Romain Teyssier- Cosmological simulations of galaxy formation
Constraints on cosmological parameters from the 6dF Galaxy Survey
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Transcript of Constraints on cosmological parameters from the 6dF Galaxy Survey
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Constraints on cosmological
parameters from the 6dF Galaxy Survey
Matthew Colless
6dFGS Workshop
11 July 2003
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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
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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.
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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
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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)
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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.
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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.
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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
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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
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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
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Recovered power spectrum
Linear PS for optimal survey, lnk=0.5 bands
Effective volume
shot noise/mode
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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.
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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)
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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
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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
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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-
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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
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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
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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
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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)
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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.