THE IA GUIDE: A BREAKDOWN OF INTRINSIC ALIGNMENT FORMALISMS

We summarize common notations and concepts in the field of Intrinsic Alignments (IA). IA refers to physical correlations involving galaxy shapes, galaxy spins, and the underlying cosmic web. Its characterization is an important aspect of modern cosmology, particularly in weak lensing analyses. This resource is both a reference for those already familiar with IA and designed to introduce someone to the field by drawing from various studies and presenting a collection of IA formalisms, estimators, modeling approaches, alternative notations, and useful references


INTRODUCTION
This resource is a condensed overview of quantities relevant for describing the intrinsic alignment (IA) of galaxies.For scientists new to the field, it is a useful starting place that contains a broad introduction to IA and helpful references with more details and derivations.It is also structured to be a quick reference for those already familiar with IA.This is not a review article and not necessarily intended to be read beginning-to-end.Sections 2-7 each contain common formalisms of an IA estimator, brief pedagogical explanations with practical advice, alternative notations, and useful references.Sections 8 and 9 summarize IA modeling and applications.Terms in teal are hyperlinked to glossary entries at the end of the document.
IA refers to correlations between galaxy shapes and between galaxy shapes and the underlying dark matter distribution (qualitatively illustrated in Figure 1).These arise naturally within our current understanding of galaxy formation, as confirmed by hydrodynamic simulations (Kiessling et al. 2015;Bhowmick et al. 2020;Samuroff et al. 2021).In the case of elliptical galaxies, shapes are elongated along the external gravitational field (Croft & Metzler 2000;Catelan et al. 2001).The shapes of spiral galaxies are typically associated with their angular momentum, which arises from the torque produced by the external gravitational field (Heavens et al. 2000;Catelan et al. 2001;Codis et al. 2015).The alignments of spiral galaxy shapes are much weaker than for ellipticals and so far have not been directly observed (Zjupa et al. 2020;Johnston et al. 2019;Samuroff et al. 2022).Some studies use galaxy spin instead of shapes to measure alignment (Lee 2011), though here we focus on shapes as they can be more directly connected to cosmic shear and are most commonly used in observational studies.Lee & Erdogdu (2007) provides a pedagogical overview of the physics and formalisms of spin alignments.
While IA can be used as a cosmological probe, historically it is most often studied as a contaminant of weak lensing.As light travels to us from distant galaxies, it is bent by the gravitational field of the large-scale structure of the Universe and thus we observe distorted galaxy images.When this effect is too small to be detected for individual galaxies, we say that we are in the weak lensing regime.The resulting shear in these galaxy shapes is known as cosmic shear, and is a primary tool used to probe cosmological parameters (Refregier 2003;Heymans et al. 2012;Hildebrandt et al. 2017;Hikage et al. 2019;Abbott et al. 2023).The correlations between observed galaxy shapes used to measure weak lensing are difficult to separate from those that arise from IA.Studies show that IA can account for a 30% error on the matter power spectrum amplitude as measured by cosmic shear (Hirata et al. 2007), making IA one of the most significant sources of systematic errors in weak lensing measurements.

Additional note: galaxy types
Throughout this guide, we refer to galaxies as "early-type" or "late-type", "blue" or "red", and "elliptical" or "spiral", depending on the reference we are following.Early-type galaxies are usually elliptical or lenticular and tend to be redder in color.Late-type galaxies are spiral and typically blue.The terminology "early" and "late" does not refer to the age of the galaxy, but to their ordering in the Hubble Sequence (also known as "The Tuning Fork") when Hubble initially thought that ellipticals evolve into spirals (Hubble 1926).Although sometimes used interchangeably, it is important to keep in mind that populations may be defined differently in different analyses.

Reviews
Here is a list of available reviews and primers on IA.A Zotero group of key IA papers can be found below1 .
• "The intrinsic alignment of galaxies and its impact on weak gravitational lensing in an era of precision cosmology " Troxel & Ishak (2015) The first comprehensive review on intrinsic alignments, presenting extensive documentation of commonly used formalisms and the role of IA in precision cosmology.
• "Galaxy alignments: An overview" Joachimi et al. (2015) Broad synopsis of IA, including physical motivations, a historical overview, and main trends.
• "Galaxy alignments: Observations and impact on cosmology" Kirk et al. (2015) Descriptions of formalisms for measuring shapes and IA tracers, an overview of IA observations, and discussion of cosmological impacts and mitigation.
• "Galaxy alignments: Theory, modelling and simulations" Kiessling et al. (2015) Detailed overview of common models and IA in N -body and hydrodynamic simulations.

ELLIPTICITY
IA studies model simulated, three-dimensional (3D) galaxy shapes as triaxial ellipsoids, and observed galaxies as their projected shape on the sky: two-dimensional (2D) ellipses.This 2D ellipticity is quantified in terms of the lengths of the major and minor axes of the ellipse (a and b, respectively, with b ≤ a) and the orientation angle of the major axis, θ, with respect to an arbitrary reference axis, as shown in Figure 2.For the detection of IA, ellipticity is typically measured relative to directions tracing the tidal field (e.g., positions of galaxy overdensities in real data, or reconstructed tidal fields in simulations), often as a function of transverse separation r p .By convention, the alignment signal is highest for very elongated shapes (larger axis ratio) that point along the direction of the tidal field.
While the formalisms below are standard, the methods of fitting shapes to observations vary across surveys and can impact the resulting IA signal.The signal is also correlated with the clustering of the galaxy sample and depends on how far along the Line of Sight (LOS) the measurement is averaged over.Therefore, it is more common to use IA correlation functions rather than a relative ellipticity, although ellipticity is a component of most estimators.

Ellipticity: 2D Formalism
There are two different ways in which the ellipticity of 2D shapes is commonly quantified.We will refer to these as ε and χ and to ellipticity generically as ϵ.In the rest of the document, ϵ can be taken to stand for either of the two ellipticity definitions.These are defined as: Both quantities are often referred to as the ellipticity, but χ is also known as the distortion (Mandelbaum et al. 2014) or the normalized polarization (Viola et al. 2014).The definition ε (but usually denoted ϵ) is often used in weak lensing studies because it is an unbiased estimator of the cosmic shear, γ.In contrast, χ must be adjusted by the responsivity R which quantifies the response of the ellipticity to an applied gravitational shear Bernstein & Jarvis (2002): (3) R = 1 − χ rms (rms is the root mean square) and is typically ≈ 0.9 depending on the galaxy sample (Singh & Mandelbaum 2016).In later parts of this guide, ϵ is used generally.It is assumed that where the χ definition is intended, it will have been adjusted for the responsivity so R is not explicitly shown.
Ellipticity is a complex quantity that can be broken up into its real and imaginary components, ϵ 1 and ϵ 2 : where The factor of 2 arises because ellipticity is a spin-2 quantity, which means that it is invariant under rotations of integer -Visualization of the real and imaginary components of ellipticity, as described in Section 2. These functionally contain the same information.ϵ 1 is maximum when a shape is highly elongated and exactly aligned with the angle that the ellipticity is defined relative to, most commonly North.ϵ 2 is maximum when the shape is aligned with π/4 away from the principal angle.multiples of π.See Figure 3.The angle θ is usually defined as East of North.Its range can be 0 − π or ± π 2 .ϵ 1 represents the orientation of an ellipse relative to the direction where θ = 0 and ϵ 2 represents the orientation relative to the direction where θ = π 4 .Note that ϵ 1 and ϵ 2 contain the same information (Figure 3).When measured relative to another galaxy or the direction of the tidal field, they are usually denoted as ϵ + and ϵ × , with the subscripts respectfully read as "plus" and "cross".ϵ + > 0 indicates an alignment along the tidal field direction, and ϵ + < 0 indicates a tangential orientation as seen in gravitational shear (Section 3).ϵ × is equivalent to ϵ 2 .On average, ϵ × is 0 for galaxies in an isotropic Universe.

