COSMOS: STOCHASTIC BIAS FROM MEASUREMENTS OF WEAK LENSING AND GALAXY CLUSTERING

The COSMOS field is a 1.64 deg² patch of sky close to the equator ($\alpha = 10\mbox{$.\!\!^{\mathrm h}$}00\mbox{$.\!\!^{\mathrm m}$}28\mbox{$.\!\!^{\mathrm s}$}$, δ = +21221). It was imaged with ACS/HST during cycles 12–13 (Scoville et al. 2007; Koekemoer et al. 2007). An intensive follow-up from UV to IR was conducted by several ground-based and space-based telescopes. In this study, we take advantage of the spatial resolution of the ACS imaging to measure the shape of the galaxies and of the multi-filter ground-based imaging to derive accurate photometric redshifts (Ilbert et al. 2009).

We use an updated version of the lensing catalog presented in Leauthaud et al. (2007). In this new catalog, raw ACS/f814w images have been corrected for the Charge Transfer Inefficiency effect using the algorithm described in Massey et al. (2010). Objects are detected with the "hot-cold" technique described in Leauthaud et al. (2007), i.e., SExtractor (Bertin & Arnouts 1996) is run twice: first to detect extended objects and then compact objects. Regions contaminated by bright stars or satellite trails are masked out. Point sources and spurious detections are discarded as detailed in Leauthaud et al. (2007). The catalog is complete up to magnitude $\mathrm{MAG\_AUTO}=26.5$ in the ACS/f814w band (hereafter I_814W).

Galaxy ellipticities are measured with the RRG algorithm described in Rhodes et al. (2000) and calibrated with the same shear calibration as in Leauthaud et al. (2007). In our final source catalog, we make the cuts on size d > 1.6 times the point-spread function size and S/N >4.5 as in Leauthaud et al. (2007). We also remove galaxies with more than one peak in their photometric redshift probability distribution function (pdf) or galaxies masked in the Subaru imaging. This way, we obtain an average redshift error σ_z/(1 + z) < 0.05. Our source catalog with good shapes and redshifts contains 210,477 galaxies, which yields a density of 36 galaxies per square arcminute. As in Massey et al. (2007b), we assign to every source galaxy an inverse variance weight w aimed at maximizing the S/N and calculate from the galaxy apparent magnitude

$$\begin{equation} w = \frac{1}{\sigma _\epsilon ^{2} +0.1}\,, \end{equation} \tag{ 1 }$$

where the observational noise (pixellation, shot noise, etc.) is well approximated by the following function of the galaxy apparent magnitude in the I_814W band:

$$\begin{equation} \sigma _\epsilon = 0.32 + 0.0014(\mathrm{MAG\_AUTO} - 20)^3\,. \end{equation} \tag{ 2 }$$

The second term 0.1 represents the variance of the intrinsic shape noise, which was found to be pretty constant in Leauthaud et al. (2007). Note that the power of 2 was omitted as a typo in Massey et al. (2007b).

2.3.1. Catalog Cuts by Redshift

In this work, we want to study the evolution of galaxy bias as a function of redshift and scale. Bias is computed from the ratio of galaxy and matter clustering in bins of redshift. To ease the interpretation of the bias results later on, we choose to define foreground galaxy samples with redshift distributions matching the redshift distribution of the matter that most efficiently perturbs the background galaxy shapes through weak lensing. Typically, the matter that lies halfway between us and the background galaxies is the one that most efficiently perturbs their shape. We show in Section 6 that a mismatch in redshift can change the estimated bias.

We create three bins B1, B2, and B3 of background sources. We set the bin limits to z = 0.3, 0.8, 1.4, and 4, so that the number of galaxies and signal to noise in each bin are similar. We assign a galaxy with redshift $z\pm \sigma _{68\%}$ to a particular redshift bin with limits z_low and z_high if $z - \sigma _{68\%} > z_\mathrm{low}$ and $z + \sigma _{68\%} < z_\mathrm{high}$. The values of z and $\sigma _{68\%}$ are estimated from the photometric redshift likelihoods. The number of galaxies per bin is reported in Table 1.

Table 1. Redshift Range and Number of Galaxies per Foreground and Background Bins

Bin	Redshift	Stellar Mass Range [h−272 M☉]		All
	Range	109 → 1010	1010 → 1011
F1	[0.17, 0.27]	1,298	644	7,920
F2	[0.32, 0.42]	3,151	1,770	13,898
F3	[0.47, 0.57]	3,215	1,456	11,256
F4	[0.60, 0.75]	7,776	3,899	22,958
F5	[0.87, 1.07]	11,828	6,223	27,911
B1	[0.3, 0.8]	...	...	63,529
B2	[0.8, 1.4]	...	...	70,157
B3	[1.4, 4.0]	...	...	59,056

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For each background bin, we compute the lensing efficiency. The lensing efficiency is a calculation based on the redshift distribution of the background galaxies in comoving coordinates p_b(w) = p_b(z)dz/dw and of the cosmological parameters. We use it as a weighting function in the following to project the three-dimensional matter power spectrum along the line of sight. The lensing efficiency as a function of the comoving distance to the lens is defined as

$$\begin{equation} g(w) = \frac{3 \Omega _m H_0^2 f_K(w)}{2 c^2 a(w)} \int _w^{w_h} dw^{\prime }\ p_b(w^{\prime }) \frac{f_K(w-w^{\prime })}{f_K{w^{\prime }}}, \end{equation} \tag{ 3 }$$

where the functions a(w) and f_K(w) are, respectively, the comoving scale factor and the comoving angular distance. We find the lensing efficiency curves to peak at redshifts z_B1 = 0.22, z_B2 = 0.37, and z_B3 = 0.51.

