Constraining Photometric Redshift Errors With Galaxy Two-point Correlation Functions
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Constraining photometric redshift errors with galaxy two-point correlations Michael Schneider UC Davis
Collaborators: Andy Connolly, Lloyd Knox, Hu Zhan
Outline The basic idea Cross-correlating the photometric sample with itself Cross-correlating with an overlapping spectroscopic sample (J. Newman) Challenges and future directions
Motivation Future dark energy surveys (DES, Pan-STARRS, LSST, EUCLID, JDEM) plan to use photometric redshifts to measure cosmic shear and galaxy correlations Hard to get fair spectroscopic training samples to the depth of the photometric sample Conventional photo-z estimation methods may leave intolerably large errors Can other calibration methods reduce the size of the fair spectroscopic training sample needed for a given photo-z error target?
Cross correlating galaxies binned by photometric redshift
astro-ph/0606098
Photo-z errors induce cross-correlations n(z)
A. Schulz Scatter
Catastrophic
bin
Z
n(z)
overlap causes correlation
z
Sensitivity of galaxy power spectrum 10-1
bins (1,1) bins (1,3), var. a13 bins (1,3), var. a31
l2 P(l) / (2π)
10-2
Auto and cross angular galaxy power spectra for: 0 < zp < 0.5 and 1 < zp < 1.5
10-3 10-4 10
Points with errors: fiducial values (with
-5
photo-z errors)
10-6 10-7
50
100
200
400 l
800
Lines:1- σ variation
Model for photo-z errors Bin galaxy number density in z and mix values between bins: dNia dzdΩ
(z, θ) =
! α
a ¯ Niα
"
1 dN (z, θ)ψα (z) a ¯ Nα dzdΩ a
#
a ¯ Niα ≡ mean number of galaxies of spectral-type a in photo-z bin
i that come from true-z bin
3
N of 10-15)
- modelled after LSST - simulation by A. Connolly
2.5 spectroscopic z
Fiducial model: - Estimate photo-z of 105 simulated galaxy colors in ugrizy filters (limited in i-band at i<25, S/
10 1
2
0.1
1.5
0.01
1 0.5 0 0
0.5
1 1.5 2 photometric z
2.5
3
Model for galaxy correlations Use Limber approximation to compute linear angular galaxy power spectrum: DM ¯α N ¯β bα bβ Pαβ (!) Cαβ (!) = !δNα δNβ " = N
constrain linear galaxy bias jointly with photo-z error parameters truncate ! range to justify Gaussian and Limber approximations
With photo-z errors: Cij (!) =
! αβ
C (!) αβ shot ¯ ¯ Niα Njβ ¯ ¯ + Cij (!) Nα Nβ
Parameter constraint forecasts photo-z bin 1
Open: red/blue split sample
10-1 10-2 10-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 4
10-1 ! / dN/dz
Filled: full sample
! / dN/dz
100
10-2
photo-z bin 2 10-1
10-1
10-2
10-2
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 5
100
100
10-1
10-1
-2
10
-3
-3
10
0 0.5 1 1.5 2 2.5 3 z
10
0 0.5 1 1.5 2 2.5 3 z photo-z bin 6
-2
10 10
photo-z bin 3
-3
0 0.5 1 1.5 2 2.5 3 z
0 0.5 1 1.5 2 2.5 3 z
2 1 ¯ ¯ ¯ - Fractional constraints on Niα ≡ Niα + Niα
- 10% prior on the galaxy bias
Bias and “red” and “blue” population constraints Red & Blue sub-populations
Galaxy bias constraints
photo-z bin 1
red blue
! / dN/dz
1.2
0.8
10-1
10-1
10-2
10-2
10-2
-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 4
0.6 0.4 0.2 0 0
0.5
1
1.5 z
2
2.5
3
photo-z bin 3
10-1
10
! / dN/dz
!((bgal(z)) / bgal(z)
1
photo-z bin 2
10
-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 5
10-1
10-1
-2
-2
10
10
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z
-3
10
0 0.5 1 1.5 2 2.5 3 z photo-z bin 6
100 10-1 10-2 0 0.5 1 1.5 2 2.5 3 z
10-3
0 0.5 1 1.5 2 2.5 3 z
