Size–Mass Relation Shows Its Colours: Contrasting Physical Imprints of Galaxy Evolution in Rest-Frame UV and Optical

Abstract

The galaxy size–mass relation (SMR) is a key scaling relation used to constrain the physical processes that build galaxy structure, yet it is almost always measured in a single rest-frame optical band, where the light traces the bulk of the old stellar mass. Tracing younger populations with flux-weighted ages of ∼100–500 Myr and low-metallicity stars, the rest-frame near-ultraviolet opens a new stellar window on this scaling relation. Because each process redistributes the light of young and old stars differently, the same mechanism shifts the slope and zero point of the SMR by different amounts in the two wavelength regimes. Here we review and synthesize the effects of main physical processes on the form of the SMR for star-forming and quiescent galaxies in the rest-UV and optical. For each process, we start from its underlying physics, the galaxy stellar masses it affects, and the light it adds/removes/rearranges, anchoring the predictions to observations and simulations. We validate the predicted imprints with forward Monte Carlo modelling. The two-wavelength view breaks several degeneracies that single-band analyses cannot, most notably between minor mergers, dry major mergers, and adiabatic expansion. These results motivate joint rest-UV and optical SMR measurements with current and upcoming wide-field imaging surveys.

1. Introduction

Galaxies follow a tight power-law correlation between effective radius $R_e$ and stellar mass $M_{*}$ since at least $z\sim 8$ [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. This size–mass relation (SMR) is commonly parametrized as

$$\log R_e=\log R_0+\alpha \log {\displaystyle \frac{M_{*}}{M_0}},$$

(1)

where $R_0$ and $\alpha$ are fitting parameters (zero point and slope, respectively), and $M_0$ is a normalization mass. Studies often choose the fiducial normalization mass to be $M_0=5\times {10}^{10}{M}_{\odot }$ so that $R_0$ approximately gives the characteristic size of a Milky Way-massed galaxy [17]. This fiducial mass also coincides with the knee of the galaxy stellar mass function ($\log (M^{★}/M_{\odot })\approx 10.7$; [18]), and lies in the regime where both star-forming and quiescent populations are sufficiently well sampled to constrain $\alpha$ and $\log R_0$ jointly.

Star-forming galaxies (SFGs) and quiescent galaxies (QGs) populate distinct loci in the size–stellar mass plane. At any given redshift and stellar mass, SFGs tend to have larger characteristic sizes than QGs except at very high masses ($M_{*}>{10}^{11}{M}_{\odot }$) [1,3,8,10]. Although the SMR slope is uniform for SFGs at $\log M_{*}/M_{\odot }\gtrsim 9$, the slope for QGs changes at a pivot stellar mass, $\log M_{*}/M_{\odot }\sim 10.4$, where the QGs exhibit a steeper relation at higher masses than lower masses [8,10,19]. The differences in the SMR parameters $(\log R_0,\alpha )$ between SFGs, low-mass QGs and high-mass QGs arise because these parameters encode the cumulative imprint of the processes that build galaxy structure—in situ star formation, bulge growth, mergers, accretion, environmental harassment, ram-pressure and tidal stripping, and the population-level effect of progenitor bias (for reviews, see [20,21,22]). Tracing the evolution of the SMRs over cosmic time is therefore a useful tool for exploring the physical processes that shape galaxies.

One of the best ways to trace galaxy evolution in the size–mass plane would be to use the mass-weighted effective radius (half-mass radius, $R_m$; [23,24]). However, measuring $R_m$ is observationally more demanding than measuring the half-light radius, $R_e$, because the former requires the mass-to-light ratio ($M/L$) as a function of radius. Furthermore, pinning down that spatially varying $M/L$ needs spatially resolved photometry over a broad wavelength range. Measuring $R_e$ from a single-band sky image, by contrast, is far more straightforward. A vast majority of observational works therefore use light-weighted $R_e$ in analysing the SMR.

However, light-weighted sizes are sensitive to the wavelength at which they are measured because different types of stars dominate the flux density in different wavelength regimes. In addition, the mapping from rest-frame to the observed band is redshift-dependent. Hence, it is important to use multi-band data to trace the SMR evolution in the same rest-frame wavelength across cosmic time. In practice, at least since the cosmic noon, observational works commonly infer the SMR in a single rest-frame optical/near-IR band, where light traces the old, mass-dominant stellar population [25,26].

