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 vs 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 zv-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 bgt10
• primary z-survey sample 2MASS galaxies to
  Ktotlt12.75
• (secondary samples Hlt13, Jlt13.75, rlt15.7, blt17)
• (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
  zv-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 b?Pgg/PmmbL2??2/Pmm,
  the linear redshift-space distortion parameter
  ???0.6/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, ? ?
  ?0.6/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-s relation
  determines the precision of the peculiar
  velocities.
• For the z-survey the operational parameters are
  sky coverage, fsky sampling fraction, a median
  depth rm
• For the v-survey the operational parameters are
  the equivalent set plus s0

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Optimal z-survey design

• BT 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 Klim11.8, a0.7, rm 255 Mpc/h
• Compare with 6dFGS, which has Klim12.75, alt0.9,
  rm150 Mpc/h

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Recovered power spectrum
Effective volume
Linear PS for optimal survey, ?lnk0.5 bands
shot noise/mode
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Parameter degeneracies

• Degeneracies can be seen by comparing derivatives
  of the PS w.r.t. the various parameters.
• Similar curves mean almost degenerate parameters.

• 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(-2?bh)

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Density field parameters - 1

• Models with Ag, ?, ?
• At kmax0.2 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?Am?m0.6.

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, ?lnk0.5 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 0.25 0.50 0.75 1.00
20 distances from Dn-?

• Distance errors dominate, and need to be
  minimized.
• Sampling should be as complete as possible.
• Large sky fractions help, but dont gain
  linearly.
• The 6dFGS v-survey should give Av to about 25.

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Velocity field parameters

• Models with Av, ?.
• At kmax0.2 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 zv-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.

z-only zv
1? contours on pairs of parameters
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Joint zv-survey constraints - 2

• For models with Ag, ?, ?, rg the errors are still
  2-3 in the first three, but lt2 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 zv
1? 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).

Errors in bands (bands shown by dots)

• 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.

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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
• Break the degeneracy between the redshift-space
  distortion parameter ??0.6/b and the galaxy-mass
  correlation parameter rg.
• Measure the four parameters Ag, ?, ? and rg with
  precisions of between 1 and 3.
• Measure the variation of rg and b with scale to
  within a few over a wide range of scales.