largely outside of the black contours in Fig. 8 (top-right-hand panel); these objects are indeed rare at low redshift, possibly having been consumed by central galaxies over cosmic time (Matharu et al. 2019; Baldry et al. 2020).

Meanwhile, Welker et al. (2017) also found that cold gas flows and consequent in situ star formation, as well as minor mergers, tended to flatten lower-mass $M_* < 10^{10.5}$ M$_\odot$ spheroidal galaxies along their minor axes, with the former potentially prompting the (re-)formation of discs among galaxies that avoid subsequent mergers. Regardless of further mergers or disc reformation, if these processes sufficiently tip the balance of $V/\sigma$, we might then classify these objects as spirals. At fixed stellar mass, spheroidal galaxies are larger in the simulation (Dubois et al. 2016),<fnref n="14" /> and expected to exist closer to filaments/nodes (Kraljic et al. 2018; Laigle et al. 2018), than typical discs at the same stellar mass; disc reformation is therefore a possible pathway for the intrinsically large spirals sourcing our ISCs.

Future efforts to dis/confirm our interpretation of ISCs in hydro-dynamical cosmological simulations could focus upon the merger histories of simulated objects, as a function of the anisotropic cosmic web environment, extending to higher-redshift snapshots to assess the interplay of mergers and morphological transitions, and the associated change in profile of ISCs. Indeed, one might seek to more finely bin galaxy samples using a metric that directly discriminates between stages of evolution, such as $V/\sigma$, age (see also Lu et al. 2020), or metallicity (La Barbera et al. 2010), or else according to galaxies’ local, anisotropic environments. Our work here should motivate a search for smooth trends in ISCs with respect to such variables, as objects evolve through the stages, and migrate through the cosmic web environments, that we have coarsely probed here.

The recently-completed Horizon Run 5 (Lee et al. 2021a) should be especially useful for answering remaining questions with regards to galaxy intrinsic size correlations in hydrodynamical cosmological simulations, with a cuboid box having $10 \times$ the volume explored here, a significant number of galaxy clusters, and improvements to subgrid physics implementation that could dispel some of our limiting concerns with regards to the behaviour of low-mass galaxies.

Before moving on, we note here some limitations of our analysis. Our usage of stellar particle virial radii as size metrics for spiral galaxies is non-optimal, as these may not correspond well to common observational spiral radii – such as half-mass/light radii, or disc scale lengths – and would be far more difficult to obtain for real data. Moreover, the correlations between FP residuals $\lambda$ and spiral galaxy properties such as effective/virial radii are causes for concern. Moving from the Masscut to the Full FP sample generally increases the significance of ISCs without dramatic changes to their form (Fig. 6), whilst also reducing the $\lambda$-property correlations (Fig. 4). However, the clustering statistics of these lower-mass objects are questionable (Hatfield et al. 2019), and this should also temper our assessment of the elliptical ISCs, where the significant Full FP $\Delta_{gg}$ signal is heavily sourced by these galaxies.

We therefore explore some extended FPs and ISCs in Appendix A, where additional FP variables serve to reduce FP scatter/tilting and $\lambda$-property correlations, for both spiral and elliptical samples. Whilst some of the resulting ISC signals lose statistical significance relative to our fiducial analysis, others gain greatly, and none are so different

in form as to be of concern. Our conclusions with respect to the variability of intrinsic sizes with environment and morphology are maintained, and we refer the reader to the appendix for more details.

## 5.4 Density-magnification contamination

We now turn to our mock estimation of the intrinsic contamination affecting a lensing magnification measurement of the projected surface density $\Sigma(r)$ (detailed in Section 4.1.1). Fig. 9 displays our predictions of the systematic contribution $\Sigma_\lambda(r)$, as estimated via galaxy size-based cosmic convergence statistics (Huff & Graves 2014), given different assumed values for the photo-z interloper fraction $f_1$ (columns). Blue (red) data points give the estimated contributions from spiral (elliptical) galaxies, with the top row featuring our Masscut FPs, and with Full FP signals on the bottom. Open points denote negative values, and the grey shading marks scales $>5 h^{-1}$ Mpc, beyond which our neglect of periodic boundary conditions renders predictions untrustworthy – quoted significances (in-panel coloured numbers, without cosmic variance) thus exclude these scales. A range of NFW profile (Navarro, Frenk & White 1996, 1997) predictions for $\Sigma(r)$, corresponding to $1\sigma$ confidence intervals constrained by Mandelbaum et al. (2006) are illustrated as orange shading – these are similar to the model depicted by Huff & Graves (2014), which was seen to accurately predict their measured $\Sigma(r)$ signal.

In the case of Masscut FP residuals, our predictions are consistent with null signals in all cases, as expected given the lack of significant $g\lambda$ signal detections within the simulation box (Fig. 6; large points). However, when relaxing the mass-selection, those intrinsic size-density contributions become significantly non-zero, with their strength modulated by the assumed value of $f_1$. For our estimation procedure, the true interloper fraction is unity; all galaxies entering the estimator (equation 21) are coincident in redshift. In the bottom-right-hand panel, one sees that the correct value for $f_1$ can mitigate most of the systematic signal, lowering the significance from $>5.5\sigma$ (~3.8$\sigma$) from spirals (ellipticals), when $f_1$ is severely underestimated, to ~1.8$\sigma$ (1.2$\sigma$).

The correction works as follows: a population of interloper objects from redshift $z_1$ is thought to be at $z_s$, and its sizes thought to trace the convergence field. The intrinsic sizes of the population are correlated with the large-scale structure at $z_1$, and thus $\Delta\log R$ (Equations 13 & 19) correlates with lens sample positions despite the lack of convergence-derived galaxy size fluctuations. The $\Delta\log R_{err}$ term (equation 18) is positive, because object redshifts have been overestimated<fnref n="15" /> $z_s > z_1$, and we are far below the redshifts where angular diameter distances $d_\alpha(z)$ turn over; thus $f_1$ weights a term that serves to make $\Delta\log R$ smaller, or more negative. When raised to an exponent to yield the estimator for $\hat{\lambda}$ ($\equiv \kappa + \tilde{\lambda}$ in a real analysis; equation 17), $f_1 \rightarrow 1$ maximizes a suppression of the systematic contribution; that is, the photo-z correction term mitigates ISCs by chance.

However, this correction involves discarding large amounts of data from small-angle regimes with the highest signal-to-noise for the convergence. It also cannot fully disentangle $\lambda$ from $\kappa$ in the presence of ISCs, as equation (17) cannot differentiate between convergence and intrinsic size variation. If the spatial average of $\Delta\log R$ is non-zero due to intrinsic correlations with the ‘foreground’ structure, then depending upon the strength of the underlying intrinsic

<footnote n="14">¹⁴We note that higher-resolution hydro-simulations produce more compact $z=2$ galaxies at fixed stellar mass, owing to improved modelling of gas flows (Chabanier et al. 2020), in better agreement with observations that show discs to be larger than spheroids at fixed stellar mass (Van Der Wel et al. 2014; Kawinwanichakij et al. 2021; Nedkova et al. 2021).</footnote>

<footnote n="15">¹⁵We note that underestimated redshifts could cause a negative correction term, which one would expect to hamper any mitigation of ISCs.</footnote>
largely outside of the black contours in Fig. 8 (top-right-hand panel); these objects are indeed rare at low redshift, possibly having been consumed by central galaxies over cosmic time (Matharu et al. 2019; Baldry et al. 2020).

Meanwhile, Welker et al. (2017) also found that cold gas flows and consequent in situ star formation, as well as minor mergers, tended to flatten lower-mass $M_* < 10^{10.5} M_\odot$ spheroidal galaxies along their minor axes, with the former potentially prompting the (re-)formation of discs among galaxies that avoid subsequent mergers. Regardless of further mergers or disc reformation, if these processes sufficiently tip the balance of $V/\sigma$, we might then classify these objects as spirals. At fixed stellar mass, spheroidal galaxies are larger in the simulation (Dubois et al. 2016),<fnref n="14" /> and expected to exist closer to filaments/nodes (Kraljic et al. 2018; Laigle et al. 2018), than typical discs at the same stellar mass; disc reformation is therefore a possible pathway for the intrinsically large spirals sourcing our ISCs.

Future efforts to dis/confirm our interpretation of ISCs in hydro-dynamical cosmological simulations could focus upon the merger histories of simulated objects, as a function of the anisotropic cosmic web environment, extending to higher-redshift snapshots to assess the interplay of mergers and morphological transitions, and the associated change in profile of ISCs. Indeed, one might seek to more finely bin galaxy samples using a metric that directly discriminates between stages of evolution, such as $V/\sigma$, age (see also Lu et al. 2020), or metallicity (La Barbera et al. 2010), or else according to galaxies’ local, anisotropic environments. Our work here should motivate a search for smooth trends in ISCs with respect to such variables, as objects evolve through the stages, and migrate through the cosmic web environments, that we have coarsely probed here.

The recently-completed Horizon Run 5 (Lee et al. 2021a) should be especially useful for answering remaining questions with regards to galaxy intrinsic size correlations in hydrodynamical cosmological simulations, with a cuboid box having 10 × the volume explored here, a significant number of galaxy clusters, and improvements to subgrid physics implementation that could dispel some of our limiting concerns with regards to the behaviour of low-mass galaxies.

Before moving on, we note here some limitations of our analysis. Our usage of stellar particle virial radii as size metrics for spiral galaxies is non-optimal, as these may not correspond well to common observational spiral radii – such as half-mass/light radii, or disc scale lengths – and would be far more difficult to obtain for real data. Moreover, the correlations between FP residuals $\lambda$ and spiral galaxy properties such as effective/virial radii are causes for concern. Moving from the Masscut to the Full FP sample generally increases the significance of ISCs without dramatic changes to their form (Fig. 6), whilst also reducing the $\lambda$-property correlations (Fig. 4). However, the clustering statistics of these lower-mass objects are questionable (Hatfield et al. 2019), and this should also temper our assessment of the elliptical ISCs, where the significant Full FP $\Delta_{gg}$ signal is heavily sourced by these galaxies.

We therefore explore some extended FPs and ISCs in Appendix A, where additional FP variables serve to reduce FP scatter/tilting and $\lambda$-property correlations, for both spiral and elliptical samples. Whilst some of the resulting ISC signals lose statistical significance relative to our fiducial analysis, others gain greatly, and none are so different

in form as to be of concern. Our conclusions with respect to the variability of intrinsic sizes with environment and morphology are maintained, and we refer the reader to the appendix for more details.

## 5.4 Density-magnification contamination

We now turn to our mock estimation of the intrinsic contamination affecting a lensing magnification measurement of the projected surface density $\Sigma(r)$ (detailed in Section 4.1.1). Fig. 9 displays our predictions of the systematic contribution $\Sigma_\lambda(r)$, as estimated via galaxy size-based cosmic convergence statistics (Huff & Graves 2014), given different assumed values for the photo-z interloper fraction $f_1$ (columns). Blue (red) data points give the estimated contributions from spiral (elliptical) galaxies, with the top row featuring our Masscut FPs, and with Full FP signals on the bottom. Open points denote negative values, and the grey shading marks scales $>5 h^{-1}$ Mpc, beyond which our neglect of periodic boundary conditions renders predictions untrustworthy – quoted significances (in-panel coloured numbers, without cosmic variance) thus exclude these scales. A range of NFW profile (Navarro, Frenk & White 1996, 1997) predictions for $\Sigma(r)$, corresponding to $1\sigma$ confidence intervals constrained by Mandelbaum et al. (2006) are illustrated as orange shading – these are similar to the model depicted by Huff & Graves (2014), which was seen to accurately predict their measured $\Sigma(r)$ signal.

