Intrinsic Alignments of Galaxies in the Illustris Simulation
We study intrinsic alignments (IA) of galaxy image shapes within the Illustris cosmic structure formation simulations. We investigate how IA correlations depend on observable galaxy properties such as stellar mass, apparent magnitude, redshift, and photometric type, and on the employed shape measurement method. The correlations considered include the matter density-intrinsic ellipticity (mI), galaxy density-intrinsic ellipticity (dI), gravitational shear-intrinsic ellipticity (GI), and intrinsic ellipticity-intrinsic ellipticity (II) correlations. We find stronger correlations for more massive and more luminous galaxies, as well as for earlier photometric types, in agreement with observations. Moreover, the correlations significantly depend on the choice of shape estimator, even if calibrated to serve as unbiased shear estimators. In particular, shape estimators that down-weight the outer parts of galaxy images produce much weaker IA signals on intermediate and large scales than methods employing flat radial weights. The expected contribution of intrinsic alignments to the observed ellipticity correlation in tomographic cosmic shear surveys may be below one percent or several percent of the full signal depending on the details of the shape measurement method. A comparison of our results to a tidal alignment model indicates that such a model is able to reproduce the IA correlations well on intermediate and large scales, provided the effect of varying galaxy density is correctly taken into account. We also find that the GI contributions to the observed ellipticity correlations could be inferred directly from measurements of galaxy density-intrinsic ellipticity correlations, except on small scales, where systematic differences between mI and dI correlations are large.
keywords:galaxies: general – gravitational lensing: weak – cosmology: theory – large-scale structure of the Universe – methods: numerical
Weak gravitational lensing by the large-scale structure, also referred to as ‘cosmic shear’, creates correlations in the observed ellipticities of distant galaxies. Measuring these correlations provides valuable information about our Universe’s matter distribution and geometry. Ongoing and planned weak lensing surveys, such as the Dark Energy Survey111http://www.darkenergysurvey.org (DES), the Kilo Degree Survey222http://www.astro-wise.org/projects/KIDS (KiDS), the Euclid333http://www.euclid-ec.org mission, and Large Synoptic Survey Telescope444http://www.lsst.org survey, are designed to constrain cosmological parameters through accurate ellipticity correlation measurements with percent-level statistical accuracy.
To fully exploit these ongoing and future surveys, one must understand the various sources of correlations in the observed galaxy ellipticities. Intrinsic alignments (IA), i.e. alignments of the intrinsic ellipticities of galaxies, are suspected to be a significant source besides gravitational lensing. Just considering these two sources, the observed ellipticity correlation has contributions from gravitational shear-gravitational shear (GG) correlations, intrinsic ellipticity-intrinsic ellipticity (II) correlations, and gravitational shear-intrinsic ellipticity (GI) correlations. II correlations may arise from physically close galaxies whose shapes are aligned due to common environmental effects such as large-scale tidal fields. GI correlations may arise when the matter structures associated with lower-redshift ‘foreground’ galaxies induce gravitational shear in the images of higher-redshift ‘background’ galaxies with a preferred direction relative to the foreground galaxy shapes.
Analytical and numerical studies suggest that intrinsic alignments contribute to the cosmic shear signal in medium and deep surveys (e.g. Croft & Metzler 2000; Heavens et al. 2000; Hirata & Seljak 2004; Heymans et al. 2006; Semboloni et al. 2008; Joachimi et al. 2013b). In observational studies, intrinsic alignments have been clearly detected and likely contribute several percent to the observed ellipticity correlations (e.g. Heymans et al. 2004; Mandelbaum et al. 2006a; Hirata et al. 2007; Mandelbaum et al. 2011; Joachimi et al. 2011; Singh et al. 2015).
Various techniques have been developed to remove the intrinsic alignment effects from weak lensing measurements, including down-weighting schemes (King & Schneider 2003; Heymans & Heavens 2003; Takada & White 2004), or nulling and boosting techniques (Joachimi & Schneider 2008, 2010). All these methods require excellent redshift information of the source galaxies, and weaken cosmological constraints due to the loss of information. Therefore it is preferable to have accurate models of intrinsic alignments, which can be used as priors in the weak lensing analysis to reduce this loss of cosmological information in lensing surveys.