Next, we create five bins of foreground galaxies. For the first three bins, we adjust their centers to match the peaks of the three lensing efficiencies. Then, we adjust their width so that the signal to noise of the galaxy–galaxy cross-correlation measurements is the largest. Too broad bins increase the mixing of angular and physical scales, and too narrow bins increase shot noise. Bins must also be broad enough, so that we can use the Limber approximation to compute the theoretical signal. Indeed, the Limber approximation breaks down at large scales for too narrow redshift bins. Simon (2007) computed the minimal bin width as a function of scale and redshift, beyond which the Limber approximation becomes inaccurate by more than 10%. To minimize the impact of this issue, we fix the bin width so that we reach 10% inaccuracy at 100 arcmin, i.e., twice the largest scale at which we compute the correlation functions. In summary, we obtain three foreground bins with limits 0.17 < z_F1 < 0.27, 0.32 < z_F2 < 0.42, and 0.47 < z_F3 < 0.57. To these three foreground bins, we add two extra bins to probe the bias at higher redshift 0.60 < z_F4 < 0.75 and 0.87 < z_F5 < 1.07. Finally, we assign the foreground galaxies to their respective bins using the same criteria we used for the background sources and compute p_f(z), the redshift distribution of the foreground bins. Again, we will use these redshift distributions as weighting functions to project the three-dimensional matter power spectrum along the line of sight and compute the different estimators presented below.

Note here three important points regarding our analysis. First, we compute the redshift distribution of the galaxies that fall in the foreground and background redshift bins p_f(z) and p_b(z) by summing their redshift posterior pdf p⁽ⁱ⁾(z). The redshift distributions p_f(z) and p_b(z) and the lensing efficiency curves are shown in Figure 1. Summing the redshift pdfs instead of building histograms of the individual redshift estimates has several advantages: (1) it helps to deal with skewed pdfs, and (2) it tells us about the probability of having galaxies outside a redshift bin. In spite of our cuts in redshift, we still find that 30% of the galaxies are probably outside the redshift limits of their respective redshift bins. If instead we choose to cut galaxies at $3\sigma _{68\%}$ from the redshift bin limits, we only decrease this probability to 28%. (3) It gives us theoretical predictions that better reproduce the measurements. Summing the pdfs produces redshift distributions with tails falling outside the redshift bin limits. As a result, we predict projected power spectra with larger amplitudes at small and large scales. They reproduce better measurements.

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Second, we must emphasize that the redshift distributions we obtain are weighting functions that we use to project the three-dimensional matter power spectrum along the line of sight and predict the behavior of several estimators. In particular, we use the lensing efficiency in place of the lensing-effective matter distribution. The latter would be the true matter redshift distribution multiplied by the lensing efficiency. Nonetheless, by averaging over many lines of sight, we expect the lensing efficiency to approximate the lensing-effective matter distribution and thus produce accurate predictions.

Third, the centers of foreground bins F4 and F5 do not perfectly match the peak of the lensing efficiency for catalog B3. In Section 6, we show how such a mismatch can alter the bias measurements. We show that the effects are particularly severe at small scales in the nonlinear regime. Fortunately for bins F4 and F5 most of the angular scales we probe correspond to physical scales in the quasi-linear and linear regime. Since the scale dependence of the bias in the linear regime is small, the effect of mismatch in redshift will be small.

2.3.2. Catalog Cuts by Flux and Stellar Mass

Schneider (1998) and van Waerbeke (1998) developed a formalism based on aperture statistics to measure the dependence of bias with scale. They define the following aperture statistics from the auto- and cross-power spectra of galaxies and matter P_n(ℓ), P_κ(ℓ), and P_nκ(ℓ), respectively,

$$\begin{equation} \langle \mathcal {N}^2(\theta)\rangle = 2\pi \int _0^{\infty } d\ell \ \ell \ P_n(\ell) \left[I(\ell \theta)\right]^2\,, \end{equation} \tag{ 8 }$$

$$\begin{equation} \big\langle M^2_\mathrm{ap}(\theta)\big\rangle = 2\pi \int _0^{\infty } d\ell \ \ell \ P_\kappa (\ell) \left[I(\ell \theta)\right]^2\,, \end{equation} \tag{ 9 }$$

$$\begin{equation} \langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle = 2\pi \int _0^{\infty } d\ell \ \ell \ P_{n\kappa }(\ell) \left[I(\ell \theta)\right]^2\,. \end{equation} \tag{ 10 }$$

The filter I(x) is defined as (Schneider et al. 1998; Hoekstra et al. 2002; Simon et al. 2007)

$$\begin{equation} I(x) = \frac{12}{\pi } \frac{J_4(x)}{x^2}, \end{equation} \tag{ 11 }$$

where J₄(x) is the Bessel function of order 4. This filter is a narrowband filter, which peaks at x ∼ 4.25. The power spectra P_n(ℓ), P_κ(ℓ), and P_κn(ℓ) are derived from the three-dimensional matter power spectrum P_m(k, w) projected along the line of sight, weighted by the functions g(w), p_f(w), or p_f(w)g(w), and altered by the bias parameters b and r. Their respective analytical expressions are

$$\begin{equation} P_n(\ell) = \int _0^{w_h} dw\, \frac{[p_f(w)]^2}{[f_K(w)]^2} b^2 P_m \left(\frac{\ell }{f_K(w)}, w \right), \end{equation} \tag{ 12 }$$

$$\begin{equation} P_\kappa (\ell) = \int _0^{w_h} dw\, \frac{[g(w)]^2}{[f_K(w)]^2} P_m \left(\frac{\ell }{f_K(w)}, w \right), \end{equation} \tag{ 13 }$$

and

$$\begin{equation} P_{\kappa n}(\ell) = \int _0^{w_h} dw\, \frac{g(w) p_f(w)}{[f_K(w)]^2} b r P_m \left(\frac{\ell }{f_K(w)}, w \right). \end{equation} \tag{ 14 }$$

We use the Limber approximation here to simplify the integrals.