In the case of Masscut FP residuals, our predictions are consistent with null signals in all cases, as expected given the lack of significant $g\lambda$ signal detections within the simulation box (Fig. 6; large points). However, when relaxing the mass-selection, those intrinsic size-density contributions become significantly non-zero, with their strength modulated by the assumed value of $f_1$. For our estimation procedure, the true interloper fraction is unity; all galaxies entering the estimator (equation 21) are coincident in redshift. In the bottom-right-hand panel, one sees that the correct value for $f_1$ can mitigate most of the systematic signal, lowering the significance from $>5.5\sigma$ (~3.8$\sigma$) from spirals (ellipticals), when $f_1$ is severely underestimated, to ~1.8$\sigma$ (1.2$\sigma$).

The correction works as follows: a population of interloper objects from redshift $z_1$ is thought to be at $z_s$, and its sizes thought to trace the convergence field. The intrinsic sizes of the population are correlated with the large-scale structure at $z_1$, and thus $\Delta\log R$ (Equations 13 & 19) correlates with lens sample positions despite the lack of convergence-derived galaxy size fluctuations. The $\Delta\log R_{err}$ term (equation 18) is positive, because object redshifts have been overestimated<fnref n="15" /> $z_s > z_1$, and we are far below the redshifts where angular diameter distances $d_\alpha(z)$ turn over; thus $f_1$ weights a term that serves to make $\Delta\log R$ smaller, or more negative. When raised to an exponent to yield the estimator for $\hat{\lambda}$ ($\equiv \kappa + \tilde{\lambda}$ in a real analysis; equation 17), $f_1 \rightarrow 1$ maximizes a suppression of the systematic contribution; that is, the photo-z correction term mitigates ISCs by chance.

However, this correction involves discarding large amounts of data from small-angle regimes with the highest signal-to-noise for the convergence. It also cannot fully disentangle $\lambda$ from $\kappa$ in the presence of ISCs, as equation (17) cannot differentiate between convergence and intrinsic size variation. If the spatial average of $\Delta\log R$ is non-zero due to intrinsic correlations with the ‘foreground’ structure, then depending upon the strength of the underlying intrinsic

<footnote n="14">¹⁴We note that higher-resolution hydro-simulations produce more compact $z=2$ galaxies at fixed stellar mass, owing to improved modelling of gas flows (Chabanier et al. 2020), in better agreement with observations that show discs to be larger than spheroids at fixed stellar mass (Van Der Wel et al. 2014; Kawinwanichakij et al. 2021; Nedkova et al. 2021).</footnote>

<footnote n="15">¹⁵We note that underestimated redshifts could cause a negative correction term, which one would expect to hamper any mitigation of ISCs.</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 9.** Predicted intrinsic contributions $\Sigma_{\lambda}(r)$ to the projected surface density $\Sigma(r)$, as measured from angular galaxy position-size correlations in the presence of catastrophic photometric redshift outliers (see Section 4.1.1). Contributions from spiral (elliptical) intrinsic sizes are shown in blue (red). Columns give different assumed values for the fraction $f_i$ of 'source' galaxies that are in reality co-located with $z \sim 0.2$ lenses, whilst rows give the systematic contributor signal estimated for more conservative Masscut Fundamental Plane samples (top; Section 2), and for Full Fundamental Plane samples (bottom). Data points and error bars correspond to the mean signals and standard deviations over 1000 bootstrap samples of the size catalogues. Coloured numbers give the significance of these estimates, computed with exclusion of the grey-shaded region ($\gtrsim 5 h^{-1}$ Mpc), and neglecting cosmic variance. The orange shaded curve illustrates a range of NFW halo profile (Navarro et al. 1996, 1997) expectations for $\Sigma(r)$, corresponding to $\pm 1\sigma$ constraints on luminous red galaxy halo masses $M_h$ and concentrations $c$ from Mandelbaum et al. (2006). The predictions for contaminants (data points) are seen to be of comparable amplitudes to the data-driven NFW model (orange curve) for $r \gtrsim 10$ Mpc, particularly when the interloper fraction $f_i$ is underestimated (for this mock measurement, the true $f_i$ is exactly unity).</figcaption></figure>

correlation, and the proportion of intrinsically correlated galaxies under consideration, even a correct determination of $f_i$ may not completely suppress $\Sigma_{\lambda}$, as can be tentatively seen in the bottom-right-hand panel of Fig. 9. It would be preferable for a magnification probe analysis to forward-model both the ISC and photo-$z$ effects, to allow for a cleaner extraction of the convergence $\kappa$.

Sheldon et al. (2004) found $\Delta\Sigma(r)$ to be < 1 $h$ M$_\odot$ pc$^{-2}$ at $\sim 10 h^{-1}$ Mpc, and $\Sigma(r)$ follows a steeper power law with respect to $r$. Thus smooth extensions of our Full FP $\Sigma_{\lambda}$ estimates to scales of order 10 Mpc and beyond$^{16}$ would be likely to impinge upon the lensing signal, as illustrated by the orange shaded curve in Fig. 9.

Interestingly, the noisier elliptical size-density correlation is of a similar amplitude to that from spirals but with an opposite sign. Given the divergent intrinsic properties of spiral and elliptical galaxies, and the current lack of a unified FP to describe joint scaling relations for their radii, one might assume that they will be kept separate for a size-based cosmic convergence analysis (Alsing et al. 2015). Analysing both samples for magnification in parallel would also provide a beneficial cross-checking mechanism, where estimates of $\Sigma(r)$ should be consistent for the same lenses when traced by sources of either morphological type (see e.g. Li et al. 2021, for a colour-split consistency analysis of cosmic shear). Our work suggests that such an analysis will be contending with the fidelity of morphology-

dependent ISC modelling for accurate determinations of $\Sigma(r)$ on large scales.

We recall here that this estimation of an intrinsic contamination to $\Sigma(r)$ derives from a single simulation snapshot at $z = 0.06$, assuming no significant redshift evolution of $f_i$, or of the density-intrinsic size correlation out to $z \sim 0.2$. These strong assumptions are convenient for this work, since we analyse only one simulation snapshot situated at $z = 0.06$, whilst optimal lens samples are to be defined at higher redshifts, allowing for more efficient lensing of yet-higher redshift background sources. Future studies of ISC phenomena in simulations should extend the redshift baseline, consider deriving light-cones, and investigate functional form(s) for $f_i(z_l, z_s, z)$ motivated by such work as Stölzner et al. (2022).

We note that an interesting route to study the impacts of ISCs in lensing magnification would be to apply the self-calibration approach of (Zhang 2010a,b). As shown for real data by Yao et al. (2020a), Yao et al. (2020b), the radial asymmetry of the gravitational lensing effect, combined with photometric redshift information, can be leveraged to extract statistically isotropic intrinsic alignment contributions from galaxy position-shear correlations in a model-independent way. This reasoning should also hold for intrinsic size correlations, and offers a promising alternative mode of study.

## 5.5 Magnification–magnification contamination

The results of fitting our phenomenological model to measured $\xi_{\lambda\lambda}$ correlations (Section 4.1.2) are shown in Fig. 6 as pink solid and dotted curves, for the Masscut and Full FP samples, respectively. We note that this simple model seems to fit the measured Full spiral FP

<footnote>$^{16}$The grey-shaded region in Fig. 9 excludes scales $\gtrsim 5 h^{-1}$ Mpc, beyond which our neglect of periodic boundaries (Section 4.1.1) is likely to become important; the amplitudes of the final two points (the last especially) are thus likely to be underestimated.</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Predicted intrinsic contributions $\Sigma_{\lambda}(r)$ to the projected surface density $\Sigma(r)$, as measured from angular galaxy position-size correlations in the presence of catastrophic photometric redshift outliers (see Section 4.1.1). Contributions from spiral (elliptical) intrinsic sizes are shown in blue (red). Columns give different assumed values for the fraction $f_i$ of 'source' galaxies that are in reality co-located with $z \sim 0.2$ lenses, whilst rows give the systematic contributor signal estimated for more conservative Masscut Fundamental Plane samples (top; Section 2), and for Full Fundamental Plane samples (bottom). Data points and error bars correspond to the mean signals and standard deviations over 1000 bootstrap samples of the size catalogues. Coloured numbers give the significance of these estimates, computed with exclusion of the grey-shaded region ($\gtrsim 5 h^{-1}$ Mpc), and neglecting cosmic variance. The orange shaded curve illustrates a range of NFW halo profile (Navarro et al. 1996, 1997) expectations for $\Sigma(r)$, corresponding to $\pm 1\sigma$ constraints on luminous red galaxy halo masses $M_{\text{h}}$ and concentrations $c$ from Mandelbaum et al. (2006). The predictions for contaminants (data points) are seen to be of comparable amplitudes to the data-driven NFW model (orange curve) for $r \gtrsim 10$ Mpc, particularly when the interloper fraction $f_i$ is underestimated (for this mock measurement, the true $f_i$ is exactly unity).</figcaption></figure>

correlation, and the proportion of intrinsically correlated galaxies under consideration, even a correct determination of $f_i$ may not completely suppress $\Sigma_{\lambda}$, as can be tentatively seen in the bottom-right-hand panel of Fig. 9. It would be preferable for a magnification probe analysis to forward-model both the ISC and photo-$z$ effects, to allow for a cleaner extraction of the convergence $\kappa$.

Sheldon et al. (2004) found $\Delta\Sigma(r)$ to be < 1 $h$ M$_{\odot}$ pc$^{-2}$ at $\sim 10 h^{-1}$ Mpc, and $\Sigma(r)$ follows a steeper power law with respect to $r$. Thus smooth extensions of our Full FP $\Sigma_{\lambda}$ estimates to scales of order 10 Mpc and beyond$^{16}$ would be likely to impinge upon the lensing signal, as illustrated by the orange shaded curve in Fig. 9.

Interestingly, the noisier elliptical size-density correlation is of a similar amplitude to that from spirals but with an opposite sign. Given the divergent intrinsic properties of spiral and elliptical galaxies, and the current lack of a unified FP to describe joint scaling relations for their radii, one might assume that they will be kept separate for a size-based cosmic convergence analysis (Alsing et al. 2015). Analysing both samples for magnification in parallel would also provide a beneficial cross-checking mechanism, where estimates of $\Sigma(r)$ should be consistent for the same lenses when traced by sources of either morphological type (see e.g. Li et al. 2021, for a colour-split consistency analysis of cosmic shear). Our work suggests that such an analysis will be contending with the fidelity of morphology-

dependent ISC modelling for accurate determinations of $\Sigma(r)$ on large scales.

We recall here that this estimation of an intrinsic contamination to $\Sigma(r)$ derives from a single simulation snapshot at $z = 0.06$, assuming no significant redshift evolution of $f_i$, or of the density-intrinsic size correlation out to $z \sim 0.2$. These strong assumptions are convenient for this work, since we analyse only one simulation snapshot situated at $z = 0.06$, whilst optimal lens samples are to be defined at higher redshifts, allowing for more efficient lensing of yet-higher redshift background sources. Future studies of ISC phenomena in simulations should extend the redshift baseline, consider deriving light-cones, and investigate functional form(s) for $f_i(z_l, z_s, z)$ motivated by such work as Stölzner et al. (2022).