Ellipticity from particles
Simulated galaxies and halos are usually modeled as triaxial ellipsoids composed of "particles".To summarize the shape of these objects, it is common to use the inertia tensor, I, which is computed by summing over the positions i, j ∈ (1, 2, 3) of N particles.To better approximate the object's position and shape, this sum is usually weighted by particle mass or luminosity (when applicable).For particles k with weights w k that sum to W , this form of the moment of inertia tensor is (Samuroff et al. 2021) It is also common to weight I ij by distance from the center of the object.This center-weighting produces the reduced inertia tensor (Chisari et al. 2015), r k is the distance between the particle k to the object's center of mass.The reduced inertia tensor can provide a better approximation of the shape of the object at its center (Joachimi et al. 2013).Equation 8is known to create a coherent bias in shapes, since the r weighting kernel is inherently round.To avoid this limitation, studies often calculate the inertia tensor on particles within a fixed volume, while iteratively rescaling the axis lengths until they converge (Schneider et al. 2012;Mandelbaum et al. 2015).
An ellipsoid can be constructed using the eigenvectors and eigenvalues of I ij , which can then be projected along the z-axis into its 2D second moments Q 11 , Q 22 , Q 12 + Q 21 (Eq.5.37 of Bartelmann & Schneider (2001)).The projected ellipticity ϵ is obtained via where Q ≡ Q ij with i, j ∈ (1, 2).Note that this assumes that ellipticity is defined by ε from Section 2.1.If the ellipticity is instead defined as χ (distortion or normalized polarization), the denominator is simply Q 11 + Q 22 -see Section 2.1 and Mandelbaum et al. (2014).
Projected Ellipticity from angular momentum Late-type galaxies are typically modeled as circular discs and their ellipticity is often assumed to be aligned with their angular momentum (tidal torquing or spin alignment), L = L x , L y , L ∥ τ .τ denotes the transpose vector(Figure 4).
To obtain the projected shape of a spiral galaxy along the LOS, or L ∥ , the orientation angle θ is given by: and the axis ratio: r edge-on describes the ratio of the disc thickness to disc diameter, which is approximately equivalent to the axis ratio for a galaxy viewed edge-on; this contribution is expected to be significant for galaxies with bulges (Joachimi et al. 2013).Assuming linear tidal torquing, a halo's spin is written as (Lee & Pen 2008) where i, j, k = 1, 2, 3 in the three spatial directions, ϵ ijk the Levi-Civita symbol and T jl the gravitational tidal shear.The latter, which is a symmetric tensor, is defined as (e.g.Blazek et al. 2011) where Φ is the gravitational potential, x i,j represents comoving Cartesian coordinates, and the indices {i, j} = {1, 2, 3} indicate the three spatial directions.Following Eq. ( 12), tidal torquing leads to quadratic alignments of galaxy shapes with the tidal shear.14) and Eq. ( 15), whereas ϕ 1 is the angle from the North Pole.

Ellipticity from tidal field
Early-type galaxies are considered to be triaxial ellipsoids whose axes align with the underlying gravitational tidal shear, T ij (Catelan et al. 2001).In order to derive the predicted galaxy ellipticities given T ij , we can project the 3D tidal field along two axes at the location of each galaxy.The convention is to project along the galaxy's North Pole distance, ϕ 1 , and right ascension, ϕ 2 (see Figure 5).The latter is the angle complementary to declination.In this setting, ϵ 1 > 0 corresponds to east-west elongation and ϵ 2 > 0 corresponds to northeast-southwest elongation.We first consider a Cartesian orthonormal basis, (x, ŷ, ẑ), at the location of each galaxy.We then rotate this basis into (n, φ1 , φ2 ), such that n is parallel to the LOS to the galaxy.These two bases are related by The next step is to intermediately define the linear combinations, Having defined the above fields, we can decompose the 3D tidal shear into the rotated basis, as (Schmidt & Jeong 2012) where T ± are the 2D ellipticities, such that (Tsaprazi et al. 2022) in the nonlinear alignment model, for example, described in Section 8.2.Here C 1 is the IA amplitude (introduced in 8.1) and (ϵ 1 , ϵ 2 ) the observed galaxy ellipticities.

Ellipticity: Additional Notations
• ϵ: frequently used rather than ε and sometimes instead of χ.
• e: sometimes used equivalently to ϵ or χ.
• ϵ T or ϵ t : the tangential component of ellipticity, equivalent to ϵ + .
• ϕ: sometimes used for the orientation angle.
• a, b: sometimes denote the length of the semi-axes of the ellipse, rather than the full axes lengths.
• q: the ratio of the ellipse axes, b/a, with b ≤ a.

Ellipticity: References
• "On the intrinsic shape of elliptical galaxies" Binggeli (1980) Historical discussion of the true and projected shapes of elliptical galaxies.
• "Means of confusion: How pixel noise affects shear estimates for weak gravitational lensing" Melchior & Viola (2012) Appendix A discusses the pros and cons of different ellipticity definitions.
• "Intrinsic alignments of galaxies in the Illustris simulation " Hilbert et al. (2017) Additional notation and definitions for 3D shapes.
• "The mass dependence of dark matter halo alignments with large-scale structure" Piras et al. (2018) Derivations for the 3D shapes of galaxies from the moment of inertia tensor.
• "Galaxy shape statistics in the effective field theory" Vlah et al. (2021) Formalism for projecting 3D shapes in forms that are convenient for numerical implementation.
• "Intrinsic alignment as an RSD contaminant", Lamman et al. (2023) Appendix A contains condensed relations for flat projection of a triaxial shape.
Fig. 6.-A diagram of a gravitational lensing system.A mass concentrated at an angular diameter distance D L from the observer lenses the light from a source located at distance D S .In the thin-lens approximation, the true angular position of the light source β with respect to the LOS is related to the observed angular position θ through the lens or ray-tracing equation, β = θ − α(θ), where α is the reduced deflection angle measured by the observer.The angle α is the deflection angle measured at the lens.Note that angles are exaggerated to aid visualization.

SHEAR
IA is often measured as a contaminant of cosmic shear (Bernstein & Jarvis 2002;Hirata & Seljak 2004) and is sometimes referred to as intrinsic shear, γ I .The coherent distortion of galaxy light by foreground mass creates a tangential shear on the sky, γ lensing , often acting in opposition to the signal from tidal alignment at large scales.As these two phenomena are difficult to distinguish observationally and are necessarily measured together, much of the IA formalism described in this and the following sections is from weak lensing.

Shear: Formalism
The observed shear signal is the combination of the intrinsic component of shapes and the component gravitationally lensed by foreground mass: This mathematical sum is an approximation for when the effects are small, as in weak lensing; see the note about shear addition below.Shear quantifies the shape of galaxies and thus is related to the ellipticity described in Section 2.1.
Lensing is often defined via the critical mean density, Σ crit , which is the maximum surface density before the light of a source is split into multiple images by a foreground mass.Σ crit is a function of fundamental constants and the radial separations involved in the lensing system, as illustrated in Figure 6.
For instance, while cosmic shear is a direct measurement of the large-scale structure between the source and observer, galaxy-galaxy lensing is the cross-correlation between source galaxies and biased tracers of the underlying matter (Sheldon et al. 2004;Heymans et al. 2021;Prat et al. 2022).It is the difference between the average surface density of galaxies within some projected separation Σ(< r p ), and the surface density at separation Σ(r p ), as a fraction of Σ crit : Variable definitions • γ: total shear, as described in Section 4.
• γ I : intrinsic shear due to tidal alignment.
• Σ: surface overdensity, projected to the plane of the sky, i.e., the mass density integrated along the LOS.
• D S : radial distance between observer and light source.
• D L : radial distance between observer and gravitational lens.
• D LS : radial distance between light source and lens.
• c: speed of light.
• γ IA : alternative notation for γ I , intrinsic shear due to tidal alignment.

Additional note: shear addition
Since shear matrices can be asymmetrical, they do not form a group under matrix multiplication, and therefore shear terms are not commutative under matrix addition.Instead, we can define a group with an addition operation, where S γ is the shear matrix, and R is the unique rotation matrix that allows Sγ 3 to be symmetric.For weak lensing in general, the shears are assumed to be small, such that using the mathematical addition is a valid approximation.However, this is not always true for IA.We refer the reader to Miralda-Escude (1991) and Bernstein & Jarvis (2002) for the derivation of the appropriate addition formalism.

Additional note: convergence
Weak lensing has two effects on observed galaxies: shear, γ, and convergence, κ, shown in Figure 7.The shear is trace-free and characterizes the anisotropic stretching of the galaxy's source image by quantifying the projection of the gravitational tidal field.Convergence measures the surface mass density and is an isotropic distortion, describing the change in the size of the lensed galaxy while maintaining a constant surface brightness.
When galaxy shapes are measured, we actually measure the reduced shear, since we measure shapes, not sizes.This is an invariant quantity that introduces a mass-sheet degeneracy (Schneider & Er 2008).Since κ ≪ 1 in the weak lensing regime, the shear is a good approximation of the reduced shear.Note that g in the above equation is not to be confused with the variable g that is used to describe galaxy positions, nor with the overdensity.Both shear and convergence contribute to the magnification, µ, which is defined as the ratio of lensed to unlensed flux: In the weak lensing regime, µ ≈ 1 + 2κ.Magnification impacts both the apparent position of galaxies and the distribution of light received from a single galaxy: in areas with positive (negative) convergence, the apparent distance between any two objects on a source plane is increased (decreased), and the captured fraction of the solid angle of light emitted from a source is amplified (reduced).Magnification thus affects the observed area number density and the selection probability of individual galaxies, thereby impacting the observed number density of objects in large-scale structure surveys.

Shear: References
• "A method for weak lensing observations" Kaiser et al. (1995) Describes the motivation for measuring shear and how to model the shear response.
• "Cosmology with cosmic shear observations: a review" Kilbinger ( 2015) More recent review of all aspects of cosmic shear.

IA CORRELATION FUNCTION NOTATION
The observed shape-density and shape-shape correlations are the result of several combinations of effects, including physical galaxy correlations and lensing.This section describes the notation most commonly used to denote these effects, while Section 5 describes methods to quantify their correlations.Here, two background galaxies are lensed by a foreground dark matter halo.The observed shape ϵ of a galaxy is a combination of its intrinsic (I) and lensed (G) components.These are correlated with the underlying dark matter density, which can be traced by galaxy shapes and positions (g).Correlations are represented by combining these notations.For example, gI is the correlation between galaxy positions and intrinsic galaxy shapes, and GI is the correlation between the lensed component of galaxy shapes and intrinsic galaxy shapes.GI correlations are important because overdensities cause IA between galaxies at the same redshift while also lensing more distant galaxies.Separating galaxies into widely-spaced tomographic bins can distinguish between the GI and II terms (Section 4.2, Figure 9).For other helpful diagrams of IA correlations, see Figure 1.6 of Fortuna (2021) and Figure 6 of Troxel & Ishak (2015).Note that this is a cartoon; galaxy shapes, orientations, and positions are only symbolic.These correlations are never measured for individual galaxies and are typically only statistically significant when measured for at least 10 4 objects.

Correlations: Formalism
The main correlated quantities relevant to IA are the intrinsic shape of galaxies, the component of the shape that is gravitationally lensed, and the position of galaxies (Figure 8).For observed correlations that are measured as a function of sky separation, these are most commonly notated as • I: intrinsic galaxy shape.
• G: lensed component of shape, also referred to as "extrinsic" shape.
• g: galaxy position (used as a tracer).
The observed shape-shape correlation ⟨ϵ i ϵ j ⟩ between galaxies in two radial bins, i and j, is the sum of every G and I combination: These effects do not necessarily sum mathematically (see the note above on shear addition).The notation ⟨X i Y j ⟩ indicates the correlation of a quantity X of a sample i relative to quantity Y of sample j (Section 5.1).For example, ⟨G i I j ⟩ is the correlation between gravitational shear of one sample relative to the intrinsic shapes of another sample.
For binning along the LOS (see Section 4.2 on tomography), if i is in a closer radial bin than j then ⟨G i I j ⟩ will be 0 for non-overlapping redshift bins since lensed shapes are created by a foreground mass.Therefore, either the second or the third term of Eq. ( 26) will be equal to zero.The first term of Eq. ( 26) is the shear correlation.This contains the majority of cosmological information, but cannot be directly measured because the observed signal includes the IA terms.The last term is the correlation between intrinsic ellipticities.Similarly, the observed galaxy shape-density correlation ⟨ϵ i n j ⟩ contains contributions from cross terms between lensed shape and density, and intrinsic shape and density, The observed galaxy number density is the sum of the intrinsic number density g and a lensing magnification component m due to foreground overdensities (see the note above on convergence).This expression can then be written as where ⟨G i g j ⟩ is usually called the galaxy-galaxy lensing signal.See Joachimi & Bridle (2010) for a cosmological analysis using a joint treatment of galaxy ellipticity, galaxy number density, and their cross-correlations.

Correlations: Additional Notations
• +: As defined in Section 2.1, the component of shape relative to the direction of the tidal field.
• ×: As defined in Section 2.1, the component of shape relative to π 4 -off the direction of the tidal field.
• γ: gravitational shear, or the correlation between lensed shapes.Also sometimes used as the total observed shape.⟨γγ⟩ is equivalent to GG.
• δ m : fractional density of matter, as opposed to overdensity traced by galaxies.Sometimes notated as just δ.
• m: sometimes used to refer to magnification from lensing, see Section 3.2.
• η e : correlation between the 3D shape of a galaxy and the position of another galaxy (Tenneti et al. 2015;Chisari et al. 2015).This is different from the lensing parameter in Section 3.
• η s : correlation between the spin direction of a galaxy and position of another galaxy (Chisari et al. 2015).
Additional note: tomography Tomography in the context of cosmological correlations refers to the technique where the redshift distribution of galaxies, n(z), is sliced into redshift bins, also referred to as tomographic bins (Figure 9).This method allows the extraction of additional information from correlations within the galaxy sample that would otherwise be projected out, for example the growth of structure as a function of time.Binning can be done so that bins are equally separated in redshift, so that there is an equal number of galaxies in each bin or some combination of the two.If correlations are measured inside one bin, they are called "auto-correlations". Correlations between galaxies in different tomographic bins are referred to as "cross-correlations".Both spectroscopic and photometric surveys utilize the tomographic technique.In spectroscopic surveys, it is possible to avoid the overlapping of redshift distributions between bins.This is not possible in photometric surveys because the redshifts are not known precisely or accurately enough.For more information on the rationale for tomography, see Hu (1999).
• "Intrinsic alignments of galaxies in the Horizon-AGN cosmological hydrodynamical simulation" Chisari et al. (2015) Defines formalisms used in correlations of 3D shapes and spins.
• "Intrinsic alignments of disc and elliptical galaxies in the MassiveBlack-II and Illustris simulations" Tenneti et al. (2016) Defines additional formalisms used in 3D correlations.

IA CORRELATION FUNCTION ESTIMATORS
In real or configuration space, correlations between the components described in Section 4 can be quantified using the correlation function ξ.This is often measured as a function of the separation of pairs along the LOS (Π) and in the transverse direction (r p for physical separation, ϑ for angular separation on the sky).Since we are dealing with pairs, these functions are called two-point correlation functions (2PCF).As opposed to the power spectra estimators described in the next sections, these correlation functions are less sensitive to survey geometry.In practice, most observations measure the projection of ξ along the LOS w p .While these projected quantities contain less information than the full 3D correlations, they are more straightforward to observe and model, particularly when used in weak lensing analyses.

IA Correlation Function: Formalism
The IA correlation function is most commonly measured using a generalized form of the Landy-Szalay (LS) estimator, which was devised to estimate galaxy clustering (Landy & Szalay 1993).This estimator accounts for systematics and has less variance than other estimators since it uses overdensity instead of density.To include information about alignment, the counts of galaxy pairs are weighted by the degree to which components are correlated (Mandelbaum et al. 2006).
The count of galaxy pairs between sample A and B is notated as AB.The count weighted by the correlation between shapes in A and the positions of B is A + B: The count weighted by the correlation between shapes in A and shapes in B is A + B + : Here, ϵ + (j|i) represents the + component of the ellipticity of galaxy j relative to the vector between it and the galaxy i (Section 2).As described in Section 2.1, 0 < ϵ + < 1 indicates radial alignment and −1 < ϵ + < 0 indicates tangential alignment.Consider a galaxy shape catalog, S, and a galaxy position catalog, D (D for density tracers).These can be from the same sample.S + and S × represent the relative + and × component of shapes, as described in Eqs. 29, 30.R D and R S are randoms for the respective datasets: generated data designed to match the survey geometry but with no correlations from large-scale clustering.The correlation functions are Note that ⟨S + ⟩ = ⟨S × ⟩ = 0, which allows us to disregard the contribution of S + R + and S × R × to first order due to systematic effects.
For observations, the angular separation of pairs on the sky ϑ is often used in place of r p .This is done when the physical transverse separation of galaxies cannot be easily determined, as in photometric surveys, or when IA is being measured as a contaminant of an angular signal.

Projected Correlation Function
Since the LOS distance is difficult to determine in photometric surveys and in the presence of redshift-space distortions (RSD), most observations measure these correlation functions integrated along the LOS Π.The projected correlation function between two properties a and b is In practice, this requires a choice of how far along the LOS to integrate (Π max ) and in what size bins (dΠ).Occasionally the projected correlation function is integrated from 0 − Π max instead of ±Π max , in which case ξ should be multiplied by 2.

3D IA correlation function
The IA correlation function can be measured in 3D space, such as ξ g+ (r p , Π) and ξ ++ (r p , Π), or the equivalent for the × component.It can also be quantified through the monopole and quadrupole components of IA correlation as a function of the redshift-space separation, s, ξ g+ (s) or ξ ++ (s) (Singh & Mandelbaum 2016;Okumura & Taruya 2023).While more complicated to measure and model, these estimators can extract significantly more information from a survey (Singh et al. 2023).

Variable definitions
• ξ: usually used to denote a correlation function.
• D: positions of sample used as a density tracer.
• S: positions of sample used for shape measurements.
• R D : positions of random points made to match the spatial distribution of galaxy sample D but with no spatial correlations.
• R S : positions of random points corresponding to S.
• S + : shapes of galaxies in shape sample.
• r p : distance transverse to the LOS.

IA Correlation Function: Additional Notations
• Alternative estimators similar to the LS estimators, are, for example (Joachimi et al. 2011), The advantages of using the LS estimator are that it is less affected by survey geometry and has a higher signal-to-noise ratio (Singh et al. 2015).
• The projected correlation function is also notated as w p .
• Here, we have used ϑ to notate angular separation on the sky to distinguish it from the galaxy orientation angle.Many studies instead use θ.

IA Correlation Function: References
• "The correlation function of galaxy ellipticities produced by gravitational lensing" Miralda-Escude (1991) Original formalism for combining shape information in correlation functions.
• "The WiggleZ Dark Energy Survey: Direct constraints on blue galaxy intrinsic alignments at intermediate redshifts" Mandelbaum et al. (2011) Definitions of the generalized LS IA estimators and helpful descriptions.
• "Galaxy Alignments: Observations and Impact on Cosmology" Kirk et al. (2015) Contains a pedagogical, in-depth discussion of different IA correlation function estimators.
• "Intrinsic alignments of SDSS-III BOSS LOWZ sample galaxies" Singh et al. (2015) and "Intrinsic Alignments of BOSS LOWZ galaxies II: Impact of shape measurement methods" Singh & Mandelbaum (2016) Contain condensed definitions of IA correlation functions with the most commonly used notation.
Additional note: estimators derived from shear correlation functions Rather than using shear correlation functions directly, other estimators derived from them may be preferred, for example, to obtain higher signal-to-noise ratios or to control the physical scales used.In weak lensing studies, derived estimators are often chosen because they separate E-modes from B-modes.Lensing does not produce B-modes so their detection in cosmic shear data could mean that the signal is affected by systematic effects such as IA.
The references below introduce three common derived estimators: complete orthogonal sets of E/B integrals (COSE-BIs), aperture mass statistics, and band powers, and include their use for IA studies.
• "Analysis of two-point statistics of cosmic shear" Schneider et al. (2002) Illustrates the use of aperture mass statistics and band powers.
• "Sources of contamination to weak lensing three-point statistics: Constraints from N-body simulations" Semboloni et al. (2008) Example of using aperture mass statistics to measure IA in simulations.
• "COSEBIs: Extracting the full E-/B -mode information from cosmic shear correlation functions" Schneider et al. (2010) Introduces COSEBIs as a method to separate E-and B-modes whilst retaining cosmological information.
• "KiDS-1000 cosmology: Cosmic shear constraints and comparison between two point statistics" Asgari et al. (2021) Discusses pros and cons of estimators.

3D IA POWER SPECTRUM
The power spectrum is perhaps the most basic statistic that can be used to describe density fields.By definition, the power spectrum is the mean square of the density fluctuation amplitudes.It is a function of wavenumber k (Fourier space) or a multipole ℓ (spherical harmonics space)2 and is the Fourier transform of the correlation function (Hikage et al. 2019).
The IA power spectrum described in this section quantifies the correlation between intrinsic galaxy shapes and the galaxy (or mass) density field in Fourier space.It is a 3D quantity that can be modeled directly or measured from simulations and is most often used to study IA directly.For a visualization of a typical IA power spectrum, see Figure 2 of Kurita et al. (2021).
We can also decompose the shear into its E-and B-modes, which are coordinate-independent quantities in Fourier space, where ϕ k is the angle measured from the first coordinate axis to k ⊥ ≡ (k 2 1 + k 2 2 ), the wave vector on the sky plane, so that ϕ k = arctan(k 1 /k 2 ).The E-mode P gE (k) and B-mode P gB (k) power spectra correspond to the real and imaginary part of P gγ , respectively.
Similarly, other types of IA power spectra are where δ m is the matter density field, and δ D is the Dirac delta function.Equivalently, we have, The B-mode power spectra, ⟨γ B γ B ⟩ = 0 for IA caused by the scalar tidal field in the linear regime.⟨gγ B ⟩ and ⟨γ E γ B ⟩ should also vanish due to the statistical parity invariance of the Universe.Note that, although E-mode and B-mode shear are defined in the 2D plane perpendicular to the LOS direction, the power spectra are functions of the 3D wave vector, k.In other words, they are functions of k = |k| and the direction of k.
• γ B (k) = −γ 1 (k) sin 2ϕ k + γ 2 (k) cos 2ϕ k , B-mode shear, that cannot be generated by the scalar field in the linear regime, used as a check for systematic errors.These γ E,B definitions correspond to equations 9-11 in Blazek et al. (2019).
• P EE (k): auto power spectrum of the E-mode shear field, i.e., the intrinsic shape correlation signal.
• P Eδ (k): cross power spectrum between the E-mode shear and the mass density field, i.e., the GI signal.
Additional note: Alternative expression for 3D IA power spectra By definition, any of the above power spectra can be expressed as where * is the complex conjugate, P (k) is any one of the 3D IA power spectra and γ denotes the intrinsic shear (or one of its components such as γ E ).
Additional note: multipole moments Multipole moments of the power spectrum (Kurita et al. 2021) can be defined as where P XY (k, µ) is one of the power spectra defined in Eqs. ( 37), ( 39) or (40), µ is the cosine of the angle between k and the LOS direction, L ℓ is the Legendre polynomial.P (0) is the monopole component, P (2) is the quadrupole component, and P (4) is the hexadecapole component.Additional multipoles are 0 in the large-scale (linear) regime.
The angular-dependent quantities can be used to quantify information about anisotropic effects, such as RSD.

Additional note: relation with the correlation function
The power spectrum can be written as Recall that the correlation function is defined as where ϕ r is the position angle of r ≡ x − x ′ on the sky plane.Thus, the relation between the correlation function and the power spectrum can be written as For observations, this expression can also be written as the redshift-space correlation function where A, B ∈ {g, +, ×} (Singh & Mandelbaum 2016).β models the effect of RSD (Jackson 1972;Kaiser 1987).β + = 0 and β × = 0 to first order in the case of shear, and β g = f /b g in the case of galaxies, where f is the structure growth rate and b g the galaxy bias (Kaiser 1984).
6.2.3D IA Power Spectrum: References • "Power spectrum of halo intrinsic alignments in simulations" Kurita et al. (2021) Formalism and first measurement in N-body simulations.
• "Power spectrum of intrinsic alignments of galaxies in IllustrisTNG" Shi et al. ( 2021) IA power spectrum of galaxies in IllustrisTNG.
• "Analysis method for 3D power spectrum of projected tensor fields with fast estimator and window convolution modeling: An application to intrinsic alignments" Kurita & Takada (2022) Methodology for measuring the IA power spectrum in observations.
• "Three-point intrinsic alignments of dark matter haloes in the IllustrisTNG simulation" Pyne et al. ( 2022) Extends the formalism to three-point statistics.

2D IA POWER SPECTRUM
In weak lensing analysis, observed galaxies are normally allocated to redshift bins, so the relevant IA power spectrum is 2D4 .It can be defined in Fourier space (P (k)) or in spherical harmonic space (the angular power spectrum C(ℓ), or simply C ℓ ).In this section, we focus on the latter.Note that the terminology is often used quite loosely and both P (k) and C(ℓ) may be referred to as the shear power spectrum.
In terms of the multipole moments of the 3D IA power spectrum, discussed in Section 6, the IA power spectrum relevant to cosmic shear analyses is the one evaluated at µ = 0, i.e., P δE (k, µ = 0) Power spectra can be expressed in terms of the corresponding correlation functions.In the case of IA, there is a family of correlation functions to choose from (see Kiessling et al. (2015) and Section 5).An example is the galaxy position-ellipticity correlation function w g+ (r p ): Here P δI (k ⊥ , z) corresponds to P δE (k ⊥ , z) in Section 6.The quantity b g is the galaxy bias, which is assumed here to be linear, scale-independent and not to depend on redshift.These assumptions are justifiable for quasi-linear scales where perturbation theory applies, and assuming that on these scales structure formation is entirely determined by gravity (Desjacques et al. 2018).At smaller scales, other considerations come into play and linear bias cannot safely be assumed (Kaiser 1984;Fry & Gaztanaga 1993).

Variable definitions
• r p : transverse separation, as described in Section 2.
• k ⊥ : wave vector perpendicular to the LOS.
• J 2 (k ⊥ r p ): a Bessel function of the first kind.
The 3D IA power spectrum can be projected along the LOS to construct the 2D projected angular IA power spectrum C ij ϵϵ (ℓ) as the sum of the gravitational lensing part GG, gravitational-intrinsic GI, and intrinsic-intrinsic contribution II: Each contribution in the above expression is described in spherical harmonic space where ℓ is the angular frequency.
Figure 10 shows the C ℓ s for each of these components.Indices (i, j) represent two redshift bins in which the correlation is taking place (see Section 4.2).Terms in Eq. ( 50) can be represented as: Section 8 discusses various methods used to model the power spectra P δI and P II .

Variable definitions
• χ is the radial comoving distance.
• χ H is the comoving distance to the horizon.
• f K (χ) is the radial function: with 1 √ |K| the curvature radius of the spatial part of spacetime.
• q i (χ) and q j (χ) are the lensing weighting functions in tomographic bins i and j, respectively: with H 0 the Hubble parameter, Ω m the matter density fraction, c the speed of light in a vacuum, and a(χ) the scale factor.
• W : The window function is often called a filter function or a survey window.As well as redshift z, it may be defined in terms of other survey parameters: position on the sky θ, etc.
• p i (χ) and p j (χ) are the redshift distributions in tomographic bins i and j, respectively, in radial comoving distance space: p i (χ) = p i (z) dz dχ and p j (χ) = p j (z) dz dχ .

Limber approximation
Power spectra on small scales (high multipoles ℓ) are computationally expensive to calculate, especially if they involve rapidly oscillating functions.In practical applications, the Limber approximation is often used (LoVerde & Afshordi 2008;Kitching et al. 2017).In this approximation, the integrand varies more slowly but the result is still accurate at small scales.
One way to understand the approximation is to write the lensing angular power spectrum, C ϕϕ ℓ , as (Lemos et al. 2017) where The Limber approximation replaces the spherical Bessel function j ℓ (kχ) with a delta function: with ν = ℓ + 1 2 = kχ.Then the shear power spectrum takes the form: The Limber approximation assumes that the variation of the kernels of the projected fluctuations is limited to scales larger than the average clustering length, which makes it invalid across all scales.This limitation, combined with the need to reduce systematic errors for future cosmological surveys, requires non-Limber methods.There have been recent efforts to move away from the Limber approximation in numerical analyses, such as in Abbott et al. (2023) and Leonard et al. (2023), which presents several alternatives to implement this calculation in a fast and reliable framework.
7.1.2D IA Power Spectra: Additional Notations • r and s: indices that denote two tomographic bins are usually i and j.Some authors use r and s instead (for example Lemos et al. (2017)).
• n r (χ) and n s (χ): often used in place of p i (χ) and p j (χ) to denote the redshift distributions in respective tomographic bins i and j.
7.2.2D IA Power Spectra: References • "Weak lensing power spectra for precision cosmology: Multiple-deflection, reduced shear and lensing bias corrections" Krause & Hirata (2010) Detailed derivation for the corrected weak lensing power spectra.
• "Cosmology from cosmic shear power spectra with Subaru Hyper Suprime-Cam first-year data" Hikage et al. (2019) Example of accounting for IA in power spectrum analysis of data from the Hyper Supreme-Cam (HSC).
• "KiDS-1000 cosmology: Cosmic shear constraints and comparison between two point statistics" Asgari et al. ( 2021) Short breakdown of the power spectra for the KiDS survey, containing a list of helpful references.
Additional note: beyond the power spectrum Power spectra are powerful probes of IA.However, 2-point statistics, such as the power spectrum or the 2PCF, fully characterize all the available information only for Gaussian and isotropic fields.Ideally, we want to ensure that high-order statistics (e.g., 3-point correlations (Schmitz et al. 2018;Pyne et al. 2022)) are taken into account when inferring the IA signal, and that uncertainties are accounted for self-consistently via forward modeling (e.g., via fieldlevel inferences (Tsaprazi et al. 2022;Porqueres et al. 2023)).With the advent of next-generation data, these methods must be advanced to smaller scales in order to increase the signal-to-noise ratio of the detections.

