Hawaii Two-0: High-redshift galaxy clustering and bias
Róbert Beck Conor McPartland Andrew Repp David Sanders István Szapudi


firstpage–lastpage
Surface-acoustic-wave-controlled optomechanically induced transparency in a hybrid piezo-optomechanical planar distributed Bragg-reflector-cavity system
    Zhong-Yang Wang
    
========================================================================================================================================================





We perform an analysis of two-point galaxy clustering and galaxy bias using Subaru Hyper-Suprime Cam (HSC) data taken jointly by the 
Subaru Strategic Program and the University of Hawaii 

in the COSMOS field. The depth of the data is similar to the ongoing Hawaii Two-0 (H20) optical galaxy survey, thus the results are indicative of future constraints from tenfold area.  

We measure the angular auto-power spectra of the galaxy overdensity in three redshift bins, defined by dropouts from the g-, r- and i-bands, and compare
them to the theoretical expectation from concordance cosmology with linear galaxy bias. We determine the redshift distribution of each bin using a standard template-based 
photometric redshift method, coupled with a self-organizing map (SOM) to quantify colour space coverage. We also investigate sources of systematic errors to inform the methodology and requirements for Hawaii Two-0.

The linear galaxy bias fit results are b_gal,g = 3.90 ± 0.33 (stat)  +0.64 
 -0.24  (sys) at redshift z ≃ 3.7,
b_gal,r = 8.44 ± 0.63 (stat)  +1.42 
 -0.72  (sys) at z ≃ 4.7, and
b_gal,i = 11.94 ± 2.24 (stat)  +1.82 
 -1.27  (sys) at z ≃ 5.9.





cosmology: large-scale structure of Universe – cosmological parameters – methods: numerical.



Surface-acoustic-wave-controlled optomechanically induced transparency in a hybrid piezo-optomechanical planar distributed Bragg-reflector-cavity system
    Zhong-Yang Wang
    
========================================================================================================================================================





§ INTRODUCTION





The standard Lambda cold dark matter (ΛCDM) cosmological model has proven extremely successful in describing various observations of the
Universe. The main observational pillars of this model have been the cosmic microwave background (CMB) at redshift z ≃ 1100, 
type Ia supernovae up to around z ≃ 1, and the large-scale structure in various wide-angle galaxy surveys (e.g., SDSS/BOSS, ; Pan-STARRS, ; DES, ), with depths reaching up to z ≃ 1.5. Quasar maps are deeper <cit.>, at the expense of being restricted to the largest scales due to shot noise.


Thus, despite enormous progress in pushing the limits of depth, currently there are few measurements available to anchor 
the ΛCDM model in the redshift range between the CMB and the deepest wide-angle surveys. In the future, 
LSST will extend to z ≃ 3 - 4 and will cover an area of 
30,000 square degrees <cit.>, while Euclid <cit.> and WFIRST <cit.> will reach z≃ 2, and 3, respectively, the latter in a smaller footprint.

The Hawaii Two-0 (H20) survey fills the redshift gap in observations — it is a 20 square degree ultra-deep optical galaxy survey with grizy broad-band photometry from the Hyper-Suprime Cam (HSC) instrument of the Subaru telescope. It has two fields of 10 square degrees each,
at the North Ecliptic Pole and Chandra Deep Field-South. The former overlaps with the Euclid Deep Calibration Field, 
thus enabling synergy between observations. The expected depths in each band for H20 are shown in Table <ref>.

Hawaii Two-0 will include broad-band measurements of galaxies up to z ≃ 7. At this redshift, the comoving distance across each 10 square degree field is roughly 500 Mpc, large enough to
include several clusters. This coverage facilitates studying galaxy evolution as it relates to environment at an unprecedented depth, while also providing a sample
large enough for cosmological study. In fact, the total volume of H20 out to z = 7 will be roughly 1.4 Gpc^3.



The most elementary parameter to characterize a galaxy sample is the linear galaxy bias b_gal. This connects the theoretically modelled 
matter overdensity δ to the measured galaxy overdensity δ_gal: δ_gal = b_galδ. More precisely, b^2 is the ratio of galaxy and dark matter power spectra under the assumption that a deterministic and linear bias holds. This is expected to be true on the large scales we consider in this paper.

