and Ω k = 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.

View Image

Figure 5: Top: The allowed range of the parameters Ω m, Ω Λ, from the WMAP 3-year data, is shown as a series of points (reproduced from [145]). The diagonal line shows the locus corresponding to a flat Universe (Ωm + Ω Λ = 1). An exactly flat Universe corresponds to H0 ∼ 70 km s–1 Mpc–1, but lower values are allowed provided the Universe is slightly closed. Bottom: Analysis reproduced from [154] showing the allowed range of the Hubble constant, in the form of the Hubble parameter h ≡ H0 / 100 km s–1 Mpc–1, and Ω m by combination of WMAP 3-year data with acoustic oscillations. A range of H0 is still allowed by these data, although the allowed region shrinks considerably if we assume that w = –1 or Ωk = 0.

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-α forest data [95], SDSS and 2dF galaxy clustering [15524] 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 (Ωk = –0.003 ± 0.006) and w (–1.04 ± 0.06), but the constraints are again likely to loosen if both w and Ωk 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. [2616752175139]).

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