Motivation Methodology and data Application to SDSS Conclusions
The flickering luminosity method
Martin Feixin collaboration with Adi Nusser (Technion) and Enzo Branchini (Roma Tre)
Institut d’Astrophysique de Paris
SSG16 Workshop,February 2nd 2016
Motivation Methodology and data Application to SDSS Conclusions
Outline
1 MotivationΛCDM modelGrowth rate of density perturbationsRSDs
2 Methodology and data
3 Application to SDSS
4 Conclusions
Motivation Methodology and data Application to SDSS Conclusions
The Universe in a nutshellThe ΛCDM cosmological model
Planck’s cosmic recipe
Image credit: ESA and the Planck Collaboration
Motivation Methodology and data Application to SDSS Conclusions
Growth rate of density perturbationsProbing the nature of cosmic acceleration
z = 6
z = 2
z = 0
δ(x, z) = D(z)δ0(x)
f (Ω) =d log Dd log a
' Ωγ(a)→ growth rate
Image credit: V. Springel / MPIA Garching
Motivation Methodology and data Application to SDSS Conclusions
Growth rate of density perturbationsProbing the nature of cosmic acceleration
Huterer et al., Astropart.Phys. 63 (2015)
Motivation Methodology and data Application to SDSS Conclusions
Growth rate of density perturbationsMeasuring the growth rate with redshift-space distortions
Peacock et al., Nature 410 (2001)Guzzo et al., Nature 451 (2008)
Motivation Methodology and data Application to SDSS Conclusions
Outline
1 Motivation
2 Methodology and dataPeculiar velocities from luminosity variationsMLE and quadratic approximation
3 Application to SDSS
4 Conclusions
Motivation Methodology and data Application to SDSS Conclusions
Peculiar velocities from LF variationsBasic concepts
• Peculiar motion introduces systematic variations in the observed luminositydistribution of galaxies (Nusser et al. 2011; Tammann et al. 1979)
M = Mobs + 5 log10DL(zobs)
DL(z)
• Linear theory (c = 1):
zobs − z1 + zobs
= V(t, r) − Φ(t, r) − ISW ≈ V(t, r)
• Maximize probability of observing galaxies given their magnitudes and redshifts:
log Ptot =∑
i
log Pi(Mi |zi,Vi) =∑
i
φ(Mi)∫ biaiφ(M)dM
, where
a/b = max /min[Mmin/max,m+/− − DM(z) − K(zobs) + Q(zobs)
]• Velocity models:
V(t, r)→ V (ξi) , V(t, r)→ V(β = f /b)
Motivation Methodology and data Application to SDSS Conclusions
Peculiar velocities from LF variationsBasic concepts
Vi = 0φ(M
)
M
Motivation Methodology and data Application to SDSS Conclusions
Peculiar velocities from LF variationsBasic concepts
Vi 6= 0φ(M
)
M
Motivation Methodology and data Application to SDSS Conclusions
Peculiar velocities from LF variationsBasic concepts
• Method independent of galaxy bias and traditional distance indicators
• For N 1, Ptot is well approximated by a Gaussian:
log Ptot(d|x) ≈ −12
(x − x0)T Σ−1(x − x0) + const, where xT =(q j, ξk
)• Build quadratic estimator to determine Ptot:
log Pi ≈ log Pi |x=x0 +∑α
∂ log Pi
∂xα
∣∣∣∣∣x=x0
xα +∑α,β
∂2 log Pi
∂xα∂xβ
∣∣∣∣∣∣x=x0
xαxβ
• Tedious and may result in complex expressions, e.g. for a Schechter LF:
φ(M) = 0.4 log (10)φ?100.4(1+α?)(M?−M) exp(−100.4(M?−M)
)
Motivation Methodology and data Application to SDSS Conclusions
Outline
1 Motivation
2 Methodology and data
3 Application to SDSSSDSS DR7 dataProof of concept: constraints on σ8Linear velocity reconstruction and growth constraints at z ∼ 1
4 Conclusions
Motivation Methodology and data Application to SDSS Conclusions
SDSS Data Release 7NYU Value-Added Galaxy Catalog (Blanton et al. 2005)
• Use r-band magnitudes (Petrosian)
• 14.5 < mr < 17.6
• −22.5 < Mobs < −17.0
• 0.02 < z < 0.22
• N ∼ 5 × 105
• Adopt pre-Planck cosmological parameters(Calabrese et al. 2013)
• Realistic mocks for testing
→ SDSS footprint→ photometric offsets between stripes→ overall tilt over the sky
Motivation Methodology and data Application to SDSS Conclusions
LF estimators“Non-parametric” spline-estimator of φ(M)
0.4
0.2
0.1r-band
α⋆ =−1.10±0.03
M⋆−5log10 h=−20.52±0.04
Q0 = 1.60±0.11
-18-19 -17
0.001
0.01
0.1
1
-20 -19 -18 -17
φ[(Mpc/h)−
3]
Mr−5log10h
-23 -22 -21
Feix et al., JCAP 09 (2014) — arXiv:1405.6710
• Normalization unimportant for our analysis
