Non-Linear Linear Alignment in Cosmology

• Non-Linear Linear Alignment (NLA) is a phenomenological intrinsic-alignment model that replaces the linear matter power spectrum with its non-linear version while preserving linear tidal coupling.
• The approach is calibrated on spectroscopic galaxy samples and effectively improves two-point cosmic-shear analyses over scales of 5–100 arcmin.
• Extensions such as NLA-β and δ-NLA address luminosity scaling and density-weighted effects, paving the way for refined intrinsic alignment modeling in simulations.

Searching arXiv for the cited NLA intrinsic-alignment papers and closely related work. Non-Linear Linear Alignment (NLA) is a phenomenological intrinsic-alignment model used in weak-lensing cosmology to describe how galaxy shapes correlate with the large-scale tidal field while replacing the linear matter power spectrum of the original linear-alignment prescription with a non-linear matter power spectrum. In this sense, the phrase “non-linear” refers to the use of $P_{\rm NL}$ or $P_\delta^{\rm nl}(k,z)$ rather than to a non-linear shape response in the baseline ansatz itself. NLA has become a standard description of intrinsic alignments in two-point cosmic-shear analyses, has been directly calibrated on spectroscopic galaxy samples, and has also been the subject of perturbative critiques showing where it breaks down and how it can be extended (Johnston et al., 2018, Blazek et al., 2017, Harnois-Déraps et al., 29 Sep 2025).

1. Definition and theoretical origin

The starting point is the “instantaneous tidal-alignment” or linear-alignment (LA) model, in which galaxy shapes—primarily luminous red or elliptical galaxies—respond linearly to the large-scale tidal field

$$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$

In the notation summarized by Blazek et al., the LA model writes

$$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$

with

$$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$

where $G(z)$ is the linear growth factor normalized as $G(0)=1$ (Blazek et al., 2017).

The NLA prescription keeps the same tidal-alignment ansatz but replaces the linear matter power spectrum by a non-linear one:

$$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$

This replacement is explicitly described as a phenomenological fix. The 2025 simulation study states that observationally “the alignments appear stronger than can be captured by purely linear theory,” and that most analyses find a better fit when $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$, for example via Halofit. That same study describes the model as yielding percent-level agreement with data on scales $5$–$P_\delta^{\rm nl}(k,z)$0 in the contexts tested there (Harnois-Déraps et al., 29 Sep 2025).

A common misconception is that the name implies a non-linear alignment law. The baseline NLA construction does not do that: it preserves linear tidal coupling and changes the matter clustering input. This distinction matters because later perturbative work identifies genuinely higher-order intrinsic-alignment operators beyond NLA.

2. Mathematical formulation and standard parameterizations

In the empirical calibration carried out with KiDS, GAMA, and SDSS, the NLA model is written directly in terms of the intrinsic–intrinsic and matter–intrinsic three-dimensional power spectra:

$P_\delta^{\rm nl}(k,z)$1

$P_\delta^{\rm nl}(k,z)$2

Here $P_\delta^{\rm nl}(k,z)$3 is a dimensionless alignment amplitude; $P_\delta^{\rm nl}(k,z)$4 normalizes the original “SuperCOSMOS” measurement; $P_\delta^{\rm nl}(k,z)$5 is the mean matter density; $P_\delta^{\rm nl}(k,z)$6 is the scale factor; $P_\delta^{\rm nl}(k,z)$7 is the linear growth factor with $P_\delta^{\rm nl}(k,z)$8; and $P_\delta^{\rm nl}(k,z)$9 is the non-linear matter power spectrum (Johnston et al., 2018).

A luminosity-dependent generalization, denoted “NLA-$$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$0,” replaces the amplitude by

$$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$1

where $$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$2 corresponds to $$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$3, and $$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$4 is the sample mean. Some analyses also introduce a redshift-evolution factor

$$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$5

In the KiDS+GAMA calibration no $$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$6-term was fitted because the redshift baseline $$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$7 was short; the summary states that effectively $$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$8 (Johnston et al., 2018).

In tomographic cosmic-shear notation, the same model can be projected into angular statistics as

$$s_{ij}(x)\equiv\left(\frac{\partial_i\partial_j}{\nabla^2}-\frac{\delta_{ij}}{3}\right)\delta(x).$$9

$$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$0

followed by

$$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$1

The 2025 implementation notes that many cosmic-shear codes equivalently write $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$2, with $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$3 absorbing the normalization and redshift dependence (Harnois-Déraps et al., 29 Sep 2025).

3. Empirical calibration in KiDS, GAMA, and SDSS

A direct calibration of NLA amplitudes was obtained using r-band shape measurements from KiDS-450 imaging over the three equatorial GAMA fields, totaling $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$4, with spectroscopic redshifts from GAMA to $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$5 at $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$6 completeness, together with an SDSS Main sample with $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$7 over $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$8. Shapes were measured with the DEIMOS moment-based method for KiDS+GAMA and with the “re-Gaussianisation” pipeline for SDSS Main. Galaxies were separated into red and blue populations with the rest-frame colour cut $$P_{II}^{\rm LA}(k,z)=F^2(z)\,P_{\rm lin}(k,z),\qquad P_{GI}^{\rm LA}(k,z)=F(z)\,P_{\rm lin}(k,z),$$9; GAMA was further split at $$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$0 into the two redshift bins Z1 and Z2, yielding Z1B, Z1R, Z2B, and Z2R subsamples (Johnston et al., 2018).

The analysis estimated the galaxy–intrinsic shear correlation $$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$1 via the M06 estimator, subtracting random–shape pairs to remove additive systematics, and projected it as

$$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$2

Galaxy clustering was estimated with the Landy–Szalay estimator and projected into $$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$3. The two observables were then fitted jointly, with $$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$4 constraining the linear galaxy bias $$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$5, on scales $$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$6 (Johnston et al., 2018).

The main parameter constraints were the following.

Sample/model	Constraint	Interpretation
Blue galaxies	$$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$7	Consistent with zero
Red galaxies	$$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$8	$$F(z)\equiv -\,C_1(z)\,\rho_{\rm crit,0}\,\Omega_{m,0}\,G^{-1}(z),$$9 detection of radial alignments
Combined sample	$G(z)$0	No colour split
Red NLA-$G(z)$1	$G(z)$2	Consistent with zero
Blue NLA-$G(z)$3	$G(z)$4	Consistent with zero

These measurements were summarized as showing no significant detection of blue-galaxy alignments, robust detections for red galaxies, and no evidence for any scaling of alignments with galaxy luminosity. Additional checks on central versus satellite galaxies revealed that large-scale GI correlations above $G(z)$5 arise almost entirely from red central galaxies, while red satellites, blue centrals, and blue satellites show no significant large-scale alignment. This suggests that colour alone does not exhaust the relevant astrophysical degrees of freedom.

4. Use in weak-lensing inference

The KiDS+GAMA calibration was explicitly translated into informative priors for cosmic-shear analyses:

$G(z)$6

These were proposed as Gaussian priors on the red and blue NLA amplitudes in weak-lensing likelihoods (Johnston et al., 2018).

Fisher forecasts for a completed KiDS-like $G(z)$7 survey, tomographically colour-split into five bins each for red and blue galaxies, showed that applying these intrinsic-alignment priors can reduce the marginalized uncertainties on

$G(z)$8

and on the dark-energy equation-of-state parameter $G(z)$9 by up to $G(0)=1$0 and $G(0)=1$1, respectively. The summary specifies that these are the maximal gains when also marginalizing over realistic photo-$G(0)=1$2 biases (Johnston et al., 2018).

The same work argues that these priors improve upon de facto wide priors that allow unrealistic levels of intrinsic-alignment contamination. At the same time, it warns that modelling alignments only through a red/blue split may be insufficient when the fraction of central and satellite galaxies varies with depth or survey footprint. A plausible implication is that informative priors on $G(0)=1$3 are most reliable when the sample definition is close to that used in the calibration.

5. Perturbative generalizations and the breakdown of NLA

Blazek et al. formulate intrinsic alignments as a perturbative expansion analogous to galaxy biasing. Up to second order, the intrinsic shear field is expanded in all spin-2, traceless combinations of the density, tidal, and velocity-shear fields:

$G(0)=1$4

In this framework, the NLA model is the subset retaining the linear tidal term while replacing the linear matter spectrum by a non-linear one (Blazek et al., 2017).

The physical assumptions of LA and NLA are stated explicitly: galaxy shapes respond linearly to the large-scale tidal field, higher-order responses are neglected, and the time delay between structure formation and galaxy alignment is ignored. The summary further states that the model is valid only on large, quasi-linear scales where non-Gaussian, nonlinear intrinsic-alignment mechanisms such as tidal torquing are subdominant; empirically, one typically restricts NLA to $G(0)=1$5–$G(0)=1$6 and $G(0)=1$7, beyond which omission of one-loop and higher corrections leads to order-unity errors (Blazek et al., 2017).