While the galaxy bias is sample-dependent and difficult to interpret on its own, it is a necessary stepping stone towards more universal
cosmological parameters. In particular, the amplitude of fluctuations, often described with the parameter σ_8, is entirely degenerate with b_gal from two-point clustering measurements.



Recent work <cit.> reveals a mild tension between the concordance value of σ_8 within ΛCDM cosmology and measurements of clustering in the local universe. 
A high-redshift constraint on σ_8 would help decide the significance of this tension.

Since H20 observations are ongoing, we chose to perform an analysis of galaxy bias on a readily available HSC data set, allowing us to test our methodology and software in preparation for the processing of actual H20 data.

Sect. <ref> describes our data sets, Sect. <ref> outlines the theoretical background, Sect. <ref> details our methods to extract the redshift distribution of our samples, Sect. <ref> describes the angular power spectra obtained and the linear galaxy bias fits, and finally Sect. <ref> summarizes and discusses our results. Appendix <ref> contains an analysis of the magnification bias that is ancillary to our main topic.




§ DATA SET






 §.§ Photometric data


An earlier HSC survey, performed jointly by the Subaru Strategic Program and the University of Hawaii, provides ultra-deep optical images in the COSMOS field <cit.>.
The data set is public, uses the same instrumentation as H20, and has a similar — albeit slightly higher — depth. Table <ref> lists the limiting magnitudes of SSP+UH in each photometric band, as well as the anticipated corresponding values for H20.
While the SSP+UH survey has full depth in only one HSC pointing (with an area of ≈ 1.8 square degrees), it was selected as the best candidate to perform our preliminary analysis for H20.



In this work, we use the reduced data and source catalogue of <cit.>, specifically the fluxes measured in a 1.5” diameter circular aperture, in the HSC grizy broad-band photometric filters.



 §.§ Dropout selection




To select galaxy sub-samples within a well-defined redshift range, we adopt the methodology of the GOLDRUSH project of the Subaru Strategic Program <cit.>. Following <cit.>, they defined colour cuts and measurement quality criteria in order to select dropout samples in the HSC g, r, i, and z bands, which they found to correspond to redshift bins around 
z ≃ 4, z ≃ 5, z ≃ 6 and z ≃ 7, respectively, with little overlap. 


We consider only the g-, r-, and i-band dropouts, since z-band dropouts by definition are detected in the y-band only and thus have a higher risk of including spurious detections. The original colour cuts for our three dropout bands were defined in <cit.> as

    g-r > 1.0 
     r-i < 1.0 
    
    g-r > 1.5 (r-i) + 0.8

for g-dropouts,

    r-i > 1.2 
     i-z < 0.7 
    
    r-i > 1.5 (i-z) + 1.0

for r-dropouts, and finally

    i-z > 1.5 
     z-y < 0.5 
    
    i-z > 2.0 (z-y) + 1.1

for i-band dropouts.

Additionally, <cit.> required that g-dropouts have signal-to-noise ratio S/N>5.0 in i-band, r-dropouts have S/N>5.0 in z-band, and i-dropouts have both S/N>5.0 in z-band and S/N>4.0 in y-band.
Finally, r-dropouts had to be undetected (specifically, S/N<2.0) in g-band, and i-dropouts needed to be undetected in both g- and r-bands. We adopt these criteria as well.

The equations above reveal that g-dropouts require photometry in the g-, r-, and i-bands, r-dropouts in the r-, i-, and z-bands, and finally i-dropouts in the i-, z-, and y-bands.
Sources that have valid photometry and are not flagged for any artifacts (e.g., satellite trail, saturation, diffraction spike; ) in these three sets of photometric bands constitute the
parent catalogs from which the respective dropout samples are selected. 
The pixels in the field that have valid photometry,
and have not been flagged for any issue are shown in Fig. <ref>.



We also performed an independent test of the validity of the colour cuts using the combined spectroscopic catalog in the COSMOS field (M. Salvato, private communication).


The well-measured spectroscopic sources (quality flag Q>=3) were cross-matched with the three dropout samples using a matching radius of 1.5”. We found 750, 63 and 2 spectroscopic matches with the g-, r- and i-dropouts, respectively. 

Tests done by <cit.>, using both template spectra and spectroscopic galaxies from VVDS, noticed no significant low-redshift contamination in their ultra-deep COSMOS sample, which is most similar to our data set. 
Contrary to this however, we found that 75.3% of the g-band matches were below z=3, with most contaminants having z<1. The left panel of Fig. <ref> shows the position of these contaminants in colour space, along with the colour cut. Clearly, the low-redshift galaxies are scattered around the original cut boundary; therefore we decided to raise the diagonal boundary by 0.4 mag, and thus the empirical colour cut we use for g-band dropouts became

    g-r > 1.0 
     r-i < 1.0 
    
    g-r > 1.5 (r-i) + 1.2,

instead of Eq. <ref>.



While the new colour cut reduces the size of the g-band dropout sample by 49.0%,  z<3 contaminants in the matched sample were limited to only 11.6% (13 out of 112 remaining matches). We note that the spectroscopic sample we matched with is not at all representative of the photometric dropout samples in terms of the distribution of observables and underlying physical parameters, due to significant biases introduced by the target selection. In fact, the spectroscopic sample is extremely biased towards low-redshift sources, as illustrated on the right panel of Fig. <ref>. Because of this, we can assume that the level of contamination in the dropout sample does not exceed a few percent with the updated cut.

A similar analysis of r- and i-band dropouts provided no evidence of such issues, albeit the sample size was very limited: 3 of 63 matches had z<4 for the r-dropouts, and neither of the 2 i-band matches were below z=5. Thus, in these bands the cuts were left unchanged.

The disparity between our findings and those of <cit.> regarding g-band dropout selection in the ultra-deep field is potentially attributable to the spectroscopic targeting of VVDS, or a difference in photometric scatter. A more thorough analysis of this issue is left to a future work. 


After applying the colour and quality cuts to the SSP+UH catalog, we obtain 36769, 3815 and 262 galaxies in the g-, r- and i-band dropout samples, respectively. The top row of Fig. <ref> shows the position of these galaxies in the sky.





§ THEORY




In this section, we briefly present the theoretical calculations performed while deriving the linear galaxy bias. (See  for a comprehensive review of galaxy bias.) Our specific notation follows <cit.> and was influenced by the equations in <cit.>.

Typically, a galaxy survey measures the n_gal(θ) projected number count of sources in a given θ direction on the sky. This quantity converts to galaxy overdensity as

    δ_gal(θ) = n_gal(θ)-n_gal(θ)/n_gal(θ),

where n_gal(θ) is the mean number count.

The linear galaxy bias b_gal is defined through the relation δ_gal = b_galδ, which assumes a simple linear 
relationship between the underlying matter distribution and that of the matter tracers, i.e. the galaxies. 
Thus, a computation of the theoretical expectation for the overall matter overdensity δ is required to find b_gal.

There are several public cosmology codes that can compute the theoretical P_δ (k,z) power spectrum of the matter overdensity via a Boltzmann equation framework,
e.g., CMBFAST <cit.>, CAMB <cit.> and CLASS <cit.>. P_δ (k,z) is the power spectrum of δ (k,z), the Fourier transform 
of the 3D overdensity field, which we wish to relate to the angular δ_gal(θ) that we measure.

Given the redshift distribution Π(z)=dN_gal/dz of the tracer sample, and the redshift-dependent linear bias b(z), we can perform a spherical projection through the expression

    C_l^gg = 2/π∫ dk   k^2 [ G_l^g(k) ] [ G_l^g(k) ] + C_Poisson,

where we have the kernel function for galaxy density 

    [G_l^g(k) ] = ∫ d τ  b(z(τ)) Π(z) dz/dτ𝒫_δ(k,z(τ)) j_l [χ(τ) k],

and the constant Poisson shot noise term

    C_Poisson = 4 π f_sky/N_gal.

In Eqs. <ref>-<ref>, τ denotes the conformal time, χ(τ)=c(τ_0-τ) is the conformal lookback distance, j_l is a spherical Bessel function of the first kind, 
f_sky represents the sky coverage fraction of the survey, and N_gal is the number of density of tracer objects (i.e. galaxies).
We define 𝒫_δ(k,z)=√(P_δ (k,z)). Also, Π(z) is normalized to unit integral.

The C_Poisson term has been added to account for the fact that, in practice, a survey measures a discrete number count of objects in a given sky pixel, which is affected by Poisson shot noise.
The autocorrelation of this noise component is positive and does not depend on the l spherical index.