The technique developed in this paper is powerful because all the power spectra are filtered by the same narrowband filter I(x). In contrast, the estimators ξ₊, ξ₋, ω(θ), and 〈γ_t〉 involve four different filters (see Appendix B). Combining them directly would lead to a mixing of modes in Fourier space and would hamper the interpretation.

The bias parameters b and r are defined as ratios of the variance aperture statistics

$$\begin{equation} b(\theta) = f_1(\theta, \Omega _m, \Omega _\Lambda) \times \sqrt{\frac{\langle \mathcal {N}^2(\theta)\rangle }{\big\langle M^2_\mathrm{ap}(\theta)\big\rangle }}\,, \end{equation} \tag{ 15 }$$

$$\begin{equation} r(\theta) = f_2(\theta, \Omega _m, \Omega _\Lambda) \times \frac{\langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle }{\sqrt{\langle \mathcal {N}^2(\theta)\rangle \big\langle M^2_\mathrm{ap}(\theta)\big\rangle }}\,. \end{equation} \tag{ 16 }$$

The functions f₁ and f₂ correct for the fact that different cosmological volumes are probed by the different statistics. They are defined as

$$\begin{equation} f_1(\theta) = \left. \sqrt{\frac{\big\langle M^2_\mathrm{ap}(\theta)\big\rangle }{\langle \mathcal {N}^2(\theta)\rangle }} \right|_{r=b=1}\,, \end{equation} \tag{ 17 }$$

$$\begin{equation} f_2(\theta) = \left. \frac{\sqrt{\langle \mathcal {N}^2(\theta)\rangle \big\langle M^2_\mathrm{ap}(\theta)\big\rangle }}{\langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle } \right|_{r=b=1}\,, \end{equation} \tag{ 18 }$$

and are computed assuming a Smith et al. (2003) nonlinear power spectrum with unbiased foreground galaxies (b = r = 1), an Eisenstein & Hu (1999) transfer function, and the WMAP7 cosmological parameters (Komatsu et al. 2011).

In Figure 2, we show the evolution of f₁ and f₂ as a function of scale and redshift. These functions are not constant in scale. The amplitude of the deviations is larger for the higher redshift samples of galaxies. In Figure 2, function f₁ for pair F1–B1 is nearly constant, whereas it varies by 25% for pair F5–B3. The scale dependence of f₁ and f₂ is due to the fact that many different physical scales are projected into the same angular bins, and the weighting functions in $\langle \mathcal {N}^2(\theta)\rangle$, 〈M²_ap(θ)〉, and $\langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle$ are different.

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Van Waerbeke (1998) showed that f₁ and f₂ do not depend on the shape of the power spectrum. We performed a set of tests to verify this statement and reached the same conclusion. We computed f₁ and f₂ with a power spectrum for the linear regime (Peacock & Dodds 1996) instead of a Smith et al. (2003) power spectrum. We altered the amplitude σ₈ and spectral index n_s by 20%, and we measured less than a 5% difference. The amplitude of the power spectrum does not have a strong influence on f₁ and f₂. Note, however, that the calibration functions f₁ and f₂ scale with Ω_m (Simon et al. 2007). We fixed it to the WMAP7 value.

Because we make measurements in bins of angular separations rather than physical separation, the bias parameters we measure are averages of the true bias parameters b(k, w) and r(k, w) in k-space and comoving distances. Their respective expressions are given by (Hoekstra et al. 2002)

$$\begin{equation} \langle b \rangle ^2 (\theta) = \frac{\int _0^{w_h} dw\ h_1(\theta,w) b^2(\frac{4.25}{\theta f_K(w)}, w)}{\int _0^{w_h} dw\ h_1(\theta,w)}\,, \end{equation} \tag{ 19 }$$

$$\begin{equation} \langle r \rangle (\theta) = \frac{\int _0^{w_h} dw\ h_3(\theta,w) r(\frac{4.25}{\theta f_K(w)}, w)}{\int _0^{w_h} dw\ h_3(\theta,w)}, \end{equation} \tag{ 20 }$$

where the weighting functions h₁(θ, w) and h₃(θ, w) are, respectively,

$$\begin{equation} h_1(\theta,w) = \left(\frac{p_f(w)}{f_K(w)} \right)^2 P_\mathrm{filter} (\theta, w)\,, \end{equation} \tag{ 21 }$$

$$\begin{equation} h_3(\theta,w) = \frac{p_f(w) g(w)}{[f_K(w)]^2} P_\mathrm{filter} (\theta, w)\,, \end{equation} \tag{ 22 }$$

and the filtered matter power spectrum is

$$\begin{equation} P_\mathrm{filter}(\theta, w) = 2\pi \int _0^\infty d\ell \ \ell \ P_m \left(\frac{\ell }{f_K(w)}, w \right) \left[ I(\ell \theta) \right]\,. \end{equation} \tag{ 23 }$$

Note that in Equations (19) and (20) the bias parameters b(k, w) and r(k, w) are estimated at scale k = 4.25/θf_K(w). In these equations, it was assumed that the bias parameters are evolving slowly with scale and their product by the I(x) filter was equivalent to the product by a Dirac δ_D(x) function. Note as well that the functions p_f(z) and p_f(z)g(z) are narrow in redshift space. As a result, very few angular scales are mixed together when bias parameters b(k, w) and r(k, w) are projected along the line of sight. Therefore, the shapes of the bias parameters 〈b〉 and 〈r〉 in real space and b(k, w) and r(k, w) in k-space are very similar.

We use the weighting functions h₁ and h₃ to relate the physical scales to the angular scales at which bias parameters are measured. The scales in physical units are obtained via the following expressions:

$$\begin{equation} \langle R_b(\theta)\rangle = 2 \pi \frac{\int _0^{w_h} dw\, h_1(\theta,w)}{\int _0^{w_h} dw\, h_1(\theta,w) \frac{4.25}{\theta f_K(w)}}, \end{equation} \tag{ 24 }$$

$$\begin{equation} \langle R_r(\theta)\rangle = 2 \pi \frac{\int _0^{w_h} dw\, h_3(\theta,w)}{\int _0^{w_h} dw\, h_3(\theta,w) \frac{4.25}{\theta f_K(w)}}. \end{equation} \tag{ 25 }$$

In Figure 3, we show the weighting functions h₁ and h₃ for our five different pairs of redshift bins. The bias we measure in the following is the averaged bias of the galaxies in the hatched regions. The h₁ and h₃ functions extend over the whole redshift range, but we only hatched the regions containing 68% of integrated signal in h₁ and h₃. We found the widths of the hatched regions to be SI.

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As described in Appendix A, aperture statistics are computed from auto- and cross-correlation functions. We use the software Athena, ⁷ which implements a tree code and computes auto- and cross-correlation functions. For the tree code, we choose an opening angle⁸ of 0.04 rad. All correlation functions are measured in 950 logarithmic bins between 0.05 and 50 arcmin, which corresponds to a sampling scale of Δϑ = 0.007ϑ. The aperture statistics and the bias parameters are also computed with this fine binning. We estimate the error bars with bootstrapping, i.e., we repeat the measurements 200 times with different galaxy samples randomly drawn from the input catalogs. For the figures, we rebin the data in coarse bins using a median binning technique. We used the median rather than the mean because the distributions of the bias parameters are not Gaussian, and so the median is more robust. However, this is a small correction since, with the mean instead, the bias parameters b and r would be only about 3% larger. To allow a straightforward comparison, we use the same binning for both the data and the simulations.

In the following, we will always compare our results to numerical predictions. We found the Smith et al. (2003) fitting formula to give a good approximation of the measurements at large scales. In contrast, at small scales, we found the fitting formula to underpredict the measurements (see, e.g., Figures 6, 8, and 9). Hilbert et al. (2009) had already noticed this discrepancy at scales ℓ > 10,000 and suggested that it could be due to the low resolution of the simulations used by Smith et al. (2003). In our case, the discrepancies could at least partly also originate from the fact that we compare power spectra of biased galaxy samples and dark matter. Indeed, power spectra of galaxies are expected to deviate from pure dark matter power spectra. Analyzing these deviations as a function of scale, redshift, and galaxy properties is in fact the whole purpose of this paper. Therefore, comparing the measured and the predicted measurements already gives us a first guess by eye of the amplitude of galaxy bias for a particular galaxy population.

Galaxy clustering. In Figure 5, we show the projected galaxy autocorrelation function ω(θ) for our five foreground redshift bins. We compute ω(θ) using the Landy & Szalay (1993) estimator defined as

$$\begin{equation} \omega (\theta) = \frac{DD}{RR} - 2\frac{DR}{RR} + 1, \end{equation} \tag{ 26 }$$

where the normalized numbers of pairs DD, RR, and DR refer to pairs of galaxy positions, random positions, and galaxy and random positions, respectively.

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By bootstrapping the data, we include shot noise in the error bars for the data and cosmic variance for the simulations. At high redshift, we calculate smaller error bars because (1) high-redshift bins contain more galaxies and (2) we probe larger volumes and cosmic variance decreases in larger volumes. In bins F4 and F5, we observe 2σ and 3σ deviations, respectively, between the measured and the predicted signals. These deviations already suggest that our flux-selected galaxy samples are biased with respect to the matter distribution (see Equations (12) and (B1)).

Galaxy autocorrelation functions have been found to closely follow a power law (PL, e.g., McCracken et al. 2007). In order to assess the evolution of galaxy clustering with redshift, we fit a PL ω(θ) = A_w(θ/1')^−δ to our measurements and report the best-fit parameters in Table 2. However, our estimator ω(θ) is mathematically known only up to an additive constant known as the integral constraint (IC), which depends on the volume and the mean galaxy density of the sample (McCracken et al. 2007). In the next section, we present how we correct for IC using the $\langle \mathcal {N}^2(\theta)\rangle$ measurements.

Table 2. Scaling and Slope of the Power-law Fit to ω(θ)

Bin	Direct Fit		IC Corrected Using $\langle \mathcal {N}^2(\theta)\rangle$
	Aw(1')	δ	Aw(1')	δ
F1	0.32	0.75	0.61 ± 0.28	0.37 ± 0.12
F2	0.21	0.93	0.50 ± 0.16	0.41 ± 0.09
F3	0.15	0.93	0.22 ± 0.05	0.64 ± 0.08
F4	0.18	0.65	0.21 ± 0.03	0.54 ± 0.05
F5	0.14	0.63	0.15 ± 0.02	0.55 ± 0.03

Notes. The IC corrected amplitudes A_w and slopes δ are obtained from the fit to $\langle \mathcal {N}^2(\theta)\rangle$ and Equation (27).

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Galaxy–galaxy lensing. In Figure 6, we show the evolution with redshift of the galaxy–galaxy lensing measurements for our five flux-selected galaxy samples. Overall, we observe our measurements to decrease with scale. To correct for systematic effects at large scales, we subtracted the signal measured around random foreground galaxies (Mandelbaum et al. 2005). To guide the eye, we overplot the predicted signal derived from the Smith et al. (2003) power spectrum with unbiased galaxies. The match is good at large scales, but we note a discrepancy at small scales. To assess the significance of this discrepancy, we compared the slopes of the data and the predictions at scales θ < 3'. Taking into account the correlation between the data points, we fitted five PL functions to the data at these scales and found their slopes to be always negative at 2σ confidence level, hence at odds with the positive slopes of the predictions. This highlights the limited accuracy of the Smith et al. (2003) power spectrum at small scales.

Shear autocorrelation. Finally, we compute the shear autocorrelation functions ξ₊ and ξ₋ for our three background catalogs B1, B2, and B3. They are presented in Figure 7. The amplitude increases between B1 and B3 mainly because the lensing efficiency in Equation (13) increases with redshift. The contribution of the structures along the line of sight is likely small, because they are mostly uncorrelated.

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In this section, we derive aperture statistics from the correlation functions. The calculations are detailed in Appendix A.

Galaxy aperture variance. The measurements of $\langle \mathcal {N}^2(\theta)\rangle$ for the five foreground bins are shown in Figure 8. For each redshift bin, we fit a PL. In contrast to ω(θ), $\langle \mathcal {N}^2(\theta)\rangle$ is not affected by IC, because the compensated filter T₊ that multiplies ω(θ) in Equation (A10) cancels out any constant, including IC (Schneider et al. 1998). Simon et al. (2007) derived an analytic relation between the parameters of a PL fit to $\langle \mathcal {N}^2(\theta)\rangle$ and the parameters of a PL fit to an IC corrected correlation function ω(θ). The slope of ω(θ) corrected for IC is given by

$$\begin{equation} f(\delta) = 0.0051 \delta ^{11.55} + 0.2769 \delta ^{3.95} + 0.2838 \delta ^{1.25}, \end{equation} \tag{ 27 }$$

where δ is the slope of the PL fit to $\langle \mathcal {N}^2(\theta)\rangle$. The amplitude of the PL fit to ω(θ) is the same as the amplitude of the PL fit to $\langle \mathcal {N}^2(\theta)\rangle$. In Table 2, we report the IC-corrected amplitudes A_w and slopes δ of the galaxy autocorrelation function ω(θ) obtained from Equation (27).

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We find that the amplitude A_w decreases with redshift, in agreement with McCracken et al. (2007), who also found smaller amplitudes for fainter galaxies in COSMOS, i.e., more likely located at higher redshift. Besides, we note a jump in amplitude between bins F3 and F2, i.e., between redshift z ∼ 0.52 and z ∼ 0.35. With respect to the slope δ, we observe a slight but not significant (less than 1σ) steepening with redshift. Such a peculiar behavior could be explained by the large-scale structure identified at redshift z ∼ 0.7 already detected in several other studies (Massey et al. 2007a; Guzzo et al. 2007; Meneux et al. 2009; de la Torre et al. 2010). We guess that more massive galaxies in this structure would increase the slope because they are typically more clustered than average.

Finally, we observe an increasing deviation (more than 5σ in bin F5) between the measurements and the signal predicted with a Smith et al. (2003) power spectrum and unbiased galaxies. This suggests that galaxies are more biased at high redshift, in agreement with expectations. For bin F4 in the last three bins, we find on average a bias of about 1.4 between the data and the predictions, and for bin F5 in the last five bins, we find a bias of about 2.6.

Galaxy-mass aperture covariance. The measurements of $\langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle$ for our five foreground redshift bins are shown in Figure 9. In agreement with the 〈γ_t〉 measurements, we again note that a Smith et al. (2003) power spectrum underpredicts the data points at small scales. We performed a χ²-test and found this disagreement to be significant in all bins at more than 3σ (we included the covariance matrices in the χ²-test). This disagreement is also present for the stellar-mass-selected galaxy samples in Figures 16 and 17. At small scales in bins F2–B2, we also note a slight change of slope at 1' scale. Again, we investigated the significance of this feature with a χ²-test but found a reduced χ² = 1.60, which is not enough to reject a simple PL model.

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An E/B mode decomposition also exists for the $\langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle$ statistics. 〈NM_x〉 quantifies the amount of B-modes in the measurement and is computed by replacing γ_t by γ_x in Equation (A11). γ_x is the galaxy–galaxy clustering signal obtained with the coordinate system rotated by 45°. This operation is commonly used to reveal B-mode contamination if an 〈NM_x〉 signal is detected. Using a χ²-test and taking into account the correlation between the data points, we find the B-modes for 〈NM_x〉 to be consistent with zero in all bins Fi–Bj at 95% confidence level, hence confirming a very low level of contamination in our analysis.

Mass aperture variance. The measurements of 〈M²_ap(θ)〉 for bins B1, B2, and B3 are shown in Figure 10. The solid line represents the predicted signal with a Smith et al. (2003) power spectrum. We performed a χ²-test between the data points and the predicted signal and found that the data points are consistent with the predicted signal at 68% confidence level for all bins B1, B2, and B3. At all scales, shape noise dominates over cosmic variance. Errors are smaller for bin B3, because the signal is larger (about 10 times larger than the signal in bin B1).

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5.3.1. Constant Bias Model

In Figure 11, we show the galaxy bias 〈b〉(R) for our sample of flux-limited galaxies. We observe that bias varies with scale and redshift, but since the data points are correlated, we use a χ²-test to quantify the significance of these variations. First, we assume the null hypothesis that bias is scale and redshift independent. To test this hypothesis, we fit a constant b₀ to the 45 data points in the five redshift bins and compute the χ² statistics

$$\begin{equation} \chi ^2 = (\mathbf {B} - b_0)^T C^{-1} (\mathbf {B} - b_0), \end{equation} \tag{ 28 }$$

where B is a vector containing the 45 data points and C is their covariance matrix. To perform the fit and find the best-fit parameters, we use the IDL AMOEBA technique (Nelder & Mead 1965) and repeat the process 100 times with different starting values. We obtain a best fit with χ² = 229. Our data points are correlated, so the effective number of degrees of freedom (dof) to perform the χ² test is less than the number of data points N = 45 minus the number of free parameters. Bretherton et al. (1999) proposed several estimators for the number of dof for correlated data. We use the estimator⁹

$$\begin{equation} \mathrm{dof} = \frac{N^2}{\sum _{ij} r^2_{ij}}, \end{equation} \tag{ 29 }$$

where r_ij is the correlation coefficient between data points b_i and b_j, and N is the number of data points. Using this estimator, we find dof = 32 − 1 = 31, where we subtracted 1 for the parameter we fit. The reduced χ² is therefore χ²/dof = 7.4, which allows us to confidently reject a constant bias model given our data.

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5.3.2. Redshift-dependent Model

Here, we discuss a test of the redshift dependence of bias. To our model, we add the following redshift dependence:

$$\begin{equation} b(z) = b_0 + b_1 z\,. \end{equation} \tag{ 30 }$$

We obtain a reduced χ²/dof = 2.5. Although still not a good fit, this model is nonetheless significantly better than the previous redshift-independent model.

So far, we have ruled out a constant bias with redshift and scale, but we did not isolate the redshift dependence or scale dependence yet. This is the purpose of the rest of this section.

5.3.3. Scale-independent Model

In Figure 12, we show the galaxy bias 〈b〉 for two stellar-mass-selected galaxy samples as a function of scale and redshift. Previous studies have shown that stellar mass is a good tracer of halo mass (see, e.g., Leauthaud et al. 2012, and references therein), which in turn parameterizes most of the bias models. In contrast, flux-selected samples are more affected by selection effects. For instance, low surface brightness extended galaxies are systematically underrepresented because they tend to evade the magnitude cuts (Meneux et al. 2009). Our stellar-mass-selected samples should therefore tell us more about halo properties.

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5.4.1. Halo Mass

First, we derive the average halo mass for our two samples of galaxies. For this purpose, we fit the bias model proposed by Tinker et al. (2010)

$$\begin{equation} b(\nu) = 1 - A\frac{\nu ^a}{\nu ^a + \delta _c^a} + B\nu ^b + C \nu ^c, \end{equation} \tag{ 31 }$$

defined in terms of the peak height ν = δ_c(z)/D(z)σ(M), where δ_c(z) is the critical density for halo collapse (Weinberg & Kamionkowski 2003), D(z) is the growth factor, and σ²(M) is the variance of the density fluctuations smoothed with a top-hat filter of size $R = (3M / 4 \pi \bar{\rho }_m)^{1/3}$. The parameters of this fitting function derive from fits to N-body simulations. They are A = 1.0 + 0.24yexp (− (4/y)⁴), a = 0.44y − 0.88, B = 0.183, b = 1.5, C = 0.019  +  0.107y  +  0.19exp (− (4/y)⁴), and c = 2.4. We define the parameter y ≡ log₁₀Δ for overdensity Δ = 200 times the cosmological mean density.

Although this bias model is not explicitly a redshift-dependent model, we make use of the fact that ν scales with redshift to test the redshift dependence of our measurements.

The Tinker et al. (2010) model is a halo bias model, i.e., it was calibrated on halo clustering measured in N-body simulations. In contrast, we measure galaxy clustering for stellar-mass-selected galaxies. Using it to infer halo mass implies the two following assumptions: (1) the dark matter mass function is constant in the range of halo mass we consider, and (2) there is only one galaxy per halo. The first assumption is valid because our galaxies are embedded in halos with peak height ν ∼ 1 and mass log₁₀(M/M_☉) ⩽ 13, and the halo mass function in this regime is almost flat (Sheth & Tormen 1999; Tinker et al. 2008). The second assumption holds in the linear regime where most of the signal comes from the clustering of central galaxies.

To overcome the problem of having data points at different redshifts, we fit a redshift-normalized peak height parameter ν₀, defined such that b(ν_i) = b(ν₀/D(z_i)), where z_i is the redshift of a data point b_i. Since this model is only valid in the linear regime, we only consider the 16 data points at scales R > 2 h⁻¹ Mpc. We find χ² = 3.13 and χ² = 4.93 for dof = 10.8 and dof = 11.2, respectively, for the low and high stellar-mass galaxy samples. Therefore, the model proposed by Tinker et al. (2010) fits well our data.

This implies that our measurements agree with an increase of bias with redshift. In addition, we can derive the halo mass of our two stellar-mass-selected galaxy samples. Indeed, the best-fit values are ν₀ = 0.77^+0.20_{− 0.31} and ν₀ = 1.01^+0.24_{− 0.18} for the low and the high stellar-mass samples, respectively. To compare to Leauthaud et al. (2012), we compute the peak height estimator at redshift z = 0.37, ν_{z = 0.37} = ν₀/D(0.37) = 0.93 and 1.21 for the low and high stellar-mass samples, respectively. This translates into 9.64 < log₁₀(M₂₀₀/h⁻¹ M_☉) < 12.29 and 11.51 < log₁₀(M₂₀₀/h⁻¹ M_☉) < 12.80, respectively. These results are in very good agreement with Leauthaud et al. (2012), when considering that our two stellar-mass-selected samples range between 10⁹ h⁻² M_☉ < M_* < 10¹⁰ h⁻² M_☉ and 10¹⁰ h⁻¹ M_☉ < M_* < 10¹¹ h⁻² M_☉. As a last check, we also fit the Sheth et al. (2001) bias model and found similar results. The data are not stringent enough to distinguish the two models.

In Figure 13, we show the evolution of bias with redshift. The data points are averages of points at scales R > 2 h⁻¹ Mpc, and errors come from the respective total covariance matrices. The total covariance matrix is the sum of the data covariance matrix and the simulation covariance matrix. The shape noise, shot noise, and cosmic variance are thus taken into account. For comparison, we overplot the bias measured by Kovač et al. (2011) with the zCOSMOS data set and by Marinoni et al. (2005) (M05) with the VLT VIMOS Deep Survey (VVDS). The zCOSMOS data set is the same as ours but with galaxies brighter than I₈₁₄ < 22.5. The bias measured by Kovač et al. (2011) follows our high stellar-mass galaxy samples, whereas the bias measured by M05 follows our low-mass samples. We attribute the difference in bias between the two measurements to differences in galaxy selection functions.

