Measuring the scale-dependence of intrinsic alignments using multiple shear estimates
We present a new method for measuring the scale-dependence of the intrinsic alignment (IA) contamination to the galaxy-galaxy lensing signal, which takes advantage of multiple shear estimation methods applied to the same source galaxy sample. By exploiting the resulting correlation of both shape noise and cosmic variance, our method can provide an increase in the signal-to-noise of the measured IA signal as compared to methods which rely on the difference of the lensing signal from multiple photometric redshift bins. For a galaxy-galaxy lensing measurement which uses LSST sources and DESI lenses, the signal-to-noise on the IA signal from our method is predicted to improve by a factor of relative to the method of Blazek et al. (2012), for pairs of shear estimates which yield substantially different measured IA amplitudes and highly correlated shape noise terms. We show that statistical error necessarily dominates the measurement of intrinsic alignments using our method. We also consider a physically motivated extension of the Blazek et al. (2012) method which assumes that all nearby galaxy pairs, rather than only excess pairs, are subject to IA. In this case, the signal-to-noise of the method of Blazek et al. (2012) is improved.
keywords:gravitational lensing: weak – methods: data analysis – cosmology: large-scale structure of Universe
As light from distant galaxies propagates through space, its trajectory is modified by the presence of massive structures, and observed galaxy images are distorted in an effect known as gravitational lensing. Weak gravitational lensing – the case in which the distortion is small and detectable only via averaging over many galaxy images – is a key cosmological observable of several upcoming surveys, including the Large Synoptic Survey Telescope (LSST; LSST Science Collaboration et al., 2009), Euclid (Laureijs et al., 2011), and the Wide-Field Infrared Survey Telescope (WFIRST; Spergel et al., 2015). The weak lensing measurements of these surveys are expected to enhance our understanding of the evolution of dark energy, the nature of gravity on cosmological scales, and other fundamental cosmological questions (see, for example, Weinberg et al. 2013). Because of the considerable decrease in statistical uncertainties expected for these next-generation weak lensing measurements, it is crucial that we understand and mitigate all systematic effects that may contaminate weak lensing observables (for a review of these effects, see Mandelbaum 2017).
Weak gravitational lensing studies typically measure two-point correlations between the shapes of source galaxies (cosmic shear) and/or between the shapes of source galaxies and the positions of foreground lens galaxies, which we call galaxy-galaxy lensing (see, for example, van Uitert et al. 2017; DES Collaboration et al. 2017). It is the latter of these, galaxy-galaxy lensing, which we consider in this work. Translating these measurements into precise cosmological constraints relies on accounting for the sub-dominant levels of correlation which arise due to other effects. In this paper, we focus on the correlation in alignment due to local gravitational effects. These astrophysical correlations are referred to as intrinsic alignments (IA). For a thorough introduction to this phenomenon, see Troxel & Ishak (2015); Joachimi et al. (2015); Kirk et al. (2015); Kiessling et al. (2015).
A common approach to dealing with this effect is to marginalize over the parameters of an IA model (for two recent examples of this approach, see van Uitert et al. 2017; DES Collaboration et al. 2017). Popular choices include the linear alignment model (Catelan et al., 2001; Hirata & Seljak, 2004), which assumes that alignment is ‘frozen in’ at early time and therefore that the IA two-point function is proportional to the linear matter power spectrum, and the related nonlinear alignment model (Bridle & King, 2007) which replaces the linear matter power spectrum with its nonlinear counterpart in an attempt to account for late-time growth of structure. However, these models, while reasonably good descriptions on larger scales, are unable to describe the intrinsic alignment correlation in the one-halo regime. Modeling and measuring IA on these smaller scales is an active field of research interest (see, for example, Schneider & Bridle 2010; Blazek et al. 2012; Singh et al. 2015; Sifón et al. 2015; Chisari et al. 2015; Blazek et al. 2015; Blazek et al. 2017), which has yet to result in a universally accepted and fully coherent model. Therefore, any insight into the scale-dependence of IA is of great value, as it enables the construction of improved models and thus the more effective mitigation of this systematic effect in weak lensing measurements.
Furthermore, existing methods for empirically constraining the IA contribution to a given galaxy-galaxy lensing measurement generally require a robust way to understand the photometric redshift (photo-z) error distribution of source galaxies (Blazek et al. 2012; Chisari et al. 2014), such as by obtaining spectroscopic redshifts of a representative subsample of source galaxies or employing cross-correlation techniques. This may be challenging for upcoming lensing surveys, which will image fainter source galaxies than ever before. It is important to quantify, for a given method of measuring intrinsic alignments, the degree to which this source of systematic uncertainty will impact IA mitigation in upcoming surveys.
