1. 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

2. 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

3. 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.

4. 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)

5. 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)

6. 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)

7. 1.1 Formation of BAO
Eisenstein, Seo & White(2007)

8. 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)

9. 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)

10. 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)

11. 1.2 Sound horizon scale
Analytically calculable sound horizon scale (Eisenstein & Hu 1998)

12. 1.3 BAO & distance measure
Standard ruler (i.e. object with known size r)
From angular diameter distance ,
Redshift separation Δz

13. 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

14. 1.3 Volume-averaged distance
Spherically averaging the BAO signal gives Where
Shoji et al. 2009

15. 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

16. 1.4 Alcock-Paczynski (AP) test
Shoji et al. 2009
Almost orthogonal to standard ruler method : gives tighter constraint on and H

17. 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

18. 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

19. 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

20. 2. 1 Imperfect coupling
Eisenstein & Hu (1998) proposed transfer function of power spectrum that takes into account imperfect coupling

21. 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

22. 2.3 BAO in 2-point correlation function
SDSS DR9 – Anderson et al.(2012)

23. 2.4 BAO in power spectrum
SDSS DR9 - Anderson et al. (2012)

24. 2.5 3rd order perturbation theory(3PT)
Describes non-linear distortion of BAO up to ~1% accuracy at z>2
Jeong & Komatsu (2006)

25. 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

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

27. 3.1 Observational Field field Spring Field (42 * 7 degree square)
Fall Field (28 * 5 degree square)

28. 3.2 Survey Geometry
In RA, DEC plane
In commoving coordinates

29. 3.2 Survey Geometry
Each shot includes 82 Integrated Field Units(IFUs)

30. 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

31. 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

32. 4.1 Mock Data
Fast way to generate density distribution that follows the original power spectrum

33. 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

34. 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

35. 4.3 Results from 2-point correlation function
Theoretical prediction – 3PT power spectrum( Jeong & Komatsu 2006)
Linear bias : b ~ 2.12

36. 4. 4 Result from Power spectrum
From 10 realizations(Chi-Ting Chiang)
3.2σ~6.7σ Detection
Significance of BAO

37. 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