Inh Jee University of Texas at Austin Eiichiro Komatsu & Karl Gebhardt Measuring Baryon Acoustic Oscillation using 2-point correlation function in HETDEX survey Inh Jee University of Texas at Austin Eiichiro Komatsu & Karl Gebhardt

ABSTRACT Introduction : What is BAO? Modeling : BAO from previous studies Observation : Hobby-Eberly Telescope Dark Energy EXperiment(HETDEX) Technical detail : Calculation of 2-point correlation function in HETDEX

1.1 Formation of BAO Let me start with this famous WMAP picture. After the inflation, the universe was field with photon, dark matter, neutrino and matter in plasma state. Each components of this ‘soup’ were perturbed due to quantum fluctuation.

1.1 Formation of BAO Eisenstein, Seo & White(2007) Imagine we have a single small density fluctuation at the center of an arbitrary coordinate. All the component, dark matter, gas(baryon), photon, and neutrino are having the same distribution assuming adiabatic initial condition. Note that the y-axis is mass profile , instead of density, such that the area under the graph will be the total mass of each species, and normalized such that it can represent fractional mass with respect to the average of each species. Eisenstein, Seo & White(2007)

1.1 Formation of BAO Eisenstein, Seo & White(2007) This plot shows the evolution of fluctuation 14,000 years after the big bang. Because neutrinos rarely interact with other species, they are free to stream radialy, without gravitational unteraction. Photon and baryon are moving together because of compton scattering of photons from electrons and coulomb interaction between baryons and electrons, and because of the huge pressure of photon the density perturbation of those two species are moving away from the center of the initial perturbation. Dark matter density perturbation stays still, collecting dark matter gravitationally from surroundings. Eisenstein, Seo & White(2007)

1.1 Formation of BAO Eisenstein, Seo & White(2007) Until 0.23Myrs after the big bang, neutrino continue to stream away, photon and baryon are tightly coupled to from a single fluid. Due to the gravitational potential of dark matter at the center, the distribution broadens. Eisenstein, Seo & White(2007)

1.1 Formation of BAO Eisenstein, Seo & White(2007)

1.1 Formation of BAO Eisenstein, Seo & White(2007) BAO bump After the recombination, photons are decoupled from baryons and now freely stream just like neutrino. Baryon remains at the position, showing an enhancement at radius of about 150 Mpc, which corresponds to the sound horizon scale of baryon at drag epoch. Now that the photon were gone, there’s no pressure that support baryon and gravitational interaction between dark matter and baryon becomes important. Sound horizon scale Eisenstein, Seo & White(2007)

1.1 Formation of BAO Eisenstein, Seo & White(2007) BAO bump As a result, distribution of dark matter and baryon becomes similar to each other, causing baryon to recover the central peak, and causing dark matter to mimic the density enhancement near sound horizon scale. Sound horizon scale Eisenstein, Seo & White(2007)

1.1 Formation of BAO Eisenstein, Seo & White(2007) BAO bump 475 Myrs after the big bang, which is about the same time when the first stars were formed, the mass distribution of dark matter and baryons are almost identical, with the enhanced density at sound horizon scale. We call this enhencement as BAO bump. Sound horizon scale Eisenstein, Seo & White(2007)

1.2 Sound horizon scale Analytically calculable sound horizon scale (Eisenstein & Hu 1998)

1.3 BAO & distance measure Standard ruler (i.e. object with known size r) From angular diameter distance , Redshift separation Δz

1.3 BAO & 2-point correlation function Definition : Probability to find another galaxy at separation when observed from position Statistical tool to probe density perturbation

1.3 Volume-averaged distance Spherically averaging the BAO signal gives Where Shoji et al. 2009

1.4 Alcock-Paczynski (AP) test Assuming isotropy of BAO, equate r from redshift separation and angular separation No need to calculate the sound horizon scale Not only sound horizon scale, but also use all scale information from observed data