Gravitational tidal fields likely cause alignments of the intrinsic shapes of galaxies and their surrounding matter on larger scales. Some intrinsic alignment models thus employ parametrized relations between the large-scale galaxy shape alignments and the statistical properties of the gravitational tidal field (e.g. Catelan et al. 2001; Crittenden et al. 2001; Hirata & Seljak 2004; Blazek et al. 2015). The parameters of these relations usually depend strongly on the galaxies’ properties and have to be determined by external data (e.g. observations, Joachimi et al. 2011), which limits the predictive power of these tidal field models.
The shapes and spins of dark matter halos and their alignments in cosmological gravity-only -body simulations can be robustly measured (e.g. Croft & Metzler 2000; Hopkins et al. 2005; Bailin & Steinmetz 2005; Bett et al. 2007). Predictions for the alignments of the luminous shapes of galaxies can then be obtained by augmenting this information with models describing the galaxy content of the halos and how the shapes of the galaxy light relate to the host dark matter halo properties (Joachimi et al. 2013a, b). This approach requires one to also specify the expected alignment of galaxies within their host halo, which is still quite uncertain (Mandelbaum et al. 2006b; Bett 2012; Schrabback et al. 2015).
Numerical simulations of cosmic structure formation that include star formation and resolve the luminous shapes of galaxies offer a more direct way to predictions for galaxy shape alignments (Chisari et al. 2015; Tenneti et al. 2015; Velliscig et al. 2015b; Tenneti et al. 2016; Chisari et al. 2016). Here we report on our study of intrinsic alignments within the Illustris simulations (Vogelsberger et al. 2014b, c). In particular, we investigate the strength of intrinsic alignments of galaxy image ellipticities, when defined such as to provide unbiased shear estimates, as a function of observable galaxy properties such as apparent magnitude, redshift, and photometric type, and the employed shape measurement method. We consider the matter density-intrinsic ellipticity (mI) correlations, galaxy density-intrinsic ellipticity (dI) correlations, gravitational shear-intrinsic ellipticity (GI) correlations, and intrinsic ellipticity-intrinsic ellipticity (II) correlations. Based on these findings, we discuss the expected contribution of intrinsic alignments to the observed ellipticity correlation in tomographic cosmic shear surveys, and also how these contributions could be estimated from measurements of galaxy density-intrinsic ellipticity correlations in a model-independent way. Furthermore, we compare the intrinsic alignments measured in the simulations to predictions from tidal alignment models.
The paper is organized as follows: We provide a brief introduction to our notation and the theory of gravitational lensing and galaxy image shapes in Section 2. The Illustris simulation and the methods we employ to extract galaxy properties and correlations from it are presented in Section 3. Section 4 deals with tests of the resulting galaxy properties such as magnitudes, colors, and shapes. In Section 5 we present our results. The main part of the paper concludes with a summary and discussion in Section 6. A more detailed discussion of correlations involving ellipticities can be found in the Appendix.
The observed shape of a galaxy image is influenced by several factors. Every galaxy has a particular intrinsic shape. Gravitational lensing may leave imprints on the observed image shapes. Various processes in the imaging apparatus (e.g. telescope aberrations or atmospheric seeing) contribute to the observed image shape. In the following, we assume that such instrumental effects have been somehow taken care of, and concentrate on gravitational lensing effects and intrinsic shapes.
2.1 Gravitational lensing
Photons emitted from sources at cosmological distances are deflected on their way toward us by the gravity of intervening matter structures. As a result of this gravitational lensing effect (see, e.g. Schneider et al. 2006, for an introduction), the observed image position of a source at redshift may thus differ from the source’s (usually unobservable) ‘true’ angular position . Spatial variations of the light deflection induce image distortions, which can be quantified to leading order by the distortion matrix
which is conventionally decomposed into the convergence , the asymmetry , and the complex shear . A rotated version of the shear (which transforms like a spin-2 quantity) may be used to define its tangential component and cross component relative to a given direction :
where denotes denotes the polar angle of the vector , and .