The rest-frame near-UV is, however, a complementary tracer: it is sensitive to a flux-weighted age of ∼300 Myr [25,27,28] and to low-metallicity stars [29,30], and thus probes a different stellar component of the same galaxies. Galaxies generally have negative colour gradients with bluer outskirts [23,24,31] and UV sizes are systematically larger than optical sizes in both SFGs and QGs [10,11]. Because each physical process in galaxy evolution transforms the SMR in its own way, the UV and optical SMRs respond to each process with varying sensitivity and, in some cases, in opposite directions. The two-wavelength comparison (UV vs. optical) therefore breaks several degeneracies that single-band analyses are subject to.

The motivation for this synthesis is twofold. First, a growing body of multi-band SMR and structural-evolution work has emerged in recent years. This includes spatially resolved colour-gradient studies [23,24,31,32,33], bulge+disk decompositions across wavelengths [34,35,36,37], and statistical UV+optical SMR measurements at $z<1$ [10,11]. The theoretical interpretations of these results illustrate how the same physical processes leave qualitatively different fingerprints in the different bands. These differences call for a unified theoretical framework in which the SMRs in rest-frame UV and optical are analysed in tandem. This can disentangle process-specific signatures far more effectively than any single wavelength band alone.

Second, recent and upcoming multi-wavelength imaging datasets are making it possible to study the evolution of the SMR at multiple rest-frame wavelengths for very large galaxy samples. Deep U-band imaging from CFHT–CLAUDS [38], combined with Subaru HSC-SSP deep optical imaging [39,40], already delivers joint U+optical photometry over ∼18.6 square degrees. The Ultraviolet Near-Infrared Optical Northern Survey (UNIONS; [41]) extends this coverage to ∼6250 square degrees of the northern sky at shallower depth. Looking ahead, the Vera C. Rubin Observatory’s LSST [42], Euclid [43], and Roman [44] will extend the joint UV+optical SMR into the low-mass regime for ${10}^6$–${10}^7$ galaxies. The Euclid NISP and Roman WFI near-IR channels in particular will additionally deliver rest-frame ∼1 $\mathsf{\mu }$m sizes for these very large samples—a regime that traces the bulk old stellar mass even more faithfully than the rest-optical and is largely insensitive to dust attenuation. Together, these datasets extend the two-wavelength comparison developed here to a UV/optical/near-IR triplet for the same galaxies.

In this paper we review and synthesize how the multi-wavelength SMR enabled by such multi-wavelength surveys can be used to discriminate between the physical processes that drive galaxy evolution. Working process by process, we ask how each mechanism redistributes UV-bright and optical-bright stellar light in individual galaxies, what trajectory this implies in the size–stellar-mass plane, and what the population-level signature is on the slope and zero point of the SMR in each wavelength regime.

This paper is organized as follows. Section 2 defines the wavelength contrast and our standard SMR baseline. Section 3 provides a process-by-process discussion, organized into a subsection for SFGs (Section 3.1) and a subsection for QGs (Section 3.2). For every process we describe (1) the process itself, (2) how it redistributes UV-bright and optical-bright stellar light in individual galaxies, (3) the trajectory of an individual galaxy in the size–mass plane, (4) the mass range in which the process is most efficient, and (5) the resulting changes in the slope and zero point of the SMR in UV versus optical. Section 4 discusses degeneracies (age–metallicity, dust, the UV upturn, and single-Sérsic versus bulge+disk biases) that complicate the empirical separation of these effects. Table 1 compiles the qualitative directions for both populations and Figure 1 and Figure 2 visualize them. We close in Section 5 with a set of testable diagnostics for ongoing and upcoming surveys.

Table 1. Predicted directions of the changes in the slope $\alpha$ and zero point $\log R_0$ of the galaxy size–mass relation in rest-frame UV and rest-frame optical, process by process, separately for star-forming galaxies and quiescent galaxies. The signs and wavelength contrasts derive from the literature synthesized in Section 3. Symbols: ↑ increase, ↓ decrease, ∼ negligible. “UV > Opt” denotes that the magnitude of the shift is larger in UV than in optical; sign columns indicate the direction in each wavelength separately. The last column grades the population-averaged amplitude of each process over $z=1\to 0$ from order-of-magnitude estimates, which combine the measured rates, timescales, affected population fractions, and size–mass-plane tracks compiled in Section 3.

