Cosmological parameters from 2dFGRS

6.2 Cosmological parameters from 2dFGRS

6.2.1 The power spectrum of 2dF Galaxies on large scales

An initial estimate of the convolved, redshift-space power spectrum of the 2dFGRS was determined by Percival et al. [72 ] for a sample of 160,000 redshifts. On scales $0.02h Mpc −1 < k < 0.15h Mpc −1$ , the data are fairly robust and the shape of the power spectrum is not significantly affected by redshift-space distortion or non-linear effects, while its overall amplitude is increased due to the linear redshift-space distortion effect (see Section 5).

Figure 24: The power spectrum of the 2dFGRS. The points with error bars show the measured 2dFGRS power spectrum measurements in redshift space, convolved with the window function. Also plotted are linear CDM models with neutrino contribution of $Ων = 0$ , $Ω ν = 0.01$ , and $Ω ν = 0.05$ (bottom to top lines). The other parameters are fixed to the concordance model. The good fit of the linear theory power spectrum at $k > 0.15h Mpc −1$ is due to a conspiracy between the non-linear gravitational growth and the finger-of-God smearing [72

]. (Figure taken from [20

].)

If one fits the $Λ$ -CDM model predictions to the 2dFGRS power spectrum (see Figure 24 ) over the above range in $k$ , one can constrain the cosmological parameters. For instance, assuming a Gaussian prior on the Hubble constant $h = 0.7 ± 0.07$ (from [22 ]), Percival et al. [72 ] obtained the 68 percent confidence limits on the shape parameter $Ω h = 0.20 ± 0.03 m$ , and a baryon fraction $Ω ∕ Ω = 0.15 ± 0.07 b m$ . For a fixed set of cosmological parameters, i.e., $n = 1$ , $Ωm = 1 − Ω Λ = 0.3$ , $2 Ωbh = 0.02$ , and $h = 0.70$ , the r.m.s. mass fluctuation amplitude of 2dFGRS galaxies smoothed over a top-hat radius of $− 1 8h Mpc$ in redshift space turned out to be $σS8g(Ls,zs) ≈ 0.94$ .

6.2.2 An upper limit on neutrino masses

The recent results of atmospheric and solar neutrino oscillations [24 , 1 ] imply non-zero mass-squared differences of the three neutrino flavours. While these oscillation experiments do not directly determine the absolute neutrino masses, a simple assumption of the neutrino mass hierarchy suggests a lower limit on the neutrino mass density parameter, $Ω = m h−2∕(94 eV ) ≈ 0.001 ν ν,tot$ . Large scale structure data can put an upper limit on the ratio $Ω ν∕Ωm$ due to the neutrino ’free streaming’ effect [33 ]. By comparing the 2dF galaxy power-spectrum of fluctuations with a four-component model (baryons, cold dark matter, a cosmological constant, and massive neutrinos) it was estimated that $Ων∕Ωm < 0.13$ (95% CL), or with concordance prior of $Ωm = 0.3$ , $Ων < 0.04$ , or an upper limit of $∼ 2 eV$ on the total neutrino mass, assuming a prior of $h ≈ 0.7$ [20 , 19 ] (see Figure 24 ). In order to minimize systematic effects due to biasing and non-linear growth, the analysis was restricted to the range $−1 0.02 < k < 0.15h Mpc$ . Additional cosmological data sets bring down this upper limit by a factor of two [79 ].

6.2.3 Combining 2dFGRS and CMB

While the CMB probes the fluctuations in matter, the galaxy redshift surveys measure the perturbations in the light distribution of particular tracer (e.g., galaxies of certain type). Therefore, for a fixed set of cosmological parameters, a combination of the two can better constrain cosmological parameters, and it can also provide important information on the way galaxies are ‘biased’ relative to the mass fluctuations,

The CMB fluctuations are commonly represented by the spherical harmonics $Cℓ$ . The connection between the harmonic $ℓ$ and $k$ is roughly

for a spatially-flat Universe. For $Ωm = 0.3$ , the 2dFGRS range $0.02 < k < 0.15h Mpc − 1$ corresponds approximately to $200 < ℓ < 1500$ , which is well covered by the recent CMB experiments.

Recent CMB measurements have been used in combination with the 2dF power spectrum. Efstathiou et al. [17 ] showed that 2dFGRS+CMB provide evidence for a positive cosmological constant $Ω Λ ∼ 0.7$ (assuming $w = − 1$ ), independently of the studies of supernovae Ia. As explained in [72 ], the shapes of the CMB and the 2dFGRS power spectra are insensitive to Dark Energy. The main important effect of the dark energy is to alter the angular diameter distance to the last scattering, and thus the position of the first acoustic peak. Indeed, the latest result from a combination of WMAP with 2dFGRS and other probes gives $h = 0.71+−00.0.043$ , $Ωbh2 = 0.0224 ± 0.0009$ , $Ωmh2 = 0.135+0−.00.00089$ , $σ8 = 0.84 ± 0.04$ , $Ωtot = 1.02 ± 0.02$ , and $w < − 0.78$ (95% CL, assuming $w ≥ − 1$ ) [79 ].

6.2.4 Redshift-space distortion

An independent measurement of cosmological parameters on the basis of 2dFGRS comes from redshift-space distortions on scales $−1 ≲ 10h Mpc$ : a correlation function $ξ (π, σ)$ in parallel and transverse pair separations $π$ and $σ$ . As described in Section 5, the distortion pattern is a combination of the coherent infall, parameterized by $β = Ω0.m6∕b$ and random motions modelled by an exponential velocity distribution function (see Equation (133 )). This methodology has been applied by many authors. For instance, Peacock et al. [66 ] derived $β (L = 0.17,z = 1.9L ) = 0.43 ± 0.07 s s ∗$ , and Hawkins et al. [30 ] obtained $β(Ls = 0.15,zs = 1.4L ∗) = 0.49 ± 0.09$ and a velocity dispersion $−1 σP = 506 ± 52 km s$ . Using the full 2dF+CMB likelihood function on the $(b,Ωm$ ) plane, Lahav et al. [42 ] derived a slightly larger (but consistent within the quoted error-bars) value, $β (Ls = 0.17,zs = 1.9L ∗) ≃ 0.48 ± 0.06$ .

6.2.5 The bi-spectrum and higher moments

It is well established that important information on the non-linear growth of structure is encoded at the high order moments, e.g., the skewness or its Fourier version, the bi-spectrum. Verde et al. [95 ] computed the bi-spectrum of 2dFGRS and used it to measure the bias parameter of the galaxies. They assumed a specific quadratic biasing model:

By analysing 80 million triangle configurations in the wavenumber range $0.1 < k < 0.5h Mpc − 1$ they found $b1 = 1.04 ± 0.11$ and $b2 = − 0.054 ± 0.08$ , in support of no biasing on large scale. This is a non-trivial result, as the analysis covers non-linear scales. Baugh et al. [5 ] and Croton et al. [14 ] measured the moments of the galaxy count probability distribution function in 2dFGRG up to order $p = 6$ (order $p = 2$ is the variance, $p = 3$ is the skewness, etc.). They demonstrated the hierarchical scaling of the averaged $p$ -point galaxy correlation functions. However, they found that the higher moments are strongly affected by the presence of two massive superclusters in the 2dFGRS volume. This poses the question of whether 2dFGRS is a ’fair sample’ for high order moments.