We note that an interesting route to study the impacts of ISCs in lensing magnification would be to apply the self-calibration approach of (Zhang 2010a,b). As shown for real data by Yao et al. (2020a), Yao et al. (2020b), the radial asymmetry of the gravitational lensing effect, combined with photometric redshift information, can be leveraged to extract statistically isotropic intrinsic alignment contributions from galaxy position-shear correlations in a model-independent way. This reasoning should also hold for intrinsic size correlations, and offers a promising alternative mode of study.

## 5.5 Magnification–magnification contamination

The results of fitting our phenomenological model to measured $\xi_{\lambda\lambda}$ correlations (Section 4.1.2) are shown in Fig. 6 as pink solid and dotted curves, for the Masscut and Full FP samples, respectively. We note that this simple model seems to fit the measured Full spiral FP

<footnote>$^{16}$The grey-shaded region in Fig. 9 excludes scales $\gtrsim 5 h^{-1}$ Mpc, beyond which our neglect of periodic boundaries (Section 4.1.1) is likely to become important; the amplitudes of the final two points (the last especially) are thus likely to be underestimated.</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 10.** Constraints upon the linear, absolute intrinsic size parameters $|B_{\text{spi/ell}}|$, as fitted (see Section 4.1.2) to the autocorrelation function $\xi_{\lambda\lambda}(r) = B^2\xi_{\delta\delta}$ (Fig. 6) of Fundamental Plane radius residuals $\lambda$ for spiral and elliptical galaxies in the Horizon-AGN simulation box. The Full spiral Fundamental Plane model amplitude is constrained to be non-zero at high significance, whilst Full elliptical and Masscut Fundamental Plane constraints are each consistent with zero at $\le 3\sigma$.</figcaption></figure>

$\xi_{\lambda\lambda}$ correlation remarkably well, extending far into the non-linear regime despite being limited scales $r > 3 h^{-1}$ Mpc. Considering the less conservative sample selection, and without a physical motivation for the model, we are reluctant to make any strong conclusions regarding the quality of this particular fit, though the fit to the Masscut spiral $\xi_{\lambda\lambda}$ is comfortably consistent. Conversely, the model prefers a significantly larger amplitude for the Full elliptical FP signal than for the Masscut signal, where the latter fit runs into the zero-prior.

Confidence contours depicting our phenomenological model constraints are shown in Fig. 10, with grey denoting the Masscut FP fits, and gold the fits to Full FP signals. Maximum likelihood points (corresponding to the curves in Fig. 6) are indicated in each case by pink lines, with solid (dotted) lines giving the Masscut (Full) FP results. The reduced$^{17}$ $\chi^2$ is 0.33 (0.45) for spiral (elliptical) $\xi_{\lambda\lambda}$ from the Masscut FP, or 0.30 when computed jointly thus accounting for the small covariance between individual $\lambda\lambda$ signals. For the Full FP signals, the reduced $\chi^2$ is 0.37 (0.55) for spiral (elliptical) $\xi_{\lambda\lambda}$, or 0.49 when computed jointly. We note that the inter- and intra-$\xi_{\lambda\lambda}$ correlations are larger in the Full FP case, and feature an anticorrelation between spiral/elliptical signals.

The results of fitting are negligibly different when performed jointly or separately. We report for the Masscut FP: $|B_{\text{spi}}| = 0.034^{+0.007}_{-0.016}$, $|B_{\text{ell}}| = 0.010^{+0.009}_{-0.007}$, and for the Full FP: $|B_{\text{spi}}| = 0.036^{+0.002}_{-0.002}$, $|B_{\text{ell}}| = 0.029^{+0.004}_{-0.010}$, where errors refer to the difference between the maximum likelihood points and the 16th, 84th percentiles (the $\sigma_{68}$ interval in the following figures) of Monte Carlo Markov Chains after an appropriate burn-in phase.

Thus our phenomenological model prefers a non-zero intrinsic size-matter density coupling $B_{\text{spi}}$ at nearly $\gtrsim 2\sigma$ in the Masscut case, and $\sim 18\sigma$ for the Full FP, where the two are comfortably consistent. Conversely, the elliptical coupling $B_{\text{ell}}$ runs into the prior $|B| \ge 0$ in the Masscut case, and is nearly 3$\sigma$ greater than zero for the Full FP. Each constitutes a good fit to the relevant signal, but we caution moderation, as this model is not physically motivated – Joachimi et al. (2015) used a very similar model and were unable to make good fits to their elliptical FP size residual autocorrelations $\lambda\lambda$. Future work should revisit these signals with a more motivated description (e.g. the EFT model for scalars; Vlah et al. 2020, or the size-luminosity halo model of Ciarlariello et al. 2015; Ciarlariello & Crittenden 2016).

As described in Section 4.1.2, we translate these model constraints into predictions of angular power spectra (Equations 22) describing the intrinsic size autocorrelation $\lambda\lambda$, and intrinsic size-convergence cross-correlation $\lambda\kappa$, for two redshift samples at $z = 0.4, 0.8$ (Joachimi et al. 2015), displaying the results in Fig. 11. The figure shows systematic $C(\ell)$'s (red/blue) corresponding to the best-fit model parameters, with the $\sigma_{68}$ intervals given by shading, in comparison with the cosmic convergence power spectrum $C_{\kappa\kappa}(\ell)$, shown in green. Results derived from Masscut (Full) FP residuals are given in the upper (lower) triangle.

The $\lambda\lambda$ correlation (solid curves) must be non-negative, and we see that the predicted amplitude of $\lambda\lambda$ signals from Masscut ellipticals is about an order of magnitude smaller than the convergence signal in the tomographic autocorrelations (middle- & bottom-left, top- & middle-right), though we remind the reader that $B_{\text{ell}}$ from the Masscut FP runs into the prior $|B| \ge 0$; these predictions are consistent with null signals.

The Masscut spiral $\lambda\lambda$ signal, however, is likely to dominate over the convergence; this from the more conservative Masscut selection is potentially concerning for future studies of cosmic convergence. Since higher redshifts are dominated by late-type galaxies, most of the signal-to-noise in a weak lensing analysis is derived from these objects, and excluding them would make the cosmological signal much more difficult to detect – perhaps especially in the case of size-based convergence, as the size dispersion is thought to be larger than the shear dispersion (Huff & Graves 2014), placing even more stringent requirements upon the sample number density.

In the tomographic cross-correlation (bottom-left & top-right), the $\lambda\lambda$ contribution weakens substantially, owing to the small overlap between redshift distributions. The dominant systematic signal now comes from the intrinsic size-convergence cross-correlations $\lambda\kappa$ (dot-dashed curves). Here, the sign of $B$ becomes important. We might assume (more securely for the Full FPs) that $B_{\text{spi}}$ is positive, and $B_{\text{ell}}$ negative, given the forms of $\Delta_{gg}$ and $\xi_{g\lambda}$, correlations seen in Fig. 6. We note, however, that our model strictly constrains only the absolute values of these parameters, and that each of the Masscut elliptical ISCs is consistent with a null detection. Thus the cross-correlation predictions are for absolute contributions $|\lambda\kappa|$.

The spiral contribution $|\lambda\kappa|$ remains likely to dominate over the cosmological signal. We display the $g\lambda$-derived constraint $B_{\text{ell}} = -0.012 \pm 0.005$ from Joachimi et al. (2015) for comparison as a grey-dashed curve with $1\sigma$ uncertainty given by shading/hatching. Our constraints upon $B_{\text{ell}}$, and corresponding predictions for $|\lambda\kappa|$, are consistent with that from Joachimi et al. (2015) at 68 per cent confidence for the Masscut FP, whilst our Full FP predicts a stronger contamination. The small $|\kappa\lambda|$ contribution (dotted curves), from foreground convergence and background ISCs, derives from a very small overlap of corresponding lensing and number kernels

<footnote>$^{17}$Taking either one parameter and three data points, thus two degrees of freedom per individual $\xi_{\lambda\lambda}$, or else two parameters and six data points, and thus four degrees of freedom for the joint computation.</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Constraints upon the linear, absolute intrinsic size parameters $|B_{\text{spi/ell}}|$, as fitted (see Section 4.1.2) to the autocorrelation function $\xi_{\lambda\lambda}(r) = B^2\xi_{\delta\delta}$ (Fig. 6) of Fundamental Plane radius residuals $\lambda$ for spiral and elliptical galaxies in the Horizon-AGN simulation box. The Full spiral Fundamental Plane model amplitude is constrained to be non-zero at high significance, whilst Full elliptical and Masscut Fundamental Plane constraints are each consistent with zero at $\le 3\sigma$.</figcaption></figure>

$\xi_{\lambda\lambda}$ correlation remarkably well, extending far into the non-linear regime despite being limited scales $r > 3 h^{-1}$ Mpc. Considering the less conservative sample selection, and without a physical motivation for the model, we are reluctant to make any strong conclusions regarding the quality of this particular fit, though the fit to the Masscut spiral $\xi_{\lambda\lambda}$ is comfortably consistent. Conversely, the model prefers a significantly larger amplitude for the Full elliptical FP signal than for the Masscut signal, where the latter fit runs into the zero-prior.

Confidence contours depicting our phenomenological model constraints are shown in Fig. 10, with grey denoting the Masscut FP fits, and gold the fits to Full FP signals. Maximum likelihood points (corresponding to the curves in Fig. 6) are indicated in each case by pink lines, with solid (dotted) lines giving the Masscut (Full) FP results. The reduced$^{17}$ $\chi^2$ is 0.33 (0.45) for spiral (elliptical) $\xi_{\lambda\lambda}$ from the Masscut FP, or 0.30 when computed jointly thus accounting for the small covariance between individual $\lambda\lambda$ signals. For the Full FP signals, the reduced $\chi^2$ is 0.37 (0.55) for spiral (elliptical) $\xi_{\lambda\lambda}$, or 0.49 when computed jointly. We note that the inter- and intra-$\xi_{\lambda\lambda}$ correlations are larger in the Full FP case, and feature an anticorrelation between spiral/elliptical signals.

The results of fitting are negligibly different when performed jointly or separately. We report for the Masscut FP: $|B_{\text{spi}}| = 0.034^{+0.007}_{-0.016}$, $|B_{\text{ell}}| = 0.010^{+0.009}_{-0.007}$, and for the Full FP: $|B_{\text{spi}}| = 0.036^{+0.002}_{-0.002}$, $|B_{\text{ell}}| = 0.029^{+0.004}_{-0.010}$, where errors refer to the difference between the maximum likelihood points and the 16th, 84th percentiles (the $\sigma_{68}$ interval in the following figures) of Monte Carlo Markov Chains after an appropriate burn-in phase.

Thus our phenomenological model prefers a non-zero intrinsic size-matter density coupling $B_{\text{spi}}$ at nearly $\gtrsim 2\sigma$ in the Masscut case, and $\sim 18\sigma$ for the Full FP, where the two are comfortably consistent. Conversely, the elliptical coupling $B_{\text{ell}}$ runs into the prior $|B| \ge 0$ in the Masscut case, and is nearly 3$\sigma$ greater than zero for the Full FP. Each constitutes a good fit to the relevant signal, but we caution moderation, as this model is not physically motivated – Joachimi et al. (2015) used a very similar model and were unable to make good fits to their elliptical FP size residual autocorrelations $\lambda\lambda$. Future work should revisit these signals with a more motivated description (e.g. the EFT model for scalars; Vlah et al. 2020, or the size-luminosity halo model of Ciarlariello et al. 2015; Ciarlariello & Crittenden 2016).

As described in Section 4.1.2, we translate these model constraints into predictions of angular power spectra (Equations 22) describing the intrinsic size autocorrelation $\lambda\lambda$, and intrinsic size-convergence cross-correlation $\lambda\kappa$, for two redshift samples at $z = 0.4, 0.8$ (Joachimi et al. 2015), displaying the results in Fig. 11. The figure shows systematic $C(\ell)$'s (red/blue) corresponding to the best-fit model parameters, with the $\sigma_{68}$ intervals given by shading, in comparison with the cosmic convergence power spectrum $C_{\kappa\kappa}(\ell)$, shown in green. Results derived from Masscut (Full) FP residuals are given in the upper (lower) triangle.

The $\lambda\lambda$ correlation (solid curves) must be non-negative, and we see that the predicted amplitude of $\lambda\lambda$ signals from Masscut ellipticals is about an order of magnitude smaller than the convergence signal in the tomographic autocorrelations (middle- & bottom-left, top- & middle-right), though we remind the reader that $B_{\text{ell}}$ from the Masscut FP runs into the prior $|B| \ge 0$; these predictions are consistent with null signals.

The Masscut spiral $\lambda\lambda$ signal, however, is likely to dominate over the convergence; this from the more conservative Masscut selection is potentially concerning for future studies of cosmic convergence. Since higher redshifts are dominated by late-type galaxies, most of the signal-to-noise in a weak lensing analysis is derived from these objects, and excluding them would make the cosmological signal much more difficult to detect – perhaps especially in the case of size-based convergence, as the size dispersion is thought to be larger than the shear dispersion (Huff & Graves 2014), placing even more stringent requirements upon the sample number density.

In the tomographic cross-correlation (bottom-left & top-right), the $\lambda\lambda$ contribution weakens substantially, owing to the small overlap between redshift distributions. The dominant systematic signal now comes from the intrinsic size-convergence cross-correlations $\lambda\kappa$ (dot-dashed curves). Here, the sign of $B$ becomes important. We might assume (more securely for the Full FPs) that $B_{\text{spi}}$ is positive, and $B_{\text{ell}}$ negative, given the forms of $\Delta_{gg}$ and $\xi_{g\lambda}$, correlations seen in Fig. 6. We note, however, that our model strictly constrains only the absolute values of these parameters, and that each of the Masscut elliptical ISCs is consistent with a null detection. Thus the cross-correlation predictions are for absolute contributions $|\lambda\kappa|$.

The spiral contribution $|\lambda\kappa|$ remains likely to dominate over the cosmological signal. We display the $g\lambda$-derived constraint $B_{\text{ell}} = -0.012 \pm 0.005$ from Joachimi et al. (2015) for comparison as a grey-dashed curve with $1\sigma$ uncertainty given by shading/hatching. Our constraints upon $B_{\text{ell}}$, and corresponding predictions for $|\lambda\kappa|$, are consistent with that from Joachimi et al. (2015) at 68 per cent confidence for the Masscut FP, whilst our Full FP predicts a stronger contamination. The small $|\kappa\lambda|$ contribution (dotted curves), from foreground convergence and background ISCs, derives from a very small overlap of corresponding lensing and number kernels

<footnote>$^{17}$Taking either one parameter and three data points, thus two degrees of freedom per individual $\xi_{\lambda\lambda}$, or else two parameters and six data points, and thus four degrees of freedom for the joint computation.</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 11.** Predicted intrinsic contributions to size-based angular convergence power spectrum estimates $C(\ell)$ derived from fits of a phenomenological model (Section 4.1.2) for intrinsic size correlations $\xi_{\lambda\lambda}$ (Fig. 6). Shown in green are the cosmic convergence spectra $\kappa\kappa$, whilst in blue (red) are the intrinsic, systematic contributions from spiral (elliptical) galaxies, with auto and cross-correlations shown for two tomographic redshift samples at $z \sim 0.4, 0.8$. Solid red/blue curves give the Fundamental Plane residual autocorrelations $\lambda\lambda$; dot-dashed curves the absolute value of the intrinsic size-convergence cross-correlation $|\lambda\kappa|$; dotted curves the reverse cross-correlation $|\kappa\lambda|$; and grey dashed curves the $|\lambda\kappa|$ prediction from Joachimi et al. (2015). The envelope around each curve gives the $\sigma_{68}$ confidence interval corresponding to the model constraint ($\pm 1\sigma$ for the Joachimi et al. 2015 prediction). Predictions from more conservative Masscut Fundamental Planes are shown in the upper triangle, whilst the Full Fundamental Plane results are given in the lower triangle. Contaminant contributions to the convergence power spectrum are seen to be comparable to the cosmological signal in several cases, and most significantly for the Full elliptical and spiral Fundamental Plane residuals.</figcaption></figure>

(equation (22); middle), and is unlikely to be of concern (similarly to the intrinsic alignment IG correlation in cosmic shear).

Each contamination prediction is strengthened when turning to the Full FP results, with spiral $\lambda\lambda$ and $|\lambda\kappa|$ signals from factors of 2–3 larger than the convergence at high significance. Meanwhile, elliptical $\lambda\lambda$ and $|\lambda\kappa|$ rise to become comparable to, or dominant over the convergence.

## 6 CONCLUSIONS

We have explored the intrinsic size correlations (ISCs) of spiral and elliptical galaxies in the $z = 0.06$ snapshot of the cosmological hydrodynamical simulation Horizon-AGN, as traced by Fundamental Plane (FP) radius residuals $\lambda$.

We defined morphological galaxy samples according to the ratio of stellar particles’ average tangential velocity and velocity dispersion, with small (large) ratios tracing spiral (elliptical) populations. Considering the clustering deficits observed for low-mass Horizon-

AGN objects by Hatfield et al. (2019), we further divided the morphological samples into a conservative, higher-mass subset, ‘Masscut’, with $M_* > 10^{10.5} M_\odot$, and ‘Full’ samples including the lower-mass galaxies (limited to $M_* > 10^{10} M_\odot$ for ellipticals).

For elliptical galaxies we followed standard practice, with the effective radius $R_{\text{eff}}$, velocity dispersion $\sigma$, and surface brightness $I$ forming the FP. We also explored FPs substituting the surface stellar mass density $\Sigma$ for the surface brightness $I$ (as done by Rosito et al. 2021, for their fiducial planes), and found that the radius residuals from this plane were not significantly correlated with the large-scale structure of the simulation, in disagreement with observational studies where such elliptical ISCs were measured at high significance (Joachimi et al. 2015; Singh et al. 2021). Unless otherwise stated, we henceforth refer to results derived from the surface brightness plane, which are far more agreeable with literature work.

We were inspired by Shen et al. (2002) to use a spiral galaxy radius estimate to tighten the scatter about the Tully–Fisher scaling of circular velocity $V_c$ with absolute magnitude $M_r$, finding the
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Predicted intrinsic contributions to size-based angular convergence power spectrum estimates $C(\ell)$ derived from fits of a phenomenological model (Section 4.1.2) for intrinsic size correlations $\xi_{\lambda\lambda}$ (Fig. 6). Shown in green are the cosmic convergence spectra $\kappa\kappa$, whilst in blue (red) are the intrinsic, systematic contributions from spiral (elliptical) galaxies, with auto and cross-correlations shown for two tomographic redshift samples at $z \sim 0.4, 0.8$. Solid red/blue curves give the Fundamental Plane residual autocorrelations $\lambda\lambda$; dot-dashed curves the absolute value of the intrinsic size-convergence cross-correlation $|\lambda\kappa|$; dotted curves the reverse cross-correlation $|\kappa\lambda|$; and grey dashed curves the $|\lambda\kappa|$ prediction from Joachimi et al. (2015). The envelope around each curve gives the $\sigma_{68}$ confidence interval corresponding to the model constraint ($\pm 1\sigma$ for the Joachimi et al. 2015 prediction). Predictions from more conservative Masscut Fundamental Planes are shown in the upper triangle, whilst the Full Fundamental Plane results are given in the lower triangle. Contaminant contributions to the convergence power spectrum are seen to be comparable to the cosmological signal in several cases, and most significantly for the Full elliptical and spiral Fundamental Plane residuals.</figcaption></figure>

(equation (22); middle), and is unlikely to be of concern (similarly to the intrinsic alignment IG correlation in cosmic shear).

Each contamination prediction is strengthened when turning to the Full FP results, with spiral $\lambda\lambda$ and $|\lambda\kappa|$ signals from factors of 2–3 larger than the convergence at high significance. Meanwhile, elliptical $\lambda\lambda$ and $|\lambda\kappa|$ rise to become comparable to, or dominant over the convergence.

# 6 CONCLUSIONS

We have explored the intrinsic size correlations (ISCs) of spiral and elliptical galaxies in the $z = 0.06$ snapshot of the cosmological hydrodynamical simulation Horizon-AGN, as traced by Fundamental Plane (FP) radius residuals $\lambda$.

We defined morphological galaxy samples according to the ratio of stellar particles’ average tangential velocity and velocity dispersion, with small (large) ratios tracing spiral (elliptical) populations. Considering the clustering deficits observed for low-mass Horizon-

AGN objects by Hatfield et al. (2019), we further divided the morphological samples into a conservative, higher-mass subset, ‘Masscut’, with $M_* > 10^{10.5} M_\odot$, and ‘Full’ samples including the lower-mass galaxies (limited to $M_* > 10^{10} M_\odot$ for ellipticals).

For elliptical galaxies we followed standard practice, with the effective radius $R_{\text{eff}}$, velocity dispersion $\sigma$, and surface brightness $I$ forming the FP. We also explored FPs substituting the surface stellar mass density $\Sigma$ for the surface brightness $I$ (as done by Rosito et al. 2021, for their fiducial planes), and found that the radius residuals from this plane were not significantly correlated with the large-scale structure of the simulation, in disagreement with observational studies where such elliptical ISCs were measured at high significance (Joachimi et al. 2015; Singh et al. 2021). Unless otherwise stated, we henceforth refer to results derived from the surface brightness plane, which are far more agreeable with literature work.

We were inspired by Shen et al. (2002) to use a spiral galaxy radius estimate to tighten the scatter about the Tully–Fisher scaling of circular velocity $V_c$ with absolute magnitude $M_r$, finding the
virial radius of stellar particles $R_{vir}$ to yield planes with manageable degrees of tilting, and scatters comparable to those of the elliptical planes. Lower-mass spirals in the Full FP sample were seen to be particularly well-described, with the tilting of the FP featuring mainly at the high-mass end.

We defined the intrinsic galaxy size fluctuation $\lambda$ as the natural logarithm of the ratio of measured-to-predicted (by the fitted FP) radii for each object, finding these to be approximately normally distributed, and having similar distributions between spiral and elliptical, Full and Masscut FPs. The coefficients and root-mean-square deviations of our residuals $\lambda$ were seen to be comparable to those of elliptical FP residuals from the literature, measured for galaxies observed in reality and in simulations.

We measured two-point galaxy correlation functions as functions of three-dimensional separations $r$, weighted by the FP residuals $\lambda$: the density-intrinsic size correlation $\xi_{g\lambda}(r)$, and the intrinsic size autocorrelation $\xi_{\lambda\lambda}(r)$. We also measured the galaxy clustering correlation function $\xi_{gg}(r)$ within subsamples selected to be intrinsically small $\lambda_-$, or intrinsically large $\lambda_+$, in order to assess the fractional difference in clustering $\Delta_{gg}(r)$ across the FP.