Pop.	Process	$\mathbf{\Delta }\mathit{\alpha }$		$\mathbf{\Delta }\log {\mathit{R}}_0$		Wavelength Contrast	Amplitude a
		UV	Opt	UV	Opt
SFG	In situ SF/inside-out disk growth	↓	↓	↑	↑	UV ≳ Opt	significant
	Bulge growth/inside-out quenching	∼	↓	∼	↓	UV < Opt	modest
	Wet major mergers	∼	∼	↓	↓	UV > Opt d	modest
	Minor mergers & accretion	↑	↑	↑	↑	UV ∼ Opt	modest
	High-velocity encounters	↑	↑	↓	↓	UV > Opt	negligible b
	Ram-pressure/tidal stripping	↑	↑ / ∼	↓	↓ / ∼	UV > Opt	negligible b
QG	Dry major mergers	↑	↑	↑	↑	UV ∼ Opt	modest
	Minor mergers & accretion	↑	↑	↑	↑	UV > Opt	significant
	Adiabatic expansion	↓	↓	↑	↑	UV ∼ Opt	negligible c
	High-velocity encounters	∼	∼	∼	∼	UV ∼ Opt	negligible b
	Tidal stripping	↑	↑ / ∼	↓	↓ / ∼	UV > Opt	negligible b
	Progenitor bias	↓	↓	↑	↑	UV < Opt	significant

Notes: The predictions are for galaxies above the QG SMR pivot mass ($\log M_{*}/M_{\odot }\gtrsim 10.4$) and SFGs above $\log M_{*}/M_{\odot }\gtrsim 9.5$. Sign symbols apply qualitatively to the steady-state effect of each process averaged over the population on which it is most efficient. Because mergers and stripping events are stochastic, all of those entries also broaden the intrinsic scatter of the SMR, in addition to shifting its slope and intercept. Throughout this table, $\Delta \log R_0$ denotes the shift in mean $\log R_e$ at the population fiducial mass ($5\times {10}^{10}{M}_{\odot }$), not the formal intercept of the linear fit extrapolated to $\log M_{*}=0$; this convention keeps the sign of $\Delta \log R_0$ aligned with the direction of size change at the masses where each process operates. ^a “Significant”: comparable to or exceeding typical SMR measurement uncertainties ($|\Delta \alpha |\gtrsim 0.05$ or $|\Delta \log R_0|\gtrsim 0.04$ dex over $z=1\to 0$); “modest”: within a factor of a few below them; “negligible”: an order of magnitude or more below them. Grades additionally discount amplitudes whose sign is not robust to parameter choices, transient or upper-bound estimates, and partial cancellation between co-acting channels. ^b Population-diluted: the per-galaxy effect is substantial, but the effect is negligible over the whole population as only galaxies in dense environments are affected; the predicted signatures are recoverable in cluster and dense-group samples. ^c Confined to the quenching epoch: shapes the arrival sizes of newly quenched galaxies (mainly at $z\gtrsim 1.5$) rather than driving continued evolution of the established quiescent population. ^d Transient: applies during the ≲1 Gyr starburst phase; the time-averaged population contrast is small, and its $z=1\to 0$ change is mildly positive (the UV zero point recovers as the merger rate declines).

Figure 1. Wavelength-dependent effects of six physical processes on the SMR of SFGs. Each row corresponds to one process and contains four panels. Before and After render a representative SFG as a barred-spiral icon (deep-blue bar and cyan spiral arms) enclosed by two effective-radius rings: a dashed cyan ring at $R_{\mathrm{UV}}$ and a solid deep-blue ring at $R_{\mathrm{opt}}$. The Process panel (header names the process) sketches the physical mechanism with purple iconography: cold-gas streams, merging galaxies, ICM wind, infalling satellites, or radial outflows. Faint grey ghosts of the Before-state rings are overlaid on the After panel. Hence, the size and morphology change is visible at a glance. The rightmost panel (SMR shift; title shown only on the top row) is a schematic of how the population size–mass relation $\log R_e=\log R_0+\alpha \log (M_{*}/M_0)$ moves: the grey line is the pre-process reference (shared across rows so the comparison isolates the differential effect of each process), and the dashed cyan and solid deep-blue lines are the post-process size–mass relations in rest-frame UV and rest-frame optical, respectively; right-anchored labels identify the lines. Signs and the relative magnitudes of the UV vs. optical shifts follow the literature synthesis of Section 3.1 and Table 1; line slopes and intercepts in the inset are illustrative, not numerical predictions.