Recently, it was shown explicitly in Singh & Mandelbaum (2016) that the use of different galaxy shape-measurement methods results in a scale-independent multiplicative change in the measured amplitude of the intrinsic alignment contribution to the galaxy-galaxy lensing signal. For the three shape-measurement methods examined there, the difference in the measured IA amplitude was on the order of tens of percent. After a series of tests which ruled out point spread function-related systematic errors as well as environmentally-dependent galaxy ellipticity gradients, the suggested explanation for this result was the presence of isophotal twisting, in which the outer radial parts of a galaxy are more aligned with the tidal field than are the inner regions. Different shear estimates are sensitive to different radial separations from the center of source images. Thus, if isophotal twisting is present, it is expected that shear estimates with sensitivity to outer regions will result in a larger intrinsic alignment signal than those with sensitivity to inner regions. This effect was earlier discussed in Schneider et al. (2013), within the context of a study of radial alignments in the Galaxy and Mass Assembly Survey. It had also been seen in simulation studies of galaxy ellipticities and intrinsic alignments (Tenneti et al., 2014, 2015; Velliscig et al., 2015a, b; Hilbert et al., 2017). Following its direct observational detection in Singh & Mandelbaum (2016), Chisari et al. (2016) proposed exploiting the effect to probe primordial non-Gaussianity, which is theorized to introduce deviations to the IA signal on large scales, where IA modeling is best-understood.
In this work, we take advantage of the finding of Singh & Mandelbaum (2016) to construct a new method for measuring the scale-dependence of IA contamination to galaxy-galaxy lensing on scales at which nonlinear and one-halo effects dominate. We consider the difference between two tangential-shear measurements using the same set of source and lens galaxies, differing only in the shear estimation method applied to sources. The lensing contribution to these signals is identical, and is thus removed by taking their difference. However, if the shear estimation methods selected are sensitive to different radial regions of the galaxy light profile, an IA portion of the signal will remain, resulting in a method to determine the IA contribution up to a constant factor and hence to measure its scale-dependence.
For this cancellation of the lensing signal to occur, the source sample associated with both shear estimates must be the same, and any residual multiplicative bias must be sub-dominant. For this reason, we suggest the use of two shear estimates of the Bayesian Fourier Domain (BFD) type (Bernstein & Armstrong 2014, Bernstein et al. 2016), adjusted to accommodate different radial weighting functions. BFD has been shown to result in sub-dominant multiplicative bias (Bernstein & Armstrong, 2014), and the use of two such similar estimates would prevent issues of different source selection cuts.
Because in our method the lensing contribution to the signal is entirely subtracted off, it does not need to be measured and then removed, meaning that this method may be especially robust to challenges in constraining the source galaxy photometric redshift errors. This method also has the potential to reduce the statistical uncertainty in the measured intrinsic alignment signal, due to the correlation of shape noise and cosmic variance between measurements made with different shear estimates. It therefore has the potential to test our small-scale alignments models in a way that results in improved data-driven models. We will investigate and quantify both of these possible advantages of this method.
This paper is organized as follows. In Section 2 we provide a brief theoretical review of the relevant galaxy-galaxy lensing observables and how they are expected to be affected by IA, then we introduce an existing method for measuring the IA contribution to galaxy-galaxy lensing observables to which we will compare (Blazek et al. 2012, hereafter B2012). We proceed, in Section 3, to present our new method for measuring IA, and we provide details on its implementation. In Section 4, we describe the two observational scenarios in which we will forecast the capabilities of our proposed method in comparison with the above-mentioned existing method. We present our main results in Section 5, in which we first describe a modification to the method of B2012 which permits fair comparison with our method. We then consider whether our method improves upon the existing method in terms of robustness to systematic uncertainties, and finally we demonstrate the power of our method in the regime in which statistical errors dominate. We discuss our findings and conclude in Section 6. Throughout this work, unless otherwise noted, we assume cosmological parameters defined by the Planck 2015 results (Ade et al. 2016): , , , (), with .
2 Theoretical background
In this section, we briefly review the theoretical basis for relevant galaxy-galaxy lensing quantities, and discuss the expected intrinsic alignment contribution to the galaxy-galaxy lensing signal. We then describe existing methods for measuring this contribution, focussing on the method of B2012 which will be used in this work as a benchmark against which to measure the performance of our new method.
2.1 Galaxy-galaxy lensing
Galaxy-galaxy lensing studies are concerned with the measurement of the cross-correlation between the shapes of background source galaxies and positions of foreground lens galaxies. Typically the measured quantity is either , the average tangential shear of source galaxies about lens galaxies, or , the differential projected surface mass density around lens galaxies (where is the projected radial distance from a lens galaxy center and a tilde indicates an observed quantity). For a single lens-source pair, and are related via:
where is the critical surface density, which depends on the separation of lens and source galaxies. It is given (in comoving coordinates) by:
where , , and are the angular diameter distances from observer to source, from observer to lens, and from lens to source, respectively. is the lens redshift. is defined as:
where is the projected surface density of matter about a lens galaxy and is the same quantity averaged within projected radial separation . can be computed theoretically via
where is the matter density in comoving units, is the line-of-sight separation, and we use to denote the correlation function of matter with lens galaxies.
