The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant

The galaxies used in this analysis were selected to $K\leq 12.9$ from the 2MASS Extended Source Catalog (2MASS XSC; Jarrett et al., 2000) and combined with redshift data from the 6dF Galaxy Survey (6dFGS; Jones et al., 2009). The 6dF Galaxy Survey is a combined redshift and peculiar velocity survey covering nearly the entire southern sky with $|b|<10^{\circ}$. It was undertaken with the Six-Degree Field (6dF) multi-fibre instrument on the UK Schmidt Telescope from 2001 to 2006. The median redshift of the survey is $z=0.052$ and the $25\%:50\%:75\%$ percentile completeness values are $0.61:0.79:0.92$. Papers by Jones et al. (2004, 2006, 2009) describe 6dFGS in full detail, including comparisons between 6dFGS, 2dFGRS and SDSS.

Galaxies were excluded from our sample if they resided in sky regions with completeness lower than 60 percent. After applying these cuts our sample contains $75\,117$ galaxies. The selection function was derived by scaling the survey completeness as a function of magnitude to match the integrated on-sky completeness, using mean galaxy counts. This method is the same adopted by Colless et al. (2001) for 2dFGRS and is explained in Jones et al. (2006) in detail. The redshift of each object was checked visually and care was taken to exclude foreground Galactic sources. The derived completeness function was used in the real galaxy catalogue to weight each galaxy by its inverse completeness. The completeness function was also applied to the mock galaxy catalogues to mimic the selection characteristics of the survey. Jones et al. (in preparation) describe the derivation of the 6dFGS selection function, and interested readers are referred to this paper for a more comprehensive treatment.

We calculated the effective volume of the survey using the estimate of Tegmark (1997)

where $n(\vec{x})$ is the mean galaxy density at position $\vec{x}$, determined from the data, and $P_{0}$ is the characteristic power spectrum amplitude of the BAO signal. The parameter $P_{0}$ is crucial for the weighting scheme introduced later. We find that the value of $P_{0}=40\,000h^{-3}\;$Mpc³ (corresponding to the value of the galaxy power spectrum at $k\approx 0.06h\;$Mpc^-1 in 6dFGS) minimises the error of the correlation function near the BAO peak.

Using $P_{0}=40\,000h^{-3}\;$Mpc³ yields an effective volume of $0.08h^{-3}\;$Gpc³, while using instead $P_{0}=10\,000h^{-3}\;$Mpc³ (corresponding to $k\approx 0.15h\;$Mpc^-1) gives an effective volume of $0.045h^{-3}\;$Gpc³.

The volume of the 6dF Galaxy Survey is approximately as large as the volume covered by the 2dF Galaxy Redshift Survey, with a sample density similar to SDSS-DR7 (Abazajian et al., 2009). Percival et al. (2010) reported successful BAO detections in several samples obtained from a combination of SDSS DR7, SDSS-LRG and 2dFGRS with effective volumes in the range $0.15-0.45h^{-3}\;$Gpc³ (using $P_{0}=10\,000h^{-3}\;$Mpc³), while the original detection by Eisenstein et al. (2005) used a sample with $V_{\rm eff}=0.38h^{-3}\;$Gpc³ (using $P_{0}=40\,000h^{-3}\;$Mpc³).

To calculate the correlation function we need a random sample of galaxies which follows the same angular and redshift selection function as the 6dFGS sample. We base our random catalogue generation on the 6dFGS luminosity function of Jones et al. (in preparation), where we use random numbers to pick volume-weighted redshifts and luminosity function-weighted absolute magnitudes. We then test whether the redshift-magnitude combination falls within the 6dFGS $K$-band faint and bright apparent magnitude limits ($8.75\leq K\leq 12.9$).

Figure 1 shows the redshift distribution of the 6dFGS $K$-selected sample (black solid line) compared to a random catalogue with the same number of galaxies (black dashed line). The random catalogue is a good description of the 6dFGS redshift distribution in both the weighted and unweighted case.

We turn the measured redshift into co-moving distance via

with

where the curvature $\Omega^{\rm fid}_{k}$ is set to zero, the dark energy density is given by $\Omega^{\rm fid}_{\Lambda}=1-\Omega^{\rm fid}_{m}$ and the equation of state for dark energy is $w^{\rm fid}=-1$. Because of the very low redshift of 6dFGS, our data are not very sensitive to $\Omega_{k}$, $w$ or any other higher dimensional parameter which influences the expansion history of the universe. We will discuss this further in Section 5.3.