In cosmological models with General Relativity as the theory of gravity, the convergence can be expressed to first order (and to a good approximation in those parts of the sky where lensing effects are weak) by a weighted projection of the matter density contrast along the line of sight:
with the source redshift-dependent geometric weight
Here, denotes the Hubble constant, denotes the cosmic mean matter density in units of the critical density, and denotes the speed of light. In these equations, we also introduce several convenient abbreviations: , , , , , and , where denotes the redshift and the comoving angular diameter distance for sources at comoving line-of-sight distance . Furthermore, denotes the relative matter overdensity at comoving transverse position555 Note that we use angular coordinates for some quantities such as the convergence , but comoving transverse coordinates for others such as the matter density contrast . , comoving line-of-sight distance , and cosmic epoch expressed by the redshift (which is tied to the line-of-sight distance for quantities at spacetime points on the observers’s backward lightcone).
The asymmetry vanishes in the first-order lensing approximation, and we will ignore it in the subsequent analysis. Furthermore, the shear field and the convergence field obey a one-to-one relation, which becomes a simple phase factor in flat harmonic space (i.e. 2D Fourier space) in the flat-sky approximation:
The shear can then also be expressed by a weighted projection:
where the matter shear contrast is related to the overdensity in two-dimensional transverse harmonic space via
The complex matter shear contrast can also be decomposed into a tangential and a cross component for a given direction on the sky:
In the following, we assume the validity of several further approximations related to weak gravitational lensing. We assume that the lens mapping for fixed source redshift is one-to-one and smooth. We also assume that on the scales of single galaxy images, the lens mapping is affine and the distortion matrix (1) is constant. Furthermore, we assume that the convergence and shear are small enough such that the reduced shear can be approximated by the shear, i.e. , and that .
2.2 Galaxy image ellipticities
The ellipticity of an observed galaxy image may be quantified with the help of moments of the observed image light distribution. For a given observed image brightness distribution , image center , and radial and flux weight function666 For simplicity, we restrict the discussion to simple radial weight functions. Weights used in practice often also depend on the local surface brightness (e.g. via a brightness threshold). , one may define the following second moments:
These can then be combined into the observed image ellipticity (Seitz & Schneider 1997)
For example, one obtains for a solid ellipse with axis ratio as image, when is chosen as the center of the ellipse and
Another common way to combine the second moments into an observed image ellipticity reads
This yields for an ellipse with axis ratio and Both ellipticity definitions contain the same information, and are related through (e.g. Bartelmann & Schneider 2001):
One may also define intrinsic image moments and intrinsic ellipticities and derived from the intrinsic light distribution one would observe in the absence of gravitational lensing:
where denotes the intrinsic center, and is the same weight function as used for the observed moments (9). The intrinsic moments can then be combined into the intrinsic image ellipticities
2.3 Shear estimators and image ellipticities
The relation between observed and intrinsic image ellipticity of a galaxy depends on the gravitational lensing distortion matrix (1) across the image, on the employed weight function , and also the employed ellipticity definition. In the case of a flat weight, , and the center of light as image center, Eqs. (9) and (13) yield ‘unweighted’ moments. The observed ellipticity (10) and the intrinsic ellipticity (14) are then related by:
An advantage of based on unweighted moments is that for , one may use the observed ellipticity as an unbiased estimator,
for the reduced shear , i.e. , where angular brackets denote averages over the galaxy intrinsic ellipticity distribution (which is assumed rotation invariant), regardless of the details of the intrinsic ellipticity distribution (Seitz & Schneider 1997).