In this work, we use CLASS (Lesgourgues, 2011) with halofit (Smith et al. 2003, Takahashi et al. 2012) via CCL (LSST DESC, 2017) to theoretically compute the nonlinear matter power spectrum, from which we obtain the two-halo term of using FFTlog (Hamilton, 2000). We assume a linear galaxy bias for the lens galaxy sample to convert to via , where we set the cross-correlation coefficient to unity. In computing the one-halo contribution, we assume a Navarro-Frenk-White (NFW) profile (Navarro et al., 1997), where we follow Mandelbaum et al. (2008) and take the concentration-mass relation to be
and the halo radius to be defined by
Together with an appropriate halo occupation distribution (HOD) model, equations 3-6 allow for the theoretical computation of and . We will specify the HOD models that we use in this work below, in Section 4.
2.2 Galaxy-galaxy lensing and intrinsic alignments
Intrinsic alignment contributions to galaxy-galaxy lensing signals arise due to correlations between shapes of source galaxies and positions of lens galaxies which are due not to lensing but to tidal gravitational effects.
Consider a generic cross-correlation of the shear of source galaxies with the positions of lens galaxies , where we have used to represent shear and to represent lens galaxy overdensity. The intrinsic alignment contamination to this cross-correlation can be expressed via
where we use to represent the true shear due to lensing, and to represent the effective contribution to the shear due to IA.
The term is generally non-zero in the case when galaxies from the source sample are in physical proximity to lens galaxies, and therefore have shapes which are correlated with the lens-galaxy positions via tidal gravitational effects. Given perfect redshift measurements for both source and lens galaxies, it would be possible to minimize or eliminate this effect by down-weighting or cutting lens-source pairs which are close along the line of sight. However, due to the large number of source galaxies employed in weak lensing measurements, in general source galaxy redshifts are determined via photometry and hence have some non-negligible uncertainty.
The general cross-correlation can be taken to represent or , and the contamination to either of these signals can be quantified via : the contribution to the measured tangential shear from IA, per contributing lens-source pair. For the forecasts which we undertake below, we will require fiducial predictions for this quantity, which are obtained via
where is the projected correlation function of the positions of lens galaxies with those of source galaxies, is the projected cross-correlation function between lens galaxy positions and source galaxy shapes, and is the comoving radial separation within which lens-source pairs are sufficiently close along the line of sight to be affected by IA. A similar equation for is given in, e.g., B2012, the difference being the additive factor of included here in the denominator. This factor is the result of the fact that we assume here that all physically associated lens-source pairs may be subject to IA (rather than excess pairs only), with being the line-of-sight separation within which pairs are expected to be physically associated. The denominator of equation 8 can be thought of as being obtained by integrating along the line-of-sight not just over , but over . We will take to be , although this value is not well-determined and represents an uncertainty in our modeling of the fiducial signal.
Non-zero intrinsic alignment signals have been observed in red galaxy populations, but not yet conclusively measured in blue galaxy populations (see, e.g., Mandelbaum et al. 2011, Kirk et al. 2015); we therefore expand equation 8 as
where and respectively represent the fraction of red and blue source galaxies amongst those which are sufficiently close in line-of-sight separation to the lens sample to be subject to intrinsic alignments. The second line of equation 9 comes from the assumption that (i.e., blue galaxies are not subject to IA at a significant level) and that blue and red galaxies cluster in the same way (which is not strictly correct but is a sufficient approximation for the purposes of our work).
The two-halo terms of these projected correlation functions are computed via (see, for example, Singh et al. 2015, where we have neglected redshift-space distortions):
where is the large-scale bias of the source galaxies, is the same for the lens galaxies, and is the nonlinear matter power spectrum. We set and to their ensemble average values as computed from an appropriately chosen HOD (the HODs used in this work for different observational scenarios are discussed in Section 4).
Equation 11 assumes the nonlinear alignment model for IA. Here, controls the amplitude of IA on scales where the two-halo term dominates, and is a normalization constant. Throughout this work we follow e.g. Joachimi et al. (2011) and set , obtained via fitting the linear alignment model to low-redshift SuperCOSMOS observations. is the combined window function of source and lens galaxy samples, given by (see for example Singh et al. 2015):
where and are the redshift distributions of the lens and source galaxies respectively.
The one-halo term of is computed using the standard halo model in combination with the relevant chosen HOD (again, given below in Section 4). The NFW profile is assumed, and concentration and virial radius are given by equation 5 and 6 respectively.
is calculated using the halo model for IA as introduced in Schneider & Bridle (2010). The relevant one-halo power spectrum is:
The parameters are fit in Schneider & Bridle (2010). In Singh et al. (2015), the parameters are adjusted to better fit the BOSS LOWZ galaxy sample; here we assume the parameters of Singh et al. (2015) and all other parameters from Schneider & Bridle (2010). can then be found via
Given the capability to theoretically calculate the one- and two-halo terms of both and , theoretical fiducial values for can be computed using equation 8. Note that we do not incorporate halo-exclusion terms in our fiducial calculations of and , but instead simply add one-halo and two-halo terms. The choice to simplify our calculations in this way has a minor effect on the shape of the fiducial signals calculated, however the magnitude of this effect is negligible relative to the variation in signal-to-noise within the IA-measurement scenarios explored below.
2.3 Existing methods for measuring intrinsic alignments
Several methods exist in the literature to measure or constrain the intrinsic alignment contribution to the galaxy-galaxy lensing signal. Because lensing measurements typically rely on source galaxies with redshifts determined photometrically, it is useful that procedures for mitigating the effect of IA take this into account. In several existing methods, source galaxies are separated into two or more bins using photometric redshifts, and the lensing and IA signals are estimated simultaneously via the assumption that the source galaxy sample(s) which are more separated in redshift from the lens galaxies will contain fewer galaxies which are subject to IA.
In Hirata et al. (2004) the intrinsic alignment contribution to an SDSS galaxy-galaxy lensing measurement was constrained, under the assumption that a source sample with higher photometric redshifts had no intrinsic alignment contribution. Methods by B2012 and Chisari et al. (2014) later relaxed this assumption, while still relying on the idea that fewer galaxies in higher photometric-redshift source samples would be subject to IA. In a similar vein, Joachimi & Schneider (2008) and Joachimi & Schneider (2010) proposed methods to null or boost the intrinsic alignment signal in cosmic shear measurements using its characteristic redshift dependence. In order to demonstrate the utility of the method which we present in this work, we will compare against the method put forward in B2012. We now describe this method in detail.
In the methodology of B2012, it is assumed that source galaxy redshifts are photometric while lens galaxy redshifts are spectroscopic. Two measurements of the galaxy-galaxy lensing quantity are then considered. The first, which we label , is for a source sample defined by the requirement that for a given lens redshift , the source (photometric) redshift satisfies , where is chosen to jointly optimize signal-to-noise of the lensing measurement and intrinsic alignment constraint on a per-survey basis. The second sample, called , is chosen in a complementary way such that . Given these two measurements of , it is possible to solve both for the lensing signal and for the contribution to the tangential shear due to IA. The expression for the latter is given by B2012:
In the form of this method originally introduced in B2012, it is assumed that only excess lens-source pairs are subject to IA (that is, only pairs which statistically contribute to a positive correlation between lenses and sources). is given by where is the photometric-redshift bias to . is the boost factor, which quantifies the presence of excess galaxies to sample , and is the average critical surface density for excess galaxies. Dependence on has been omitted in equation 16 for clarity.
, , and are explicitly given for a generic source sample by:
where in equation 17, the second equality makes use of the fact that the true value of is approximately zero for source galaxies at or very near the lens redshift. Sums in equations 17, 18, and 19, and similar sums below, should be interpreted as being over lens-source pairs with source galaxies in the relevant sample. A label of ‘lens’ indicates a sum over all such pairs, while a label of ‘excess’ implies a sum over only those pairs which are statistically in excess of what would be present in the case without clustering. A label of ‘rand’ indicates that the sum should be performed over a set of ‘lenses’ which are sampled randomly from the same redshift distribution and window function as the true set of lenses.
We note particularly that equation 17 for requires a sum over the product of , the estimated critical surface density (using photometric redshifts for source galaxies) and , the inverse of the true critical surface density (using spectroscopic redshifts for source galaxies). This sum is in practice therefore taken over only the subsample of lens-source pairs for which sources have spectroscopic redshifts. We assume that in the case in which a representative spectroscopic subsample of sources is unavailable, a re-weighting method (see, for example, Lima et al. 2008) is used to approximate a representative subsample as closely as possible. Such a method will be imperfect when the spectroscopic subsample completely neglects parts of the source galaxy parameter space (in terms of e.g. color or magnitude), or if the rate of redshift failure at a fixed point in color-magnitude space depends on redshift. In such cases, will be subject to a systematic uncertainty.
As stated, the objective of the B2012 method to which we will compare is to solve simultaneously for both and . Therefore, the weights, , are chosen in that work to optimize the signal-to-noise of :
where is the contribution due to shape-noise and is the measurement error associated with the source of lens-source pair . This choice of weights is sub-optimal for the estimation of IA, due in part to the downweighting of nearby lens-source pairs. We do not advocate its use with our proposed method, and will introduce a different weighting scheme below for this purpose. We nevertheless use the weights of equation 20 when making calculations in the B2012 method to enable comparison with the measurements of that work. One might then ask whether comparisons made between our method, with a more-optimal weighting scheme, and the B2012 method, with this less-optimal scheme, are meaningful. To address this issue, we have checked the effect of using a modified version of the B2012 method which is formulated in terms of tangential shear and uses the redshift-independent weights introduced below in equation 23. We find no significant changes to our conclusions in this scenario (although small quantitative changes do occur). We mention a further implication of this alternative weighting scheme for the B2012 method in Section 5.3.
