---
abstract: 'Weak gravitational lensing is one of the most promising cosmological probes to constrain dark matter, dark energy and the nature of gravity at cosmic scales. Intrinsic alignments (IA) of galaxies have been recognized as one of the most serious systematic effects facing gravitational lensing. Such alignments must be isolated and removed to obtain a pure lensing signal. Furthermore, the alignments are related to the processes of galaxy formation, so their extracted signal can help in understanding such formation processes and improving their theoretical modeling. We report in this letter the first detection of the gravitational shear–intrinsic shape (GI) correlation and the intrinsic shape–galaxy density (Ig) correlation in a photometric redshift survey using the self-calibration method. These direct measurements are made from the KiDS-450 photometric galaxy survey with a significance of 2.74$\sigma$ in the third bin for the Ig correlation, and 2.73$\sigma$ for the GI cross-correlation between the third and fourth bins. The self-calibration method uses the information available from photometric surveys without needing to specify an IA model and will play an important role in validating IA models and IA mitigation in future surveys such as LSST, Euclid and WFIRST.'
author:
- 'Eske M. Pedersen$^{1}$'
- 'Ji Yao$^{1,2}$'
- 'Mustapha Ishak$^{1}$'
- 'Pengjie Zhang$^{2}$'
bibliography:
- 'bibliography.bib'
title: 'First detection of the GI-type of intrinsic alignments of galaxies using the self-calibration method in a photometric galaxy survey'
---

[***Introduction.***]{} In the last few decades, cosmology has entered a flourishing era of high precision made possible by the advancement of astronomical surveys and missions. These will continue to provide large volume, high quality observational data that will allow the scientific community to put stringent constraints on cosmological models of the universe. With such an abundance of data, it has become clear that the challenges facing modern cosmology lie in systematic uncertainties associated with the data rather than statistical ones.

One of the most powerful cosmological probes of large-scale structure and matter in the universe is weak gravitational lensing, also known as cosmic shear. Weak gravitational lensing is the physical phenomenon where images of billions of background galaxies are distorted and harmonically aligned by the foreground dark matter and galaxies. These distorted images encode valuable cosmological information about the intervening cosmos that light traveled through. Depending on the position of the sources, lenses and the observer, gravitational lensing occurs: in a strong regime giving, astonishing multiple images; an intermediate regime, giving arcs and arclets; and a weak regime, giving small distortions of the images of background galaxies. For more details see the reviews [@1992grle.book.....S; @2015RPPh...78h6901K] and references therein.

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\[fig:IAmodel\]

The effect in the weak regime is tiny but overwhelmingly abundant and is collected by surveys using statistical methods to build a powerful signal to constrain cosmological model parameters. Weak lensing is sensitive to the amount and distribution of matter in the universe as well as the parameters of the dark energy driving the acceleration of the universe. Weak lensing also probes the growth rate of large scale structures in the universe which allows it to test the theory of gravity at cosmological scales. A number of weak lensing surveys such as CFHTLens, KiDS-450, and Dark Energy Survey have already delivered – in combination with other probes – very tight constraints on the amount of matter, the amplitude of matter clustering, and equation of state of dark energy, see e.g. [@2012MNRAS.427..146H; @2017MNRAS.465.1454H; @2018PhRvD..98d3528T]. Weak lensing is thus found to be one of the most promising cosmological probes, and a number of ambitious surveys are being built and scheduled to start taking data in the upcoming decade, including LSST, Euclid, and WFIRST. Again, all these surveys will be dominated by systematic uncertainties and the scientific community is working on such systematics as uncertainties on photometric redshifts, intrinsic alignments of galaxies, baryonic effects, and modeling of non-linear regimes, among others, for more details see, for example, the reviews and references therein.