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We found bias in bins F3–B3 for the high stellar-mass galaxy sample to be larger than expected. Although this data point still agrees with the bias model predictions at 95% confidence level, it shall be discussed further. Indeed at a similar redshift, Kovač et al. (2011) also identified a significant overdensity of galaxies that could explain the large bias value, but Finoguenov et al. (2007) identified very few massive groups. These two inconsistent observations are puzzling. A halo model like the one proposed by Leauthaud et al. (2011) could help us understand better the peculiar properties of the galaxies at this redshift, in particular the fraction of satellite galaxies.

5.4.2. Scale-dependent Model

In Section 5.3, we showed that an SD bias model was preferred but still not significantly better than an SI model. Nonetheless, we observe by eye that bias systematically decreases at small scales, and the turndown seems to start at scale R ∼ 2 h⁻¹ Mpc, which could correspond to the transition between the one-halo and the two-halo term, already identified in several previous papers. According to Zehavi et al. (2005), the behavior of bias at this transition scale is due to pairs of satellite galaxies. With a halo model applied to simulated data, Kravtsov et al. (2004) showed that a significant amount of pairs with satellite galaxies was crucial for a smooth transition. Coupon et al. (2011), using the CFHT Legacy Survey, and Peng et al. (2011), with SDSS data, also found that the fraction of satellite galaxies depends more on their local over-density than on the halo mass or redshift. Since overdensities increase at late time, the number of satellite galaxies increases in low-redshift samples of galaxies. A change of slope in our data would therefore indicate a change of the satellite fraction in the galaxy samples.

In order to measure the significance of an evolution of the slope with redshift, we parameterize it as α(z) = α₁ + α₂z. We also introduce a turn-down scale R_TD beyond which bias evolves linearly. The final SD bias model we fit to the data is

$$\begin{equation} \vspace*{-2pt} b(R,z) = b_\mathrm{lin}(\nu) \left\lbrace \begin{array}{@{}lr}\dsty\left(\frac{R}{R_\mathrm{TD}} \right)^{\alpha _1 + \alpha _2 z} & R < R_\mathrm{TD} \\ \ms 1 & R \ge R_\mathrm{TD}\end{array} \right., \end{equation} \tag{ 32 }$$

where we use the redshift-dependent bias model b_lin(ν) of Tinker et al. (2010) to describe the bias in the linear regime. We find χ²/dof = 0.7 and χ²/dof = 1.0 for the low and high stellar-mass-selected samples, respectively. The best-fit parameters are reported in Table 3.

Table 3. Best-fit Parameters for the Scale and Redshift-dependent Model (Equation (32)) Applied to the 10⁹ h⁻² M_☉ < M* < 10¹⁰ h⁻² M_☉ (Low) and 10¹⁰ h⁻² M_☉ < M* < 10¹¹ h⁻² M_☉ (High) Stellar-mass-selected Samples

	Low	High
RTD h−1 Mpc	2.6 ± 1.2	1.0+0.8− 0.2
α1	0.42+0.32− 0.21	0.63 ± 0.18
α2	−0.17+0.44− 0.41	−0.44 ± 0.24
log M200/h−1 M☉	11.7+0.6− 1.3	12.4+0.2− 2.9
χ2/dof	0.7	1.0

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The difference in χ² between the SI and SD models Δχ² = χ²_SI/dof_SI − χ²_SD/dof_SD = 0.79 is not large enough to completely rule out the SI bias model.

The best-fit parameters of the SD model merit some particular attention. First, the scale at which bias turns down is detected to be between 0.8 h⁻¹ Mpc and 3.8 h⁻¹ Mpc at 68% CL. In contrast to what was observed in SDSS (Johnston et al. 2007), the data suggest R_TD to be marginally larger for less massive galaxies, but we would need more bins in stellar mass to confirm this tendency.

Second, the slope of bias at scales below R_TD is detected to be larger than zero, but at less than 2σ CL. We measured α₁ = 0.42^+0.32_{− 0.21} and α₁ = 0.63 ± 0.18 for the low and high stellar-mass samples, respectively. Regarding its evolution with redshift, we found α₂ = −017^+0.44_{− 0.41} for the low stellar-mass galaxy sample, hence no significant evolution, but α₂ = −0.44 ± 0.24 for the high stellar-mass-selected sample, hence a slight flattening of the slope at small scales at higher redshift bins.

It is expected that the steepness of the slope is related to the occurrence of satellite galaxies in our sample, with more satellite galaxies producing shallower slopes. The coincident fact that the turn-down scale is larger and the slope is shallower for the low stellar-mass sample suggests a larger amount of satellite galaxies in our low stellar-mass-selected sample.

In Figure 14, we show the correlation coefficient 〈r〉 for our flux-limited galaxy samples. Our measurements agree with r = 1 at all scales and all redshifts. We nonetheless observe some trend in the data that might arise from possible artifacts in the method (see Section 6). To highlight them, we applied the method to the simulated data and indeed found a trend with a signal greater than 1 at small scale and lower than 1 at large scale. We show the correlation coefficient and errors obtained from the simulations as pink regions. Consequently, the trends we observe in our data points can be explained by artifacts in the method. The pink hatched regions mark the scales where r is consistent with 1 in the simulations. To compute the correlation factor for a particular galaxy sample, we only average the data points in these regions.

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We found no significant deviation from r = 1, although in bins F1–B1, we obtain r = 0.58 ± 0.28. Hoekstra et al. (2001) and Simon et al. (2007) found the correlation coefficient 〈r〉 < 1 at 8σ, highlighting some stochasticity in the bias relations of their galaxy samples. In our case, we can only say that our flux-selected and stellar-mass-selected samples are good tracers of the underlying mass distribution given the error bars (see also Figures 16 and 17 for the results with the stellar-mass samples).

