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Introduction to Astronomy and Astrophysics ILecture 12

Yogesh Wadadekar

Aug-Sep 2019

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LSS optical surveys compared

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Pencil beam surveys e.g. Hubble Ultra Deep Field

These surveys map out evolution of field galaxies and LSS out toz ∼ 1− 2 and beyond.

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Ergodic principle crucial for galaxy evolution

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VIPERS survey

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Large Area Surveys at other wavelengths

Xray: ROSAT all sky survey (RASS)GALEX: all sky imaging survey (AIS)near-IR: 2MASS all sky surveyMid-infrared: WISE survey in 4 mid-infrared bandsFar Infrared: IRAS, COBE/WMAP/PLANCKRadio: NVSS, FIRST, TGSS

Pencil beam surveys at other wavelengths are also very numerous.But all these are less useful than surveys involving opticalspectroscopy. Why?

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Two-point (auto) correlation function

ξ(r), defined as an “excess probability” of finding another galaxy at adistance r from some galaxy, relative to a uniform random distributionFor small values of r this is well fit by a power law ξ(r) = (r/r0)

−γ . Thebest fit r0 is 5 h−1 Mpc and γ ∼ 1.8γ and r0 are functions of various galaxy properties; clustering inclusters is stronger. The slope also steepens at r/r0 & 2

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2DF auto correlation function

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Question

Can the two point auto correlation function have a negative value?

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How to estimate ξ(r)

Simplest estimator: count the number of data-data pairs, 〈DD〉, andthe equivalent number in a randomly generated (Poissonian) catalog,〈RR〉:

ξ(r) =〈DD〉〈RR〉

− 1 (1)

A better estimator not affected by edge effects is:

ξ(r) =〈DD〉 − 2〈RD〉+ 〈RR〉

〈RR〉(2)

where 〈RD〉 is the number of data random pairs (Landy & Szalay1993).

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How to estimate ξ(r)

Simplest estimator: count the number of data-data pairs, 〈DD〉, andthe equivalent number in a randomly generated (Poissonian) catalog,〈RR〉:

ξ(r) =〈DD〉〈RR〉

− 1 (1)

A better estimator not affected by edge effects is:

ξ(r) =〈DD〉 − 2〈RD〉+ 〈RR〉

〈RR〉(2)

where 〈RD〉 is the number of data random pairs (Landy & Szalay1993).

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Another way to estimate ξ(r)

is via the overdensity in a particular cell relative to the average density

δi(r) =Ni − 〈Ni〉〈Ni〉

(3)

The ξ(r) is the expectation value

ξ(x1,x2) = 〈δ(x1)δ(x2)〉 (4)

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Are bright galaxies more clustered than faint ones?

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Are red galaxies more clustered than blue ones?

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Two-point cross correlation function

Corellate two populations - e.g. are galaxies clustered aroundquasars?

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Three point (auto) correlation function

ζ = 〈δ1δ2δ3〉

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Angular correlation function

If only 2D information is available you can use the angularauto-correlation function - w(θ) = (θ/θ0)

−β If all galaxies are at aboutthe same distance, β = γ − 1.

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Various correlation functions

Two point auto correlation functionTwo point cross correlation functionTwo point angular correlation functionThree point correlation function

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Methods of probing the LSS

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Correlation function and power spectrum

The overdensity field is: δ(x) = n(x)〈n〉 − 1

Then the following Fourier pairs can be defined:δ(x) = 1

(2π)3

∫d3keikxδ(k)

δ(k) =∫

d3xe−ikxδ(x) where k = 2π/λ is the wave number.Power spectrum is defined as: P(k) = |δ(k)|2If 2 point correlation function is the expectation of the overdensity fieldthen the power spectrum is its Fourier pair. The two are equivalent.

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LCRS correlation function and power spectrum

Correlation function is easier to measure, but we need power spectrumto compare with theory.

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Power spectrum and CDM model

Tegmark et al. (2004)IUCAA-NCRA Grad School 21 / 36

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Cluster correlation function

Clusters are more strongly correlated than individual galaxies and richones are more clustered than poor ones. Why?

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Biasing

See Kaiser (1984) and Bardeen et al. (1996)

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Biasing in galaxies

Bardeen et al. (1996) show that for a Gaussian distribution of initialmass density fluctuations, the peaks which first collapse to formgalaxies will be more clustered than the underlying mass distribution.