MODELING
Many IA models have been developed over the past two decades and even the earlier ones are still useful.Advancements reflect the increasing understanding and importance of IA and the need to extend theory to smaller scales and a wider range of alignment mechanisms.Models are often created for specific scales and galaxy types, the main two being spiral and elliptical.As noted in Section 1, how these morphologies relate to ELG/LRGs is a source of discussion and the distinction between "red" and "blue" galaxy samples is survey-and simulation-dependent.
This section briefly summarizes common models with the aim of introducing typical modeling formalisms; it is not a complete overview of the field.In Sections 8.1 and 8.2 we discuss the linear and nonlinear alignment models.These were developed first and are the most frequently used to date.Section 8.3 expands on aspects of the intrinsic alignment amplitude parameter which is a feature of these models.Sections 8.4 to 8.6 introduce three more wellestablished approaches to modeling IA: the Tidal Alignment and Tidal Torquing model, effective field theory, and the halo model.Section 8.7 briefly highlights some more recent models.Section 8.8 summarizes the observational status of the models.Finally, Section 8.9 introduces self-calibration, a model-agnostic technique.

Linear Alignment model (LA)
This approach assumes that the ellipticity of a galaxy is linearly related to the gravitational potential at the time the galaxy formed.Before we start, it is useful to define two redshifts relevant for the LA and other models: • z IA : the redshift at which the alignment is set.
• z obs : the redshift at which the IA is observed.
Since the details of galaxy formation and evolution are not well understood, two scenarios are proposed for the redshift z IA at which the alignment is produced.
• Instantaneous Alignment: In this case the alignment redshift is assumed to be the same as the observed redshift, z IA = z obs , and the amplitude of the alignment, A IA , is modeled as a simple linear function of redshift.
where C 1 is a normalization constant (a contribution due to the response to the tidal field) and ρ m,0 is the matter density at z = 0 (today).
• Early Alignment: In the early alignment scenario, the redshift at which the IA signal is observed is later than the redshift at which the alignment was set, z obs < z IA .If it is assumed that the halo (or galaxy) first forms during the matter domination era (primordial alignment) then A IA does not depend on z IA and evolves as where D(z) has been set to unity at matter domination: D(z) = (1 + z)D(z), and D(z) is the growth factor.Alternatively, it is often assumed that alignment happens later than this because it is affected by more recent mergers and accretion.Then the amplitude is given by: Finally, in each of these cases, the IA power spectra are: where P L (k, z) is the linear matter power spectrum.
• "Intrinsic alignment-lensing interference as a contaminant of cosmic shear" Hirata & Seljak (2004) Development of the LA model.

Nonlinear Alignment model (NLA)
This is an empirical model that replaces the linear power spectrum used in the LA model (Eqs.62 and 63) with the full nonlinear matter power spectrum, P NL (k, z), while preserving the assumption that density perturbations are described by the Poisson equation.Physically, the latter assumption fails in the nonlinear regime of structure formation due to gravitational evolution.However, this model has been widely adopted due to its ability to accurately reproduce ellipticity correlations for red elliptical galaxies down to ∼ 6 h −1 Mpc.The NLA model is often enhanced to include other sample dependencies, for example, on redshift or galaxy luminosity, usually captured by additional power laws.
To incorporate a luminosity L dependence, the IA amplitude A IA is expressed as a power law of the general form: where A is a prefactor, L 0 is a pivot luminosity and α L is a luminosity scaling.Furthermore, the IA amplitude form can be extended to account for the redshift evolution as with α z a redshift scaling parameter.This parameter is often denoted η or, if the redshift contribution is separated into low-and high-redshift parts, η low−z and η high−z .This should not be confused with the definition in Section 2.3 or in Section 4. For more details, consult Chisari et al. (2016).

Practical uses
Another parameterization of the redshift-and luminosity-dependent IA amplitude frequently used in analyses is: where A 0 is a prefactor and z 0 is a pivot redshift.C 1 , ρ m,0 and D(z) are described in Section 8.1.Since the signal is not observed in blue galaxies, the amplitude is corrected for the fraction of red galaxies and the redshift scaling can be separated into "low redshift" and "high redshift" contributions.The details can be found in Krause et al. (2016).
8.2.1.NLA model: References • "Dark energy constraints from cosmic shear power spectra: Impact of intrinsic alignments on photometric redshift requirements" Bridle & King (2007) First use of the NLA model.The authors argue that this model might not be "closer to the truth", but that it matches the data slightly better than the LA model.
• "Constraints on intrinsic alignment contamination of weak lensing surveys using the MegaZ-LRG sample" Joachimi et al. (2011) Frequently cited paper giving commonly used value for the constant C 1 in the LA/NLA models (though the original source of the value can be traced back via Bridle & King (2007), Hirata & Seljak (2004) and Brown et al. (2002)).
• "The impact of intrinsic alignment on current and future cosmic shear surveys" Krause et al. (2016) Detailed description of the luminosity-and redshift-dependent IA amplitude.

Alignment Amplitude
The alignment amplitude A IA in these models is essentially a free bias parameter that relates a measurement of the local tidal field to the amplitude, or strength, of the IA signal.When cosmological parameters are estimated from weak lensing, A IA is often treated as a "nuisance" parameter to be marginalized over.
As discussed in Sections 8.1 and 8.2, the amplitude is expected to evolve with redshift and also to depend on galaxy/halo mass.The redshift dependence is thought to arise because galaxies traverse evolving regions of tidal shear (advection) (Schmitz et al. 2018) or because halo mass accretion is stronger at higher redshifts (Asgari et al. 2023).
In simulations A IA has been found to depend on several galaxy properties including mass, luminosity (increasing with both), color, and morphology (stronger for red/elliptical than blue/spiral) (Samuroff et al. 2021).Observations have found A IA to be consistent with zero for blue galaxies (Johnston et al. 2019), suggesting a lack of tidal alignment, but robust detections have been made for red galaxies, e.g., Fortuna et al. (2021b).
All these dependencies are difficult to quantify in practice.Because of this, the redshift scaling is often separated into "low redshift" and "high redshift" contributions.Alternatively, the alignment amplitude is simply set to be constant across the redshift range.The amplitude may also be corrected for the fraction of red galaxies, or the galaxies can be divided into "red" and "blue" sub-samples which are analyzed separately (for example Samuroff et al. (2019)).As discussed in Section 8.2, in observational studies luminosity is often used as a proxy for mass so that we have A IA (z, L).A further complication is that the observed alignment amplitude has been found to depend on the method used to measure galaxy shapes (Singh & Mandelbaum 2016).