In the literature, the assumption of linear growth is often made,
introducing the D(z(τ)) growth factor to describe the redshift dependence of the matter power spectrum, 
yielding the expression P_δ (k,z) = D^2(z(τ)) P_δ (k).
This way, P_δ (k) can be moved outside [G_l^g(k) ] in Eq. <ref> to speed up the calculation. We do not make this assumption in our work.

When dealing with small angular scales (e.g., l>20), the Limber approximation <cit.> is often adopted to considerably reduce computational cost by simplifying the Bessel functions. 
Since our survey area is rather small, we can safely make this assumption. Under the Limber approximation, Eq. <ref> becomes

    C_l^gg =  ∫ dτ 1/cχ^2(τ) P_δ(k,z(τ)) b^2(z(τ)) Π^2(z) (dz/dτ)^2 + 
     + 4 π f_sky/N_gal.


Instead of using the general, redshift-dependent form for the linear galaxy bias, we assume and fit a single bias value for each dropout band, therefore b(z(τ))=b_gal.

C_l^gg is the theoretical spherical autocorrelation power spectrum of the galaxy overdensity, and it scales with b^2. By measuring the δ_gal(θ) overdensity
map in our survey, and computing its autocorrelation, we get C̃_l^gg, the empirical spherical power spectrum. We can then simply fit b by scaling C_l^gg to C̃_l^gg.

We note that in magnitude-limited samples, gravitational lensing magnification by foreground matter can provide a contribution to the observed δ_gal(θ) overdensity. We show in Appendix <ref> that such magnification bias is negligible in our data set.

We use PolSpice <cit.> to calculate empirical spherical power spectra, as it can handle heavily masked fields.
Also, we use PyCAMB[http://camb.readthedocs.io/en/latest/], a Python wrapper for CAMB, to compute P_δ (k,z).
We have developed the SpheriCosmo[https://github.com/beckrob/SpheriCosmo] Python package both to wrap the required functionality in PolSpice and PyCAMB,
and to compute, in a convenient manner, the Bessel and Limber spherical power formulas for matter, the integrated Sachs-Wolfe effect, and the auto- and cross-correlations for weak lensing and lensing magnification.

For all calculations in this paper, we adopted the cosmological parameters of <cit.>, specifically the following: 
H_0=67.74, Ω_b h^2=0.0223, Ω_c h^2=0.1188, τ = 0.066, n_s=0.9667, σ_8 = 0.8159, 
spatially flat geometry, and no contribution from tensor modes (i.e. Ω_k=0 and r_0.002=0). 
Additionally, a single massive neutrino of mass m_ν=0.06  eV was assumed.




§ REDSHIFT DISTRIBUTIONS




As discussed in Sect. <ref>, the Π(z) redshift distribution of the matter tracer sample is required to compute Eq. <ref>. Thus, we need to estimate the 
redshift distribution of the g-band, r-band, and i-band dropout galaxy samples.

To perform this task, we turn to photometric redshift (photo-z) estimation. A wide variety of methods have been published in the literature, which can broadly be categorized as either machine learning 
<cit.> 
or spectral template fitting <cit.> approaches. Refer to <cit.> for comparisons of different methods.

At the high redshifts which we are probing, the spectroscopic coverage is very limited. Thus, machine learning photo-z methods are effectively ruled out, as they rely on a spectroscopic training set
that should cover the space of physical parameters. Instead, we selected EAZY[https://github.com/gbrammer/eazy-photoz], a public template fitting photo-z code <cit.> 
to perform the redshift estimation.

We ran EAZY in two configurations, the first with the default set of spectral templates and settings, denoted by EAZY-def, and the second using the updated templates of the newest 
code release, denoted by EAZY-new. The redshift grid spanned z=0.001 - 8.0 with a step size of 0.01 / (1+z). We use the z_peak maximum likelihood redshift output.



 §.§ Monte Carlo sampling


The r- and i-dropout samples are relatively small in size (3815 and 262 galaxies), which means that, taken directly, their redshift histograms would be a rather crude sampling of the underlying
Π(z) distribution. One option would be to simply smooth the histograms by the estimated redshift inaccuracy; however, that would artificially blur the sharp boundaries expected in such 
dropout samples <cit.>.



We chose to instead perform a Monte Carlo sampling of the colour space, randomly generating fluxes for each source by sampling from Gaussian distributions with the same
mean and standard deviation as each measured flux and flux error. Photometric errors are a major obstacle in obtaining accurate photo-z-s, as they enhance degeneracies between different galaxy types at
different redshifts <cit.>. By augmenting our galaxy samples in this manner, our goal is to better represent their colour space footprint, and at the same time
take into account photometric errors in the photo-zs without arbitrarily modifying the redshift distribution itself.

In practice, we performed the Monte Carlo flux sampling on the parent catalog of each dropout sample, and only afterwards applied the dropout colour and quality cuts (see Sect. <ref>).
This was done to simulate galaxies randomly scattering across the colour cut boundaries. Over 5 million samples were generated for each dropout band to ensure the colour space is properly covered, 
and the results are stable.





 §.§ Self-organizing maps


As spectral template fitting photo-z approaches are comparatively slow, we followed the approach of <cit.> and used self-organizing maps (SOMs) to quantize the large number of 
Monte Carlo samples into a much smaller number of colour space cells. We trained SOMs on the parent catalog of each dropout sample and projected from the 5D magnitude space into a two-dimensional 100 × 100 
rectangular grid of SOM cells. To perform this computation, we used the SOMPY[https://github.com/sevamoo/SOMPY] Python package.


The SOM introduces another source of randomness into the results, as the training process involves random starting points for the cells, and the training data is also processed in random order.
To ensure the stability of the results, we trained 20 different SOMs for each dropout sample, and the final redshift distributions have been averaged over these instances.



 §.§ Redshift results




With the SOM projection done, we only need to run EAZY on the centrepoint of each SOM cell (in 5D magnitude space, converted to fluxes), 
and then the photo-z of each cell is weighted by the number of Monte Carlo samples that fell into that cell when creating the redshift histogram.

As described above, the whole process, including the random sampling, has been repeated for 20 SOM instances, and the SOM-wise redshift histograms have been averaged for every redshift bin.

The resulting final redshift histograms for the EAZY-def and EAZY-new configurations, for the three dropout galaxy samples,
are shown in Fig. <ref>. For reference, we also show the redshift histograms from <cit.> for their similar dropout samples.
While the histograms for a given dropout band are largely similar, it is clear that the choice of templates (and, more broadly, methodology) can lead to significant redshift bias and change in the shape of the histograms.  



Furthermore, despite the random sampling, the peculiarities of a template set can lead to relatively sharp peaks. 
Based on our tests of the dropout cuts in Sect. <ref>, and on the fact that the peaks change both shape and position
with the choice of templates, we do not believe the sharp peaks are physical.
For this reason, we introduce a small amount of smoothing to the two sets of histograms, using a Gaussian kernel of σ=0.01 / (1+z). The smoothed redshift histograms, denoted EAZY-def-sm and EAZY-new-sm, 
are also shown in Fig. <ref>.



§ GALAXY AUTOCORRELATION RESULTS




We next computed δ_gal(θ) for HealPix[http://healpix.sourceforge.net/] <cit.> pixelized
maps of the g-, r- and i-band dropout galaxy samples, shown in the bottom row of Fig. <ref>. Specifically, we used the
HealPy[https://github.com/healpy/healpy] Python wrapper, choosing a HealPix resolution of NSIDE=1024.

We note that the HealPix pixels in  Fig. <ref> are ≈ 3600 times larger in area than the original pixels 
of the detection mask in Fig. <ref>. We consider HealPix pixels with less than a 40 % valid detection area as masked,
and within non-masked pixels the object counts were weighted in accordance with their valid area to calculate the overdensity.



We then utilized the PolSpice package (via SpheriCosmo, see Sect. <ref>) to compute the C̃_l^gg
empirical spherical autocorrelation power spectra of the three overdensity maps. Because of the small survey area of SSP+UH, 
apodization
of θ=1.374^∘ was required to ensure numerical stability <cit.>.



Our HealPix resolution choice of NSIDE=1024 allows us to safely perform an analysis up to a spherical index of l≃2000.
At higher values of l, the scales are small enough that, even at the high redshift of our samples, the non-linear component
of the matter power spectrum would start to dominate. As the modelling of the matter power spectrum is more complex in non-linear regime,
we 
terminated our analysis at l_max=2000.

To obtain the minimum l, corresponding to the largest scales we can probe, the obvious limitation is the small area of the survey.
An angular separation of 1^∘ roughly corresponds to l≃180, and therefore we selected l_min=200 as the largest
scale such that enough galaxy pairs exist in the two-point correlation.

A side effect of the apodization performed by PolSpice is that nearby l values in the power spectrum 
become correlated and thus can no longer be considered independent measurements. 
To account for this fact, we bin C̃_l^gg into l-bands of width Δ l = 300, weighting each l uniformly,
and computing the error of the binned measurement from the block-average of the covariance matrix reported by PolSpice.

In summary, our analysis focuses on 6 l-bands of width 300, covering the range l=200-2000. 
The raw and binned autocorrelation power spectra appear in Fig. <ref>.


In addition to C̃_l^gg, we determined the C_l^gg theoretical autocorrelation power spectra using Eq. <ref>, as implemented in SpheriCosmo.
We calculated the spectra for each of the four redshift distributions described in Sect. <ref>: 
EAZY-def, EAZY-def-sm, EAZY-new and EAZY-new-sm. Additionally, for reference, we performed the computations
for the <cit.> redshift distributions, labelled Ono2018.
The theoretical spectra were binned in the same way as the empirical spectra. 

Finally, for each redshift setup, the best-fitting b_gal linear bias was computed using the Levenberg-Marquardt method 
<cit.>, fitting the
binned  C_l^gg values to the binned C̃_l^gg values.

We show the bias fit results for all redshift setups in Fig. <ref>. Theoretical curves corresponding to both the non-linear (which uses a halo model) 
and linear 3D matter power spectra from CAMB are plotted, but we report the results only for the non-linear model.



The autocorrelation curve shapes of the different models are barely different, except for their amplitude, and thus the fitted b_gal.
One discernible disparity is the amount of extra power the non-linear model predicts, especially at high l values for the EAZY-new configuration and the g-band dropouts.
The non-linear, small scales are mainly introduced by the sharp z ≃ 0.4 peak of the redshift histogram, as predicted by the photo-z method (see Fig. <ref>). 
Without a spectroscopic sample which is representative of our dropout catalogs, we currently have no reasonable method of better constraining the relative strength of these low-z
contaminant peaks.

The fitted b_gal values of Fig. <ref> demonstrate that the particular choices made when deriving the Π(z) distributions can
significantly affect the results. In fact, the systematic bias caused by the Π(z) setup is as large as, or even larger than the statistical uncertainty of the
measurements, even with the relatively small sample sizes available in the SSP+UH survey.

We attempt to quantify the systematic bias in our results by artificially adding plausible amounts of photo-z bias to our most reasonable photo-z configuration.



 §.§ Systematic effects of photo-z bias


The EAZY-new-sm photo-z setup has been chosen as our fiducial Π(z) distribution in the following test, for several reasons. 
First, it yielded the median of the five b_gal values in our fits for the i-band and r-band dropouts (and was one away from the median for g-dropouts).
Second, it utilizes the newest, most complete set of template spectra available with EAZY. 
Third, while the distribution shows some unphysical “peakiness”, it has been somewhat mitigated by the applied extra smoothing.

Starting with this fiducial distribution, we applied an artificial redshift bias of Δ z / (1+z) = ± 0.02 and Δ z / (1+z) = ± 0.04 to the EAZY-new-sm Π(z) distributions and re-ran
the computations for the b_gal fits. This photo-z bias range covers, and indeed exceeds, the overall bias expectation for EAZY <cit.>.

The resulting redshift distributions and fitted b_gal values are shown in  Fig. <ref>. There is a clear trend in b_gal due to the redshift bias.



We summarize the results of this test using the largest observed statistical error value for the overall statistical error, 
and the difference between the highest (lowest) observed b_gal and the original galaxy bias as the positive (negative) systematic bias. 
We get b_gal,g = 3.90 ± 0.30 (stat) ± 0.15(sys) for the g-dropouts,
b_gal,r = 8.44 ± 0.57 (stat)  +0.29 
 -0.28  (sys) for the r-dropouts, and 
b_gal,i = 11.94 ± 2.06 (stat)  +0.38 
 -0.41  (sys) for the i-dropouts.

While the above numbers do represent the systematic effect of redshift bias, the test assumes that other details of the photo-z distribution are correct.
To more thoroughly take into account potential systematics, we determine the difference between b_gal of the fiducial Π(z) and the largest (smallest) b_gal-value of any other redshift setup; we then add that difference to the positive (negative) systematic error.
Again, the overall statistical error was chosen to be the largest observed such value for a given dropout sample.