Unweighted moments and ellipticities derived from them are susceptible to image noise in the outer image regions [due to the factor in the integral in Eq. (9)]. Moreover, unweighted moments and ellipticities are strongly affected by errors in the separation of the galaxy light from the light of neighboring galaxies. Ellipticity estimation from real galaxy images thus often employs down-weighting of the outer regions of the image. The drawback of such radially weighted moments is that they complicate the relation between observed and intrinsic ellipticities. For a general choice of ellipticity definition and weight function , and sufficiently small reduced shear , the observed ellipticity as a function of reduced shear can be approximated by a linear response in :
where the intrinsic ellipticity , and the shear polarizability
If can be estimated with sufficient accuracy (e.g. from the observed galaxy image itself) and is invertible, one may obtain an unbiased shear estimator by applying as a shear polarizability correction factor:
If the average shear polarizability for the observed galaxy population is known, another unbiased shear estimator can be constructed that employs as polarisability correction:
The above image ellipticity-based shear estimators can be treated in a uniform manner,777 For simplicity, we assume and .
with appropriate definitions for the observed ellipticity and intrinsic ellipticity . For example, Eq. (17) suggests
with and from unweighted moments. For computed from unweighted moments, one obtains (e.g. Bernstein & Jarvis 2002):888 This equation usually does not hold when is computed from moments for a non-uniform radial weight function.
A simplified version of the estimator by Kaiser et al. (1995, KSB) yields:
with and from moments with a Gaussian radial weight function, and estimated from the individual observed galaxy images.
As for the shear, the ellipticities and can be decomposed into tangential and cross components with respect to a given angular direction or comoving transverse direction :
with , , , and .
2.4 Density and intrinsic shape correlations
The intrinsic shapes of galaxies may have a preferred orientation towards nearby other galaxies and matter overdensities. The preferred orientation of galaxies towards matter overdensities may be quantified by the correlation999 We employ the symbol to denote two-point correlation functions of spacetime fields to avoid potential confusion with angular correlations of fields on the sky denoted with the symbol .
Here, denotes the expectation for a statistical ensemble of realizations of the galaxy population and matter density fields for a given cosmological model, and denotes the tangential component of the intrinsic ellipticity of a galaxy at spacetime position relative to the transverse direction . The first argument of indicates that due to statistical isotropy assumed here, the correlation function depends on the transverse separation only through its magnitude but not its direction. To be general, we also consider the possibility of different redshifts and for the two fields entering the correlations.
One can observe galaxy ellipticities only at positions where there is a galaxy to measure a shape from. Thus, correlation functions of practical relevance feature only in conjunction with a factor , where denotes the relative galaxy number overdensity of the galaxy population with shape information, e.g.
A related correlation substitutes the matter density contrast in Eq. (28) by the number density contrast of a suitable density tracer population such as a galaxy population with sufficiently known galaxy bias:
In case the density tracers follow a simple linear deterministic local bias model, i.e. , the above correlations obey
Correlations between the shapes of galaxies may be quantified by
|and linear combinations of these:|
where denotes the cross component of relative to the transverse direction .
Analogous (cross)correlations may be defined for the intrinsic ellipticities of two different galaxy populations and between intrinsic galaxy ellipticities and the shear density contrast. Of particular relevance for intrinsic alignments in cosmic shear (see Section 2.6) are the correlations and .
When denoting correlations of quantities defined as functions of redshift and angular position (instead of redshift and comoving position), we follow the above pattern. For example, correlations of observed ellipticities read
where denotes the tangential/cross component of the observed ellipticity of a galaxy at sky position and redshift relative to the angular direction on the sky.
Expectation values for the ellipticity correlation estimators discussed in this work involve correlations of projected density and ellipticity fields. Such correlations can often be expressed in Limber-type approximations (Limber 1953) that feature projected correlation functions (see Appendix A). For the above correlations, one may define corresponding projected correlations by:101010In certain cases (e.g. when the correlations do not decrease sufficiently fast with increasing line-of-sight separation), one may wish to include a non-uniform l.o.s. weighting function in the projection integral.