Now we measure the separation between all galaxy pairs in our survey and count the number of such pairs in each separation bin. We do this for the 6dFGS data catalogue, a random catalogue with the same selection function and a combination of data-random pairs. We call the pair-separation distributions obtained from this analysis $DD(s),RR(s)$ and $DR(s)$, respectively. The binning is chosen to be from $10h^{-1}$ Mpc up to $190h^{-1}$ Mpc, in $10h^{-1}$ Mpc steps. In the analysis we used $30$ random catalogues with the same size as the data catalogue. The redshift correlation function itself is given by Landy & Szalay (1993):

where the ratio $n_{r}/n_{d}$ is given by

and the sums go over all random ($N_{r}$) and data ($N_{d}$) galaxies. We use the inverse density weighting of Feldman, Kaiser & Peacock (1994):

with $P_{0}=40\,000h^{3}$ Mpc^-3 and $C_{i}$ being the inverse completeness weighting for 6dFGS (see Section 2.1 and Jones et al., in preparation). This weighting is designed to minimise the error on the BAO measurement, and since our sample is strongly limited by sample variance on large scales this weighting results in a significant improvement to the analysis. The effect of the weighting on the redshift distribution is illustrated in Figure 1.

Other authors have used the so called $J_{3}$-weighting which optimises the error over all scales by weighting each scale differently (e.g. Efstathiou, 1988; Loveday et al., 1995). In a magnitude limited sample there is a correlation between luminosity and redshift, which establishes a correlation between bias and redshift (Zehavi et al., 2005). A scale-dependent weighting would imply a different effective redshift for each scale, causing a scale dependent bias.

Finally we considered a luminosity dependent weighting as suggested by Percival, Verde & Peacock (2004). However the same authors found that explicitly accounting for the luminosity-redshift relation has a negligible effect for 2dFGRS. We found that the effect to the 6dFGS correlation function is $\ll 1\sigma$ for all bins. Hence the static weighting of eq. 7 is sufficient for our dataset.

We also include an integral constraint correction in the form of

where $ic$ is defined as

The function $RR(s)$ is calculated from our mock catalogue and $\xi_{\rm model}(s)$ is a correlation function model. Since $ic$ depends on the model of the correlation function we have to re-calculate it at each step during the fitting procedure. However we note that $ic$ has no significant impact to the final result.

Figure 2 shows the correlation function of 6dFGS at large scales. The BAO peak at $\approx 105h^{-1}\;$Mpc is clearly visible. The plot includes model predictions of different cosmological parameter sets. We will discuss these models in Section 5.2.

To obtain reliable error-bars for the correlation function we use log-normal realisations (Coles & Jones, 1991; Cole et al., 2005; Kitaura et al., 2009). In what follows we summarise the main steps, but refer the interested reader to Appendix A in which we give a detailed explanation of how we generate the log-normal mock catalogues. In Appendix B we compare the log-normal errors with jack-knife estimates.

Log-normal realisations of a galaxy survey are usually obtained by deriving a density field from a model power spectrum, $P(k)$, assuming Gaussian fluctuations. This density field is then Poisson sampled, taking into account the window function and the total number of galaxies. The assumption that the input power spectrum has Gaussian fluctuations can only be used in a model for a density field with over-densities $\ll 1$. As soon as we start to deal with finite rms fluctuations, the Gaussian model assigns a non-zero probability to regions of negative density. A log-normal random field $LN(\vec{x})$, can avoid this unphysical behaviour. It is obtained from a Gaussian field $G(\vec{x})$ by

which is positive-definite but approaches $1+G(\vec{x})$ whenever the perturbations are small (e.g. at large scales). Calculating the power spectrum of a Poisson sampled density field with such a distribution will reproduce the input power spectrum convolved with the window function. As an input power spectrum for the log-normal field we use

where $A=b^{2}(1+2\beta/3+\beta^{2}/5)$ accounts for the linear bias and the linear redshift space distortions. $P_{\rm lin}(k)$ is a linear model power spectrum in real space obtained from CAMB (Lewis et al., 2000) and $P_{\rm nl}(k)$ is the non-linear power spectrum in redshift space. Comparing the model above with the 6dFGS data gives $A=4$. The damping parameter $k_{*}$ is set to $k_{*}=0.33h$ Mpc^-1, as found in 6dFGS (see fitting results later). How well this input model matches the 6dFGS data can be seen in Figure 9.

We produce $200$ such realisations and calculate the correlation function for each of them, deriving a covariance matrix

Here, $\xi_{n}(s_{i})$ is the correlation function estimate at separation $s_{i}$ and the sum goes over all $N$ log-normal realisations. The mean value is defined as

The case $i=j$ gives the error (ignoring correlations between bins, $\sigma_{i}^{2}=C_{ii}$). In the following we will use this uncertainty in all diagrams, while the fitting procedures use the full covariance matrix.

The distribution of recovered correlation functions includes the effects of sample variance and shot noise. Non-linearities are also approximately included since the distribution of over-densities is skewed.

In Figure 3 we show the log-normal correlation matrix $r_{ij}$ calculated from the covariance matrix. The correlation matrix is defined as

where $C$ is the covariance matrix (for a comparison to jack-knife errors see appendix B).

The model of linear redshift space distortions introduced by Kaiser (1987) is based on the plane parallel approximation. Earlier surveys such as SDSS and 2dFGRS are at sufficiently high redshift that the maximum opening angle between a galaxy pair remains small enough to ensure the plane parallel approximation is valid. However, the 6dF Galaxy Survey has a maximum opening angle of $180^{\circ}$ and a lower mean redshift of $z\approx 0.1$ (for our weighted sample) and so it is necessary to test the validity of the plane parallel approximation. The wide angle description of redshift space distortions has been laid out in several papers (Szalay et al., 1997; Szapudi, 2004; Matsubara, 2004; Papai & Szapudi, 2008; Raccanelli et al., 2010), which we summarise in Appendix C.

We find that the wide-angle corrections have only a very minor effect on our sample. For our fiducial model we found a correction of $\Delta\xi=4\cdot 10^{-4}$ in amplitude at $s=100h^{-1}$ Mpc and $\Delta\xi=4.5\cdot 10^{-4}$ at $s=200h^{-1}$ Mpc, (Figure 17 in the appendix). This is much smaller than the error bars on these scales. Despite the small size of the effect, we nevertheless include all first order correction terms in our correlation function model. It is important to note that wide angle corrections affect the correlation function amplitude only and do not cause any shift in the BAO scale. The effect of the wide-angle correction on the unweighted sample is much greater and is already noticeable on scales of $20h^{-1}$ Mpc. Weighting to higher redshifts mitigates the effect because it reduces the average opening angle between galaxy pairs, by giving less weight to wide angle pairs (on average).

There are a number of non-linear effects which can potentially influence a measurement of the BAO signal. These include scale-dependent bias, the non-linear growth of structure on smaller scales, and redshift space distortions. We discuss each of these in the context of our 6dFGS sample.

As the universe evolves, the acoustic signature in the correlation function is broadened by non-linear gravitational structure formation. Equivalently we can say that the higher harmonics in the power spectrum, which represent smaller scales, are erased (Eisenstein, Seo & White, 2007).

The early universe physics, which we discussed briefly in the introduction, is well understood and several authors have produced software packages (e.g. CMBFAST and CAMB) and published fitting functions (e.g Eisenstein & Hu, 1998) to make predictions for the correlation function and power spectrum using thermodynamical models of the early universe. These models already include the basic linear physics most relevant for the BAO peak. In our analysis we use the CAMB software package (Lewis et al., 2000). The non-linear evolution of the power spectrum in CAMB is calculated using the halofit code (Smith et al., 2003). This code is calibrated by $n$-body simulations and can describe non-linear effects in the shape of the matter power spectrum for pure CDM models to an accuracy of around $5-10\%$ (Heitmann et al., 2010). However, it has previously been shown that this non-linear model is a poor description of the non-linear effects around the BAO peak (Crocce & Scoccimarro, 2008). We therefore decided to use the linear model output from CAMB and incorporate the non-linear effects separately.

All non-linear effects influencing the correlation function can be approximated by a convolution with a Gaussian damping factor $\exp[-(rk_{*}/2)^{2}]$ (Eisenstein et al., 2007; Eisenstein, Seo & White, 2007), where $k_{*}$ is the damping scale. We will use this factor in our correlation function model introduced in the next section. The convolution with a Gaussian causes a shift of the peak position to larger scales, since the correlation function is not symmetric around the peak. However this shift is usually very small.

All of the non-linear processes discussed so far are not at the fundamental scale of $105h^{-1}$ Mpc but are instead at the cluster-formation scale of up to $10h^{-1}$Mpc. The scale of $105h^{-1}$ Mpc is far larger than any known non-linear effect in cosmology. This has led some authors to the conclusion that the peak will not be shifted significantly, but rather only blurred out. For example, Eisenstein, Seo & White (2007) have argued that any systematic shift of the acoustic scale in real space must be small ($\apprle 0.5\%$), even at $z=0$.

However, several authors report possible shifts of up to $1\%$ (Guzik & Bernstein, 2007; Smith et al., 2008, 2007; Angulo et al., 2008). Crocce & Scoccimarro (2008) used re-normalised perturbation theory (RPT) and found percent-level shifts in the BAO peak. In addition to non-linear evolution, they found that mode-coupling generates additional oscillations in the power spectrum, which are out of phase with the BAO oscillations predicted by linear theory. This leads to shifts in the scale of oscillation nodes with respect to a smooth spectrum. In real space this corresponds to a peak shift towards smaller scales. Based on their results, Crocce & Scoccimarro (2008) propose a model to be used for the correlation function analysis at large scales. We will introduce this model in the next section.

To model the correlation function on large scales, we follow Crocce & Scoccimarro (2008) and Sanchez et al. (2008) and adopt the following parametrisation¹¹1note that $r=s$, the different letters just specify whether the function is evaluated in redshift space or real space.:

Here, we decouple the scale dependency of the bias $B(s)$ and the linear bias $b$. $G(r)$ is a Gaussian damping term, accounting for non-linear suppression of the BAO signal. $\xi(s)$ is the linear correlation function (including wide angle description of redshift space distortions; eq. 44 in the appendix). The second term in eq 15 accounts for the mode-coupling of different Fourier modes. It contains $\partial\xi(s)/\partial s$, which is the first derivative of the redshift space correlation function, and $\xi^{1}_{1}(r)$, which is defined as

with $j_{1}(x)$ being the spherical Bessel function of order $1$. Sanchez et al. (2008) used an additional parameter $A_{\rm MC}$ which multiplies the mode coupling term in equation 15. We found that our data is not good enough to constrain this parameter, and hence adopted $A_{\rm MC}=1$ as in the original model by Crocce & Scoccimarro (2008).

In practice we generate linear model power spectra $P_{\rm lin}(k)$ from CAMB and convert them into a correlation function using a Hankel transform

where $j_{0}(x)=\sin(x)/x$ is the spherical Bessel function of order $0$.

The $*$-symbol in eq. 15 is only equivalent to a convolution in the case of a 3D correlation function, where we have the Fourier theorem relating the 3D power spectrum to the correlation function. In case of the spherically averaged quantities this is not true. Hence, the $*$-symbol in our equation stands for the multiplication of the power spectrum with $\tilde{G}(k)$ before transforming it into a correlation function. $\tilde{G}(k)$ is defined as

with the property

The damping scale $k_{*}$ can be calculated from linear theory (Crocce & Scoccimarro, 2006; Matsubara, 2008) by

where $P_{\rm lin}(k)$ is again the linear power spectrum. $\Lambda$CDM predicts a value of $k_{*}\simeq 0.17h\;$Mpc^-1. However, we will include $k_{*}$ as a free fitting parameter.

The scale dependance of the 6dFGS bias, $B(s)$, is derived from the GiggleZ simulation (Poole et al., in preparation); a dark matter simulation containing $2160^{3}$ particles in a $1h^{-1}\;$Gpc box. We rank-order the halos of this simulation by $V_{\rm max}$ and choose a contiguous set of $250\,000$ of them, selected to have the same clustering amplitude of 6dFGS as quantified by the separation scale $r_{0}$, where $\xi(r_{0})=1$. In the case of 6dFGS we found $r_{0}=9.3h^{-1}\;$Mpc. Using the redshift space correlation function of these halos and of a randomly subsampled set of $\sim{10^{6}}$ dark matter particles, we obtain

which describes a $1.7\%$ correction of the correlation function amplitude at separation scales of $10h^{-1}\;$Mpc. To derive this function, the GiggleZ correlation function (snapshot $z=0$) has been fitted down to $6h^{-1}\;$Mpc, well below the smallest scales we are interested in.

The effective redshift of our sample is determined by

where $N_{b}$ is the number of galaxies in a particular separation bin and $w_{i}$ and $w_{j}$ are the weights for those galaxies from eq. 7. We choose $z_{\rm eff}$ from bin $10$ which has the limits $100h^{-1}$ Mpc and $110h^{-1}$ Mpc and which gave $z_{\rm eff}=0.106$. Other bins show values very similar to this, with a standard deviation of $\pm 0.001$. The final result does not depend on a very precise determination of $z_{\rm eff}$, since we are not constraining a distance to the mean redshift, but a distance ratio (see equation 25, later). In fact, if the fiducial model is correct, the result is completely independent of $z_{\rm eff}$. Only if there is a $z$-dependent deviation from the fiducial model do we need $z_{\rm eff}$ to quantify this deviation at a specific redshift.

Along the line-of-sight, the BAO signal directly constrains the Hubble constant $H(z)$ at redshift $z$. When measured in a redshift shell, it constrains the angular diameter distance $D_{A}(z)$ (Matsubara, 2004). In order to separately measure $D_{A}(z)$ and $H(z)$ we require a BAO detection in the 2D correlation function, where it will appear as a ring at around $105h^{-1}$ Mpc. Extremely large volumes are necessary for such a measurement. While there are studies that report a successful (but very low signal-to-noise) detection in the 2D correlation function using the SDSS-LRG data (e.g. Gaztanaga, Cabre & Hui, 2009; Chuang & Wang, 2011, but see also Kazin et al. 2010), our sample does not allow this kind of analysis. Hence we restrict ourself to the 1D correlation function, where we measure a combination of $D_{A}(z)$ and $H(z)$. What we actually measure is a superposition of two angular measurements (R.A. and Dec.) and one line-of-sight measurement (redshift). To account for this mixture of measurements it is common to report the BAO distance constraints as (Eisenstein et al., 2005; Padmanabhan & White, 2008)

where $D_{A}$ is the angular distance, which in the case of $\Omega_{k}=0$ is given by $D_{A}(z)=D_{C}(z)/(1+z)$.

To derive model power spectra from CAMB we have to specify a complete cosmological model, which in the case of the simplest $\Lambda$CDM model ($\Omega_{k}=0,w=-1$), is specified by six parameters: $\omega_{c},\omega_{b},n_{s},\tau$, $A_{s}$ and $h$. These parameters are: the physical cold dark matter and baryon density, ($\omega_{c}=\Omega_{c}h^{2},\;\omega_{b}=\Omega_{b}h^{2}$), the scalar spectral index, ($n_{s}$), the optical depth at recombination, ($\tau$), the scalar amplitude of the CMB temperature fluctuation, ($A_{s}$), and the Hubble constant in units of $100\;$km s^-1Mpc^-1 ($h$).

Our fit uses the parameter values from WMAP-7 (Komatsu et al., 2010): $\Omega_{b}h^{2}=0.02227$, $\tau=0.085$ and $n_{s}=0.966$ (maximum likelihood values). The scalar amplitude $A_{s}$ is set so that it results in $\sigma_{8}=0.8$, which depends on $\Omega_{m}h^{2}$. However $\sigma_{8}$ is degenerated with the bias parameter $b$ which is a free parameter in our fit. Furthermore, $h$ is set to $0.7$ in the fiducial model, but can vary freely in our fit through a scale distortion parameter $\alpha$, which enters the model as

This parameter accounts for deviations from the fiducial cosmological model, which we use to derive distances from the measured redshift. It is defined as (Eisenstein et al., 2005; Padmanabhan & White, 2008)

The parameter $\alpha$ enables us to fit the correlation function derived with the fiducial model, without the need to re-calculate the correlation function for every new cosmological parameter set.

At low redshift we can approximate $H(z)\approx H_{0}$, which results in

Compared to the correct equation 25 this approximation has an error of about $3\%$ at redshift $z=0.1$ for our fiducial model. Since this is a significant systematic bias, we do not use this approximation at any point in our analysis.