1.4 Alcock-Paczynski (AP) test Shoji et al. 2009 Almost orthogonal to standard ruler method : gives tighter constraint on and H

1.5 Redshift-Space Distortion(RSD) Peculiar motion changes measured line-of-sight position of a galaxy from initial position Kaiser effect on large scales due to bulk flow Finger-of-God effect on small scales due to virialized random motion

1.5 Redshift-Space Distortion(RSD) Shoji et al. 2009 RSD affects the AP test by causing anisotropy on BAO signal Marginalize over RSD will give

2. 1 Imperfect coupling Silk damping(Silk 1986) reduces the amplitude of BAO on small scales Velocity overshoot(Sunyaev & Zeldovich 1970) changes the frequency of BAO => sound horizon scale does not exactly match the BAO bump Silk damping : photon diffuses from baryon + Compton drag of baryons from overdense -> underdense region Velocity overshoot : Kinematic motion of baryons at the drag epoch

2. 1 Imperfect coupling Eisenstein & Hu (1998) proposed transfer function of power spectrum that takes into account imperfect coupling

2.2 Non-linear degradation of BAO Non-linear density field evolution Redshift-space distortion Galaxy bias => Broaden & shift the BAO peak Amount of shift

2.3 BAO in 2-point correlation function SDSS DR9 – Anderson et al.(2012)

2.4 BAO in power spectrum SDSS DR9 - Anderson et al. (2012)

2.5 3rd order perturbation theory(3PT) Describes non-linear distortion of BAO up to ~1% accuracy at z>2 Jeong & Komatsu (2006)

3. Hobby-Eberly Telescope Dark Energy Experiment(HETDEX) Final goal : constrain dark energy equation of state in redshift range 1.9<z<3.5 Area : 42*7 degree square (V ~10 Gpc^3) Tracers : Lyman alpha emitters Expectation : ~ 400k galaxies from each low(1.9<z<2.5) and high 2.5<z<3.5) redshift BAO as a standard ruler

3. Hobby-Eberly Telescope Dark Energy Experiment(HETDEX) Espectation

3.1 Observational Field field Spring Field (42 * 7 degree square) Fall Field (28 * 5 degree square)

3.2 Survey Geometry In RA, DEC plane In commoving coordinates

3.2 Survey Geometry Each shot includes 82 Integrated Field Units(IFUs)

3.2 Survey Geometry Shots and IFUs are not completely filling the volume Large field – impossible to apply flat sky approximation Large redshift range : conic shape window function => need to consider survey geometry carefully

4. 1 Mock Data Log-normal realization (Donghui Jeong) Log-transformed field of the density field From a given power spectrum, calculate its log-transformed field power spectrum Throw a Gaussian random variable and choose variables that follows by acceptance-rejection algorithm

4.1 Mock Data Fast way to generate density distribution that follows the original power spectrum

4.2 Landy-Szalay(LS) estimator Discretely distributed galaxies -> pair count Simplest estimator is biased, and variance is large Landy & Szalay suggested a modified estimator for the calculation of 2-point correlation function Landy & Szalay 1993

4.3 k-dimensional tree pair count Based on a python code by Nicolas Canac Construct binary tree Chose a level to be the node level Do range search between a pair of nodes Moore et al. 2000 Node level

4.3 Results from 2-point correlation function Theoretical prediction – 3PT power spectrum( Jeong & Komatsu 2006) Linear bias : b ~ 2.12

4. 4 Result from Power spectrum From 10 realizations(Chi-Ting Chiang) 3.2σ~6.7σ Detection Significance of BAO

SUMMARY Photon-baryon interaction causes BAO Sound horizon scale imprinted on BAO can be used as a standard ruler Imperfect coupling and non-linear degradation of BAO should be taken into account 2-point correlation function : LS estimator - pair count based on kd-tree algorithm 3.2σ~6.7σ detection significance of BAO under HETDEX geometry using power spectrum