Cosmic Variance and Luminosity Function Fitting

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


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


High z galaxies Hundreds of galaxies have been detected in

recent years at z>4:HDFGOODSUDF, UDF05Subaru deep fields….


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?


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


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)


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


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|)(|


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


Relative 1 uncertainty in number counts

Pencil beam

~Cubic volume


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


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)


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)


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


Total fractional counts error for i-dropouts in GOODS


~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!


Total fractional error for z and J-dropouts


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!


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)


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


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


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)?


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.


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



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


LF fitting using LSS renormalization Significant artificial steepening introduced in

presence of a deficit of brigh objects in the deep field


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)


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)


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

Documents

Cosmic Variance and Luminosity Function Fitting