3 Measuring intrinsic alignments with multiple shear estimates
Having now provided theoretical background, we introduce our new method to measure the scale-dependence of the intrinsic alignment contribution to the galaxy-galaxy lensing signal.
We consider two estimates of the tangential shear obtained via different methods, which we will call and . These are given as:
where all sums are over lens-source pairs, is the tangential shear due to lensing for a given lens-source pair , and is the contribution to the shear signal due to IA for lens-source pair . is once again the boost factor, which is included to ensure that the tangential shear is normalized in the standard way. and are, for each shear estimate, the residual multiplicative bias remaining after calibration. is the constant by which the measured IA amplitudes are offset one from the other. For reference, as we define here, Singh & Mandelbaum (2016) find for the three method pairs formed by de Vaucouleurs shapes, isophotal shapes, and re-Gaussianization. These pairings of shear estimates are not expected to be optimal; we provide their values simply for reference and would expect more optimal methods to yield smaller values (as discussed below). Note that our definition of is slightly different from the ratio of IA amplitude values used to describe this effect in Singh & Mandelbaum (2016).
In contrast to the method described in Section 2.3, our objective is not to simultaneously estimate both and , but simply to measure the scale-dependence of . We therefore choose weights differently than in the above case:
This is a typical choice of weights in measurements of lensing tangential shear. While not being explicitly designed to optimize for , this choice is more optimal than e.g. the weights of equation 20 because it dispenses with the factor of , which down-weights the very lens-source pairs expected to be subject to IA. It is nevertheless likely that there exist more optimal weight choices than equation 23, and therefore the forecasts conducted below may not reflect the highest possible signal-to-noise available via the proposed method.
The success of our proposed method is dependent upon using the same weighting scheme when computing and . Therefore, if the two shear estimates in question result in different values of , or different per-galaxy values of , it is necessary to use a version of equation 23 which adopts the same value of these for both methods (e.g. by adopting their average value). This ensures the required cancellation when taking the difference of and , as we do now.
Terms involving and are no longer present in equation 24. We have assumed that calibration is performed via a method in which any such residual uncertainty ( or above) is demonstrably sub-dominant (see, for example, Bernstein et al. 2016; Sheldon & Huff 2017). If this is not the case, two additional terms would be added to the right-hand-side of equation 24, representing an additive bias: one proportional to and one proportional to . We estimate that in order to limit this additive bias to at most -percent the value of , for a galaxy-galaxy lensing measurement involving sources from LSST and lenses from the Dark Energy Spectroscopic Instrument (see Section 4 below for details on this observational scenario), we would require that the individual absolute values of and not exceed .
We consider a sample of lens-source pairs which is defined by the requirement that for a given lens-source pair, lens redshift and source photometric redshift are sufficiently close that we would naively expect them to be physically associated. As mentioned above, we will take this maximum line-of-sight separation to be in this work. The quantity which we aim to measure, up to a constant factor, is : the tangential shear due to intrinsic alignments per contributing lens-source pair. To obtain this quantity from equation 24, we must multiply both sides by an additional factor, which ensures that we divide by the sum of weights of contributing pairs only:
where we have defined :
and the label ‘rand, close’ indicates that the sum should be taken over pairs in which source and lens are sufficiently close as measured by spectroscopic redshift that we would expect them to be intrinsically aligned, and the ‘lenses’ are drawn randomly from the lens redshift distribution and window function.
can evidently not be computed on a pair-by-pair basis, as we do not expect to have access to the spectroscopic redshift of all source galaxies. Instead it should be computed statistically via and , where the former is the number density of sources with respect to spectroscopic redshift, and the latter is the probability of a source galaxy with measured photometric redshift having true (spectroscopic) redshift , or vice-versa. In this way, is given by:
where and are, respectively, the maximum and minimum spectroscopic redshifts at which we would expect source galaxies to be intrinsically aligned, for a given lens redshift . is the number density of lens galaxies with respect to (spectroscopic) redshift, and integrals without explicit limits should be taken as being over the integration variable’s full range.
Equation 27 makes explicit the fact that because encodes information about true source galaxy redshifts, it must be calculated using and as estimated from the spectroscopic subsample of sources. Much like in the B2012 method above, in the case of an inadequately representative spectroscopic subsample or imperfect re-weighting, may therefore become a source of systematic error.
Equation 28 is the fundamental expression of our method. It allows us to measure the intrinsic alignment contribution to galaxy-galaxy lensing up to a poorly-known constant in order to gain information about its scale-dependence. We now prepare to test how well this method is expected to perform compared to the existing method described in Section 2.3.
4 Observational scenarios for forecasting
To evaluate the effectiveness of our proposed method for measuring the scale-dependence of , we select two observational scenarios in which to forecast expected performance.
The first of these, which we will call the ‘SDSS’ case, assumes that lens and source galaxies are both drawn from Sloan Digital Sky Survey (SDSS) data. For the other, which we will call the ‘LSST+DESI’ case, we consider a scenario which combines data from two upcoming surveys, with sources from the Large Synoptic Survey Telescope (LSST) and lenses from the Dark Energy Spectroscopic Instrument Luminous Red Galaxy sample (DESI LRGs). These two choices ensure that we can both compare our predictions to the actual measurement of by B2012 (in the SDSS case), and explore how our proposed method may perform for a next-generation measurement (in the LSST+DESI scenario).
In the SDSS case, lens galaxies are assumed to be from the SDSS LRG sample as selected in B2012 (see also Kazin et al. 2010 and their ‘DR7-Dim’ sample), with a surface density of (Mandelbaum et al. 2013). The median redshift of this sample is (Kazin et al. 2010). Source galaxies are assumed to be from the sample described in Reyes et al. (2012), with an effective surface density of . The per-component rms distortion of the source sample is ; with responsivity , this results in (Reyes et al. 2012). The overlapping sky area of these lens and source samples is taken as 7131 (B2012).
The LSST+DESI observational scenario assumes lens galaxies from the anticipated DESI sample of LRGs, which we assume to have a surface density of (DESI Collaboration et al., 2016). The effective redshift of the sample is taken as (estimated from Figure 3.8 of DESI Collaboration et al. 2016). Source galaxies in this scenario are taken to be from the final LSST lensing sample, with an expected effective surface density of , and a per-component rms ellipticity of , which with the ellipticity definition of Chang et al. (2013) is equivalent to . These source and lens samples would be expected to have an overlapping sky area of 3000 (Schmidt et al., 2014).
Further assumptions and details associated with each of these observational scenarios are as follows:
Redshift distribution of lenses:
The number density of the SDSS LRG sample is shown as a function of redshift in Figure 2 of Kazin et al. (2010). We smooth with a box filter and perform a conversion to appropriate units to obtain the redshift distribution of SDSS LRGs. The expected redshift distribution of the DESI LRG sample is given in Figure 3.8 of DESI Collaboration et al. (2016). Both distributions are plotted in the left-hand panel of Figure 1.
Redshift distribution of sources:
The redshift distribution (in terms of spectroscopic redshifts ) of source galaxies is given for the SDSS source sample as:
where and (Nakajima et al., 2012). For the LSST source sample, the distribution is given by:
where , and (Chang et al., 2013). In both cases, is appropriately normalized over the redshift range once convolved with the photometric redshift model. The assumed source redshift distributions are plotted in the right-hand panel of Figure 1.
Model for photometric redshifts:
In both observational scenarios under consideration, source galaxy redshifts are photometrically determined. The source galaxy redshift distribution in terms of photometric redshift is given by:
We choose a simple Gaussian model for in both observational scenarios:
where is taken to be for the SDSS source sample (B2012) and for the LSST source sample (Chang et al., 2013).
and are the fiducial amplitudes of the one-halo and two-halo terms of , respectively. Singh et al. (2015) provides a fit to a power law in luminosity for each of these quantities:
where is the r-band luminosity and is the pivot luminosity corresponding to an absolute r-band magnitude of . Parameters are fit in Singh et al. (2015) using data from the SDSS BOSS LOWZ sample, and found to be , , , and . We take the best-fit values of these parameters as their fixed values in estimating and for our fiducial calculations.
In order to then determine the appropriate values of and for the scenarios under consideration, we use a Schechter luminosity function (Schechter, 1976) and integrate equations 33 and 34 over luminosity. Following Krause et al. (2016), we use Schechter function parameters for red galaxies from Loveday et al. (2012) and Faber et al. (2007). We assume the limiting -band apparent magnitude of the SDSS shape sample to be (see Figure 3 of Reyes et al. 2012), and the same quantity for the LSST lensing sample to be (LSST Science Collaboration et al., 2009). As a result, we find and for the SDSS scenario, and , for the LSST+DESI case.