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Large Scale Structure of galaxies

A range of structures: galaxies ( 10 kpc), groups ( 0.3 - 1 Mpc),clusters ( few Mpc), superclusters ( 10 - 100 Mpc)Redshift surveys are used to map LSS > 2× 106 galaxies nowLSS quantified through 2-point correlation function, well fit by apower-law: γ ∼ 1.8, r0 ∼ few Mpc. Equivalent description: powerspectrum P(k)CDM model fits the data over a very broad range of scalesObjects of different types have different clustering strengths andmore massive structures cluster more strongly

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Question

There is one major component of the baryonic large scale structurethat we have ignored so far

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Intergalactic medium (IGM) - baryons betweengalaxies

Its density evolution follows the LSS formation, and the potentialwells defined by the DM, forming a web of filaments, the “CosmicWeb”An important distinction is that this gas unaffiliated with galaxiessamples the low-density regions, which are still in a linear regimeGas falls into galaxies, where it serves as a replenishment fuel forstar formation. Likewise, enriched gas is driven from galaxiesthrough the radiatively and SN powered galactic winds, whichchemically enriches the IGMChemical evolution of galaxies and IGM thus track each other

How to observationally detect the IGM?

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Absorption line systems

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Quasar spectrum

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Luminosity functions

The Luminosity Function specifies the relative number of galaxies ateach luminosity.The Luminosity function is a convolution of many effects:

primordial density fluctuationsprocesses that destroy/create galaxiesprocesses that change one type of galaxy into another (egmergers, stripping)processes that transform mass into light

Observed LFs are fundamental observational quantities. Successfultheories of galaxy formation/evolution must reproduce them.

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The luminosity function

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Schecter Luminosity Function

In 1974 Press and Schechter calculated the mass distribution ofclumps emerging from the young universe, and in 1976 Paul Schechterapplied this function to fit the luminosity distribution of galaxies in Abellclusters.

φ(L)dL = n∗

(LL∗

)αexp

(− L

L∗

)d(

LL∗

)(5)

Function has two parts and three parameters.

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Schecter Luminosity function

L∗ : luminosity that separates the low and high luminosity parts;L∗ ∼ 1010LB�h−2, or MB,∗ ∼ −19.7 + 5 log(h)At low luminosity, (L < L∗): We have a power law with α ∼ −0.8 to-1.3 (“flat” to “steep”) lower luminosity galaxies are more common.At high luminosity, (L > L∗): We have an exponential cutoff, veryluminous galaxies are very raren∗ : is a normalization, set at L∗ n∗ ∼ 0.02h3 Mpc−3 for the totalgalaxy population. Depending on context, n∗ can be a number; anumber per unit volume; or a probability. Note the implicitdependence on Hubble constant, via h3.

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What each parameter does

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Different equivalent forms of the luminosity function

φ(L) per dL, (which is usually plotted log(φ) vs log L). φ(M) per dMwhere M is absolute magnitude, so this is effectively d(log L).Sometimes the cumulative LF is given: N > L or N < M. So pleasecheck the axes on your plots.Observationally, it is also important tospecify:

whether the LF is for specific Hubble Types, or integrated over allTypeswhether the LF is for Field galaxies or Cluster galaxies (orwhatever the environment is)the value of H0, since φ varies as h3 while L or M vary as h−2

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How to measure the luminosity function? in Clusters

All cluster galaxies are at the same distance.

1 bin galaxies by apparent magnitude, down to some limit, to getφ(m)

2 use cluster redshift (distance) to get φ(M)

3 Fit a Schechter function to φ(M) by minimizing χ2 to obtain M∗and α.

Complications arise principally from trying to eliminatefore/back-ground field galaxy contamination: here galaxy velocities areuseful. Also dwarfs are often too faint to measure (except BCDs)because they have low SB. We need to apply statistical corrections toN(m) using field samples.

IUCAA-NCRA Grad School 36 / 36

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How to measure the luminosity function? in Clusters

All cluster galaxies are at the same distance.1 bin galaxies by apparent magnitude, down to some limit, to getφ(m)

2 use cluster redshift (distance) to get φ(M)

3 Fit a Schechter function to φ(M) by minimizing χ2 to obtain M∗and α.

Complications arise principally from trying to eliminatefore/back-ground field galaxy contamination: here galaxy velocities areuseful. Also dwarfs are often too faint to measure (except BCDs)because they have low SB. We need to apply statistical corrections toN(m) using field samples.

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