• Two-parameter Schechter function doesquite well
• To reduce errors, adopt more flexible formfor φ(M)
• Model φ(M) as a spline with samplingpoints φ j(M) for M j < M < M j+1
• Advantage: smoothness, nice analyticproperties for integrals / derivatives)
• Parameterize luminosity evolution:
e(z) = Q0(z − z0) + O(z2
)
Motivation Methodology and data Application to SDSS Conclusions
Proof of conceptRedshift-binned velocity model
• Expand binned velocity field in SHs:
V(t, r)→ V(r), V(r) =∑l,m
almYlm(r), 0.02 < z1 < 0.07 < z2 < 0.22
• Determine Ptot with quadratic estimator
• Marginalize over LF parameters and construct posterior for Cl =⟨|alm |
2⟩
by applyingBayes’ theorem:
P (Cl) ∝∫
P (d|alm) P (alm|Cl) dalm
• Assume alm as normally distributed
• For a ΛCDM model prior, Cl = Cl (ck):
Cl =2π
∫dkk2PΦ(k)
∣∣∣∣∣∣∫
drW(r)(
l jlr− k jl+1
)∣∣∣∣∣∣2
Motivation Methodology and data Application to SDSS Conclusions
Constraints on σ8Results from SDSS data analysis (lmax = 5 in two redshift bins)
σ8 = 1.61±0.38
σ8 = 1.52±0.37
σ8 = 1.55±0.40
σ8 = 1.08±0.53
σ8 = 1.01±0.45
σ8 = 1.06±0.51
both bins low-z bin only
0 0.5 1 1.5 2
0
1
2
3
4
5
0 0.5 1 1.5 2
∆χ2
σ8 σ8
Feix et al., JCAP 09 (2014) — arXiv:1405.6710
σ8 ≈ 1.1 ± 0.4 σ8 ≈ 1.0 ± 0.5
Motivation Methodology and data Application to SDSS Conclusions
SDSS Data Release 7NYU Value-Added Galaxy Catalog (Blanton et al. 2005)
• Use r-band magnitudes (Petrosian)
• 14.5 < mr < 17.6
• −22.5 < Mobs < −17.0
• 0.02 < z < 0.22→ 0.06 < z < 0.12
• N ∼ 5 × 105 → 2 × 105
• Adopt pre-Planck cosmological parameters(Calabrese et al. 2013)
• Realistic mocks for testing
→ SDSS footprint→ photometric offsets between stripes→ overall tilt over the sky
Motivation Methodology and data Application to SDSS Conclusions
Building the velocity field of SDSS galaxiesLinear velocity reconstruction (Nusser & Davis 1994; Nusser et al. 2012)
• Assume β = f /b = const over sample range
• Smooth redshift-space density field on a scale Rs ∼ 10h−1 Mpc
• Problem in “spherical harmonics” space:
1s2
dds
(s2 dΦlm
ds
)−
11 + β
l(l + 1)Φlm
s2 =β
1 + β
(δ
glm −
d log Sds
dΦlm
ds
)• Boundary conditions: set δ = 0 outside data volume (zero-padding)
• Must exclude monopole and dipole terms
• Models robust w.r.t. small-scale issues, e.g. details of galaxy bias
• Assign galaxy velocities for discrete β-values→ likelihood analysis
Motivation Methodology and data Application to SDSS Conclusions
Constraints on fσ8 at z ∼ 0.1Results for SDSS mock catalogs (lmax = 150)
0 10 20 30 40 50 60
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
λ [deg]
cosη ∆M
0.03
0.04
Feix et al., PRL 115, 011301 (2015) — arXiv:1503.05945
f σ8 = 0.49±0.22(l > 5)
f σ8 = 0.48±0.19(l > 1)
0
5
20
25
0.4 0.8 1.20
10
15
f σ8
Nm
ock
Motivation Methodology and data Application to SDSS Conclusions
Constraints on fσ8 at z ∼ 0.1Results from SDSS data analysis (lmax = 150)
f σ8 = 0.37±0.13 (l > 1)
f σ8 = 0.56±0.25 (l > 5)
0
1
2
3
4
0.4 0.6 0.8
5
0.2
∆χ2
f σ8
Feix et al., PRL 115, 011301 (2015) — arXiv:1503.05945
2MRS/SFI++2MRS
This work6dFGS
SDSS MGS2dFGRSWiggleZSDSS LRG
0.5
0.7
0.6
0.4
0.3
0.2
0 0.05 0.1 0.2 0.250.15
fσ8
z
Motivation Methodology and data Application to SDSS Conclusions
Outline
1 Motivation
2 Methodology and data
3 Application to SDSS
4 Conclusions
Motivation Methodology and data Application to SDSS Conclusions
Conclusions
• ML estimators extracting the large-scale velocity field through spatial modulations inthe observed LF of galaxies offer a powerful and complementary alternative tocurrently used methods
• New growth measurements are in agreement with the results from Planck
• Luminosity-based constraints on the growth rate at z ∼ 0.1 are both compatible andconsistent with those coming from RSD analyses of similar datasets
• Consistency is striking in view of the different possible systematic biases associatedwith the different methods
• Luminosity-based techniques are less sensitive to nonlinear corrections than thetwo-point statistics which enter the analysis of RSDs
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