2.5 Observed density-ellipticity correlations
Important information on the intrinsic alignment of galaxies may be obtained by estimating the correlation between the number density of a galaxy ‘density sample’ (used to trace the overall matter density) and the observed ellipticities of a galaxy ‘shape sample’. Consider galaxies with observed sky positions in a survey with solid angle , and redshifts drawn from an underlying redshift distribution serving as density sample, and galaxies with observed ellipticities , observed positions and redshift distribution serving as shape sample. The projected galaxy density-ellipticity correlation as a function of comoving transverse separation may be then estimated by (e.g. Mandelbaum et al. 2006a):
Here, and denote weights, denotes the tangential component of the observed ellipticity of galaxy in the shape sample relative to the direction towards galaxy of the density sample, and the and denote positions obtained by randomly distributing the ellipticity/density sample galaxy positions within the survey area. The bin window function
where denotes the bin width (which we assume small compared to scales on which correlations change noticeably).
For these contributions (see Appendix B for a derivation),
denotes the effective area for a bin at radius and density sample galaxies at redshift , and
denotes the source redshift weighted geometric weight.
If the density or shape sample’s redshift distribution varies little over the range where the correlation between tracer galaxy density and intrinsic ellipticity is markedly different from zero to permit a Limber-type approximation for the dI term, one obtains:
The estimator (34) is very similar to estimators commonly used in galaxy-galaxy lensing. A notable difference is the normalization (36), which sums over random positions instead of the actual galaxy positions. This feature makes the normalization insensitive to correlations between the density and shape sample’s galaxy densities.
2.6 Observed ellipticity correlations
Measured correlations of observed galaxy image ellipticities are used in gravitational lensing studies to obtain constraints on the spatial correlations of the gravitational shear. However, the observed ellipticity correlation may also contain contributions from intrinsic shape correlations. As part of most weak lensing survey analyses, the ellipticity correlation between two (possibly identical) sets of galaxies is estimated as a function of image separation . Assume each set contains galaxies with observed ellipticities , observed angular positions inside a survey field with area , and known probability distribution for their redshifts . A common estimator reads
Here, the denote statistical weights. Furthermore, denotes the tangential/cross component of the observed ellipticity of galaxy relative to the direction towards galaxy .
Assuming Eq. (22) holds, the expectation can then be separated into four terms:
The gravitational shear-shear (GG), gravitational shear-intrinsic ellipticity (GI), intrinsic ellipticity-gravitational shear (IG), and intrinsic ellipticity-intrinsic ellipticity (II) contributions are given by (see Appendix C for a derivation):
Here, denotes the overdensity of the galaxy density field underlying the distribution of the galaxies . The general geometric factor is given by Eq. (4). The effective geometric factor for the source galaxy sample is given by Eq. (43) with replacing .
For redshift distributions sufficiently broad to permit a Limber-type approximation, the normalization and the II term can also be expressed in terms of projected correlations:
If density cross-correlations can be neglected, e.g. when the two sets of galaxies are well separated in redshift, the normalization and GG term reduce to:
The above expressions simplify considerably further if all correlations with galaxy overdensities as factors are neglected (see Appendix C). Although it has been pointed out (e.g. by Hirata & Seljak 2004; Valageas 2014; Blazek et al. 2015) that these galaxy density correlations are important for the understanding of observed ellipticity correlations, such correlations are often not explicitly taken into account in works on intrinsic alignment (Joachimi et al. 2015; Kirk et al. 2015; Kiessling et al. 2015).111111 In various studies, however, , , etc. are measured even though their notation may suggest , etc.
2.7 More on notation
Our naming scheme for correlations differs from commonly used notations in the literature on intrinsic alignments. A systematic notation such as ours facilitates the discussion of intrinsic alignments and observed ellipticity correlations. Translation between the notations is straightforward in many cases. For example, the correlation called in several works (e.g. in Joachimi et al. 2015) becomes in our notation.
Our full naming scheme is somewhat lengthy in some cases (in particular, when used as labels in plots). For correlations involving the matter density field (m), the matter shear field (G), a galaxy density sample (d), or a galaxy shape sample (I), we introduce shorter names: