Constraints on cosmological parameters from the 6dF Galaxy Survey

Constraints on cosmological

parameters from the 6dF Galaxy Survey

Matthew Colless

6dFGS Workshop

11 July 2003

What can the 6dFGS tell us?

• Strong constraints on cosmological parameters result from combining the wide range of existing datasets: 2dFGRS/SDSS, WMAP, distant SNe, Lyman forest, weak lensing…

• Given this plethora of data, what can the 6dFGS add?

• Specifically, what advantage does the combination of redshift and peculiar velocity information give?

• The answers presented here are based on…

“Prospects for galaxy-mass relations from the 6dF Galaxy Redshift & Peculiar Velocity

survey” Dan Burkey & Andy Taylor

z-surveys and v-surveys

• Galaxy redshift surveys: simple, quick and easy (ha!) so can be very large, but…– unknown ‘bias’ linking galaxies to the matter

distribution;– z-space distortion mixes Hubble expansion and

peculiar velocities (both positive and negative consequences).

• Peculiar velocity surveys are the best way to map the matter distribution, but…– measuring v’s is difficult and time-consuming;– only works nearby, so surveys must cover large

areas;– hence v-surveys are generally small (~1000

objects), or eclectic compilations of different samples and methods.

The 6dF Galaxy Survey

• The 6dFGS is designed to be the first of a new generation of combined z+v-surveys, combining… A NIR-selected redshift survey of the local universe. A peculiar velocity survey using Dn- distances.

• Survey strategy… – survey whole southern sky with |b|>10°– primary z-survey sample: 2MASS galaxies to

Ktot<12.75

– (secondary samples: H<13, J<13.75, r<15.7, b<17)– (additional samples: sources from radio, X-ray,

IRAS…)– v-survey sample: ~15,000 brightest early-type

galaxies

The Fisher information matrix

• The information in a survey of a random field (r) parameterised by ; if the field is Gaussian, then

• where the power spectrum P is defined by

• and the effective volume of the survey is

• The covariance of the discretely sampled field is

• For P(k) the uncertainty is:

(Fisher matrix)

Properties of the Fisher matrix

• The Fisher matrix, F, …– has the conditional error for a parameter on its

diagonal;– gives the marginalized error for the ith parameter

as

– gives the correlation between measured parameters as

– the variance in maximum likelihood (minimum variance) parameter estimates is the marginalized error from F.

– for multiple fields, the covariance matrices of each can be combined to give a joint Fisher matrix.

Application to surveys

• Burkey & Taylor use the Fisher matrix methods to estimate the uncertainties in estimating cosmological parameters from z- and v-surveys and z+v-surveys.

• The fields are the z-space density perturbations and the radial gradient of the radial peculiar velocities.

• The auto- and cross-power spectra of these fields are specified by: the matter power spectrum Pmm(k), the bias parameter bPgg/Pmm=bL

2+2/Pmm, the linear redshift-

space distortion parameter Ω0.6/b, the Hubble constant H0.

Parameters of model• The cosmological parameters used to specify

the cosmological model are:– the amplitude of the galaxy power spectrum, Ag =

b Am

– the power spectrum shape parameter, = mh

– the redshift-space distortion parameter, Ω0.6/b

– the mass density in baryons, b (or b = bh2)

– the correlation between luminous and dark matter, rg

• Parameters not considered are:– the index of the primordial mass spectrum, n (= -

1)

– the small-scale pairwise velocity dispersion, v

Parameters of survey• The parameters of the survey itself enter

through the noise terms:– the level of shot noise is determined by the

number density of galaxies, ng(r), in the z- and v-surveys;

– the fractional error in the Dn- relation determines the precision of the peculiar velocities.

• For the z-survey the operational parameters are sky coverage, fsky; sampling fraction, ; median depth rm

• For the v-survey the operational parameters are the equivalent set plus 0

Optimal z-survey design

• B&T first employ this machinery to determine the depth of a redshift survey that minimizes the error in Ag in fixed time.

