modeling the 3-point correlation function felipe marin department of astronomy & astrophysics...

Modeling the 3-point correlation function

Modeling the 3-point correlation function

Felipe MarinDepartment of Astronomy &

AstrophysicsUniversity of Chicago

arXiv:0705.0255

Felipe MarinDepartment of Astronomy &

AstrophysicsUniversity of Chicago

arXiv:0705.0255

06/01/2007 Great Lakes Cosmology Workshop 8

Collaborators:Collaborators:

Josh Frieman

(KICP-Chicago & Fermilab)

Josh Frieman

(KICP-Chicago & Fermilab)

Bob Nichol

(ICG, Portsmouth)

Bob Nichol

(ICG, Portsmouth)

Risa Wechsler

(KICP-Chicago, now Stanford)

Risa Wechsler

(KICP-Chicago, now Stanford)


Correlation functions on LSSCorrelation functions on LSS

Galaxy surveys show us that the (luminous) matter does not distribute very smoothly in the Universe

From cosmological N-body simulations, we can see that is also not the case for the dark matter

How do we get a more quantitative insight? Can we infer DM clustering using galaxies?

Galaxy surveys show us that the (luminous) matter does not distribute very smoothly in the Universe

From cosmological N-body simulations, we can see that is also not the case for the dark matter

How do we get a more quantitative insight? Can we infer DM clustering using galaxies?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

A. Kravtsov

M. Tegmark


N-point statisticsN-point statistics One way to achieve this is using spatial N-point correlation functions: measure how more

likely is to have certain configurations of N-points in a particular field than in a random distribution.

For instance, we can describe the probability that two objects (galaxies, dark matter particles, DM halos, etc.) are found at a distance r:

One way to achieve this is using spatial N-point correlation functions: measure how more likely is to have certain configurations of N-points in a particular field than in a random distribution.

For instance, we can describe the probability that two objects (galaxies, dark matter particles, DM halos, etc.) are found at a distance r:

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P = n2(1+ ξ (r)) This defines the two-point correlation function. Along with

its Fourier counterpart, the Power Spectrum, have been measured in simulations and galaxy surveys, CMB,etc.

This defines the two-point correlation function. Along with its Fourier counterpart, the Power Spectrum, have been measured in simulations and galaxy surveys, CMB,etc.


The need for a more complete descriptionThe need for a more complete description

It is possible that two distributions have the same 2-point statistics, but they look completely different! 2-point statistics just describe completely only Gaussian Fields. They do not take into account non-spherical morphologies

It is possible that two distributions have the same 2-point statistics, but they look completely different! 2-point statistics just describe completely only Gaussian Fields. They do not take into account non-spherical morphologies

Sefusatti & Scoccimarro 2005


The three-point correlation functionThe three-point correlation function

The next order correlation is the three-point correlation function (3PCF): The probability to find 3 objects in a certain triangle configuration:

The next order correlation is the three-point correlation function (3PCF): The probability to find 3 objects in a certain triangle configuration:

€

P = n3(1+ ξ (r1,r2) + ξ (r2,r3) + ξ (r3,r1) + ζ (r1,r2,r3))

1

3

€

Q(r,u,α ) =ζ (r1,r2,r3)

ξ 2(r1,r2) + ξ 2(r2,r3) + ξ 2(r3,r1)

2

The value of the 3PCF depends on the overall scale of the triangle, as well as on its shape.

Useful to define the reduced 3PCF:

The value of the 3PCF depends on the overall scale of the triangle, as well as on its shape.

Useful to define the reduced 3PCF:

u r

r


N-body simulations & mock galaxy catalogsN-body simulations & mock galaxy catalogs

We want to measure & compare the 3PCF of dark matter and galaxies in real & redshift space.

We use high-resolution N-body simulations run using ART code (Kravtsov et al. ’97,’04) with concordance LCDM parameters. Can detect dark matter halos of galactic size

Two boxes: L120 with 120 Mpc/h on the side & L200 with 200 Mpc/h on the side

Redshift space: long-distance observer approximation: peculiar velocities distortions

We want to measure & compare the 3PCF of dark matter and galaxies in real & redshift space.

We use high-resolution N-body simulations run using ART code (Kravtsov et al. ’97,’04) with concordance LCDM parameters. Can detect dark matter halos of galactic size

Two boxes: L120 with 120 Mpc/h on the side & L200 with 200 Mpc/h on the side

Redshift space: long-distance observer approximation: peculiar velocities distortions

Kravtsov et al 04

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xs = xr + vx,rH(z)−1


From DM to galaxiesFrom DM to galaxies

Kravtsov et al (2004), Conroy, Wechsler & Kravtsov (2006) : Vmax, of a DM halo is a good indicator of the stellar mass and henceforth, of the luminosity of a galaxy.

In order to get luminosities, for both L120 & L200 boxes, the r-band SDSS luminosity function is matched to the cumulative velocity function at the redshift of observation n(>Vmax,now)

Colors are assigned using the observed relation between local density and SDSS color (Wechsler et al. 2004, Tasitsiomi et al 2004).

Kravtsov et al (2004), Conroy, Wechsler & Kravtsov (2006) : Vmax, of a DM halo is a good indicator of the stellar mass and henceforth, of the luminosity of a galaxy.