IA Amplitude: Additional Notation
Although A IA is a common notation, the amplitude is sometimes denoted as A or A I .However, one must be careful as they often represent different quantities depending on the model.Additionally, superscripts may be used to indicate the amplitude of subsamples e.g., A R IA , A B IA for red and blue samples.For models with more than one IA amplitude the notation A 1 , A 2 , . . . is often used.In this case, A 1 may also be used for the single-amplitude model.8.3.2.IA Amplitude: References • "Redshift and luminosity evolution of the intrinsic alignments of galaxies in Horizon-AGN" Chisari et al. (2016) A detailed work on the IA amplitude dependency on redshift and luminosity.
• "KiDS+GAMA: Intrinsic alignment model constraints for current and future weak lensing cosmology " Johnston et al. (2019) Dependency of IA amplitude on color in observations.
• "KiDS-1000: Constraints on the intrinsic alignment of luminous red galaxies" Fortuna et al. (2021b) Luminosity and redshift dependence of IA for LRGs.
• "Advances in Constraining Intrinsic Alignment Models with Hydrodynamic Simulations" Samuroff et al. ( 2021) Comprehensive analysis comparing multiple simulations and dependence of IA on galaxy properties.The LA and NLA models are strictly thought to apply only to elliptical galaxies.TATT was introduced to account for the alignment of spiral galaxies whose configuration depends on angular momentum rather than being pressuresupported.The basic idea is to express a galaxy's intrinsic shape γ I ij as an expansion of the trace-free tidal field tensor s ij to second order in the linear density field: where δ and s respectively describe the density and tidal fields.Importantly, the TATT model does not include all possible terms of the perturbative bias expansion to second order (Blazek et al. 2019).
The first term of Eq. 67 corresponds to tidal alignment as in the LA/NLA models.The second is "density weighting" and arises because alignment can only be measured where there is a galaxy.The third term is tidal torquing.
Tidal alignment is described by C1 where C 1 is the same normalization constant as in the LA and NLA models.A similar formula applies to C1δ , which can be parameterized as C1δ = b 1 C1 , where b 1 is called the linear bias.A 1 is equivalent to A IA in the LA/NLA models.
As for the LA model (Section 8.1), adjustments can be made to allow for different assumptions about the redshift at which alignment is set.
The tidal torquing parameter, C2 , is given by where A 2 is a second alignment amplitude.The factor 5 is an approximation related to the different IA power spectrum in the pure tidal alignment and tidal torquing cases.
• "Advances in constraining intrinsic alignment models with hydrodynamic simulations" Samuroff et al. (2021) 2PCF measurements from simulations comparing the NLA and TATT models (and comparing simulations).This contains a helpful discussion of the models.

Effective Field Theory (EFT) model
EFT extends the modeling of IA to smaller scales in a more theoretically rigorous way than the NLA model.It is similar to Eulerian perturbation theory which has previously been applied to scalar fields such as counts of galaxies (for example Baumann et al. (2012)).In the case of IA we instead have a tensor field, but in other respects the method is similar.On large scales, the matter density distribution is treated as an effective fluid, while the small-scale physics are decoupled and packed into a set of free parameters with values that can be constrained through either simulations or observations.
For IA, the EFT approach involves first capturing the physical interactions that give rise to intrinsic galaxy shape correlations within the galaxies' rest frame.These correlations are then expressed in terms of local gravitational observables.This accounts for the statistical properties of galaxy shapes resulting from gravitational effects and allows for a systematic approach to modeling the corresponding IA.
The shape of a galaxy can be defined by a trace-free symmetric tensor g ij based on its light distribution.The shape tensor is then connected to local gravitational observables by a bias expansion where [O] are renormalized operators and b O are corresponding bias parameters.This is exactly the same approach as for a scalar field.The next step is to determine which operators should be included to any given order in perturbation theory.Since these include higher spatial derivatives of the density and tidal fields, the modeling of IA is extended from linear scales to the quasi-linear regime.
In the case of IA, the final step is to project the obtained statistical information onto the sky, effectively mapping the correlations onto the observed galaxy shapes (Vlah et al. 2020).The EFT model can characterize the auto-spectrum of IA B-modes with high accuracy (Bakx et al. 2023) and has been used to predict the B-mode IA of clusters (Georgiou et al. 2023).
• "Galaxy shape statistics in the effective field theory" Vlah et al. ( 2021) Projection onto the observed sky.
• "Effective field theory of intrinsic alignments at one loop order: a comparison to dark matter simulations" Bakx et al. (2023) Comparison of EFT of IA to simulations.

Halo model
The halo model of IA (Schneider & Bridle 2010) is an alternative approach that builds on the halo model of galaxy clustering (Cooray & Sheth 2002).In this model, all matter is assumed to reside in dark matter halos and the galaxy power spectrum is expressed as the sum of a one-halo term corresponding to the (small-scale) correlation of galaxies within a single dark matter halo, and a two-halo term corresponding to the (large-scale) correlation of galaxies in different halos.
The same approach can be taken for the IA power spectra so that From now on we consider only P GI (k).In all cases equivalent expressions can be derived for P II (k).
Crucially, this formalism enables different modeling of the alignment of satellite and central galaxies, which are thought to contribute to the IA power spectra in different ways and on different scales.It can be assumed that centrals contribute mainly to the two-halo term, that they have the same ellipticity and orientation as their parent halos, and that this large-scale behavior can be described by the linear alignment model.Moreover, the two-halo term can easily be split to allow different amplitudes for the alignment contribution of red and blue populations: where the fractions of red and blue centrals, f red c and f blue c , can be determined using a model of the halo occupation distribution.
The alignment of satellites has been found to be weaker and more complex.At small scales, satellites are often assumed to point radially toward the center of their host halo.At large scales there is evidence that their alignment vanishes, suppressing the overall signal.The halo model can take this complexity into account.
Under the assumption of a spherical halo, the inter-halo alignment between centrals and satellites is zero on average, and thus the one-halo term describes the alignment of satellites with each other and the local matter field: where n(M ) is the halo mass function, f s is the fraction of satellite galaxies, which can be written as a function of redshift, ⟨N s |M ⟩ is the halo occupation distribution of satellites, Û (M, k) is the normalized mass density profile, and γI is the density-weighted average of the satellite intrinsic ellipticity.An additional dependence on radius can also be included.
The halo model has been shown to fit observations in the range 0.3 − 1.5 h −1 Mpc (Singh et al. 2015), and on larger scales can be complemented by the LA or the NLA models.Although its formalism relies on physical assumptions, such as the symmetry of the halos or the alignment of satellites inside a halo, and it suffers from the lack of constraints for fainter galaxies, this model performs well in the context of weak lensing corrections for current surveys.

Variable definitions
• P 2h GI (k): two-halo term of the gravitational-intrinsic power spectrum.
• f red c , f blue c : fractions of red and blue central galaxies.
• P 1h GI (k): one-halo term of the gravitational-intrinsic power spectrum.
• f s : fraction of satellite galaxies.
• ⟨N s |M ⟩: halo occupation distribution of satellites.
• ns : mean number density of galaxies.
• "A halo model for intrinsic alignments of galaxy ellipticities" Schneider & Bridle (2010) The first IA halo model.
• "The halo model as a versatile tool to predict intrinsic alignments" Fortuna et al. (2021a) Development of a halo model for IA.
• "The halo model for cosmology: A pedagogical review" Asgari et al. (2023) Introduction to the halo model with some discussion of IA.

Other IA models
Additional modeling approaches have been proposed to address weaknesses in existing models and to take advantage of increased computational resources.This is a rapidly-moving field and so here we only mention three examples.
• Semi-analytic models: These assign shapes and orientations to galaxies placed in dark-matter-only simulations.The approach is not new, but a recent example using up-to-date simulations is Hoffmann et al. (2022).
• HYMALAIA (Maion et al. 2023): This is a hybrid Lagrangian model that combines a perturbative bias expansion in Lagrangian space with the fully nonlinear displacement field from N-body simulations.It has been found to be more accurate than TATT and to be valid up to smaller scales while having fewer free parameters than EFT models.
• Lagrangian perturbation theory (Chen & Kokron 2024): Similar to the Eulerian EFT model in Vlah et al. (2020) but with a different treatment of long-wavelength linear displacements and an alternative formulation of the bias scheme.

Modeling: Observational status
The assumption of the linearity of tidal shear breaks down on small scales, where linear models under-predict correlations.Therefore, alignment models that are more informed about nonlinear structure growth are required for accurate modeling of galaxy shapes.Chisari et al. (2014) reported that the linear alignment model behaves well only down to 10 h −1 Mpc, and Singh et al. (2015) found that the nonlinear alignment model accurately models the ellipticities of luminous red galaxies down to 6 h −1 Mpc.The TATT model attempts to extend scales down to 2 h −1 Mpc (Samuroff et al. 2022), though does not describe the alignment of halos at quasi non-linear regimes as the full EFT model does (Bakx et al. 2023).Finally, the halo model has been successfully used in the range 0.3 − 1.5 h −1 Mpc (Singh et al. 2015), yet also lacks the ability to fully capture correlations in the nonlinear regime.

Self-calibration
When weak lensing data is used to constrain cosmological parameters, an alternative to directly modeling IA and marginalizing over parameters is to follow a model-agnostic approach through self-calibration.
The approach is based on the insight that in photometric surveys it is possible to identify features of the GG (lensing) power spectrum that distinguish it from the GI and II spectra.Zhang (2010a) considered the dependence of the power spectra on the difference ∆z P between the photometric redshifts of two tomographic bins and showed that the relationship for GG can be expected to be quite different from that for GI or II.This is essentially because of the different weighting functions in the expressions for the power spectra (see Section 7).It is then possible to derive scaling relations for the IA power spectra in terms of ∆z P and non-IA observables.Thus the IA elements of the observed power spectrum can be removed without making any assumptions about the form of the IA and without needing any data external to the survey (for example to set priors on IA model parameters).
The articles below are selected examples of self-calibration, ranging from its initial proposal to its application to recent survey data.
• "Self-calibration for three-point intrinsic alignment autocorrelations in weak lensing surveys " Troxel & Ishak (2012) Useful overview and extends the concept to three-point statistics.
• "Self-calibration method for II and GI types of intrinsic alignments of galaxies" Yao et al. (2019) Review of methodology and forecast for LSST.
• "KiDS-1000: Cross-correlation with Planck cosmic microwave background lensing and intrinsic alignment removal with self-calibration" Yao et al. (2023) Comparison of self-calibration with other methods and application to KiDS data.

IA APPLICATIONS
Although galaxy IA has been intensively studied as one of the most important systematic effects for weak lensing cosmology, it can also bias RSD measurements and has the potential to be a novel probe in cosmology.Here we list a selection of papers that describe some of these applications.As a direct result of galaxy formation mechanics, in principle IA can also be used to explore the relationship between the formation of galaxies and their dark matter environment.However, currently, higher precision measurements and more detailed modeling are necessary before IA is useful for constraining galaxy formation, particularly on small scales.hydrodynamic simulations: Simulations of galactic and structure formation that contain baryons and feedback.Done on smaller scales than N-body simulations which are dark matter only.Page 3 late-type galaxies: These are usually spiral galaxies, as described by the Hubble Sequence (Hubble 1926) and are sometimes referred to as simply "blue" galaxies.Pages 3, 6 Legendre polynomial: Legendre polynomials are solutions of the second-order differential equation (1 − x 2 )y ′′ − 2xy ′ + n(n + 1)y = 0 .

Fig. 2 .
Fig. 2.-The quantities a, b and θ that define the shape and orientation of an ellipse.The dotted line indicates an arbitrary reference axis.

Fig
Fig.3.-Visualization of the real and imaginary components of ellipticity, as described in Section 2. These functionally contain the same information.ϵ 1 is maximum when a shape is highly elongated and exactly aligned with the angle that the ellipticity is defined relative to, most commonly North.ϵ 2 is maximum when the shape is aligned with π/4 away from the principal angle.

Fig. 4 .
Fig.4.-The setup for obtaining the projected shape of a spiral galaxy.In linear theory, the angular momentum vector L of the galaxy is aligned along the direction of tidal stretching.The projected axis ratio, b/a, is a function of L and the ratio of the disk's intrinsic thickness to its diameter (not shown here).
Fig. 5.-Representation of the Northern Celestial Hemisphere.The observer is indicated by O and a given observed galaxy by galaxy.The purple basis indicates the rotated basis defined in Eq. (14) and Eq.(15), whereas ϕ 1 is the angle from the North Pole.

Fig. 7 .
Fig. 7.-A cartoon demonstrating the difference between convergence and shear.Convergence isotropically changes the apparent size of a galaxy while conserving surface brightness, demonstrated here by a fainter color.Shear elongates the galaxy.

Fig. 8 .
Fig.8.-A schematic of correlations relevant to IA and weak lensing studies, shown along the LOS.Here, two background galaxies are lensed by a foreground dark matter halo.The observed shape ϵ of a galaxy is a combination of its intrinsic (I) and lensed (G) components.These are correlated with the underlying dark matter density, which can be traced by galaxy shapes and positions (g).Correlations are represented by combining these notations.For example, gI is the correlation between galaxy positions and intrinsic galaxy shapes, and GI is the correlation between the lensed component of galaxy shapes and intrinsic galaxy shapes.GI correlations are important because overdensities cause IA between galaxies at the same redshift while also lensing more distant galaxies.Separating galaxies into widely-spaced tomographic bins can distinguish between the GI and II terms (Section 4.2, Figure9).For other helpful diagrams of IA correlations, see Figure1.6 ofFortuna (2021) and Figure6ofTroxel & Ishak (2015).Note that this is a cartoon; galaxy shapes, orientations, and positions are only symbolic.These correlations are never measured for individual galaxies and are typically only statistically significant when measured for at least 10 4 objects.

Fig
Fig. 9.-An illustrative example of binning used to measure cosmic shear.The initial samples are sliced into three tomographic bins by redshift (real surveys typically use more).The redshift distributions shown on the vertical and horizontal axes can be from the same or different samples.The boxes are number-and color-coordinated to show the six different combinations for correlations.If the correlations are done within the same bin, they are referred to as auto-correlations.Correlations between different bins are called cross-correlations.

Fig. 10 .
Fig. 10.-Angular power spectra for the different contributions given in Eq. (50).The C GI ℓ curve shows the absolute value for the component, as it is an anti-correlation.This plot was generated using the python package pyccl 4 (Chisari & Dvorkin 2013), assuming the nonlinear alignment model described in Section 8.2.

8. 4 .
Tidal Alignment and Tidal Torquing model (TATT) photometric: In the context of cosmology, this refers to galaxy properties (most commonly redshifts) that were estimated through imaging color, as opposed to spectroscopy.Pages 12, 14, 27 pivot luminosity: The value of luminosity at which the luminosity function goes from one regime to another.This is set to correspond to the absolute magnitude of -22.Page 22