Using the model introduced above we performed fits to $18$ data points between $10h^{-1}\;$Mpc and $190h^{-1}\;$Mpc. We excluded the data below $10h^{-1}\;$Mpc, since our model for non-linearities is not good enough to capture the effects on such scales. The upper limit is chosen to be well above the BAO scale, although the constraining contribution of the bins above $130h^{-1}\;$Mpc is very small. Our final model has $4$ free parameters: $\Omega_{m}h^{2}$, $b$, $\alpha$ and $k_{*}$.

The best fit corresponds to a minimum $\chi^{2}$ of $15.7$ with $14$ degrees of freedom ($18$ data-points and $4$ free parameters). The best fitting model is included in Figure 2 (black line). The parameter values are $\Omega_{m}h^{2}=0.138\pm 0.020$, $b=1.81\pm 0.13$ and $\alpha=1.036\pm 0.062$, where the errors are derived for each parameter by marginalising over all other parameters. For $k_{*}$ we can give a lower limit of $k_{*}=0.19h\;$Mpc^-1 (with $95\%$ confidence level).

We can use eq. 25 to turn the measurement of $\alpha$ into a measurement of the distance to the effective redshift $D_{V}(z_{\rm eff})=\alpha D_{V}^{\rm fid}(z_{\rm eff})=456\pm 27\;$Mpc, with a precision of $5.9\%$. Our fiducial model gives $D_{V}^{\rm fid}(z_{\rm eff})=440.5\;$Mpc, where we have followed the distance definitions of Wright (2006) throughout. For each fit we derive the parameter $\beta=\Omega_{m}(z)^{0.545}/b$, which we need to calculate the wide angle corrections for the correlation function.

The maximum likelihood distribution of $k_{*}$ seems to prefer smaller values than predicted by $\Lambda$CDM, although we are not able to constrain this parameter very well. This is connected to the high significance of the BAO peak in the 6dFGS data (see Section 7). A smaller value of $k_{*}$ damps the BAO peak and weakens the distance constraint. For comparison we also performed a fit fixing $k_{*}$ to the $\Lambda$CDM prediction of $k_{*}\simeq 0.17h\;$Mpc^-1. We found that the error on the distance $D_{V}(z_{\rm eff})$ increases from $5.9\%$ to $8\%$. However since the data do not seem to support such a small value of $k_{*}$ we prefer to marginalise over this parameter.

The contours of $D_{V}(z_{\rm eff})-\Omega_{m}h^{2}$ are shown in Figure 4, together with two degeneracy predictions (Eisenstein et al., 2005). The solid line is that of constant $\Omega_{m}h^{2}D_{V}(z_{\rm eff})$, which gives the direction of degeneracy for a pure CDM model, where only the shape of the correlation function contributes to the fit, without a BAO peak. The dashed line corresponds to a constant $r_{s}(z_{d})/D_{V}(z_{\rm eff})$, which is the degeneracy if only the position of the acoustic scale contributes to the fit. The dashed contours exclude the first data point, fitting from $20-190h^{-1}$ Mpc only, with the best fitting values $\alpha=1.049\pm 0.071$ (corresponding to $D_{V}(z_{\rm eff})=462\pm 31\;$Mpc), $\Omega_{m}h^{2}=0.129\pm 0.025$ and $b=1.72\pm 0.17$. The contours of this fit are tilted towards the dashed line, which means that the fit is now driven by the BAO peak, while the general fit (solid contours) seems to have some contribution from the shape of the correlation function. Excluding the first data point increases the error on the distance constraint only slightly from $5.9\%$ to $6.8\%$. The value of $\Omega_{m}h^{2}$ tends to be smaller, but agrees within $1\sigma$ with the former value.

Going back to the complete fit from $10-190h^{-1}\;$Mpc, we can include an external prior on $\Omega_{m}h^{2}$ from WMAP-7, which carries an error of only $4\%$ (compared to the $\approx 15\%$ we obtain by fitting our data). Marginalising over $\Omega_{m}h^{2}$ now gives $D_{V}(z_{\rm eff})=459\pm 18\;$Mpc, which reduces the error from $5.9\%$ to $3.9\%$. The uncertainty in $\Omega_{m}h^{2}$ from WMAP-7 contributes only about $5\%$ of the error in $D_{V}$ (assuming no error in the WMAP-7 value of $\Omega_{m}h^{2}$ results in $D_{V}(z_{\rm eff})=459\pm 17\;$Mpc).