We found that elliptical galaxies with intrinsically small effective radii $\lambda_-$ are significantly more clustered ($5.6\sigma$) over scales $\sim 0.5–17 h^{-1}$ Mpc than their large counterparts in the simulation, in agreement with the findings of Joachimi et al. (2015) and Singh et al. (2021). However, for our fiducial planes, the significance of this signal is diminished ($0.7\sigma$) upon removal of the lower-mass elliptical galaxies with $10^{10} < M_* / M_{\odot} < 10^{10.5}$.

We explored the dependence of FP residuals, and other elliptical galaxy properties, on the isotropic density of their local environments, estimated according to the galaxy number density contrast $\delta_{g,S}$ in spheres of radius $S$. We saw that the conservative stellar mass threshold removes compact, fainter, lower-dispersion ellipticals from high-density environments, thereby flattening the density gradient between intrinsically small $\lambda_-$ and large $\lambda_+$ samples, and rendering $\Delta_{gg}$ insignificant.

We detected significantly positive $\Delta_{gg}$ ($5.9\sigma$), $\xi_{g\lambda}$ ($5.2\sigma$), and $\xi_{\lambda\lambda}$ ($9\sigma$) signals for the Full spiral FP, showing that, conditional on the presence of lower-mass spirals, intrinsically large spirals are significantly more clustered, and exist in regions of higher density, than small spirals. However, $\Delta_{gg}$ and $\xi_{g\lambda}$ weaken to insignificance upon removal of the lower-mass $M_* < 10^{10.5} M_{\odot}$ subset ($0.6\sigma$ and $1.6\sigma$, respectively), whilst the Masscut $\xi_{\lambda\lambda}$ remains statistically significant, at $3.4\sigma$. This suggests that intrinsic fluctuations in spiral galaxy virial radii are consistently spatially correlated in the simulation, as reckoned by our fiducial spiral FP.

Analysis of the environmental dependence of spiral galaxies' intrinsic sizes and other properties revealed that the mass-selection, affecting $\sim 75$ per cent of the sample, preferentially removes absolutely, and intrinsically small-radius objects from low-density environments. The consequence for $\Delta_{gg}$ and $\xi_{g\lambda}$ is that the signal-to-noise is greatly diminished, whilst the autocorrelation $\xi_{\lambda\lambda}$ relies less upon a broad range of environments for its significance.

We proposed a tentative interpretation of our measured correlations and intrinsic size-isotropic density relations in the context of the findings of Welker et al. (2017). We suggested that compact ellipticals in the simulation form by wet mergers of spiral galaxies, which trigger gas compaction and destruction of the disc structure whilst galaxies migrate from filaments into nodes. Conversely, we speculated that large spirals are the result of elliptical objects grown large through mergers on their approach to filament centres and nodes, which then regress to disc-like morphologies under cold gas flows and minor mergers. We recommend that future work undertake

a detailed consideration of the anisotropic cosmic web environment as it influences accretion mechanisms, and thus the galaxy size-mass-morphology distribution, across cosmic time.

We also conducted a mock estimation of the ISC contamination $\Sigma_{\lambda}(r)$ affecting a convergence-based measurement of the projected surface mass density $\Sigma(r)$, as traced by FP residuals (Huff & Graves 2014). Assuming our $z \sim 0$ ISCs to be representative out to $z \sim 0.2$, we estimated the contribution to $\Sigma(r)$ that could arise due to overestimation of source redshifts at the lens plane, such that 'background' galaxies exhibit size fluctuations that are correlated with the foreground structure, in the absence of any gravitational lensing.

We found that the contamination is likely to be of comparable amplitude to the cosmological signal for large separations $r \gtrsim 10$ Mpc, but subdominant on smaller scales. The spiral- and elliptical-derived contaminations are of similar form but opposite sign, offering a promising route to cross-check the modelling of galaxy type-dependent ISCs under a colour-split magnification analysis (see Li et al. 2021, for a colour-split consistency analysis in the context of cosmic shear).

We constructed a simple phenomenological model for the type-specific intrinsic size fields, assuming them to be linearly proportional to the density field (Alsing et al. 2015; Joachimi et al. 2015) and thus each described by single parameters $B_{\text{spi}}$, $B_{\text{ell}}$, such that $\xi_{\lambda\lambda} = B^2 \xi_{\delta\delta}$. Fitting to our measured $\xi_{\lambda\lambda}$ correlations, we found a strong tendency for a non-zero spiral intrinsic size-density coupling $|B_{\text{spi}}|$ as reckoned by the Full FP sample, weakening to a $\sim 2\sigma$ non-zero preference after the mass-selection, but having a consistent amplitude. For the fit to the Full elliptical FP sample $\xi_{\lambda\lambda}$, we found $|B_{\text{ell}}| > 0$ at almost $3\sigma$, whilst the Masscut constraint is fully consistent with a zero amplitude, having run into the prior.

We converted these models into expectations for intrinsic–intrinsic $C_{\lambda\lambda}(\ell)$, and intrinsic-convergence $|C_{\lambda\kappa}(\ell)|$ angular power spectra that could contaminate the cosmological convergence signal $C_{\kappa\kappa}(\ell)$, assuming that our low-redshift ISCs can be extrapolated to $z \sim 0.4, 0.8$. For the Masscut FP case, we saw that the resulting systematic $\lambda\lambda$, $\lambda\kappa$ contributions from spirals are likely to dominate over the convergence signal at 68 per cent confidence. This increases to a highly significant dominance for the Full FP case, where $\lambda\kappa$ ($\lambda\lambda$) are factors of $\sim 2$ ($\sim$several) larger than $\kappa\kappa$ with small uncertainties.

Meanwhile, we saw that elliptical $\lambda\kappa$ contributions are consistent with those predicted by Joachimi et al. (2015) for the Masscut case, and thus comparable to the convergence signal. The corresponding $\lambda\lambda$ contribution is somewhat weaker, perhaps comparable with $\kappa\kappa$ at the upper-end of the 68 per cent confidence interval – however, the 95 per cent lower-bound is comfortably consistent with a null signal for all Masscut elliptical predictions. The Full FP-derived constraint upon the linear amplitude of elliptical ISCs exceeds that found by Joachimi et al. (2015) at 68 per cent confidence, but is consistent at 95 per cent. Thus the elliptical $\lambda\lambda$, $\kappa\lambda$ contributions predicted from the Full FP are at least comparable to the convergence $\kappa\kappa$, and likely to be dominant at 68 per cent confidence, having amplitudes 2–3 times larger than the cosmological signal. Our results therefore suggest that the impact of intrinsic size correlations upon lensing magnification statistics is likely to require modelling and marginalization, in order to avoid biases in cosmological parameter inference.

We note that our FP residuals showed variably significant correlations with FP variables and other galaxy properties, including measured radii, stellar mass, and surface area, with spirals showing the strongest $\lambda$-property correlations. We investigated these tendencies in the appendix, considering extended FPs with additional variables: the
virial radius of stellar particles $R_{vir}$ to yield planes with manageable degrees of tilting, and scatters comparable to those of the elliptical planes. Lower-mass spirals in the Full FP sample were seen to be particularly well-described, with the tilting of the FP featuring mainly at the high-mass end.

We defined the intrinsic galaxy size fluctuation $\lambda$ as the natural logarithm of the ratio of measured-to-predicted (by the fitted FP) radii for each object, finding these to be approximately normally distributed, and having similar distributions between spiral and elliptical, Full and Masscut FPs. The coefficients and root-mean-square deviations of our residuals $\lambda$ were seen to be comparable to those of elliptical FP residuals from the literature, measured for galaxies observed in reality and in simulations.

We measured two-point galaxy correlation functions as functions of three-dimensional separations $r$, weighted by the FP residuals $\lambda$: the density-intrinsic size correlation $\xi_{g\lambda}(r)$, and the intrinsic size autocorrelation $\xi_{\lambda\lambda}(r)$. We also measured the galaxy clustering correlation function $\xi_{gg}(r)$ within subsamples selected to be intrinsically small $\lambda_-$, or intrinsically large $\lambda_+$, in order to assess the fractional difference in clustering $\Delta_{gg}(r)$ across the FP.

We found that elliptical galaxies with intrinsically small effective radii $\lambda_-$ are significantly more clustered ($5.6\sigma$) over scales $\sim 0.5-17 h^{-1}$ Mpc than their large counterparts in the simulation, in agreement with the findings of Joachimi et al. (2015) and Singh et al. (2021). However, for our fiducial planes, the significance of this signal is diminished ($0.7\sigma$) upon removal of the lower-mass elliptical galaxies with $10^{10} < M_* / M_{\odot} < 10^{10.5}$.

We explored the dependence of FP residuals, and other elliptical galaxy properties, on the isotropic density of their local environments, estimated according to the galaxy number density contrast $\delta_{g,S}$ in spheres of radius $S$. We saw that the conservative stellar mass threshold removes compact, fainter, lower-dispersion ellipticals from high-density environments, thereby flattening the density gradient between intrinsically small $\lambda_-$ and large $\lambda_+$ samples, and rendering $\Delta_{gg}$ insignificant.

We detected significantly positive $\Delta_{gg}$ ($5.9\sigma$), $\xi_{g\lambda}$ ($5.2\sigma$), and $\xi_{\lambda\lambda}$ ($9\sigma$) signals for the Full spiral FP, showing that, conditional on the presence of lower-mass spirals, intrinsically large spirals are significantly more clustered, and exist in regions of higher density, than small spirals. However, $\Delta_{gg}$ and $\xi_{g\lambda}$ weaken to insignificance upon removal of the lower-mass $M_* < 10^{10.5} M_{\odot}$ subset ($0.6\sigma$ and $1.6\sigma$, respectively), whilst the Masscut $\xi_{\lambda\lambda}$ remains statistically significant, at $3.4\sigma$. This suggests that intrinsic fluctuations in spiral galaxy virial radii are consistently spatially correlated in the simulation, as reckoned by our fiducial spiral FP.

Analysis of the environmental dependence of spiral galaxies' intrinsic sizes and other properties revealed that the mass-selection, affecting $\sim 75$ per cent of the sample, preferentially removes absolutely, and intrinsically small-radius objects from low-density environments. The consequence for $\Delta_{gg}$ and $\xi_{g\lambda}$ is that the signal-to-noise is greatly diminished, whilst the autocorrelation $\xi_{\lambda\lambda}$ relies less upon a broad range of environments for its significance.

We proposed a tentative interpretation of our measured correlations and intrinsic size-isotropic density relations in the context of the findings of Welker et al. (2017). We suggested that compact ellipticals in the simulation form by wet mergers of spiral galaxies, which trigger gas compaction and destruction of the disc structure whilst galaxies migrate from filaments into nodes. Conversely, we speculated that large spirals are the result of elliptical objects grown large through mergers on their approach to filament centres and nodes, which then regress to disc-like morphologies under cold gas flows and minor mergers. We recommend that future work undertake

a detailed consideration of the anisotropic cosmic web environment as it influences accretion mechanisms, and thus the galaxy size-mass-morphology distribution, across cosmic time.

We also conducted a mock estimation of the ISC contamination $\Sigma_{\lambda}(r)$ affecting a convergence-based measurement of the projected surface mass density $\Sigma(r)$, as traced by FP residuals (Huff & Graves 2014). Assuming our $z \sim 0$ ISCs to be representative out to $z \sim 0.2$, we estimated the contribution to $\Sigma(r)$ that could arise due to overestimation of source redshifts at the lens plane, such that 'background' galaxies exhibit size fluctuations that are correlated with the foreground structure, in the absence of any gravitational lensing.