Cross correlating with an overlapping spectroscopic sample
See J. Newman paper: http://astron.berkeley.edu/~jnewman/xcorr/xcorr.pdf
Model for galaxy correlations From A. Schulz Moriond talk la u g n DA
2
c s s o r cr
n o i t a l o r re
-
ss o r c 3D
ti a l e r cor
on
n o i t c f un
Ph o to
ct e l e s e t r ic
io
o i t c n n fu
m
At large (linear) scales assume:
Now observable In previous notation: phot ! b spec α ¯ Niα spec Cαβ (!) Ciβ (!) = b α α
n
Monte Carlo tests (J. Newman) Assumptions: Gaussian photo-z errors (fit for 2 parameters) No bias evolution (so no degeneracy) 25k spec. galaxies per unit z 10 phot. galaxies per arcmin^2 clustering of photometric sample independent of z
How many spectra do we need? J. Newman
Near-term spec. samples J. Newman Blue: SDSS + AGES + VVDS + DEEP2+1700 galaxies/unit z at high z Red: add zCOSMOS + PRIMUS + WiggleZ + 5000 galaxies/unit z at high z
Test with N-body simulations (A. Schulz) Populate 1 (Gpc/h)^3 box with galaxies using HOD No z evolution of correlations or bias
Boxside (Gpc/h)
Boxside (Gpc/h)
Complications Galaxy bias: redshift evolution nonlinear bias Magnification bias Intrinsic l.o.s. correlations between narrow z-bins Sample variance Cosmology dependence Practical method for reconstruction
Restricting the number of parameters
! / dN/dz
photo-z bin 1
photo-z bin 3
10-1
10-1
10-1
10-2
10-2
10-2
10-3
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 4
10-1 ! / dN/dz
photo-z bin 2
10-2 10-3
0 0.5 1 1.5 2 2.5 3 z
0 0.5 1 1.5 2 2.5 3 z photo-z bin 5
100
100
10-1
10-1
10-2
10-2
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z
0 0.5 1 1.5 2 2.5 3 z photo-z bin 6
0 0.5 1 1.5 2 2.5 3 z
2.0
grizY, i < 24.3
!!
!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!
0.9
! !
0.8
!
0.7
!
!
0.6
(Collister & Lahav 2004, Banerji et al. 2007)
!
!
0.5
Sims. from M. Banerji website
Cum. proportion of variance Cum. prop. of variance
http://zuserver2.star.ucl.ac.uk/~mbanerji/DESdata/
1.0
PC decomposition of error distributions
!
1.5
0
10
20
40
Mode number
0.0
0.2
0.4
Effect on DE constraints?
!0.4
!0.2
Eigenfunctions eigenfunctions
1.0 0.5 0.0
phot. z
30
0.0
0.5
1.0 spec. z
1.5
2.0 0.0
0.5
1.0 z
1.5
2.0
Constraints on bias?
Add weak lensing measurements Fit with HOD model (Blake, Collister, & Lahav) Add 3-point correlations (McBride & Connolly, Ashley & Brunner)
Conclusions Amount of “leakage” of galaxies between photo-z bins due to catastrophic errors can be constrained to ~10% of the number of galaxies in each bin if galaxy bias is known. Priors on the galaxy bias are necessary to constrain the photo-z error parameters. Separation of the galaxy sample according to spectral type may significantly improve the photo-z error parameter constraints. Cross-correlating with a spatially overlapping spectroscopic sample may provide even tighter constraints on the photo-z errors. The sizes of the required spectroscopic training samples are not yet determined. Might be able to jointly constrain galaxy bias. Need to test with realistic mocks or data!
Multipole ranges in galaxy power spectra photo-z range
!min (z)
!max (z)
0.0 - 0.5 0.5 - 1.0 1.0 - 1.5 1.5 - 2.0 2.0 - 2.5 2.5 - 3.0
7 23 45 71 103 140
114 458 1018 1875 3195 5186
Fiducial model for “red” and “blue” galaxy spectral types Total dN/dz normalized to 65 galaxies per sq. arcmin.
30
total red blue
25
Red and blue dN/dz’s are ad-hoc Use Cooray 2006 CLF models for red and blue biases
dN/dzd!
