Cosmic Variance and Luminosity Function Fitting
Michele TrentiMichele Trenti
August 8, 2007
In collaboration with Massimo Stiavelli and the UDF05 team
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Outline
Large scale structure and galaxy number counts
Cosmic variance and luminosity function fitting:Number countsuncertainty M* and dependence on environment
Quantifying luminosity function evolution
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Context Ultimate goal is to get a reliable measure of
the galaxy luminosity function (LF) and to quantify its errorA measure has little meaning without proper error
bars, both random and systematic
LF fundamental measure for: Global star formation historyGalaxy assembly process At z6: Reionization history of the Universe
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
High z galaxies Hundreds of galaxies have been detected in
recent years at z>4:HDFGOODSUDF, UDF05Subaru deep fields….
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Field to field variations
Cosmic volume probed by these high z surveys is however limitedtypically tens to hundreds of
arcmin2
tiny fraction of the sky! How does the result
depend on the pointing chosen, that is what is the distribution of the expected number counts of galaxies?
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Field to field variations
Universe is not homogenous on small scales!
E.g.: UDF V or i dropouts volume is 104 (Mpc/h)3
This volume contains only 1015 M/h○ Large Scale Structure is
importantSignificant uncertainty in
the number counts due to galaxy clustering
SDSS Cosmic Web
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Cosmic variance Number counts
distribution in Galaxy surveys does not follow Poisson
We define cosmic variance the excess relative variance over Poisson noise:
NN
NNv
12
22
2
Simulated number counts distribution for i-dropouts in the UDF
Trenti & Stiavelli (2007)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Cosmic variance and the total error budget
The total 1 fractional error (vr) in the number counts is given by combining:Cosmic variance (intrinsic property of galaxy
population)Poisson noise associated to the observed
counts (includes observational incompletness):
obsvr Nv 122
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Estimating cosmic variance:Analytical approach
The cosmic variance is related to the two point correlation function (r) of the sample (e.g. Somerville et al. 2004):
Depends on clustering properties ((r)) and on the geometry of the survey (volume integral)
V
Vv
rdrd
rdrdrr
21
21212|)(|
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Cosmic variance and survey geometry
Spherical volumes have the largest variations in number counts:The volume may easily sit on
top of overdensities/ underdensities
Pencil beam surveys for LBG galaxies probe a variety of environments: z=1 320 Mpc/h at z=6.1Uncertainty is reduced
Trenti & Stiavelli (2007)
Relative 1 uncertainty in number counts
Pencil beam
~Cubic volume
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Estimating cosmic variance:Cosmologic simulations
Analytical approach inexpensive but limited to the variance of the counts distribution
Counts may have strong skew and non gaussian tails
Cosmological simulations computationally expensive but provide synthetic catalogsFull probability distribution of countsIn addition: allow us to explore fitting of the LF
from the mock catalogs
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Mock Catalogs from Cosmological Simulations Cosmological simulation
with 300 million particles, 128Mpc/h box≈1010 M/h halos resolved
Dark matter halos populated using HOD models
Luminosity-Mass relation based on Cooray (2005)
Pencil beam traced through the box
Redshift evolution taken into account (snapshots spaced by z=0.125)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Mock Catalogs from Cosmological Simulations For Lyman Break
galaxies selection z≈1pencil beam is 300Mpc/h
it wraps around the box, spaced by >15Mpc/h
negligible correlation (rlin<0.01) introduced in the counts
Different HOD models give similar p(N) at fixed <N>minor changes in average
bias of galaxies even changing detection probability by factor 2
V dropouts counts in two combined UDF05 fields
Adapted from Oesch et al. (2007)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Total fractional error for V and i-dropouts, ACS field of view
Trenti & Stiavelli (2007)Typical deep field has >25% uncertainty in number counts, 2.5-3 times larger than Poisson
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Total fractional counts error for i-dropouts in GOODS
Trenti & Stiavelli (2007)
~18% uncertainty!
GOODS N+S fields, ~ 320 arcmin2
GOODS N+S fields have ~30 times UDF area, but not as deep
Detected objects are more luminous more massivemore clustered, higher
bias Cosmic variance still
high despite larger area!
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Total fractional error for z and J-dropouts
Trenti & Stiavelli (2007)
Significant total fractional errorvr> 50%
Independent fields beat cosmic variance:6 independent deep
NICMOS fields (already existing) better than one deep WFC3 field, despite smaller area!
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Luminosity function and environment Does the luminosity function
depend on the environment? First order dependence in
normalization: * proportional to the galaxy
number counts Does the shape of the LF (that is
and M*) also depend on number counts? Fundamental question to
properly address claims of evolution of the LF shape over redshift
L
Faint end: power law, slope
Bright end: exponential
L* (Typical luminosity)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Shape of the luminosity function and LSS from our mock catalogs
LF from synthetic V-drop catalogs, 1 ACS field, UDF depth from Trenti & Stiavelli (2007)
M* is fainter in underdense fields
(consistent with the local universe, see SDSS LF in voids – Hoyle et al. 2005)
independent of environment
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
LF fitting: M*- degeneracy and binning
BINNED UNBINNED
LF from synthetic V-drop catalogs, 1 ACS field, UDF depth Well known degeneracy between and M* is present Smaller uncertainty when Maximum Likelihood is used: binning
leads to information loss
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Combining fields: luminosity function fitting Combining independent fields helps beating cosmic
variance Fields at different depths provide optimal use of
telescope time: large area to constraint bright end of LFultradeep field to constraint faint end of FLfor example: combination of GOODS+UDF
But… Is the resulting LF sensitive to fitting method used?Is there an optimal method to derive the LF and to “correct
for” cosmic variance (e.g. see Bouwens et al. 2006)?
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
An attempt to correct for LSS Bouwens et al. (2006) assume that Large Scale
Structure can be measured from bright detections Correction on normalization of deep fields for
i-dropouts based on GOODS counts:Degradation of deeper fields to GOODS depthRe-Normalization of the faint end of the LF based on the
ratio of degraded counts over expected counts from GOODS.
Is this justified?We need to investigate the faint-bright counts relation!
Note however, that as of July (Bouwens et al. 2007), they no longer consider this method the preferred choice for LF fitting.
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Bright-Faint counts relation Assume a linear faint-
bright counts relation:<Nft> = + <Nbr>
In a uncorrelated world:=1, 0When <Nft> >> <Nbr>
field to field variations in faint counts cannot be corrected○ no information from Nbr
=1
Faint (UDF) – Bright (GOODS) i-drop counts, uncorrelated Poisson World
Trenti & Stiavelli (2007)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Bright-Faint counts relation LSS correlates
bright and faint counts, but not completely 0, > 1
Bouwens et al. 2006 assume total correlation, that is = 0Artificial steepening
of the faint end in underdense fields!
LSS
=0
Faint (UDF) – Bright (GOODS) i-drop counts, LSS Mock Catalog
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
LF fitting using LSS renormalization Significant artificial steepening introduced in
presence of a deficit of brigh objects in the deep field
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
LF fitting using Maximum Likelihood Normalization is left free between fields at different
depthsUnbiased measure of the LF slopeM* has residual dependence on counts (but physical origin)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Conclusions I Cosmic variance introduces significant
uncertainty in galaxy number counts in deep field surveysDominant over Poisson noise for typical deep
surveys:○ UDF and GOODS have similar cosmic variance at
their respective depthsGOODS area larger but UDF deeper, so bias is smaller
Sparse coverage beats cosmic variancebut contiguous fields are useful beyond LF
determination (e.g. weak lensing)
August, 8 2007 STScI Summer PostDoc Talks Michele Trenti
Conclusions II Cosmic variance introduces uncertainty in the
shape of the luminosity function M* measured in underdense LBG fields is
fainter (like in local voids) Degeneracy between M* and Systematic errors are important: hard to
assess changes in LF of < 0.15 (68% cl) Naïve “renormalization” for large scale
structure may introduce significant biasUnbinned data analysis with free * optimal for
recovering information