We found that the contamination is likely to be of comparable amplitude to the cosmological signal for large separations $r \gtrsim 10$ Mpc, but subdominant on smaller scales. The spiral- and elliptical-derived contaminations are of similar form but opposite sign, offering a promising route to cross-check the modelling of galaxy type-dependent ISCs under a colour-split magnification analysis (see Li et al. 2021, for a colour-split consistency analysis in the context of cosmic shear).

We constructed a simple phenomenological model for the type-specific intrinsic size fields, assuming them to be linearly proportional to the density field (Alsing et al. 2015; Joachimi et al. 2015) and thus each described by single parameters $B_{\text{spi}}$, $B_{\text{ell}}$, such that $\xi_{\lambda\lambda} = B^2 \xi_{\delta\delta}$. Fitting to our measured $\xi_{\lambda\lambda}$ correlations, we found a strong tendency for a non-zero spiral intrinsic size-density coupling $|B_{\text{spi}}|$ as reckoned by the Full FP sample, weakening to a $\sim 2\sigma$ non-zero preference after the mass-selection, but having a consistent amplitude. For the fit to the Full elliptical FP sample $\xi_{\lambda\lambda}$, we found $|B_{\text{ell}}| > 0$ at almost $3\sigma$, whilst the Masscut constraint is fully consistent with a zero amplitude, having run into the prior.

We converted these models into expectations for intrinsic–intrinsic $C_{\lambda\lambda}(\ell)$, and intrinsic-convergence $|C_{\lambda\kappa}(\ell)|$ angular power spectra that could contaminate the cosmological convergence signal $C_{\kappa\kappa}(\ell)$, assuming that our low-redshift ISCs can be extrapolated to $z \sim 0.4, 0.8$. For the Masscut FP case, we saw that the resulting systematic $\lambda\lambda$, $\lambda\kappa$ contributions from spirals are likely to dominate over the convergence signal at 68 per cent confidence. This increases to a highly significant dominance for the Full FP case, where $\lambda\kappa$ ($\lambda\lambda$) are factors of $\sim 2$ ($\sim$several) larger than $\kappa\kappa$ with small uncertainties.

Meanwhile, we saw that elliptical $\lambda\kappa$ contributions are consistent with those predicted by Joachimi et al. (2015) for the Masscut case, and thus comparable to the convergence signal. The corresponding $\lambda\lambda$ contribution is somewhat weaker, perhaps comparable with $\kappa\kappa$ at the upper-end of the 68 per cent confidence interval – however, the 95 per cent lower-bound is comfortably consistent with a null signal for all Masscut elliptical predictions. The Full FP-derived constraint upon the linear amplitude of elliptical ISCs exceeds that found by Joachimi et al. (2015) at 68 per cent confidence, but is consistent at 95 per cent. Thus the elliptical $\lambda\lambda$, $\kappa\lambda$ contributions predicted from the Full FP are at least comparable to the convergence $\kappa\kappa$, and likely to be dominant at 68 per cent confidence, having amplitudes 2–3 times larger than the cosmological signal. Our results therefore suggest that the impact of intrinsic size correlations upon lensing magnification statistics is likely to require modelling and marginalization, in order to avoid biases in cosmological parameter inference.

We note that our FP residuals showed variably significant correlations with FP variables and other galaxy properties, including measured radii, stellar mass, and surface area, with spirals showing the strongest $\lambda$-property correlations. We investigated these tendencies in the appendix, considering extended FPs with additional variables: the
projected surface area for spirals, and the stellar mass for ellipticals. These extended FPs exhibited rms scatters 1.4–1.7 times smaller than the fiducial versions, and were able to soften or negate many of the $\lambda$-property correlations.

ISCs measured for the extended FP residuals were seen to be qualitatively consistent with those from the fiducial FPs, with the detection significance rising to >3σ for the Masscut elliptical & spiral $\Delta_{gg}$, and Full elliptical $\xi_{g\lambda}$, $\xi_{\lambda\lambda}$ signals. The exception was the Masscut spiral $\xi_{\lambda\lambda}$ signal, which was only detected at 2σ according to the extended FP. Thus the observed spiral intrinsic size autocorrelation may have been contaminated by $\lambda$-property correlations within the Masscut FP sample, and our corresponding phenomenological model predictions for the systematic contributions $C_{\lambda\lambda}(\ell)$, $|C_{\lambda\lambda}(\ell)|$ should be considered as upper limits.

A future analysis of intrinsic size correlations in hydrodynamical simulations can improve upon this work first by considering more observationally motivated planes, e.g. using the disc scale-length for spiral FPs (Shen et al. 2002), and by carefully optimizing the planes for the avoidance of FP residual-galaxy property correlations that might create spurious signals or complicate interpretations.

An extension to higher-redshift snapshots could assess the evolution of the ISC effect, and test the validity of the strong assumptions and extrapolations made here. Additionally making use of precise knowledge of the underlying, anisotropic dark structure of the cosmic web, combined with galaxy merger trees and larger simulated volumes (e.g. Horizon Run 5; Lee et al. 2021a), the primary drivers of ISCs could be identified and incorporated into modelling.

More physically motivated prescriptions for ISCs include a unified linear model for spirals and ellipticals (Ghosh et al. 2021), and an effective field theory model (Vlah et al. 2020), and the calibration of these would be greatly aided by the identification of a unified Fundamental Plane describing both morphological galaxy types (Ferrero et al. 2021). Such improvements to the modelling of intrinsic galaxy size correlations could shed new light upon the formation and evolution of galaxies in distinct environments within the cosmic web, as well as facilitating direct forecasting of cosmological parameter biases due to ISC contamination of lensing magnification statistics.

## ACKNOWLEDGEMENTS

The authors acknowledge useful discussions with Andrej Dvornik, Robert Reischke, Corentin Cadiou, and Christophe Saulder. This work is part of the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by Ministry of Education, Culture and Science, Netherlands (OCW). This work has made use of the HPC resources of CINES (Jade and Occigen supercomputer) under the time allocations 2013047012, 2014047012, and 2015047012 made by GENCI. This work is partially supported by the Spin(e) grants ANR-13-BS05-0005 (http://cosmicorigin.org) of the French Agence Nationale de la Recherche and by the ILP LABEX (under reference ANR-10-LABX-63 and ANR11-IDEX-0004-02). Part of the analysis of the simulation was performed on the DiRAC facility jointly funded by STFC, BIS, and the University of Oxford. We thank S. Rouberol for running smoothly the Horizon cluster for us.

## DATA AVAILABILITY

Simulation products can be shared upon reasonable request via the collaboration's webpage: https://www.horizon-simulation.org/data.html or by getting in touch with the corresponding author.

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projected surface area for spirals, and the stellar mass for ellipticals. These extended FPs exhibited rms scatters 1.4–1.7 times smaller than the fiducial versions, and were able to soften or negate many of the $\lambda$-property correlations.

ISCs measured for the extended FP residuals were seen to be qualitatively consistent with those from the fiducial FPs, with the detection significance rising to >3σ for the Masscut elliptical & spiral $\Delta_{gg}$, and Full elliptical $\xi_{g\lambda}$, $\xi_{\lambda\lambda}$ signals. The exception was the Masscut spiral $\xi_{\lambda\lambda}$ signal, which was only detected at 2σ according to the extended FP. Thus the observed spiral intrinsic size autocorrelation may have been contaminated by $\lambda$-property correlations within the Masscut FP sample, and our corresponding phenomenological model predictions for the systematic contributions $C_{\lambda\lambda}(\ell)$, $|C_{\lambda\lambda}(\ell)|$ should be considered as upper limits.

A future analysis of intrinsic size correlations in hydrodynamical simulations can improve upon this work first by considering more observationally motivated planes, e.g. using the disc scale-length for spiral FPs (Shen et al. 2002), and by carefully optimizing the planes for the avoidance of FP residual-galaxy property correlations that might create spurious signals or complicate interpretations.

An extension to higher-redshift snapshots could assess the evolution of the ISC effect, and test the validity of the strong assumptions and extrapolations made here. Additionally making use of precise knowledge of the underlying, anisotropic dark structure of the cosmic web, combined with galaxy merger trees and larger simulated volumes (e.g. Horizon Run 5; Lee et al. 2021a), the primary drivers of ISCs could be identified and incorporated into modelling.

More physically motivated prescriptions for ISCs include a unified linear model for spirals and ellipticals (Ghosh et al. 2021), and an effective field theory model (Vlah et al. 2020), and the calibration of these would be greatly aided by the identification of a unified Fundamental Plane describing both morphological galaxy types (Ferrero et al. 2021). Such improvements to the modelling of intrinsic galaxy size correlations could shed new light upon the formation and evolution of galaxies in distinct environments within the cosmic web, as well as facilitating direct forecasting of cosmological parameter biases due to ISC contamination of lensing magnification statistics.

## ACKNOWLEDGEMENTS

The authors acknowledge useful discussions with Andrej Dvornik, Robert Reischke, Corentin Cadiou, and Christophe Saulder. This work is part of the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by Ministry of Education, Culture and Science, Netherlands (OCW). This work has made use of the HPC resources of CINES (Jade and Occigen supercomputer) under the time allocations 2013047012, 2014047012, and 2015047012 made by GENCI. This work is partially supported by the Spin(e) grants ANR-13-BS05-0005 (http://cosmicorigin.org) of the French Agence Nationale de la Recherche and by the ILP LABEX (under reference ANR-10-LABX-63 and ANR11-IDEX-0004-02). Part of the analysis of the simulation was performed on the DiRAC facility jointly funded by STFC, BIS, and the University of Oxford. We thank S. Rouberol for running smoothly the Horizon cluster for us.

## DATA AVAILABILITY

Simulation products can be shared upon reasonable request via the collaboration's webpage: https://www.horizon-simulation.org/data.html or by getting in touch with the corresponding author.

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## APPENDIX A: EXTENDED FUNDAMENTAL PLANES

We outline here the extended Fundamental Planes described in Section 5.3, which are intended to reduce the correlations between FP residuals $\lambda$ and galaxy properties, shown for the fiducial planes in Fig. 4. To that end, we include a fourth variable in each FP: the stellar mass $M_*$ for ellipticals, and the projected surface area $A$ for spirals. The extended planes are thus given as

$$ \log R_{\text{eff}} = a \log \sigma + b \log I + d \log M_{*} + c, \quad (A1) $$

for ellipticals, and

$$ \log R_{\text{vir}*} = \alpha M_r + \beta \log V_c + \delta \log A + \gamma, \quad (A2) $$

for spirals. The resulting coefficients are given in Table A1, where one sees that the velocity dispersion slope is heavily flattened and reversed in sign, also picking up a much larger scatter such that $a$ is consistent with zero, whilst the new stellar mass slope $d$ is positive. For spirals, absolute magnitude and circular velocity slopes $\alpha, \beta$ are softened, and the new surface area slope is positive.

The agreement between Masscut and Full FP coefficients is improved for $a, b, \alpha$, and $\beta$, with $\alpha$ becoming consistent at $1\sigma$, and $\beta$ at $2\sigma$, whilst intercepts $c, \gamma$ are almost unchanged. The additional coefficients $d, \delta$ are also seen to be consistent at $1\sigma$ between the Masscut and Full planes. The rms scatters of extended FP residuals $\sigma_{FP}$ are reduced by factors of ~1.5 for ellipticals, and ~1.6 for spirals, compared with the fiducial FPs (Table 1). Each extended plane also shows a significantly reduced tilt with respect to its fiducial counterpart, which can be clearly seen for the Full elliptical FP in Fig. A1. The gradients of linear fits to each extended plane (green dashed line(s) in Fig. A1, or in Fig. 3 for the fiducial planes) are consistent with unity at $<1\sigma$ for ellipticals, $<2\sigma$ for Masscut spirals, and $<3\sigma$ for the Full spiral sample.