20 15 10 5 0 0
0.5
1
1.5 z
2
2.5
3
Collaborators: Andy Connolly, Lloyd Knox, Hu Zhan
Outline The basic idea Cross-correlating the photometric sample with itself Cross-correlating with an overlapping spectroscopic sample (J. Newman) Challenges and future directions
Motivation Future dark energy surveys (DES, Pan-STARRS, LSST, EUCLID, JDEM) plan to use photometric redshifts to measure cosmic shear and galaxy correlations Hard to get fair spectroscopic training samples to the depth of the photometric sample Conventional photo-z estimation methods may leave intolerably large errors Can other calibration methods reduce the size of the fair spectroscopic training sample needed for a given photo-z error target?
Cross correlating galaxies binned by photometric redshift
astro-ph/0606098
Photo-z errors induce cross-correlations n(z)
A. Schulz Scatter
Catastrophic
bin
Z
n(z)
overlap causes correlation
z
Sensitivity of galaxy power spectrum 10-1
bins (1,1) bins (1,3), var. a13 bins (1,3), var. a31
l2 P(l) / (2π)
10-2
Auto and cross angular galaxy power spectra for: 0 < zp < 0.5 and 1 < zp < 1.5
10-3 10-4 10
Points with errors: fiducial values (with
-5
photo-z errors)
10-6 10-7
50
100
200
400 l
800
Lines:1- σ variation
Model for photo-z errors Bin galaxy number density in z and mix values between bins: dNia dzdΩ
(z, θ) =
! α
a ¯ Niα
"
1 dN (z, θ)ψα (z) a ¯ Nα dzdΩ a
#
a ¯ Niα ≡ mean number of galaxies of spectral-type a in photo-z bin
i that come from true-z bin
3
N of 10-15)
- modelled after LSST - simulation by A. Connolly
2.5 spectroscopic z
Fiducial model: - Estimate photo-z of 105 simulated galaxy colors in ugrizy filters (limited in i-band at i<25, S/
10 1
2
0.1
1.5
0.01
1 0.5 0 0
0.5
1 1.5 2 photometric z
2.5
3
Model for galaxy correlations Use Limber approximation to compute linear angular galaxy power spectrum: DM ¯α N ¯β bα bβ Pαβ (!) Cαβ (!) = !δNα δNβ " = N
constrain linear galaxy bias jointly with photo-z error parameters truncate ! range to justify Gaussian and Limber approximations
With photo-z errors: Cij (!) =
! αβ
C (!) αβ shot ¯ ¯ Niα Njβ ¯ ¯ + Cij (!) Nα Nβ
Parameter constraint forecasts photo-z bin 1
Open: red/blue split sample
10-1 10-2 10-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 4
10-1 ! / dN/dz
Filled: full sample
! / dN/dz
100
10-2
photo-z bin 2 10-1
10-1
10-2
10-2
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 5
100
100
10-1
10-1
-2
10
-3
-3
10
0 0.5 1 1.5 2 2.5 3 z
10
0 0.5 1 1.5 2 2.5 3 z photo-z bin 6
-2
10 10
photo-z bin 3
-3
0 0.5 1 1.5 2 2.5 3 z
0 0.5 1 1.5 2 2.5 3 z
2 1 ¯ ¯ ¯ - Fractional constraints on Niα ≡ Niα + Niα
- 10% prior on the galaxy bias
Bias and “red” and “blue” population constraints Red & Blue sub-populations
Galaxy bias constraints
photo-z bin 1
red blue
! / dN/dz
1.2
0.8
10-1
10-1
10-2
10-2
10-2
-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 4
0.6 0.4 0.2 0 0
0.5
1
1.5 z
2
2.5
3
photo-z bin 3
10-1
10
! / dN/dz
!((bgal(z)) / bgal(z)
1
photo-z bin 2
10
-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 5
10-1
10-1
-2
-2
10
10
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z
-3
10
0 0.5 1 1.5 2 2.5 3 z photo-z bin 6
100 10-1 10-2 0 0.5 1 1.5 2 2.5 3 z
10-3
0 0.5 1 1.5 2 2.5 3 z
Cross correlating with an overlapping spectroscopic sample
See J. Newman paper: http://astron.berkeley.edu/~jnewman/xcorr/xcorr.pdf