In order to get luminosities, for both L120 & L200 boxes, the r-band SDSS luminosity function is matched to the cumulative velocity function at the redshift of observation n(>Vmax,now)

Colors are assigned using the observed relation between local density and SDSS color (Wechsler et al. 2004, Tasitsiomi et al 2004).

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Vmax =GM(< r)

rmax

Conroy,Wechsler & Kravtsov 06

€

ngal (> Li) = nhalo(>Vmax,i)


Results: DM vs halos Equilateral trianglesResults: DM vs halos Equilateral triangles

Reduced 3PCF for DM particles & halos Jack-knife error bars Halos strongly biased w.r.t. DM particles Strong scale dependence in real space, strong redshift evolution on small scales Features of Q very washed out in redshift space

Reduced 3PCF for DM particles & halos Jack-knife error bars Halos strongly biased w.r.t. DM particles Strong scale dependence in real space, strong redshift evolution on small scales Features of Q very washed out in redshift space

MWFN 2007


Why real-space Q so different from redshift-space Q?Why real-space Q so different from redshift-space Q?

Big effect in observation, in Galaxy biasing Big effect in observation, in Galaxy biasing


Results: DM vs. halosShape dependenceResults: DM vs. halosShape dependence

We measured Q(r,u=2,), for = 0 -180 degrees

Blue line: Biased dark matter Q… see later…

The amplitude of Q() is higher at elongated configurations: U-shape.

We measured Q(r,u=2,), for = 0 -180 degrees

Blue line: Biased dark matter Q… see later…

The amplitude of Q() is higher at elongated configurations: U-shape.

MWFN 2007


Luminosities & ColorsLuminosities & Colors Kayo et al. (2004): little or no

dependence of Q for SDSS galaxies in color and luminosity, for equilateral triangles

Our results agree in general: need more volume?

Kayo et al. (2004): little or no dependence of Q for SDSS galaxies in color and luminosity, for equilateral triangles

Our results agree in general: need more volume?

MWFN 2007


Comparison w/SDSS results.Comparison w/SDSS results.

We compare the 3PCF of our boxes to the recent measurements of the SDSS 3PCF by Nichol et al (2006)

There’s a good agreement within the errors in general with our L120 box results using Vmax,now

Here we use a much lower resolution than in our previous results: then features of Q() are severely attenuated.

We compare results with bigger resolution: errors do not get much higher.

We compare the 3PCF of our boxes to the recent measurements of the SDSS 3PCF by Nichol et al (2006)

There’s a good agreement within the errors in general with our L120 box results using Vmax,now

Here we use a much lower resolution than in our previous results: then features of Q() are severely attenuated.

We compare results with bigger resolution: errors do not get much higher.


Galaxy Biasing with 3PCFGalaxy Biasing with 3PCF

The different 3PCF from Dark Matter and galaxies reflect differences in spatial distributions: galaxy bias

Higher-order statistics can provide constrains. On large scales, where rms overdensities are small

compared to unity, we can adopt a local bias model. This will affect the values of the correlation functions as well:

The different 3PCF from Dark Matter and galaxies reflect differences in spatial distributions: galaxy bias

Higher-order statistics can provide constrains. On large scales, where rms overdensities are small

compared to unity, we can adopt a local bias model. This will affect the values of the correlation functions as well:

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δgal = b1δDM +b2

2δDM

2 ⇒ ξ gal = b12ξDM and Qgal =

1

b1

QDM +b2

b1

⎛

⎝ ⎜

⎞

⎠ ⎟


Galaxy Biasing: resultsGalaxy Biasing: results Adopting c1=b1 and c2=b2/b1, using the JK

error covariance matrix we get constrains in these parameters from the 3PCF, 2PCF & overdensities.

The three methods have a good agreement in real space, giving c1~1.2 & c2 ~ -0.2

In redshift space the agreement is not as good, but constrains from 3PCF are consistent with 2dF results: c1~0.9& c2 ~ -0.3

Adopting c1=b1 and c2=b2/b1, using the JK error covariance matrix we get constrains in these parameters from the 3PCF, 2PCF & overdensities.

The three methods have a good agreement in real space, giving c1~1.2 & c2 ~ -0.2

In redshift space the agreement is not as good, but constrains from 3PCF are consistent with 2dF results: c1~0.9& c2 ~ -0.3


Summary Summary

The 3PCF for both galaxies and dark matter has a strong dependence on scale and shape.

The redshift space 3PCF is strongly attenuated w.r.t. the real space 3PCF.

The galaxy reduced 3PCF shows little dependence on luminosity & color.

Our model predictions are in good agreement with the last SDSS measurements

On scales of order 10 Mpc/h, a local bias scheme is in reasonable agreement with galaxy and DM distributions.

The 3PCF for both galaxies and dark matter has a strong dependence on scale and shape.

The redshift space 3PCF is strongly attenuated w.r.t. the real space 3PCF.

The galaxy reduced 3PCF shows little dependence on luminosity & color.

Our model predictions are in good agreement with the last SDSS measurements

On scales of order 10 Mpc/h, a local bias scheme is in reasonable agreement with galaxy and DM distributions.

modeling the 3-point correlation function felipe marin department of astronomy & astrophysics...

Documents

great lakes cosmology

correlation functions

spatial n

stanford slide

order correlation

certain configurations

tegmark slide

u r r slide