Figure 2. As Figure 1 but for QGs. The QG host is rendered as a smooth red gradient ellipsoid; the size rings are dashed darkorange for $R_{\mathrm{UV}}$ and solid red for $R_{\mathrm{opt}}$. Process iconography includes merging galaxies, accreted satellites, post-feedback adiabatic outflows, and tidal perturbers; the bottom row (progenitor bias) is rendered differently because the effect is population-level rather than individual-galaxy: the Before panel shows two groups of galaxies at the same $M_{*}$ (small QGs at left, larger SFGs at right) and the After panel shows the enlarged QG sample after the SFG cohort quenches and joins it. Hence, the mean $R_e$ at fixed $M_{*}$ rises without any individual galaxy growing. The SMR-shift convention is identical to Figure 1 with the QG-family colours substituted (dashed orange = UV, solid red = optical). Signs and wavelength contrasts follow Section 3.2 and the lower half of Table 1.

Throughout the text, we use “UV”/“near-UV” to mean the rest-frame near-UV at ${\lambda }_{\mathrm{rest}}\approx 3000$ Å and “optical” to mean rest-frame optical at ${\lambda }_{\mathrm{rest}}\approx 5000$ Å. The two choices are motivated by stellar-population physics. The ∼3000 Å is short enough to be dominated by main-sequence A and late-B stars (flux-weighted age ∼${10}^{8.5}$ yr) yet long enough to lie above the “UV upturn” contribution from extreme-horizontal-branch and post-AGB stars in old populations at $\lambda \lesssim 2500$ Å (Section 4). The ∼5000 Å samples the FGK main-sequence and the giant-branch stars that hold most of the stellar mass, and it is also the wavelength range at which the great majority of published SMRs are measured, enabling direct comparability with the literature.

We note that the rest-frame near-IR (∼1–1.6 $\mathsf{\mu }$m) is a much better tracer of total stellar mass as it is weighted toward K and M dwarfs and giants and much less affected by dust. Therefore, the predictions we develop here for the optical regime apply a fortiori to the near-IR, with IR–UV wavelength contrasts that should be comparable to or larger than those of the optical–UV. Similarly, our interpretations of the rest-frame UV can be easily extended to other star formation tracers, such as the H$\alpha$ emission line.

2. Framework: The Two-Wavelength SMR

Rest-frame UV light is dominated by main-sequence A and late-B stars and by the outer envelopes of intermediate-mass stars [25,26,27], while the rest-frame optical is dominated by the ensemble of FGK main-sequence and giant-branch stars that hold most of the stellar mass. For a single stellar population this difference translates into a flux-weighted age of ∼${10}^{8.5}$ yr in the UV versus ∼${10}^{9.5}$ yr in the optical [25,26]. For a composite population with ongoing or recent star formation, the UV light is preferentially produced in regions of high specific star-formation rate (sSFR), while the optical light tracks the spatial distribution of the bulk of the old stellar mass.

Two consequences follow from this wavelength contrast. First, any process that changes where the most intense star formation occurs (such as in situ growth, inside-out quenching, harassment-induced starbursts, wet mergers; [45,46,47,48]) will reshape the UV light profile—and therefore the UV-derived $R_e$—more than the optical profile. Second, any process that adds, removes, or kinematically rearranges the bulk old stellar mass, which alters its radial population mix (dry major mergers, adiabatic expansion, deep tidal stripping of the relaxed component; [49,50,51]), will reshape the optical and UV profiles by comparable amounts. One further channel modulates this dichotomy: at fixed old age, metal-poor populations remain UV-brighter [29,30]. Any process that builds up or strips the metal-poor, ex situ outer envelopes of QGs (minor mergers/accretion and tidal stripping; Section 3.2.2 and Section 3.2.5) carry a UV-side bias even in the absence of young stars. Even before any process acts, a typical galaxy’s negative colour gradient makes its UV size exceed its optical size (we return to this topic in Section 4. The differential response of the UV and optical SMRs to a given physical process is what we denote here as its “wavelength contrast”. In Section 3 we work through twelve such processes; their predicted wavelength contrasts are compiled in Table 1 and visualized in Figure 1 (for SFGs) and Figure 2 (for QGs).