The one-loop corrections computed in that work include quadratic alignment and density-weighting terms and are reported to contribute $G(0)=1$8 corrections to the total intrinsic-alignment signal at $G(0)=1$9. In an LSST-like forecast with five tomographic bins, $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$0, area $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$1, and $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$2, a data vector generated with the full one-loop intrinsic-alignment model but analyzed with NLA yields

$$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$3

described as a $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$4 excursion. Allowing the NLA amplitude to scale as $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$5 reduces the bias to

$$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$6

but does not eliminate it. By contrast, fitting the full perturbative intrinsic-alignment model gives

$$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$7

with bias $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$8, while marginalizing over $$P_{II}^{\rm NLA}(k,z)=F^2(z)\,P_{\rm NL}(k,z),\qquad P_{GI}^{\rm NLA}(k,z)=F(z)\,P_{\rm NL}(k,z).$$9 and $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$0 increases the uncertainty by only $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$1 relative to the no-intrinsic-alignment analysis (Blazek et al., 2017).

These results are often interpreted as evidence that NLA is adequate as a large-scale effective model for current two-point analyses but not as a complete description for Stage IV weak-lensing inference.

6. Simulation-level implementations and extended model families

A later simulation study embeds NLA directly into weak-lensing simulations constructed from the Outer-Rim N-body light cone, using $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$2 particles in a $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$3 box and $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$4 curved-sky mass shells up to $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$5 on Healpix $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$6. For each shell, the projected tidal field is obtained from the density map, the B-mode is set to zero, and $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$7 maps are reconstructed. The resulting tidal-field maps are smoothed with a Gaussian beam of width $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$8, with tests at $P_\delta^{\rm lin}\to P_\delta^{\rm NL}$9 and $5$0, and then downgraded to $5$1 (Harnois-Déraps et al., 29 Sep 2025).

Mock source galaxies are either placed randomly on the octant, corresponding to the NLA assumption of no density weighting, or are Poisson-sampled from the smoothed shells with mean proportional to $5$2 for linear-bias tracers, or from halo-occupation-distribution catalogues for non-linear-bias tracers used in $5$3-NLA. Intrinsic ellipticities are assigned by interpolating the tidal field at each galaxy position and then combined with simulated reduced shear $5$4 through

$5$5

The NLA implementation is validated against CosmoSIS predictions using fully non-linear $5$6: agreement is reported at the $5$7–$5$8 level for $5$9 in $P_\delta^{\rm nl}(k,z)$00 and $P_\delta^{\rm nl}(k,z)$01 in $P_\delta^{\rm nl}(k,z)$02 when $P_\delta^{\rm nl}(k,z)$03 (Harnois-Déraps et al., 29 Sep 2025).

Within this simulation framework, NLA is compared to several related models. The extended NLA model, also called e-NLA or $P_\delta^{\rm nl}(k,z)$04-NLA, multiplies the NLA ellipticity by $P_\delta^{\rm nl}(k,z)$05:

$P_\delta^{\rm nl}(k,z)$06

The tidal-torque (TT) model instead uses a quadratic coupling,

$P_\delta^{\rm nl}(k,z)$07

with $P_\delta^{\rm nl}(k,z)$08. The study also considers $P_\delta^{\rm nl}(k,z)$09-TT and HOD-TATT variants. It reports that the $P_\delta^{\rm nl}(k,z)$10-NLA model has by far the largest impact on most non-Gaussian probes, at times more than twice the strength of the NLA; that minima, void profiles, and the lensing PDF are the best probes for model rejection because large differences between models appear in under-dense regions; and that the third-order aperture mass statistic $P_\delta^{\rm nl}(k,z)$11 and the integrated three-point functions are particularly sensitive to the source-clustering term when low-redshift data are included, often exceeding a $P_\delta^{\rm nl}(k,z)$12 impact on the data vector (Harnois-Déraps et al., 29 Sep 2025).

In the same work, nested-sampling likelihood analyses varying $P_\delta^{\rm nl}(k,z)$13 recover $P_\delta^{\rm nl}(k,z)$14 and values of $P_\delta^{\rm nl}(k,z)$15 and $P_\delta^{\rm nl}(k,z)$16 consistent with the true Outer-Rim cosmology, within sampling variance, when the simulated data are generated with NLA and analyzed with the corresponding NLA theory. Opening the full TATT parameter space introduces the expected degeneracy between $P_\delta^{\rm nl}(k,z)$17 and the quadratic amplitude $P_\delta^{\rm nl}(k,z)$18, but $P_\delta^{\rm nl}(k,z)$19 remains stable. This suggests that NLA remains a robust phenomenological description on the scales used by current two-point lensing analyses, even as higher-order statistics and next-generation surveys motivate more general intrinsic-alignment models.