Undoubtedly, one of the most serious systematic effects that weak lensing surveys face is the so-called Intrinsic Alignments (IA) of galaxies that act as a contaminant to the weak gravitational lensing signal. Galaxies in the universe are not randomly aligned but rather possess an intrinsic alignment due to how they formed and the environment they formed into. More detail can be found in, for example, [@2015PhR...558....1T; @2015SSRv..193....1J] and references therein. indeed these IA generate additional signals that contaminate the pure cosmic gravitational shear and significantly affect the values of cosmological parameters. Studies have shown, e.g. [@2007NJPh....9..444B], that IA, if not accounted for in weak lensing cosmological analyses, leads to biases (shifts) of up to 30% in the amplitude parameter of matter fluctuations in the universe and up to 50% in the equation of state of dark energy.

To complicate the issue, there are two types of IA that require different methods of mitigation. First, a collection of galaxies formed around a massive dark matter structure will tend to be radially aligned toward such a structure. This type of IA is called the Intrinsic shape–Intrinsic shape correlation and is referred to as II. The other type of intrinsic alignment is slightly more subtle and comes from the fact that the same massive matter structure will not only radially align a galaxy close to it but also lenses the image of a background galaxy. This creates an anti-correlation between the images of the two galaxies on the observed sky. This effect is called the gravitational shear–intrinsic shape correlation and is referred to as GI (or IG) signal. The two effects, II and IG, are illustrated in Fig. \[fig:IAmodel\].

The scientific community working on weak lensing cosmology and the communities working on preparing software pipelines for upcoming photometric surveys have a strong need of efficient methods to mitigate and control the IA nuisance effect. While the effect of the II signal of IA can be reduced by not considering pairs of galaxies close to each other along the line of sight (i.e. not the same redshift bins), the GI signal cannot be reduced in the same way as it is present at long distances. One method used to try to account for GI is to assume a model of IA with a few parameters and then add those parameters to the cosmological analysis such that the parameters can be constrained from the photometric survey data. This technique relies on the knowledge and specification of an IA model which is still an area of active development itself; see, e.g., [@2017arXiv170809247B; @2019arXiv191008085V]. Another proposed mitigation method is the nulling technique that uses different redshift dependencies of lensing and intrinsic alignments but it was found to throw away too much of precious lensing signal [@2010arXiv1009.2024J]. A third scheme that was proposed in [@Zhang:2008pw] for the 2-point correlations and later re-studied and extended to 3-point correlations in [@Troxel:2011za] is called the self-calibration method. As we describe in the next section, we use all the observed correlations between shapes and densities of galaxies in a photometeric survey and put them into a procedure that will separate the GG and GI signals. This separation is based on using the dependencies of GG and GI on the respective positions of the sources and lenses in small redshift bins but still allowing the use of the whole redshift extent of the survey. Ref. [@2017JCAP...10..056Y] showed how such a method can mitigate biases on the dark energy parameters. Therefore, self-calibration complements the marginalization method as it does not rely on the specification of an IA model. It allows one to extract the GI signal that can be then subtracted from the GG signal before performing cosmological analyses. Additionally, self-calibration provides the extracted GI signal that can be fit to models of IA and help study and improve such models.

In this letter, we report first detections of intrinsic shape–gravitational shear (IG) and intrinsic shape–galaxy density (Ig) in a photometric redshift survey using the self-calibration method where no IA model has been assumed. We provide a concise description of intrinsic alignment, the self-calibration method, the steps that directly lead to the detections, and the results obtained. A more detailed description of the Ig part of the results and related methods, as well as other developments can be found in a companion paper [@JietAl2019].\
[***Intrinsic alignments of galaxies and basic elements of the self-calibration method.***]{} In photometric galaxy surveys, the total measured shear is given by $\gamma^{obs}=\gamma^G+\gamma^I+\gamma^N$, where the superscript G stands for gravitational shear, I for intrinsic alignment, and N for shot noise. Thus, the observed angular cross-correlation, $<\gamma^{obs,i},\gamma^{obs,j}>$, between two redshift bins $i$ and $j$ includes: a GG term that corresponds to the genuine gravitational shear signal; GI, II, and IG terms that represent intrinsic alignment components; and a noise term. This can be written in terms of the corresponding shape-shape power spectrum as follows: $$C^{\gamma\gamma}_{ij}(\ell)=C^{GG}_{ij}(\ell)+C^{IG}_{ij}(\ell)+C^{GI}_{ij}(\ell)+C^{II}_{ij}(\ell)+\delta_{ij}C^{GG,N}_{ii}.$$ Fig. \[fig:IAmodel\] shows the physical mechanisms behind the correlation giving the terms $C^{II}_{ij}(\ell)$ and $C^{IG}_{ij}(\ell)$. Note that we use here the convention that IG term represents the intrinsic alignment signal and that the GI term should become negligible.

The components $C^{GI}_{ij}(\ell)$ and $C^{II}_{ij}(\ell)$ can be minimized by choosing bins with $i<j$. $C^{GI}_{ij}(\ell)$ will be minimal due to G being in front of I so no such correlation can be present, while the $C^{II}_{ij}(\ell)$ term is negligible since it is present only for close galaxies but not between galaxies in distinct bins.

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   [![image](IGcorr34.png){width="48.00000%"}]{}   [![image](GammaIgGg_3.png){width="48.00000%"}]{}
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Now, the self-calibration is used to separate the two remaining terms, $C^{GG}_{ij}(\ell)$ and $C^{IG}_{ij}(\ell)$. First, in the small redshift bin approximation, a scaling relation was derived to relate the IG term to the Ig term [@Zhang:2008pw]: $$\vspace{-0.2cm}
    \label{eq:scaling}
    C_{ij}^{IG}\left(\ell\right) \simeq \frac{W_{ij}\Delta_i}{b_i\left(\ell\right)} C_{ii}^{Ig}\left(\ell\right)
    \vspace{-0.2cm}$$ where $W_{ij}$ is the weighted lensing kernel, $\Delta_i$ is the effective width of the $i$th bin, and $b_i$ is the galaxy bias in the $i$th bin.

We use the self-calibration method including the Hankel transform of Eq. \[eq:scaling\] to measure the $w_{GI}$ correlation signal in the KiDS-450 photometric survey. Following the approach outlined in [@Zhang:2008pw; @Troxel:2011za], we start by defining the selection function $(S)$, which selects only pairs of galaxies with photometric redshifts $z_G^P < z_g^P$, for the photometric bin. This leads to defining the parameter $Q$ for how well we can actually distinguish the galaxy shear–galaxy density signal Gg with or without the selection function: $$\vspace{-0.2cm}
    \label{eq:Q}
    Q\left( \ell \right) = \frac{\left.C^{Gg}_{ii}\left(\ell\right)\right|_S}{C^{Gg}_{ii}\left(\ell\right)}$$ To calculate this we use the following spectra: $$\begin{aligned}
    C_{ii}^{Gg}\left(\ell\right) &= \int_0 ^\infty \frac{W_i\left(\chi\right)n_i\left(\chi\right)}{\chi^2}b_g P_\delta \left(k=\frac{\ell}{\chi};\chi\right) d\chi \label{eq:CGg}\\
    \left.C^{Gg}_{ii}\left(\ell\right)\right|_S &= \int_0 ^\infty \frac{W_i\left(\chi\right)n_i\left(\chi\right)}{\chi^2}b_g P_\delta \left(k=\frac{\ell}{\chi};\chi\right) \eta_i\left(z\right)d\chi\label{eq:CGgS}
$$ where $W_i$ is the lensing efficiency, $n_i$ is the true redshift distribution, $\chi$ is the co-moving distance, $b_g$ is the galaxy bias that we assume is approximately constant over the bin, $P_\delta$ is the matter power-spectra, and $\eta_i$ is a function of the selection function that was defined in [@Zhang:2008pw] as:

$$\eta_i(z) =\frac 
    {2\int_{z^P_{ i, \rm min}}^{z^P_{ i, \rm max}}dz^P_{G}\int_{z^P_{ i, \rm min}}^{z^P_{ i, \rm max}}dz^P_g\int_{0}^{\infty}dz_G W_L(z,z_G)p(z_G|z^P_G)p(z|z^P_g)S(z^P_G,z^P_g)n^P_i(z^P_G)n^P_i(z^P_g)}
    {\int_{z^P_{ i, \rm min}}^{z^P_{ i, \rm max}}dz^P_{G}\int_{z^P_{ i, \rm min}}^{z^P_{ i, \rm max}}dz^P_g\int_{0}^{\infty}dz_G W_L(z,z_G)p(z_G|z^P_G)p(z|z^P_g)n^P_i(z^P_G)n^P_i(z^P_g)}  \label{eta}$$

where $W_L$ is the lensing kernel, the superscript $P$ denotes photometric redshift, $p(z_G|z^P_G)$ is the photometric probability distribution function (PDF), $n^P_i$ are the photometric redshift distribution in the $i$th tomographic bin, and $S$ is the selection function: $$\label{selection}
    S(z^P_G,z^P_g)=\begin{cases}
    1 &\text{for $z^P_G<z^P_g$}\\
    0 &\text{otherwise.}
    \end{cases}$$ For this work we have assumed that the PDF is Gaussian of the form: $$p(z|z^P) = \frac{1}{\sqrt{2\pi}\sigma_z\left(1+z\right)}\exp\left( -\frac{\left(z-z^P\right)^2}{2\left(\sigma_z\left(1+z\right)\right)^2}\right)$$ In this paper we have used $\sigma_z = 0.082$. With these tools in mind, we can move on to the separation of the correlation functions. In [@Zhang:2008pw] the work is done in $\ell$ space, but here we will instead focus on real space, to do this we define a constant $\hat{Q}_i$ as the average of $Q\left(\ell\right)$ over a reasonable range of $\ell$. With this, we can then perform a Hankel transform as outlined in [@Joudaki:2017zdt] to get: $$\begin{aligned}
    w^{Ig}\left(\theta\right) &= \frac{\left.w^{\gamma g}\right|_S \left(\theta\right) - \hat{Q}_i w^{\gamma g}\left(\theta \right)}{1-\hat{Q}_i}\\
    w^{Gg}\left(\theta\right) &= \frac{ w^{\gamma g}\left(\theta \right)-\left.w^{\gamma g}\right|_S \left(\theta\right)}{1-\hat{Q}_i}\end{aligned}$$ The terms here can be obtained via the Treecorr code [@Jarvis:2003wq], and the $\hat{Q}_i$ can be obtained separately for each bin as seen in the right panel of Fig \[fig:IAmodel\].\
\
[***Detection of GI-type intrinsic alignment using the self-calibration method.*** ]{} We have designed two pipelines for the separation of the $Ig$ and $Gg$ signals, as a way to cross-validate our results. Pipeline-1 was designed as a tool using AstroPy and SciPy [@scipy] to calculate the integrals needed to obtain the Qs as given in . The correlations shown in the right panel of Fig. \[fig:IG-signal\] for this pipeline are calculated using version 3 of Treecorr [@Jarvis:2003wq], with jackknife-regions obtained using the tiles from KiDS-450 [@2017MNRAS.465.1454H]. A first separation of the Ig correlation using the self-calibration method was obtained with Pipeline-1 [@Yao_dissertation] in the third redshift bin of KiDS-450 data.