To make this differential response transparent, in each panel of Figure 1 and Figure 2 a grey reference line shows an idealized SMR before the physical process acts on the population; the post-process SMRs in rest-UV (dashed) and rest-optical (solid) are overlaid in the colours of the population (cyan/blue for SFGs, orange/red for QGs). In reality, the UV and optical SMRs of the unmodified population would themselves differ at the ≲0.1 dex level, but using a single shared baseline across panels isolates the differential effect of the processes on the SMR in two wavelengths, which is the quantity of interest.

In this synthesis, we focus on the well-measured, well-sampled part of each SMR, namely $\log M_{*}/M_{\odot }\gtrsim 9.5$ for SFGs and $\log M_{*}/M_{\odot }\gtrsim 10.4$ (the pivot mass) for QGs. We reach below these limits only where a process acts specifically there, as with the dwarf and sub-$M^{★}$ systems shaped by environmental stripping.

3. Process-by-Process Synthesis

In this section we work through twelve physical processes that shape galaxy structure—six for star-forming galaxies in Section 3.1 (illustrated in Figure 1) and six for quiescent galaxies in Section 3.2 (illustrated in Figure 2)—and ask, for each, how its effect on the size–mass relation differs between rest-frame UV and rest-frame optical. Within each population the six processes span three physical categories: in situ growth and quenching, mergers and accretion, and environmental transformation. The qualitative wavelength contrasts for all twelve processes are compiled in Table 1.

To establish which of these processes matter at the population level, we accompany each with a qualitative amplitude estimate over a common $z=1\to 0$ baseline, which is obtained by combining its measured occurrence rate, timescale, affected population fraction, and size–mass-plane track from the literature, and cross-checked with simple Monte Carlo population models (see Appendix A for details). The resulting amplitude grades appear in the last column of Table 1.

A recurring result of these estimates is that several processes carry robust per-galaxy signatures (high-velocity encounters, ram-pressure and tidal stripping) yet appear strongly diluted in a fit to the global galaxy population, because only a small fraction of galaxies inhabit the dense clusters and groups where they operate. This dilution is an artefact of mixing environments in a single fit rather than a sign that the processes are weak. Once the sample is split by environment, these same processes become considerable in the dense-environment satellite subpopulation. The channels that act on the general population (in situ growth, mergers and accretion, progenitor bias), by contrast, dominate the evolution of the fitted relations for the full sample.

3.1. Star-Forming Galaxies

3.1.1. In-Situ Star Formation and Inside-Out Disk Growth

Cosmologically averaged star formation in disk-dominated SFGs proceeds inside-out: the radial profile of star formation rate (SFR) surface density is shallower than that of stellar-mass surface density (equivalently, the sSFR rises outward). Star formation deposits new stars at progressively larger radii with cosmic time. This is supported observationally by spatially resolved $H\alpha$ maps from 3D-HST [45,52,53], by stacked CANDELS profiles [54,55,56], by IFS surveys such as MaNGA and SAMI [57,58,59], and by colour-gradient analyses [23,24]. Analytic “equilibrium” models also predict the same inside-out behaviour. In these models the SFR is set by the balance between cosmological gas inflow1, star formation, and outflows. Gas accreted at later times carries higher specific angular momentum and therefore settles and forms stars at progressively larger radii, while the central regions are assembled early from the low-angular-momentum gas accreted first, supplemented by gas transported inward through the disk [65,66].