A selection of FP residual property correlations are shown in Fig. A2, where grey curves give the fiducial relations, reproduced from Fig. 4, and coloured curves give those for the extended FPs – each is renormalized here by the standard deviation $\sigma(\lambda)$, to aid with the comparison. One sees that the extended FPs either reduce the $\lambda$-property correlations or leave them unchanged. Worrisome persistent correlations are those with $R_{\text{vir}*}, M_*$ for spirals, and $R_{\text{eff}}$ for ellipticals – future work should explore further methods for the mitigation of these correlations, such that derived intrinsic size correlations will be as free from contamination as possible.

We measure ISCs for the extended FPs as described in Section 4, and find signals broadly consistent with those measured for the

<figure><img src="image_12.png" /><figcaption>**Figure A1.** The Full elliptical sample extended Fundamental Plane (analogous to top-left-hand panel of Fig. 3), given by equation (A1). The 1:1 relation is shown in cyan, a linear fit to all points on the plane, and its gradient $m$, are shown in green (Section 3.2), and contours illustrating 68 and 95 per cent of $M_* < 10^{10.5} M_{\odot}$ objects (Section 2) are shown in red. The addition of stellar mass as a fourth Fundamental Plane variable is seen to effectively reduce the scatter and tilting of the plane.</figcaption></figure>

fiducial FPs. Extended FP signals, however, exhibit generally greater significance of detection, with both the Masscut and Full elliptical $\Delta_{gg}$ signals becoming highly significant at $>8\sigma$, having similar form and amplitude to the fiducial Full FP signal in Fig. 6. The Full elliptical FP $\xi_{g\lambda}$ also becomes statistically significant (3$\sigma$) whilst maintaining its form and amplitude, as does the Full elliptical $\xi_{\lambda\lambda}$ signal (3.3$\sigma$) though with a slight reduction in amplitude, particularly at small scales. The Masscut elliptical $\xi_{g\lambda}, \xi_{\lambda\lambda}$ signals also maintain their form whilst rising in significance, but remain at $<2\sigma$.

For spirals, the extended Masscut FP $\Delta_{gg}$ is detected at $3.5\sigma$, at a slightly higher (but consistent) amplitude than the fiducial signal. Other spiral signals are very similar in form and significance between the fiducial and extended FPs, with the exception of $\xi_{\lambda\lambda}$. For the extended Masscut and Full FPs, the significance of detection of $\xi_{\lambda\lambda}$ drops by $3.4\sigma \rightarrow 2\sigma$, and $9\sigma \rightarrow 5.2\sigma$, respectively, accompanied by reductions in amplitude by factors of ~3, whilst the shapes of the signals are maintained. It may then be that the fiducial $\xi_{\lambda\lambda}$ signals are partially contaminated by correlations between FP residuals $\lambda$ and galaxy properties; for example $R_{\text{eff}}$, or the surface area and derived quantities $I, \Sigma$, each of which is largely erased by the extended FP (Fig. A2). We leave a detailed investigation of these trends to future studies of spiral Fundamental Planes.

For now, we remark that the net effect of the extended FPs upon $\xi_{\lambda\lambda}$ signals is to lower their amplitudes (except for Masscut ellipticals, which are in either case consistent with a null detection). This causes agreement to improve between our phenomenological model fits to elliptical $\xi_{g\lambda}$ (Section 4.1.2), and those of Joachimi et al. (2015) to the projection of $\xi_{g\lambda}$ in observational data, for both Masscut and Full FPs,

<table><caption>**Table A1.** Sample details and extended Fundamental Plane constraints for elliptical and spiral galaxy samples defined in the Horizon-AGN simulation. Columns are the same as in Table 1, but now including a fourth variable $d|\delta$ for the extended Fundamental Planes described in Appendix A, and without the Lens samples, which are not fitted with Fundamental Planes.</caption><thead><tr><th>FP/lens sample</th><th><i>N</i></th><th>⟨<i>L</i>⟩/<i>L</i><sub>vir</sub></th><th><i>a</i> | <i>α</i></th><th><i>b</i> | <i>β</i></th><th><i>d</i> | <i>δ</i></th><th><i>c</i> | <i>γ</i></th><th>σ<sub>FP</sub></th></tr></thead><tbody><tr><td>Elliptical FP</td><td>6254</td><td>0.55</td><td>−0.146 ± 0.309</td><td>−0.348 ± 0.055</td><td>0.421 ± 0.092</td><td>0.737 ± 0.013</td><td>0.0453</td></tr><tr><td>Elliptical FP (&gt;10<sup>10.5</sup> M<sub>☉</sub>)</td><td>3684</td><td>0.84</td><td>−0.184 ± 0.424</td><td>−0.354 ± 0.072</td><td>0.436 ± 0.142</td><td>0.838 ± 0.017</td><td>0.0495</td></tr><tr><td>Spiral FP</td><td>26215</td><td>0.24</td><td>−0.145 ± 0.017</td><td>−0.485 ± 0.105</td><td>0.331 ± 0.029</td><td>−1.497 ± 0.006</td><td>0.0507</td></tr><tr><td>Spiral FP (&gt;10<sup>10.5</sup> M<sub>☉</sub>)</td><td>6394</td><td>0.59</td><td>−0.182 ± 0.035</td><td>−0.891 ± 0.229</td><td>0.291 ± 0.054</td><td>−1.294 ± 0.013</td><td>0.0557</td></tr></tbody></table>
Yao J., Shan H., Zhang P., Kneib J.-P., Jullo E., 2020b, ApJ, 904, 135
York D. G. et al., 2000, AJ, 120, 1579
Zandivarez A., Martínez H. J., 2011, MNRAS, 415, 2553
Zhang P., 2010a, MNRAS, 406, L95
Zhang P., 2010b, ApJ, 720, 1090
Zucca E. et al., 2009, A&A, 508, 1217

APPENDIX A: EXTENDED FUNDAMENTAL PLANES

We outline here the extended Fundamental Planes described in Section 5.3, which are intended to reduce the correlations between FP residuals $\lambda$ and galaxy properties, shown for the fiducial planes in Fig. 4. To that end, we include a fourth variable in each FP: the stellar mass $M_*$ for ellipticals, and the projected surface area A for spirals. The extended planes are thus given as

$$ \log R_{\text{eff}} = a \log \sigma + b \log I + d \log M_{*} + c, \quad (A1) $$

for ellipticals, and

$$ \log R_{\text{vir}*} = \alpha M_r + \beta \log V_c + \delta \log A + \gamma, \quad (A2) $$

for spirals. The resulting coefficients are given in Table A1, where one sees that the velocity dispersion slope is heavily flattened and reversed in sign, also picking up a much larger scatter such that a is consistent with zero, whilst the new stellar mass slope d is positive. For spirals, absolute magnitude and circular velocity slopes $\alpha$, $\beta$ are softened, and the new surface area slope is positive.

The agreement between Masscut and Full FP coefficients is improved for a, b, $\alpha$, and $\beta$, with $\alpha$ becoming consistent at $1\sigma$, and $\beta$ at $2\sigma$, whilst intercepts c, $\gamma$ are almost unchanged. The additional coefficients d, $\delta$ are also seen to be consistent at $1\sigma$ between the Masscut and Full planes. The rms scatters of extended FP residuals $\sigma_{FP}$ are reduced by factors of ~1.5 for ellipticals, and ~1.6 for spirals, compared with the fiducial FPs (Table 1). Each extended plane also shows a significantly reduced tilt with respect to its fiducial counterpart, which can be clearly seen for the Full elliptical FP in Fig. A1. The gradients of linear fits to each extended plane (green dashed line(s) in Fig. A1, or in Fig. 3 for the fiducial planes) are consistent with unity at $<1\sigma$ for ellipticals, $<2\sigma$ for Masscut spirals, and $<3\sigma$ for the Full spiral sample.

A selection of FP residual property correlations are shown in Fig. A2, where grey curves give the fiducial relations, reproduced from Fig. 4, and coloured curves give those for the extended FPs – each is renormalized here by the standard deviation $\sigma(\lambda)$, to aid with the comparison. One sees that the extended FPs either reduce the $\lambda$-property correlations or leave them unchanged. Worrisome persistent correlations are those with $R_{\text{vir}*}$, $M_*$ for spirals, and $R_{\text{eff}}$ for ellipticals – future work should explore further methods for the mitigation of these correlations, such that derived intrinsic size correlations will be as free from contamination as possible.

We measure ISCs for the extended FPs as described in Section 4, and find signals broadly consistent with those measured for the

<figure><img src="image_14.png" /><figcaption>**Figure A1.** The Full elliptical sample extended Fundamental Plane (analogous to top-left-hand panel of Fig. 3), given by equation (A1). The 1:1 relation is shown in cyan, a linear fit to all points on the plane, and its gradient $m$, are shown in green (Section 3.2), and contours illustrating 68 and 95 per cent of $M_* < 10^{10.5} M_{\odot}$ objects (Section 2) are shown in red. The addition of stellar mass as a fourth Fundamental Plane variable is seen to effectively reduce the scatter and tilting of the plane.</figcaption></figure>

fiducial FPs. Extended FP signals, however, exhibit generally greater significance of detection, with both the Masscut and Full elliptical $\Delta_{gg}$ signals becoming highly significant at $>8\sigma$, having similar form and amplitude to the fiducial Full FP signal in Fig. 6. The Full elliptical FP $\xi_{g\lambda}$ also becomes statistically significant ($3\sigma$) whilst maintaining its form and amplitude, as does the Full elliptical $\xi_{\lambda\lambda}$ signal ($3.3\sigma$) though with a slight reduction in amplitude, particularly at small scales. The Masscut elliptical $\xi_{g\lambda}$, $\xi_{\lambda\lambda}$ signals also maintain their form whilst rising in significance, but remain at $<2\sigma$.

For spirals, the extended Masscut FP $\Delta_{gg}$ is detected at $3.5\sigma$, at a slightly higher (but consistent) amplitude than the fiducial signal. Other spiral signals are very similar in form and significance between the fiducial and extended FPs, with the exception of $\xi_{\lambda\lambda}$. For the extended Masscut and Full FPs, the significance of detection of $\xi_{\lambda\lambda}$ drops by $3.4\sigma \rightarrow 2\sigma$, and $9\sigma \rightarrow 5.2\sigma$, respectively, accompanied by reductions in amplitude by factors of ~3, whilst the shapes of the signals are maintained. It may then be that the fiducial $\xi_{\lambda\lambda}$ signals are partially contaminated by correlations between FP residuals $\lambda$ and galaxy properties; for example $R_{\text{eff}}$, or the surface area and derived quantities I, $\Sigma$, each of which is largely erased by the extended FP (Fig. A2). We leave a detailed investigation of these trends to future studies of spiral Fundamental Planes.