Model for galaxy correlations From A. Schulz Moriond talk la u g n DA
2
c s s o r cr
n o i t a l o r re
-
ss o r c 3D
ti a l e r cor
on
n o i t c f un
Ph o to
ct e l e s e t r ic
io
o i t c n n fu
m
At large (linear) scales assume:
Now observable In previous notation: phot ! b spec α ¯ Niα spec Cαβ (!) Ciβ (!) = b α α
n
Monte Carlo tests (J. Newman) Assumptions: Gaussian photo-z errors (fit for 2 parameters) No bias evolution (so no degeneracy) 25k spec. galaxies per unit z 10 phot. galaxies per arcmin^2 clustering of photometric sample independent of z
How many spectra do we need? J. Newman
Near-term spec. samples J. Newman Blue: SDSS + AGES + VVDS + DEEP2+1700 galaxies/unit z at high z Red: add zCOSMOS + PRIMUS + WiggleZ + 5000 galaxies/unit z at high z
Test with N-body simulations (A. Schulz) Populate 1 (Gpc/h)^3 box with galaxies using HOD No z evolution of correlations or bias
Boxside (Gpc/h)
Boxside (Gpc/h)
Complications Galaxy bias: redshift evolution nonlinear bias Magnification bias Intrinsic l.o.s. correlations between narrow z-bins Sample variance Cosmology dependence Practical method for reconstruction
Restricting the number of parameters
! / dN/dz
photo-z bin 1
photo-z bin 3
10-1
10-1
10-1
10-2
10-2
10-2
10-3
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z photo-z bin 4
10-1 ! / dN/dz
photo-z bin 2
10-2 10-3
0 0.5 1 1.5 2 2.5 3 z
0 0.5 1 1.5 2 2.5 3 z photo-z bin 5
100
100
10-1
10-1
10-2
10-2
10-3
10-3
0 0.5 1 1.5 2 2.5 3 z
0 0.5 1 1.5 2 2.5 3 z photo-z bin 6
0 0.5 1 1.5 2 2.5 3 z
2.0
grizY, i < 24.3
!!
!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!
0.9
! !
0.8
!
0.7
!
!
0.6
(Collister & Lahav 2004, Banerji et al. 2007)
!
!
0.5
Sims. from M. Banerji website
Cum. proportion of variance Cum. prop. of variance
http://zuserver2.star.ucl.ac.uk/~mbanerji/DESdata/
1.0
PC decomposition of error distributions
!
1.5
0
10
20
40
Mode number
0.0
0.2
0.4
Effect on DE constraints?
!0.4
!0.2
Eigenfunctions eigenfunctions
1.0 0.5 0.0
phot. z
30
0.0
0.5
1.0 spec. z
1.5
2.0 0.0
0.5
1.0 z
1.5
2.0
Constraints on bias?
Add weak lensing measurements Fit with HOD model (Blake, Collister, & Lahav) Add 3-point correlations (McBride & Connolly, Ashley & Brunner)
Conclusions Amount of “leakage” of galaxies between photo-z bins due to catastrophic errors can be constrained to ~10% of the number of galaxies in each bin if galaxy bias is known. Priors on the galaxy bias are necessary to constrain the photo-z error parameters. Separation of the galaxy sample according to spectral type may significantly improve the photo-z error parameter constraints. Cross-correlating with a spatially overlapping spectroscopic sample may provide even tighter constraints on the photo-z errors. The sizes of the required spectroscopic training samples are not yet determined. Might be able to jointly constrain galaxy bias. Need to test with realistic mocks or data!
Multipole ranges in galaxy power spectra photo-z range
!min (z)
!max (z)
0.0 - 0.5 0.5 - 1.0 1.0 - 1.5 1.5 - 2.0 2.0 - 2.5 2.5 - 3.0
7 23 45 71 103 140
114 458 1018 1875 3195 5186
Fiducial model for “red” and “blue” galaxy spectral types Total dN/dz normalized to 65 galaxies per sq. arcmin.
30
total red blue
25
Red and blue dN/dz’s are ad-hoc Use Cooray 2006 CLF models for red and blue biases
dN/dzd!
20 15 10 5 0 0
0.5
1
1.5 z
2
2.5
3
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