Pipeline-2 was designed to be compatible with future surveys as well. For the Q calculation, we use the Core Cosmology library (CCL) [@Chisari:2018vrw] for calculating the linear power-spectra needed in and . Single and double integrals, was calculated using SciPy [@scipy], to solve triple integrals we use Monte-Carlo integration to obtain a reliable result in a reasonable time, using the SciKit-Monaco code [^1]. Treecorr 4.0.8 [@Jarvis:2003wq] with jackknife regions obtained via the K-means algorithm designed by E. Sheldon [^2] were utilized for correlation functions. For the detection we used a fixed-size, random catalog containing $10^8$ objects generated with the help of Healpix\_Util [^3]. A measurement of the Ig correlation was obtained with a significance of 2.74$\sigma$ and 1.53$\sigma$ in the third and fourth bin of the KiDS-450 photometric galaxy survey, respectively. The two independent pipelines have been used to extract the Ig correlations separately.

We used the scaling relation to obtain a 2.73$\sigma$ measurement of the IG signal by cross-correlating bin 3 and bin 4. The IG result is depicted in the left panel of Fig. \[fig:IG-signal\]. For this, we used the Hankel transform of Eq. , where the scaling coefficient can be assumed to have very little variation throughout the bin with a constant averaged value. We used error-propagation as described in the appendix of Ref. [@Yao:2017dnt] to obtain the errors on IG correlation. To get an estimate of error on the galaxy bias, we have calculated the average bias over the bin as $b_{i\text{avg}} = \int b_i(z) n_i (z) dz$, using the galaxy bias model suggested in [@Yao:2018pgk]. We considered the error between this average and the bias obtained for the model over the bin width. We have also included the numerical errors on the $W_{ij}$ and $\Delta_i$. We find here a negative IA signal from the self-calibration method in the third and forth bins, using the best redshift estimate of [@2017MNRAS.465.1454H] from the Bayesian Photometric Redshift (BPZ) Code and stacking the PDF’s. This is in agreement with the negative IA amplitude found in KIDS450 [@2017MNRAS.465.1454H] where they used the marginalization method and their stacked estimated redshift distribution from BPZ method. But we concentrated here the analysis to the individual third bin and fourth bin and their cross-correlation where we obtained a better signal to noise than in the first and second bin.\
\
[***Conclusion.***]{} A first detection of Intrinsic shape–Gravitational shear (IG) and the Intrinsic shape–galaxy density (Ig) in a photometric redshift survey using the self-calibration method is reported. This was measured with a 2.74$\sigma$ and 1.53$\sigma$ significance in the third and fourth bin of the KiDS-450 photometric survey for the Ig correlations. The significance of the IG cross-correlation between the two bins is 2.73$\sigma$. The negative IA signal we find here from the self-calibration method in the third and forth bins, using the BPZ-determined best redshift estimate of [@2017MNRAS.465.1454H] is in agreement with the negative sign IA amplitude found there using the marginalization approach and the BPZ method. We focused here on the third and fourth bins in KiDS-450 and their cross-correlation where we found a better IA signal to noise than in the first and second bins. The self-calibration method has the advantage of not requiring the specification of an intrinsic alignment model. On the contrary, when an IA signal is extracted, it can be used to test and validate such models. It is worth noting that two independent pipelines have been used to derive the results for Ig correlations and were found to be in good agreement. These results also confirm that the self-calibration method works and shows that it provide a means of extracting intrinsic alignment signals from important future photometric surveys such as LSST, Euclid and WFIRST.\
\
[***Acknowledgment.***]{} We thank Anish Agashe, Hendrik Hildebrandt, Mike Jarvis, Lindsay King, Huanyuan Shan, Michael A. Troxel, and Haojie Xu for useful discussions. EP thanks his wife Shelbi Parker for proofreading the manuscript. MI acknowledges that this material is based upon work supported in part by the U.S. National Science Foundation under grant AST-1517768 and the U.S. Department of Energy, Office of Science, under Award Number DE-SC0019206. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas for providing HPC resources that have contributed to the research results reported within this paper. URL: http://www.tacc.utexas.edu.

[^1]: https://pypi.org/project/scikit-monaco/

[^2]: https://github.com/esheldon/kmeans\_radec

[^3]: https://github.com/esheldon/healpix\_util