For now, we remark that the net effect of the extended FPs upon $\xi_{\lambda\lambda}$ signals is to lower their amplitudes (except for Masscut ellipticals, which are in either case consistent with a null detection). This causes agreement to improve between our phenomenological model fits to elliptical $\xi_{g\lambda}$ (Section 4.1.2), and those of Joachimi et al. (2015) to the projection of $\xi_{g\lambda}$ in observational data, for both Masscut and Full FPs,

<table><caption>**Table A1.** Sample details and extended Fundamental Plane constraints for elliptical and spiral galaxy samples defined in the Horizon-AGN simulation. Columns are the same as in Table 1, but now including a fourth variable $d|\delta$ for the extended Fundamental Planes described in Appendix A, and without the Lens samples, which are not fitted with Fundamental Planes.</caption><thead><tr><th>FP/lens sample</th><th>N</th><th>⟨L⟩/L<sub>pix</sub></th><th>a | α</th><th>b | β</th><th>d | δ</th><th>c | γ</th><th>σ<sub>FP</sub></th></tr></thead><tbody><tr><td>Elliptical FP</td><td>6254</td><td>0.55</td><td>-0.146 ± 0.309</td><td>-0.348 ± 0.055</td><td>0.421 ± 0.092</td><td>0.737 ± 0.013</td><td>0.0453</td></tr><tr><td>Elliptical FP (&gt;10<sup>10.5</sup> M<sub>☉</sub>)</td><td>3684</td><td>0.84</td><td>-0.184 ± 0.424</td><td>-0.354 ± 0.072</td><td>0.436 ± 0.142</td><td>0.838 ± 0.017</td><td>0.0495</td></tr><tr><td>Spiral FP</td><td>26215</td><td>0.24</td><td>-0.145 ± 0.017</td><td>-0.485 ± 0.105</td><td>0.331 ± 0.029</td><td>-1.497 ± 0.006</td><td>0.0507</td></tr><tr><td>Spiral FP (&gt;10<sup>10.5</sup> M<sub>☉</sub>)</td><td>6394</td><td>0.59</td><td>-0.182 ± 0.035</td><td>-0.891 ± 0.229</td><td>0.291 ± 0.054</td><td>-1.294 ± 0.013</td><td>0.0557</td></tr></tbody></table>
<figure><img src="image_13.png" /></figure>

**Figure A2.** Intrinsic sizes $\lambda$, as estimated from Fundamental Plane residuals (Section 3), normalized by their respective standard deviations $\sigma(\lambda)$ (to ensure a clear comparison between planes with different scatters), versus a selection of galaxy properties from Fig. 4, which displayed non-zero correlations for the fiducial planes. Those relations are reproduced here in grey, whilst colours give the equivalent relations for the extended planes. Rows give, respectively, the Masscut spiral sample, the Full spiral sample, the Masscut elliptical sample, and the Full elliptical sample. In each panel, the extended plane is seen to reduce the degree of correlation between plane residuals $\lambda$ and the respective galaxy property, or else leave it unchanged.

<figure><img src="image_14.png" /></figure>

**Figure A3.** The Masscut elliptical sample extended Fundamental Plane, coloured according to the local galaxy density contrast $\delta_{g,S}$ estimated in spheres of radius $S = 1 \text{ h}^{-1} \text{ Mpc}$ (analogous to top-left-hand panel of Fig. 8). The 1:1 relation, the linear fit to all points on the plane (Section 3.2), and the contours illustrating the distribution of $M_* < 10^{10.5} \text{ M}_\odot$ objects (Section 2), are reproduced from Fig. 3, here in black. Annotations $\lambda_-$ and $\lambda_+$ denote the sides of the plane corresponding to intrinsically small and large objects, respectively. The local density gradient from intrinsically small $\lambda_-$ to large $\lambda_+$ objects is stronger for the extended than for the fiducial FP (Fig. 8; bottom-left-hand panel), and results in a significantly negative detection of $\Delta_{gg}$ (equation 12); that is, intrinsically small ellipticals defined on the extended Masscut FP are significantly more clustered than their large counterparts, which was not seen for the fiducial Masscut plane.

where the latter becomes consistent at 68 per cent confidence. The $|B_{sp}|$ constraint from the Masscut $\xi_{\lambda\lambda}$ signal, meanwhile, becomes consistent with zero; the predictions for contamination of $C_{\kappa\kappa}$ in the upper panels of Fig. 11 should thus be taken with moderation, and considered as somewhat pessimistic. For the Full FPs, the picture is largely unchanged, but for the reductions in amplitudes. The extended Full FPs therefore predict elliptical, rather than spiral, ISCs to be the dominant systematic contribution to $C_{\kappa\kappa}$, but with a larger uncertainty than the spiral contribution. Each of the extended FP elliptical, spiral, $\lambda\lambda$, and $\lambda\kappa$ contributions are within a factor ~2 of the cosmological signal.

The increased significance of the Masscut $\Delta_{gg}$ signal for elliptical galaxies can be seen in Fig. A3 to derive from the clearer segregation of intrinsically large ellipticals into regions of lower

<figure><img src="image_15.png" /></figure>

**Figure A4.** Predicted intrinsic contributions $\Sigma_{\lambda}(r)$ to the projected surface density $\Sigma(r)$, as measured from angular galaxy position-size correlations in the presence of catastrophic photometric redshift outliers, as in Fig. 9 (see Section 4.1.1), but now computed with residuals $\lambda$ from the Masscut extended Fundamental Plane, and showing only the interloper fraction $f_i = 0$ panel. The inclusion of the spiral surface area as a fourth Fundamental Plane variable results in a greatly increased significance for the spiral $\Sigma_{\lambda}(r)$ contribution.

density (as characterized within spheres of radius $S = 1 \text{ h}^{-1} \text{ Mpc}$), compared to that seen in Fig. 8. Similar trends are seen on large isotropic smoothing scales for spiral galaxies, consistent with our suggestions that intrinsic size gradients are apparent across different environments (Section 5.3) for ellipticals (nodes versus filaments) and spirals (filament centres versus outskirts and the field), and the reduced tilting of both planes also aids with clearer intrinsic size segregation.

Lastly, we note that the extended FPs yield small differences in our mock measurements of the ISC contamination $\Sigma_{\lambda}(r)$ of the projected surface density $\Sigma(r)$, as measured by FP residuals (Huff & Graves 2014). Masscut signals are seen to be of generally higher significance, and whilst the elliptical contribution remains at $\le 0.5\sigma$, the spiral contribution rises to almost $3\sigma$ when the interloper fraction $f_i$ (Section 4.1.1) is catastrophically underestimated, as can be seen in Fig. A4. For the Full FPs, significances are more similar to the fiducial case when $f_i$ is seriously underestimated, but drop faster as the true value $f_i = 1$ is approached, at which point both
<figure><img src="image_15.png" /></figure>

**Figure A2.** Intrinsic sizes $\lambda$, as estimated from Fundamental Plane residuals (Section 3), normalized by their respective standard deviations $\sigma(\lambda)$ (to ensure a clear comparison between planes with different scatters), versus a selection of galaxy properties from Fig. 4, which displayed non-zero correlations for the fiducial planes. Those relations are reproduced here in grey, whilst colours give the equivalent relations for the extended planes. Rows give, respectively, the Masscut spiral sample, the Full spiral sample, the Masscut elliptical sample, and the Full elliptical sample. In each panel, the extended plane is seen to reduce the degree of correlation between plane residuals $\lambda$ and the respective galaxy property, or else leave it unchanged.

<figure><img src="image_16.png" /></figure>

**Figure A3.** The Masscut elliptical sample extended Fundamental Plane, coloured according to the local galaxy density contrast $\delta_{g,S}$ estimated in spheres of radius $S = 1 \ h^{-1} \ \text{Mpc}$ (analogous to top-left-hand panel of Fig. 8). The 1:1 relation, the linear fit to all points on the plane (Section 3.2), and the contours illustrating the distribution of $M_* < 10^{10.5} \ M_{\odot}$ objects (Section 2), are reproduced from Fig. 3, here in black. Annotations $\lambda_-$ and $\lambda_+$ denote the sides of the plane corresponding to intrinsically small and large objects, respectively. The local density gradient from intrinsically small $\lambda_-$ to large $\lambda_+$ objects is stronger for the extended than for the fiducial FP (Fig. 8; bottom-left-hand panel), and results in a significantly negative detection of $\Delta_{gg}$ (equation 12); that is, intrinsically small ellipticals defined on the extended Masscut FP are significantly more clustered than their large counterparts, which was not seen for the fiducial Masscut plane.

where the latter becomes consistent at 68 per cent confidence. The $|B_{sp}|$ constraint from the Masscut $\xi_{\lambda\lambda}$ signal, meanwhile, becomes consistent with zero; the predictions for contamination of $C_{\kappa\kappa}$ in the upper panels of Fig. 11 should thus be taken with moderation, and considered as somewhat pessimistic. For the Full FPs, the picture is largely unchanged, but for the reductions in amplitudes. The extended Full FPs therefore predict elliptical, rather than spiral, ISCs to be the dominant systematic contribution to $C_{\kappa\kappa}$, but with a larger uncertainty than the spiral contribution. Each of the extended FP elliptical, spiral, $\lambda\lambda$, and $\lambda\kappa$ contributions are within a factor ~2 of the cosmological signal.

The increased significance of the Masscut $\Delta_{gg}$ signal for elliptical galaxies can be seen in Fig. A3 to derive from the clearer segregation of intrinsically large ellipticals into regions of lower

<figure><img src="image_17.png" /></figure>

**Figure A4.** Predicted intrinsic contributions $\Sigma_{\lambda}(r)$ to the projected surface density $\Sigma(r)$, as measured from angular galaxy position-size correlations in the presence of catastrophic photometric redshift outliers, as in Fig. 9 (see Section 4.1.1), but now computed with residuals $\lambda$ from the Masscut extended Fundamental Plane, and showing only the interloper fraction $f_i = 0$ panel. The inclusion of the spiral surface area as a fourth Fundamental Plane variable results in a greatly increased significance for the spiral $\Sigma_{\lambda}(r)$ contribution.

density (as characterized within spheres of radius $S = 1 \ h^{-1} \ \text{Mpc}$), compared to that seen in Fig. 8. Similar trends are seen on large isotropic smoothing scales for spiral galaxies, consistent with our suggestions that intrinsic size gradients are apparent across different environments (Section 5.3) for ellipticals (nodes versus filaments) and spirals (filament centres versus outskirts and the field), and the reduced tilting of both planes also aids with clearer intrinsic size segregation.

Lastly, we note that the extended FPs yield small differences in our mock measurements of the ISC contamination $\Sigma_{\lambda}(r)$ of the projected surface density $\Sigma(r)$, as measured by FP residuals (Huff & Graves 2014). Masscut signals are seen to be of generally higher significance, and whilst the elliptical contribution remains at $\le 0.5\sigma$, the spiral contribution rises to almost $3\sigma$ when the interloper fraction $f_i$ (Section 4.1.1) is catastrophically underestimated, as can be seen in Fig. A4. For the Full FPs, significances are more similar to the fiducial case when $f_i$ is seriously underestimated, but drop faster as the true value $f_i = 1$ is approached, at which point both
contributions drop to <1σ significance. Our conclusions are thus
maintained in the sense that an underestimation of $f_i$ could lead to
significant contamination of $\Sigma(r)$ at large scales, whilst we might be
more optimistic about the possibility of correcting for the intrinsic
contributions with accurate characterizations of $f_i$. More exploration

of these potential contaminants with optimized Fundamental Planes
could be of great value.

This paper has been typeset from a LaTeX file prepared by the author.
contributions drop to <1σ significance. Our conclusions are thus
maintained in the sense that an underestimation of $f_i$ could lead to
significant contamination of $\Sigma(r)$ at large scales, whilst we might be
more optimistic about the possibility of correcting for the intrinsic
contributions with accurate characterizations of $f_i$. More exploration

of these potential contaminants with optimized Fundamental Planes
could be of great value.

This paper has been typeset from a LaTeX file prepared by the author.