In Figure 5 we plot the ratio $D_{V}(z)/D^{low-z}_{V}(z)$ as a function of redshift, where $D^{low-z}_{V}(z)=cz/H_{0}$. At sufficiently low redshift the approximation $H(z)\approx H_{0}$ is valid and the measurement is independent of any cosmological parameter except the Hubble constant. This figure also contains the results from Percival et al. (2010).

Rather than including the WMAP-7 prior on $\Omega_{m}h^{2}$ to break the degeneracy between $\Omega_{m}h^{2}$ and the distance constraint, we can fit the ratio $r_{s}(z_{d})/D_{V}(z_{\rm eff})$, where $r_{s}(z_{d})$ is the sound horizon at the baryon drag epoch $z_{d}$. In principle, this is rotating Figure 4 so that the dashed black line is parallel to the x-axis and hence breaks the degeneracy if the fit is driven by the BAO peak; it will be less efficient if the fit is driven by the shape of the correlation function. During the fit we calculate $r_{s}(z_{d})$ using the fitting formula of Eisenstein & Hu (1998).

The best fit results in $r_{s}(z_{d})/D_{V}(z_{\rm eff})=0.336\pm 0.015$, which has an error of $4.5\%$, smaller than the $5.9\%$ found for $D_{V}$ but larger than the error in $D_{V}$ when adding the WMAP-7 prior on $\Omega_{m}h^{2}$. This is caused by the small disagreement in the $D_{V}-\Omega_{m}h^{2}$ degeneracy and the line of constant sound horizon in Figure 4. The $\chi^{2}$ is $15.7$, similar to the previous fit with the same number of degrees of freedom.

We can also fit for the ratio of the distance between the effective redshift, $z_{\rm eff}$, and the redshift of decoupling ($z_{*}=1091$; Eisenstein et al., 2005);

with $(1+z_{*})D_{A}(z_{*})$ being the CMB angular comoving distance. Beside the fact that the Hubble constant $H_{0}$ cancels out in the determination of $R$, this ratio is also more robust against effects caused by possible extra relativistic species (Eisenstein & White, 2004). We calculate $D_{A}(z_{*})$ for each $\Omega_{m}h^{2}$ during the fit and then marginalise over $\Omega_{m}h^{2}$. The best fit results in $R=0.0324\pm 0.0015$, with $\chi^{2}=15.7$ and the same $14$ degrees of freedom.

Focusing on the path from $z=0$ to $z_{\rm eff}=0.106$, our dataset can give interesting constraints on $\Omega_{m}$. We derive the parameter (Eisenstein et al., 2005)

which has no dependence on the Hubble constant since $D_{V}\propto h^{-1}$. We obtain $A(z_{\rm eff})=0.526\pm 0.028$ with $\chi^{2}/\rm d.o.f.=15.7/14$. The value of $A$ would be identical to $\sqrt{\Omega_{m}}$ if measured at redshift $z=0$. At redshift $z_{\rm eff}=0.106$ we obtain a deviation from this approximation of $6\%$ for our fiducial model, which is small but systematic. We can express $A$, including the curvature term $\Omega_{k}$ and the dark energy equation of state parameter $w$, as

with

and

Using this equation we now linearise our result for $\Omega_{m}$ in $\Omega_{k}$ and $w$ and get

For comparison,  Eisenstein et al. (2005) found

based on the SDSS LRG DR3 sample. This result shows the reduced sensitivity of the 6dFGS measurement to $w$ and $\Omega_{k}$.

We now use the 6dFGS data to derive an estimate of the Hubble constant. We use the 6dFGS measurement of $r_{s}(z_{d})/D_{V}(0.106)=0.336\pm 0.015$ and fit directly for the Hubble constant and $\Omega_{m}$. We combine our measurement with a prior on $\Omega_{m}h^{2}$ coming from the WMAP-7 Markov chain results (Komatsu et al., 2010). Combining the clustering measurement with $\Omega_{m}h^{2}$ from the CMB corresponds to the calibration of the standard ruler.

We obtain values of $H_{0}=67\pm 3.2\;$km s^-1Mpc^-1 (which has an uncertainty of only $4.8\%$) and $\Omega_{m}=0.296\pm 0.028$. Table 1 and Figure 6 summarise the results. The value of $\Omega_{m}$ agrees with the value we derived earlier (Section 5.3).