Halo occupation distributions:
When computing the fiducial value of , as well as the cosmic variance terms of the covariance matrices for and (see Appendix A), we must specify a halo occupation distribution (HOD) for both the lens and the source samples. We use this to calculate one-halo terms of two-point functions, as well as to obtain the large-scale galaxy bias, and hence the two-halo term. For the SDSS LRGs, we use the HOD fit to this same galaxy sample in Reid & Spergel (2009), which yields a large scale bias value of . For both the SDSS and LSST source samples, we use the HOD developed in Zu & Mandelbaum (2015). This HOD has the benefit of accepting as input the number density of galaxies, allowing its use for both source samples. The galaxy bias of both source samples, according to this HOD, is . For the DESI LRGs, lacking a better option, we employ an HOD fit to the SDSS BOSS CMASS sample (More et al., 2015) when computing one-halo terms. This is not ideal as it is fit to a different sample and does not accept number density of galaxies as input. However, as we will show below, most of the power of our proposed method is not deep within the one-halo regime, so we do not expect this sub-optimal choice to have a significant effect on our overall results. Recognizing that this HOD is not a perfect choice, we do not in this case obtain a large-scale bias value from it, but rather from the expression , given in DESI Collaboration et al. (2016), which at the effective redshift of the DESI LRGs results in . Where possible, we compare our calculation of mean central and satellite galaxy occupation numbers to those computed with the Halotools package (Hearin et al., 2017) and find agreement.
Note also that in computing quantities such as power spectra and correlation functions using these HODs, it is important to pair them with theoretical quantities (e.g. the halo mass function) which are calculated using appropriate values of cosmological parameters. We must use the parameter values which were either fixed in fitting HOD parameters or simultaneously fit with HOD parameters. Otherwise, the observables we calculate will not match the sample in question. Therefore, in this context only, we divert from our default cosmological parameters to use parameter values as given in the references above for each HOD, in order to best replicate observable quantities.
Measurement noise, :
We take the measurement noise, , required to compute the weight of each lens-source pair as in equation 23, to be related to the average per-source-galaxy signal-to-noise via (Bernstein & Jarvis, 2002):
We take the average to be for the SDSS scenario. For LSST, we take (Chang et al., 2013), equivalent to an average per-galaxy signal-to-noise of .
The boost factor, defined above in equation 18, is the ratio of the sum of weights over all lens-source pairs in a given sample, to the sum of weights of the same sample with randomly distributed lenses. In other words, it quantifies the degree to which correlation augments the number of galaxy pairs in the sample. To compute the boost for the various samples required for our analysis, we assume the power law form:
which can be seen to be a good approximation in, for example, B2012, and results from the fact that the boost is essentially proportional to the projected galaxy correlation function.
We determine the required proportionality factor for each sample of lens-source pairs. We do so by evaluating the expression (see for example B2012)
with . and are, respectively, the upper and lower photometric cuts which define the source sample at for a given lens redshift. Evidently, we could compute the boost at all using this method, but we choose to use the power law form of equation 36, as it captures the scale-dependence of the boost sufficiently well for our purposes.
For the method developed in B2012 and reviewed in Section 2.3, we require a value of , which defines the photometric redshift range of each source sample. In the SDSS scenario, we follow B2012 and choose . In the case of LSST+DESI, we choose to optimize the signal-to-noise of in the B2012 method, and find that the optimal choice is .
To obtain , we once again employ Schechter luminosity functions with parameters fit by Loveday et al. (2012) and Faber et al. (2007) as in Krause et al. (2016). In this case, we consider one luminosity function with parameters fit to red galaxies only, as above, and another with parameters fit to a full sample including red and blue galaxies. To obtain the red fraction in each case, we simply take the ratio of the integrated luminosity function for red galaxies to the integrated luminosity function for all galaxies, and find the value averaged over the line-of-sight separation on which we expect pairs to be physically associated. We find for SDSS, and for LSST.
We now describe the results of forecasting constraints on the intrinsic alignment contamination to galaxy-galaxy lensing using our method, as compared to the method of B2012. We first describe an extension to the method of B2012 which will enable a fair comparison, then discuss the impact of systematic uncertainties associated with an inadequately representative spectroscopic subsample of sources. We finally present the type of measurement which may be possible in a scenario in which statistical uncertainties are dominant.
5.1 Incorporating all physically-associated galaxies
In order to make a fair comparison between the constraining power of our proposed method and the method of B2012, we revisit the derivation of equation 16 under the assumption that all physically associated galaxies may be subject to IA (rather than only excess galaxies). In this scenario, equation 16 becomes:
This result contains the new terms and , where was introduced in equation 26, as well as the modified term , which is the average critical density over all physically-associated lens-source pairs for sample . is given (for a generic source galaxy sample) by:
and depend on sums over pairs for which the true (rather than photometric) redshift of the source galaxy is close enough to that of the lens that the pair is considered physically associated. Therefore, and must be computed via integration over the source redshift distribution, and may also be subject to systematic error in the case of an inadequately representative spectroscopic subsample of sources or an imperfect re-weighting scheme. As this updated version of the B2012 method contains six terms which are in principle subject to this type of systematic uncertainty, it is reasonable to consider whether our proposed method, for which only is subject to this type of error, may be more robust to this effect. We will address this question in Section 5.2.