Because star formation proceeds inside-out, new young stars appear preferentially in the outer disk, where they radiate strongly in the rest-UV. This freshly added light broadens the UV emission, and the UV $R_e$ grows rapidly. The rest-optical traces the older and more centrally concentrated disk, and its $R_e$ grows more slowly, lagging modestly behind the UV. However, secular radial migration works in the opposite direction to inside-out growth, carrying old, optically bright but UV-faint stars outward through repeated scattering by spiral arms [67,68,69]. This builds an extended outer disk that grows the optical size slightly and narrows the usual excess of UV size over optical.

In the size–mass plane an individual SFG therefore moves up and to the right, with a steeper trajectory if the sizes are measured based on rest-frame UV rather than optical light profiles. Because the main-sequence sSFR decreases with stellar mass [70,71,72,73], the fractional size growth is larger for SFGs with lower masses. Thus, the low-mass end of the relation is lifted more than the high-mass end and the slope of the SMR flattens. The flattening is, however, of comparable magnitude in the two bands: the offset between the growth track and the relation is nearly band-independent, because the ratio of the recent star formation extent to the stellar extent itself increases with stellar mass [3,53], compensating for the heavier UV weighting of the newly formed stars. Net effect: both slopes flatten by comparable, modest amounts; the UV zero point rises at least as fast as the optical zero point. Thus, the baseline UV–optical characteristic size offset of the population (∼0.1 dex) is maintained or grows slowly over time.

3.1.2. Bulge Growth and Inside-Out Quenching

In SFGs, bulges are built initially by star formation. Gas driven inward by secular bar and spiral-arm torques [69,74] and by gas-rich minor mergers [75,76] fuels a compact central starburst that grows a dense stellar core. At high redshift, the same central starburst is driven still more strongly by wet compaction of gas-rich disks [62,77,78,79], typically near $M_{*}\sim {10}^{9.5}{M}_{\odot }$ at $z\sim 2$–4, whereas at $z<1$ it is the secular and gas-rich merger channels that dominate. Once this core becomes dense enough, it stabilizes the residual gas disk against fragmentation and shuts off star formation from the inside out (morphological quenching; [36,80,81,82]). Whereas bar and spiral torques mainly grow a disky pseudobulge, the merger and compaction channels build the old, dispersion-supported classical bulge whose stars come to dominate the galaxy’s central optical light.

Old, metal-rich classical bulges [83,84,85] contribute disproportionately in the optical. They are dimmer in UV both because of their stellar age and because the sites of ongoing star formation are pushed outward, into the disk [46,86,87]. As the bulge grows, the global single-Sérsic optical $R_e$ of the host galaxy shrinks (the central concentration increases and the outer disk is de-emphasized), while the UV $R_e$ shrinks much less or, in the inside-out-quenching limit, can briefly remain steady or grow as star formation migrates outward [10,32,55,88].

In the size–mass plane an individual galaxy therefore moves down and slightly to the right in the optical (modest central mass growth, shrinking global $R_e$) but nearly horizontally in the UV. However, bulges are more prominent in massive SFGs than in low-mass systems [89,90], and the central 1 kpc stellar-mass surface density ${\Sigma }_1$, a measure of central concentration, is highest in the most massive disks [10,87,91]. Bulge-driven compaction therefore acts preferentially at $\log M_{*}/M_{\odot }\gtrsim 10.5$. Net effect: the optical zero point drops while the UV zero point is nearly unchanged; the optical slope flattens measurably while the UV slope is nearly unchanged; and the UV–optical size difference of SFGs grows because the central concentration becomes higher in the optical than in the UV.

3.1.3. Wet Major Mergers

Gas-rich major mergers (mergers between galaxies of comparable masses; mass ratio above 1:4) between SFGs torque cold gas to the centre on ≲${10}^9$ yr timescales. Such an event ignites a compact nuclear starburst, building a young classical bulge, and leaving the remnant more centrally concentrated than either progenitor [48,92,93,94,95,96,97,98].

The nuclear starburst is intensely UV-bright but also dust-obscured [99,100,101], while the longer-lived optical light remains dominated by the disrupted progenitor disks. The UV light therefore tracks the compact nuclear starburst more closely than does the extended optical envelope. Hence, the UV $R_e$ drops further than the optical $R_e$ [94,96,98,102]. However, in the most heavily obscured phases, dust can suppress the nuclear UV and temporarily weaken this signature (Section 4).

