Redshift Selection and Mask Effect

For this study, we restrict our analysis to the K_sgalaxies with redshifts of z = 1− 3.

The upper limit of z = 3 is included to maximize the overlap in redshift space with the SMG sample so that we could obtain significant correlation signal, while the lower bound of z = 1 is to prevent the correlation signal from being biased toward low redshifts where SMGs are rare.

It should be noted that we do not include the probability distribution function (PDF) of redshift in the whole work, since the use of redshift PDF complicates the work. We use spectroscopic redshift only when provided, otherwise the single best-derived photometric redshifts from the catalogs are used for the K_sgalaxies instead.

To perform cross-correlation analysis in the field, we need to apply the same selec-tion mask for all populaselec-tions regarded. For the correlaselec-tion analysis, we require random catalogs of galaxies at random positions across the fields. GOODS fields contain several bright stars with large holes, where very few galaxies are detected in the catalogs. We create a mask according to it, and apply the mask to random catalogs, SMGs, and the K_s galaxies so that the positions of the random galaxies would be unbiased with respect to the SMG and K_sgalaxy samples, thus the correlation analysis would be unaffected by the holes. The final images are as shown in Fig. 2.6 on page 15, with an area∼ 100 arcmin² (radius∼ 6^′). The resulting GOODS-N photometric catalog comprises a total of 2978 sources while 12% have z_spec , and GOODS-S comprises 2407 sources while 20% have z_spec. Furthermore, SMGs sample have reduced to 75 and 66 in GOODS-N and GOODS-S, respectively. We summarize the number of galaxies used in Table 2.1.

Field GOODS-N GOODS-S

Galaxy SMG K_sgalaxies SMG K_sgalaxies

Original size 208 15750 146 18713

Final size 75 2978 66 2407

Table 2.1: Galaxy samples used in this study. We choose SMGs with S850 > 2mJy and σ < 0.5mJy.

Infrared Ksnormal galaxies (5σ) are chosen to reside in the same area coverage, with 1 < z < 3.

Figure 2.6: (a) GOODS-N (b) GOODS-S

Two-dimensional distribution of SMGs and infrared normal K_sgalaxies in GOODS fields that are used in our analysis. The SMGs shown represent the subset of the full samples SMGs that have matched the noise and threshold cut. The Ksgalaxies are chosen to reside at z = 1−3 with 5 σ detection. The SMGs are shown here individually with red circles while galaxies are in gray points. The blank areas represent the regions which are excluded from the analysis, i.e., around bright stars where few galaxies have been detected.

Chapter 3

Correlation Analysis

In this section, we outline our methods for measuring the angular cross-correlation between SMGs and galaxies, the autocorrelation of the galaxies, the absolute bias and characteristic DM halo mass.

3.1 Correlation Method

To analyze the clustering properties of galaxy populations, we evaluate the two-point correlation function. The two-point correlation function can be measured by counting the number of unique galaxy pairs as a function of separation and comparing the resulting dis-tribution to that of a catalog of random points with the same number density and subject to the same observing geometry. Because we detect galaxies on a two-dimensional surface, we use the angular correlation function, a projection of the three-dimensional spatial cor-relation function (Peebles 1980). The two-point corcor-relation function provides us a robust way of tracing the dependence of large-scale structure on galaxy properties and evolution through redshift. Several estimators for the angular two-point correlation functions are available, we use the Landy and Szalay (1993) estimator for observed angular correla-tion funccorrela-tion ( hereafter ω_obs), which have shown to have minimum variance and bias, as described by

ωobs(θ) = DD(θ)− 2DR(θ) + RR(θ)

RR(θ) , (3.1)

where DD(θ), DR(θ) and RR(θ) are the galaxy-galaxy, galaxy-random and random-random normalized pair counts, respectively.

However, since the angular correlation is the excess probability of finding a data pair versus finding a random pair, as the data pairs decrease over distance the normalized number of random pairs is greater than the number of data pairs, ω_obs cannot be positive for all θ. Therefore, in field of finite size, estimators of the correlation function based on pair counts are subject to the integral constraint, which can be expressed as (Groth and Peebles 1977)

∫ ∫

ωobs(θ12) dΩ1dΩ2 ≃ 0, (3.2)

where θ₁₂ is the angle between the solid angle elements dΩ₁ and dΩ₁ and the integrals are over the survey area. The size of this bias increases with the clustering strength and decreases with field size; in our very small field studies, it is a significant effect and a correction must be made. The integral constraint correction is approximately constant and equal to the fractional variance of galaxy counts in a field,

IC≈ 1

⟨Ngal⟩ + ω_Ω, (3.3)

where the first term on the right is the Poisson variance and the second accounts for the additional variance caused by clustering (Peebles 1980),

ω_Ω = 1 Ω²

∫ ∫

ω(θ₁₂)dΩ₁dΩ₂, (3.4)

In this study we consider the latter term ωΩonly since it dominates the integral constraint.

ω_Ω is dependent on the intrinsic clustering of galaxies, normally by adopting some form for ω(θ). We use the formalism of Roche and Eales (1999),

ω_Ω =

∑RR(θ)ω(θ)

∑RR(θ) . (3.5)

Numerous studies have shown that most galaxy populations obey power law

approx-imation on the angular correlation function ( hereafter ACF ),

Adopting power law form of ACF, the resulting correlation returns

ω_obs = A

The second term in the bracket is the geometric feature of field studied, which we name C hereafter if mentioned. The uncertainty in the Landy and Szalay estimator can be estimated by assuming that DD(θ) has Poisson variance, in this case

σ_obs(θ) = 1 + ω(θ)

√DD(θ). (3.8)

Derive the ACF on small sample (number of SMG∼ 10²) is expected to produce very large statistical errors, reducing our ability to derive well-constrained clustering properties (Chen et al. 2016). However, we can apply a closely related correlation function: the two-point cross-correlation function (CCF), by using the larger sample of Ks-band selected galaxies in the same field. We cross-correlate the target sample galaxies (D_s) with the tracer galaxies (D_t), as follows:

ω_obs(θ) = D_sD_t(θ)− DsR(θ)− DtR(θ) + RR(θ)

RR(θ) , (3.9)

where both data sets are normalized by the total pair counts. By cross-correlating a small target sample (SMGs) with a large tracer population (K_s tracer galaxies, we explain the subset selection in next section), we significantly increase the number of pairs, reaching greatly reduced statistical uncertainties, compared to directly derive the ACF of SMGs alone.