We have found no standard halo model to predict the scale dependence of the correlation coefficient r. In contrast, Neyrinck et al. (2005) propose a galaxy–halo model that does. In their model, they assume that halos attach to galaxies, in contrast to the standard halo model where galaxies attach to halos. By construction, assuming that all halos have the same density profile, the correlation coefficient is always r = 1. In the case in which halos have different density profiles, the correlation factor becomes SD. At large scales, many halos are averaged over, resulting in a mean density profile. The matter distribution therefore matches the galaxy distribution and r = 1. At scales smaller than the minimum intergalactic distance, r drops below 1 because of the many different density profiles. Finally, assuming that the inner parts of the density profiles are similar, r raises back to 1.

In light of this model, we attempt to identify a dip in bins F1–B1 and F2–B2 at scale R ∼ 2 h⁻¹ Mpc. Hoekstra et al. (2002) and Simon et al. (2007) also obtained such a dip at scale R ∼ 1 h⁻¹ Mpc with much higher significance. We fit a constant r = 1 to our data points in the hatched areas and measured χ² = 3.45 in bins F1–B1 (dof = 2.75, >68% CL at χ² > 3.20) and χ² = 2.20 in bins F2–B2 (dof = 3.06, >68% CL at χ² > 3.57). A constant model r = 1 therefore still provides a good fit to the data.

Figure 15 shows the correlation matrix of the bias (including cosmic variance) for the flux-limited galaxy sample F1–B1. The data points are significantly correlated between 1' and 10' for the bias parameter 〈b〉 and less correlated for the correlation coefficient. The large amount of correlation at the smallest scales θ < 04 is due to numerical artifacts (see Section 6).

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Note as well that the large correlation between the angular bins for parameter 〈b〉 only shows up when the covariance matrix derived from simulations is added to the covariance matrix derived from the data. Otherwise, the amount of correlation is weak because of the low signal to noise in the data. The signal to noise in the simulated data is much higher, hence the stronger signal in the correlation matrix.

It is well known that aperture statistics, obtained through integration of other estimators, are very sensitive to integration limits. Kilbinger et al. (2006) found that the signal is underestimated by more than 10% at 12 times the lower integration limit, i.e., 06 in our case. On the other hand, at large scales, the signal is only valid up to half the upper scale limit (see Equations (9) and (8)). We used simulations to assess the systematic effects produced by aperture statistics. In Figure 16 and 17, our simulations show that the correlation coefficient 〈r〉 is overestimated by 10% to 50% at scales θ < 1', depending on the redshift bin. For the bias parameter 〈b〉, deviations from b = 1 occur at scales θ < 07.

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In the simulations, we also observed that the cross-correlation signals $\langle \mathcal {N}(\theta)M_\mathrm{ap}(\theta) \rangle$ and 〈γ_t〉 are biased low at large scales. Subtracting the signal measured around random foreground did not help in recovering r = 1 at large scales. We found this effect to be less important as the number of foreground galaxies decreases with respect to the number of background galaxies. This can be seen by comparing the red dashed lines in Figures 16 and 17. Although this effect is of minor consequence on our measurements given the size of the error bars, we tried to understand it. It is possible to demonstrate analytically that a constant additive systematic will cancel between sources 90° separated. When foreground galaxies do not have sources all around them or the additive systematic is not constant, the average 〈γ_t〉 is biased low. In the SDSS, Mandelbaum et al. (2005) found that subtracting the signal measured around random foreground galaxies was effectively removing this noise. However, our simulations do not include shape noise, and therefore the signal we measure at large scales cannot be due to shape noise.

Another source of systematic error could be related to the tree code we use to compute the correlation functions. The documentation for the tree code Athena mentions that choosing a too large opening angle can smear out the ellipticities of galaxies at large scales. Indeed, we find that if we increase the opening angle, then the effect worsens, but it does not improve with values smaller than the one we use. Because this systematic error is subdominant to our measurements, we decided not to further attempt its correction.

In this section, we show that a mismatch in redshift between peaks of the p_f(z) and the lensing efficiency curves g(z) can alter the bias measurements. We looked into this issue because this is the case in bins F4–B3 and F5–B3. To understand the origin of the problem, we should recall that in our method, the bias is the product of a function f₁ and a measurement $\langle \mathcal {N}^2(\theta)\rangle / \langle M^2_\mathrm{ap}(\theta)\rangle$.

On the left panel of Figure 18, we compute the ratios $\langle \mathcal {N}^2(\theta)\rangle / \langle M^2_\mathrm{ap}(\theta)\rangle$ and the inverse of the functions f₁ for different combinations of foreground and background sample bins. We find that between bins F1–B1 and F1–B3 $\langle \mathcal {N}^2(\theta)\rangle / \langle M^2_\mathrm{ap}(\theta)\rangle$ decreases by 37% and 33% at large and small scales, respectively, whereas function f₁ decreases by 78% and 64%. As a result, our estimations of bias change.

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We do expect the signal to decrease because 〈M²_ap(θ)〉 increases with redshift, but we do not expect the measured and the predicted signal to change by different amounts. According to Equation (19), the bias measurements should only depend on the foreground redshift distribution, and not on the choice of the background galaxy sample and associated lensing efficiency. This is of course assuming that 〈M²_ap(θ)〉 and the matter power spectrum grow linearly, and the lensing-effective matter redshift distribution is well approximated by the lensing efficiency. For very wide surveys where many lines of sight are averaged over, the lensing efficiency might be a good approximation of the effective matter distribution, but in COSMOS, especially at low redshift, this might not be the case. On the other hand, at small scales in the nonlinear regime, the power spectrum might not grow linearly. In Figure 18, smaller scales are more affected by bin mismatch.

Given the limited size of the COSMOS field, we therefore posit that cosmic variance or shape measurements could yield these discrepancies because the volumes probed by 〈M²_ap(θ)〉 are too small. These types of discrepancies warn us that for future surveys we should try to match as closely as possible the mean redshift of the foreground galaxies and the peak of the lensing efficiency, in order to limit the effect of cosmic variance.