Consider, though, for the moment, a scenario in which the systematic error due to source galaxy redshift uncertainty is negligible (an assumption which will be justified below). Uncertainties are then a combination of statistical errors and systematic errors associated with the boost. We can compare the expected constraints on from the original version of the method (encapsulated in equation 16) and this modified version (described in equation 38). The forecast signal-to-noise in each case is displayed in Figure 2, for both observational scenarios described in Section 4.
We see that the extension to the method of B2012 introduced here improves the signal-to-noise, in particular for the LSST+DESI scenario. This can be understood by noting that the boost factor is subject to a non-negligible systematic error due to effects such as variation of the density of lenses as a result of observational conditions and fluctuations in large-scale clustering (B2012). The boost, representing as it does the weighted ratio of all lens-source pairs to non-excess lens-source pairs, goes to unity on large scales, and so the fractional error associated with the boost increases arbitrarily on these scales, as discussed in B2012. The addition of , which is constant with projected radial separation, ensures that the equivalent term in equation 38 never grows arbitrarily large, controlling this error and resulting in the improvement seen in Figure 2. When comparing our proposed method to the method of B2012, for the remainder of this work, we use the modification described by equation 38.
5.2 Redshift-related systematic uncertainties
Due to the faint limiting magnitude of future lensing surveys such as LSST, potential systematic errors due to inadequately-characterized photo-z uncertainty are a concern (Newman et al., 2015). Quantities affected by this source of error enter our method only via (see equation 28), while in the case of the B2012 method, , , and all may be subject to this source of uncertainty. We now compare our method to that of B2012 in terms of robustness to this type of systematic error.
We first determine the maximum level of redshift-related systematic error which is of practical interest for the measurement of IA contamination to the galaxy-galaxy lensing signal. This is set by the fact that the mitigation of IA is only relevant when the lensing uncertainty itself is not dominated by redshift-related systematics. We therefore work in the regime where the fraction of the total integrated uncertainty on which is due to redshift-related systematic error is less than 50%. We find that in order for this to be true, assuming that the source sample is representative and typical for a measurement, the maximum fractional systematic error on must be less than in the SDSS case and for LSST+DESI. These values in fact also represent upper limits on the redshift-related systematic error of the other six quantities of interest (, , , ), because each of these is sensitive to near-lens sources, for which the photo-z calibration is expected to be best.
We are interested in how the total redshift-related systematic uncertainty on the IA signal compares to its statistical uncertainty, for our method and for the B2012 method. To investigate this, we consider the ratio of (integrated redshift-related systematic error) to (integrated statistical error); a smaller value of this ratio indicates greater robustness to redshift-related systematic uncertainty, while unity indicates that statistical uncertainty and redshift-related systematic uncertainty are equally important. We examine this ratio as a function of the fractional error on each of , , , and in turn (fixing the error on the others to zero), and find a power law relation, with slopes given in Table 1. To be explicit, the power law takes the form
where is one of , , , or . We see in Table 1 that within the B2012 method, the overall importance of redshift-related systematic uncertainty is most sensitive to the level of systematic uncertainty on and . This is sensible: the B2012 method incorporates the difference of the two large quantities and ; uncertainty in either of these terms will result in a relatively large error on their much-smaller difference.
We use this set of power-law relationships to easily calculate the ratio when each of , , , or in turn takes its maximum tolerable fractional error (with errors on the remaining quantities fixed at zero). These ratio values are listed in Table 2. We see there that applying the maximum interesting level of redshift-related systematic uncertainty to any individual quantity is insufficient to cause this source of error to dominate over statistical uncertainty.
Even when the redshift-related systematic error on takes its maximum value of interest, statistical error heavily dominates our method, with for SDSS and for LSST+DESI. For the B2012 method, we examine the worst case scenario in which the redshift-related systematic errors of , , and all take their maximum values simultaneously, and find the corresponding maximum value of to be for SDSS and for LSST+DESI. Although these values are significantly larger than in our method, and thus we see that the B2012 method is less robust to redshift-related systematic errors in the worst-case scenario, statistical error remains dominant.
5.3 Statistical uncertainty
As we have shown in Section 5.2, the uncertainty on in our proposed method (and in the method of B2012) cannot realistically be dominated by redshift-related systematic errors. With this in mind, we now examine the relative performance of these methods when statistical errors dominate.
We compute the signal-to-noise of the intrinsic alignment measurement from each method, assuming only statistical uncertainty and a small contribution due to systematic error on the boost factor, (B2012) (see Appendix A for how the required covariance matrices are calculated in each scenario). Note that for both methods, we assume that statistical uncertainty on the boost is negligible; we have tested this assumption and found that including this source of error yields less than a change in the total uncertainty. For our method, we compute the signal-to-noise as a function of , the ratio of intrinsic alignment amplitudes between the two shape measurement methods, and of the correlation between the shape-noise of the shear estimates, which we call . As stated above, for the shear estimation methods found in Singh & Mandelbaum (2016<