Thus, our final, more conservative estimate for the linear galaxy bias is b_gal,g = 3.90 ± 0.33 (stat)  +0.64 
 -0.24  (sys) for g-band dropout
galaxies, with median redshift z ≃ 3.7; b_gal,r = 8.44 ± 0.63 (stat)  +1.42 
 -0.72  (sys) for r-band dropout galaxies at z ≃ 4.7;
and b_gal,i = 11.94 ± 2.24 (stat)  +1.82 
 -1.27  (sys) for i-band dropout galaxies at z ≃ 5.9.

With this choice of summarizing, we may be overestimating the systematic error by “double counting” the redshift bias, as the different Π(z) distributions might be 
biased around a central value. On the other hand, we may not be taking into account all potential idiosyncrasies of redshift distributions, and thus could be underestimating the
systematic error.

In the future, this question could be reasonably resolved by verifying our photo-z distribution with a statistically representative sample of spectroscopic redshifts.




§ DISCUSSION AND CONCLUSION





In this paper, we have presented a measurement of linear galaxy bias at high redshifts, using dropout galaxy
catalogs in the SSP+UH survey.

The g-band dropout sample at z ≃ 3.7 yields b_gal,g = 3.90 ± 0.33 (stat)  +0.64 
 -0.24  (sys), 
the r-band dropout sample at z ≃ 4.7 yields b_gal,r = 8.44 ± 0.63 (stat)  +1.42 
 -0.72  (sys); 
and the i-dropout sample at z ≃ 5.9 yields b_gal,i = 11.94 ± 2.24 (stat)  +1.82 
 -1.27  (sys).

The value of the galaxy bias depends strongly on the specifics of the sample selection and, in particular, on the depth of the potential wells occupied by the galaxies we measure.
Our b_gal results are somewhat larger than expected
<cit.>, i.e.
the galaxies selected by our colour cuts correspond to higher density regions on average, especially in the case of i-dropouts. Beyond the fact that these galaxies are situated in higher mass halos, we cannot draw further conclusions due to the uncertainties in the redshift distributions.


Our work will facilitate performing a similar analysis in the 20 square degree area of the upcoming Hawaii Two-0 survey. With an area approximately ten times larger,
the statistical error in the measurement is expected to be a factor of ≈√(10) lower <cit.>.


We have identified the largest source of our systematic error as the determination of the Π(z) redshift distribution of the respective
galaxy samples. In the future, we will obtain spectroscopic redshifts for a subset of our data with Keck DEIMOS <cit.>. This will enable a more precise calibration of our photometric redshifts and the redshift distributions.


At present, we expressed our results in terms of the linear galaxy bias, assuming concordance ΛCDM cosmology. With the larger H20 data set and better calibrated photometric redshifts, we can constrain cosmology, in particular measure  σ_8 at high redshifts. Since the galaxy power spectrum constrains σ_8 × b_gal in the linear regime, we need to add at least another measurement. The possibilities include: galaxy overdensity–weak lensing correlations; constrain the bias itself using counts-in-cells distributions <cit.>, and then fit σ_8;  compute
higher-order statistics, going to three-point correlations instead of using two-point correlations only <cit.>. 


In summary, we found that within the context of Planck concordance cosmology, a linear bias model adequately explains the clustering of galaxies at z ≃ 3-6.5. With 10 times more data in the near future and better redshift calibration, the Hawaii Two-0 survey will produce high-redshift constraints on cosmological parameters and galaxy formation.




§ ACKNOWLEDGEMENTS


The authors wish to sincerely thank Peter Capak for valuable discussions concerning the SOM method. 

IS and RB acknowledge support from the National Science Foundation (NSF) award 1616974. AR gratefully acknowledges support by NASA Headquarters under Grant 80NSSC18K108 of the NASA Earth and Space Science Fellowship program. DS, IS and CM gratefully acknowledge support from NASA/JPL grants 1596038, 1608337, 1623921 and 1633586. DS and CM also acknowledge support from NSF grant 1716994.

In this work we made use of the COSMOS
master spectroscopic catalog, available within the collaboration and kept updated by Mara Salvato.


The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. 



This paper is based on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by Subaru Telescope and Astronomy Data Center at National Astronomical Observatory of Japan. Data analysis was in part carried out with the cooperation of Center for Computational Astrophysics, National Astronomical Observatory of Japan.




 
mn2e







§ AN ANALYSIS OF MAGNIFICATION BIAS




The δ_gal(θ) galaxy overdensity that is observed in magnitude-limited galaxy samples can
be modulated by an effect known as magnification bias. Foreground matter density can enhance (or diminish) observed source counts via weak gravitational lensing,
by pushing sources above (or below) the detection limit. Here we provide an analysis of how magnification bias affects our results. 

First, we give a short summary of how the modelling described in Sect. <ref> changes due to the inclusion of 
magnification bias. The formulae are based on the works of <cit.>, refer to these for more theoretical details.

Instead of measuring the galaxy overdensity directly, the overdensity we observe can be written as

    δ_O(θ) = δ_gal(θ) + δ_μ(θ),

where O denotes observed, and μ denotes the magnification contribution.

Accordingly, the empirical spherical autocorrelation power spectrum of this observed overdensity becomes C̃_l^OO, and the theoretical model for it is

    C_l^OO = C_l^gg + 2 C_l^gμ + C_l^μμ.


The gμ and μμ terms can be computed through the analogues of Eq. <ref>,

    C_l^gμ = 2/π∫ dk   k^2 [ G_l^g(k) ] [ G_l^μ(k) ]

and

    C_l^μμ = 2/π∫ dk   k^2 [ G_l^μ(k) ] [ G_l^μ(k) ],

where we introduced the kernel function for lensing magnification

    [G_l^μ(k) ] = (5s-2)3 H_0^2 Ω_m/2 c×
    ×∫ d τ  g(z(τ)) (1+z(τ)) 𝒫_δ(k,z(τ)) j_l [χ(τ) k].


Above, H_0 is the Hubble constant at present time, Ω_m denotes the cosmological mass density parameter, c is the speed of light,

    s = d log_10 N (<m)/dm|_m_0

is the slope of the galaxy number count function at the limiting magnitude m_0, and finally

    g(z)=χ(z) ∫_z^∞ dz' χ(z')-χ(z)/χ(z')Π(z)

is the lensing weight function.



Again, we adopt the Limber approximation <cit.> to speed up the computation, yielding

    C_l^gμ = (5s-2)3 H_0^2 Ω_m/2 c∫ dτ 1/cχ^2(τ) 
    P_δ(k,z(τ)) ×
    × b(z(τ)) Π(z) (dz/dτ) g(z(τ)) (1+z(τ))

and

    C_l^μμ = (5s-2)^2(3 H_0^2 Ω_m/2 c)^2 ∫ dτ 1/cχ^2(τ) 
    P_δ(k,z(τ)) ×
    × g^2(z(τ)) (1+z(τ))^2.


Examining the components of C_l^OO in Eq. <ref>, it is clear that C_l^gg
scales with b_gal^2 (see Eq. <ref>), C_l^gμ scales with b_gal, and C_l^μμ does not depend on the galaxy bias.

We evaluated the theoretical expressions via SpheriCosmo, and performed the linear galaxy bias fit
on the dropout samples using our fiducial EAZY-new-sm Π(z) distribution.

The results are b_gal,g = 3.89 ± 0.29, b_gal,r = 8.44 ± 0.55
and b_gal,i = 11.93 ± 2.00 (statistical error only) for the three dropout galaxy samples.
These values are indistinguishable from the original EAZY-new-sm results (see Fig. <ref>), which did not take into account magnification bias.

The reason why the results are unaffected by magnification bias can be illustrated by comparing the
components of C_l^OO, shown in Fig. <ref>. It is clear that C_l^μμ and C_l^gμ
are orders of magnitude smaller than C_l^gg at all considered l values.



We note that C_l^gμ and C_l^μμ scale with the factor (5s-2) and (5s-2)^2.
The number count slope was determined to be s=0.11 for g-dropouts, s=0.13 for r-dropouts, and s=0.16 for i-dropouts, based on the i, z and y magnitude distributions, respectively.
The value of s depends on the exact choice of limiting magnitude at the edge of the sample, but the credible interval for all dropout samples certainly does not extend beyond the range [0.0,0.3].
Within this range, (5s-2) is small enough that the magnification contribution remains inconsequential.

We conclude that, at the precision currently allowed by our data, the effect of magnification bias is negligible.