• Other things being equal, want largest possible fsky

• If Klim 5 log rm - 0.255 optimum hemisphere survey has Klim=11.8, =0.7, rm= 255 Mpc/h

• Compare with 6dFGS, which has Klim=12.75, <0.9, rm=150 Mpc/h

fsky=0.25

fsky=0.75

fsky=0.5

fsky=1.0

=1

Recovered power spectrum

Linear PS for optimal survey, lnk=0.5 bands

Effective volume

shot noise/mode

Parameter degeneracies

• Ag, , and rg are all ~constant and so ~degenerate.

and b are also similar (both relate to damping of the PS); the effective shape is eff = exp(-2bh)

• Degeneracies can be seen by comparing derivatives of the PS w.r.t. the various parameters.

• Similar curves mean almost degenerate parameters.

Density field parameters - 1

• Models with Ag, ,

• At kmax~0.2 h/Mpc (limit set by non-linear clustering) the uncertainties are 2-3% on all three parameters.

• Correlations are:– very strong between

and (a change in amplitude can be mimicked by a change in scale);

– moderate between Ag and , with Ag~Amm

0.6.

Fractional marginalized uncertainties

Correlations

Maximum wavenumber (k/h Mpc-1)

Density field parameters - 2

• Models with Ag, , , rg

• Ag, are unaffected (errors of 2-3%), but uncertainties on , rg are much larger (~35%)

• This is due to the strong correlation between and rg, which results because both parameters affect the normalization of the galaxy PS

Fractional marginalized uncertainties

Correlations

Peculiar velocity power spectrum

• Expected 6dFGS 3D velocity PS, lnk=0.5 bands (+effective volume)

• Larger errors reflect smaller size of survey and 1D peculiar velocities

• Effective volume for each mode is also shown

Optimal v-survey design

• ‘Optimal’ survey minimizes the error in Av in given time

• For various fixed fsky, the figure shows the error in Av in terms of the single free parameter, the degenerate variable 0/1/2.

fsky

0.250.500.751.00

• Distance errors dominate, and need to be minimized.• Sampling should be as complete as possible.• Large sky fractions help, but don’t gain linearly.

• The 6dFGS v-survey should give Av to about 25%.

6dFGS

20% distances from Dn-

Velocity field parameters

• Models with Av, .

• At kmax~0.2 h/Mpc (limit set by non-linear clustering) the uncertainties are ~25% on both parameters.

• Av and are strongly anti-correlated (change in normalization can be mimicked by a shift in scale).

Fractional marginalized uncertainties

Correlation

Joint z+v-survey

constraints - 1• Combine z- and v-survey

data and estimate joint constraints from overall Fisher matrix.

• For models with Ag, , the errors are still 2-3% in all three.

• This is very similar to z-survey, as v-survey does not break the main Ag- degeneracy.

1 contours on pairs of parameters

z-only z+v

Joint z+v-survey

constraints - 2• For models with Ag, , , rg

the errors are still 2-3% in the first three, but <2% in rg.

• Ag, are unchanged by v-survey and has degraded slightly (due to residual correlation with rg).

• The v-survey greatly improves the joint constraint on and rg, which are now only relatively weakly correlated.

z-only z+v1 contours on pairs of parameters

Scale constraints on rg and b

• Do the bias or the galaxy/mass correlation vary with scale?

• Figure shows errors on band estimates of rg and b (each assuming the other is fixed).

• If b is fixed, variations in rg can be measured at 5-10% level.

• If rg is fixed, variations in b can be measured at the few % level over a wide range of scales.

Errors in bands (bands shown

by dots)

Conclusions• In terms of constraining cosmological

parameters, the major advantage of the 6dFGS is combining the redshift and peculiar velocity surveys to…

1. Break the degeneracy between the redshift-space distortion parameter =0.6/b and the galaxy-mass correlation parameter rg.

2. Measure the four parameters Ag, , and rg with precisions of between 1% and 3%.

3. Measure the variation of rg and b with scale to within a few % over a wide range of scales.