The flickering luminosity method Martin Feix in collaboration with Adi Nusser (Technion) and Enzo Branchini (Roma Tre) Institut d Astrophysique de Paris SSG16 Workshop, February 2nd 2016
Outline 1 Motivation ΛCDM model Growth rate of density perturbations RSDs 2 Methodology and data 3 Application to SDSS 4 Conclusions
The Universe in a nutshell The 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 perturbations Probing the nature of cosmic acceleration z=6 z=2 z=0 δ(x, z) = D(z)δ0 (x) f (Ω) = d log D ' Ωγ (a) growth rate d log a Image credit: V. Springel / MPIA Garching
Growth rate of density perturbations Probing the nature of cosmic acceleration Huterer et al., Astropart.Phys. 63 (2015)
Growth rate of density perturbations Measuring the growth rate with redshift-space distortions Peacock et al., Nature 410 (2001) Guzzo et al., Nature 451 (2008)
Outline 1 Motivation 2 Methodology and data Peculiar velocities from luminosity variations MLE and quadratic approximation 3 Application to SDSS 4 Conclusions
Peculiar velocities from LF variations Basic concepts Peculiar motion introduces systematic variations in the observed luminosity distribution of galaxies (Nusser et al. 2011; Tammann et al. 1979) Linear theory (c = 1): M = M obs + 5 log 10 D L (z obs ) D L (z) z obs z 1 + z obs = V(t, r) Φ(t, r) ISW V(t, r) Maximize probability of observing galaxies given their magnitudes and redshifts: φ(m i ) log P tot = log P i (M i z i, V i ) = bi, where i i φ(m)dm a i a/b = max / min [ M min/max, m +/ DM(z) K(z obs ) + Q(z obs ) ] Velocity models: V(t, r) V ({ξ i }), V(t, r) V(β = f /b)
Peculiar velocities from LF variations Basic concepts V i = 0 φ (M) M
Peculiar velocities from LF variations Basic concepts V i 0 φ (M) M
Peculiar velocities from LF variations Basic concepts Method independent of galaxy bias and traditional distance indicators For N 1, P tot is well approximated by a Gaussian: log P tot (d x) 1 2 (x x 0) T Σ 1 (x x 0 ) + const, where x T = ( {q j }, {ξ k } ) Build quadratic estimator to determine P tot : log P i log P i log P i x=x0 + x α α x α + x=x0 α,β 2 log P i x α x β x α x β x=x0 Tedious and may result in complex expressions, e.g. for a Schechter LF: φ(m) = 0.4 log (10)φ 10 0.4(1+α )(M M) exp ( 10 0.4(M M) )
Outline 1 Motivation 2 Methodology and data 3 Application to SDSS SDSS DR7 data Proof of concept: constraints on σ 8 Linear velocity reconstruction and growth constraints at z 1 4 Conclusions
SDSS Data Release 7 NYU Value-Added Galaxy Catalog (Blanton et al. 2005) Use r-band magnitudes (Petrosian) 14.5 < m r < 17.6 22.5 < M obs < 17.0 0.02 < z < 0.22 N 5 10 5 Adopt pre-planck cosmological parameters (Calabrese et al. 2013) Realistic mocks for testing SDSS footprint photometric offsets between stripes overall tilt over the sky
LF estimators Non-parametric spline-estimator of φ(m) φ[(mpc/h) 3 ] 1 0.1 r-band 0.1 0.4 0.01 0.2 0.001-23 -22-21 -19-18 -17 Q0 = 1.60 ±0.11 M 5log 10 h = 20.52 ±0.04 α = 1.10 ±0.03-20 -19-18 -17 M r 5log 10 h Normalization unimportant for our analysis Two-parameter Schechter function does quite well To reduce errors, adopt more flexible form for φ(m) Model φ(m) as a spline with sampling points {φ j (M)} for M j < M < M j+1 Advantage: smoothness, nice analytic properties for integrals / derivatives) Parameterize luminosity evolution: e(z) = Q 0 (z z 0 ) + O ( z 2) Feix et al., JCAP 09 (2014) arxiv:1405.6710
Proof of concept Redshift-binned velocity model Expand binned velocity field in SHs: V(t, r) Ṽ(ˆr), Ṽ(ˆr) = a lm Y lm (ˆr), 0.02 < z 1 < 0.07 < z 2 < 0.22 l,m Determine P tot with quadratic estimator Marginalize over LF parameters and construct posterior for C l = a lm 2 by applying Bayes theorem: P ({C l }) P (d {a lm }) P ({a lm } {C l }) da lm Assume {a lm } as normally distributed For a ΛCDM model prior, C l = C l ({c k }): C l = 2 dkk 2 P Φ (k) π ( ) l jl 2 drw(r) r k j l+1
Constraints on σ 8 Results from SDSS data analysis (l max = 5 in two redshift bins) 5 4 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 χ 2 3 2 σ 8 1.1 ± 0.4 σ 8 1.0 ± 0.5 1 0 both bins 0 0.5 1 1.5 2 low-z bin only 0 0.5 1 1.5 2 Feix et al., JCAP 09 (2014) arxiv:1405.6710 σ 8 σ 8
SDSS Data Release 7 NYU Value-Added Galaxy Catalog (Blanton et al. 2005) Use r-band magnitudes (Petrosian) 14.5 < m r < 17.6 22.5 < M obs < 17.0 0.02 < z < 0.22 0.06 < z < 0.12 N 5 10 5 2 10 5 Adopt pre-planck cosmological parameters (Calabrese et al. 2013) Realistic mocks for testing SDSS footprint photometric offsets between stripes overall tilt over the sky
Building the velocity field of SDSS galaxies Linear 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 R s 10h 1 Mpc Problem in spherical harmonics space: ( d s 2 dφ lm ds ds 1 s 2 ) 1 l(l + 1)Φ lm 1 + β s 2 = β 1 + β ( δ g lm d log S 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 ) dφ lm ds
Constraints on f σ 8 at z 0.1 Results for SDSS mock catalogs (l max = 150) 0.6 0.4 0.04 0.03 0.02 25 20 f σ 8 = 0.48 ± 0.19 (l > 1) f σ 8 = 0.49 ± 0.22 (l > 5) 0.2 0.01 cosη 0-0.2-0.4 0-0.01-0.02 M Nmock 15 10-0.6-0.03-0.04 0 10 20 30 40 50 60 λ [deg] Feix et al., PRL 115, 011301 (2015) arxiv:1503.05945 5 0 0 0.4 0.8 1.2 f σ 8
Constraints on f σ 8 at z 0.1 Results from SDSS data analysis (l max = 150) 5 4 f σ 8 = 0.37 ± 0.13 (l > 1) f σ 8 = 0.56 ± 0.25 (l > 5) 0.7 0.6 2MRS/SFI++ 2MRS 6dFGS This work SDSS MGS 2dFGRS WiggleZ SDSS LRG 3 0.5 χ 2 2 f σ8 0.4 1 0.3 0 0.2 0.2 0.4 0.6 0.8 f σ 8 0 0.05 0.1 0.15 0.2 0.25 z Feix et al., PRL 115, 011301 (2015) arxiv:1503.05945
Outline 1 Motivation 2 Methodology and data 3 Application to SDSS 4 Conclusions
Conclusions ML estimators extracting the large-scale velocity field through spatial modulations in the observed LF of galaxies offer a powerful and complementary alternative to currently 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 and consistent with those coming from RSD analyses of similar datasets Consistency is striking in view of the different possible systematic biases associated with the different methods Luminosity-based techniques are less sensitive to nonlinear corrections than the two-point statistics which enter the analysis of RSDs