To combine our measurement with the latest CMB data we use the WMAP-7 distance priors, namely the acoustic scale

the shift parameter

and the redshift of decoupling $z_{*}$ (Tables 9 and 10 in Komatsu et al. 2010). This combined analysis reduces the error further and yields $H_{0}=68.7\pm 1.5\;$km s^-1Mpc^-1 ($2.2\%$) and $\Omega_{m}=0.29\pm 0.022$ ($7.6\%$).

Percival et al. (2010) determine a value of $H_{0}=68.6\pm 2.2$ km s^-1Mpc^-1 using SDSS-DR7, SDSS-LRG and 2dFGRS, while Reid et al. (2010) found $H_{0}=69.4\pm 1.6$ km s^-1Mpc^-1 using the SDSS-LRG sample and WMAP-5. In contrast to these results, 6dFGS is less affected by parameters like $\Omega_{k}$ and $w$ because of its lower redshift. In any case, our result of the Hubble constant agrees very well with earlier BAO analyses. Furthermore our result agrees with the latest CMB measurement of $H_{0}=70.3\pm 2.5$ km s^-1Mpc^-1 (Komatsu et al., 2010).

The SH0ES program (Riess et al., 2011) determined the Hubble constant using the distance ladder method. They used about $600$ near-IR observations of Cepheids in eight galaxies to improve the calibration of $240$ low redshift ($z<0.1$) SN Ia, and calibrated the Cepheid distances using the geometric distance to the maser galaxy NGC 4258. They found $H_{0}=73.8\pm 2.4$ km s^-1Mpc^-1, a value consistent with the initial results of the Hubble Key project Freedman et al. ($H_{0}=72\pm 8$ km s^-1Mpc^-1; 2001) but $1.7\sigma$ higher than our value (and $1.8\sigma$ higher when we combine our dataset with WMAP-7). While this could point toward unaccounted or under-estimated systematic errors in either one of the methods, the likelihood of such a deviation by chance is about $10\%$ and hence is not enough to represent a significant discrepancy. Possible systematic errors affecting the BAO measurements are the modelling of non-linearities, bias and redshift-space distortions, although these systematics are not expected to be significant at the large scales relevant to our analysis.

To summarise the finding of this section we can state that our measurement of the Hubble constant is competitive with the latest result of the distance ladder method. The different techniques employed to derive these results have very different potential systematic errors. Furthermore we found that BAO studies provide the most accurate measurement of $H_{0}$ that exists, when combined with the CMB distance priors.

One key problem driving current cosmology is the determination of the dark energy equation of state parameter, $w$. When adding additional parameters like $w$ to $\Lambda$CDM we find large degeneracies in the WMAP-7-only data. One example is shown in Figure 7. WMAP-7 alone can not constrain $H_{0}$ or $w$ within sensible physical boundaries (e.g. $w<-1/3$). As we are sensitive to $H_{0}$, we can break the degeneracy between $w$ and $H_{0}$ inherent in the CMB-only data. Our assumption of a fiducial cosmology with $w=-1$ does not introduce a bias, since our data is not sensitive to this parameter and any deviation from this assumption is modelled within the shift parameter $\alpha$.

We again use the WMAP-7 distance priors introduced in the last section. In addition to our value of $r_{s}(z_{d})/D_{V}(0.106)=0.336\pm 0.015$ we use the results of Percival et al. (2010), who found $r_{s}(z_{d})/D_{V}(0.2)=0.1905\pm 0.0061$ and $r_{s}(z_{d})/D_{V}(0.35)=0.1097\pm 0.0036$. To account for the correlation between the two latter data points we employ the covariance matrix reported in their paper. Our fit has $3$ free parameters, $\Omega_{m}h^{2}$, $H_{0}$ and $w$.

The best fit gives $w=-0.97\pm 0.13$, $H_{0}=68.7\pm 2.8$ km s^-1Mpc^-1 and $\Omega_{m}h^{2}=0.1380\pm 0.0055$, with a $\chi^{2}/\rm d.o.f.=1.3/3$. Table 2 and Figure 7 summarise the results. To illustrate the importance of the 6dFGS result to the overall fit we also show how the results change if 6dFGS is omitted. The 6dFGS data improve the constraint on $w$ by $24\%$.

Finally we show the best fitting cosmological parameters for different cosmological models using WMAP-7 and BAO results in Table 3.