In the size–mass plane the remnant moves down and to the right (stellar mass roughly doubles in a single event, $R_e$ decreases). The galaxy major-merger rate at $z<1$ is weakly mass-dependent above $\log M_{*}/M_{\odot }\sim 10$ [103,104,105], while the cold-gas fraction—and with it the dissipative shrinkage per event—decreases with stellar mass [94,106,107,108]. These two mass dependences act on the fitted slope in opposite directions (gas-rich low-mass remnants shrink more per event, while the higher event rate of massive galaxies transports compacted remnants up-mass) and largely cancel, and hence, the slope of the SMR is barely affected; the dominant effect is on the zero point. Two population-level effects further temper the wavelength signature. Firstly, the UV-compact starburst phase lasts only ≲0.5–1 Gyr. Therefore, few SFGs exhibit it at any given epoch (and heavy nuclear obscuration can temporarily reverse it; Section 4). Secondly, a fraction of remnants quenches rapidly and exits the star-forming sample altogether, transferring the most compact merger products to the quiescent relation [109,110]. Net effect: both slopes are nearly unchanged, both zero points fall, and the time-averaged zero-point drop is only slightly larger in UV than in optical.

3.1.4. Minor Mergers and Accretion

A minor merger (mass ratio below 1:4) delivers gas and stars to different places. The gas sinks to the centre and feeds the bulge [75,76,111]. In contrast, the satellite’s stars are collisionless and sink slowly. These accreted stars settle in the outer envelope building a diffuse stellar halo and increase the size [49,112,113,114,115,116,117,118]. Unlike a dry merger that drives only the envelope channel (Section 3.2.2), a gas-rich merger, typical of SFGs, drives both. Dry mergers are approximately dissipationless (i.e., their mechanical energy is conserved rather than radiated away). A combination of the virial theorem with this energy conservation gives the scaling relation for the final product: $R_e\propto M_{*}^2$ [49,115]. SFG disks, on the other hand, fall short of this $R_e\propto M_{*}^2$ track because gas is diverted to the centre and the diffuse debris adds little light-weighted $R_e$ (ex situ fractions ∼5–15%; [119]).

In SFGs, the wavelength contrast is modest, because minor mergers grow the UV and optical $R_e$ by comparable amounts. When a satellite is disrupted, its stars are tidally stripped and spread smoothly through the host’s outer envelope, which is a diffuse addition of light at large radii rather than discrete clumps. Light added at large radii enlarges $R_e$ in both the UV and the optical. What sets the contrast between the two bands is the colour of that added envelope light relative to the host’s existing outskirt, not how bright the outskirt already is. In a QG, the outskirt is old and red, and the accreted stars are relatively bluer. They add proportionally more UV light at large radii, and the UV $R_e$ grows far more than the optical (Section 3.2). In an SFG, by contrast, the outer disk is already blue and the added light either matches it in colour or is slightly redder. Hence, the two bands then grow alike. The contrast, if any, runs opposite to the QG case, because the infalling gas feeds a compact blue starburst that draws the UV $R_e$ inward and leaves the optical $R_e$ marginally the larger of the two [75,76,111].

Minor-merger-driven growth is more efficient for high-mass hosts, both because their merger rate is higher [88,120,121,122] and because their accreted-to-in situ ratio rises with mass (shown most clearly for massive centrals by [123]; and extended to disk-host morphologies by [119]; see also [118,124]). Hence, this process steepens the slope of the SFG SMR. Net effect: a small but mass-dependent increase in zero point and slope, in both wavelengths to a similar degree.

3.1.5. High-Velocity Encounters

High-velocity flyby encounters with other cluster members and the cluster potential (historically termed galaxy harassment) heat stellar disks, strip loosely bound material, and can drive gas inwards to ignite a short-lived central starburst [47,125,126,127,128,129,130]. Unlike a merger, the galaxies move too fast to coalesce, and it is the cumulative effect of many such impulsive shocks that reshapes the galaxy.

The UV traces the loosely bound young outer disk, which the encounters strip first. That truncation makes the UV profile more compact, and with no fresh gas to replenish it, the smaller UV size persists. A tidally triggered nuclear starburst can shrink the UV size further still, but only transiently, fading within a few hundred Myr [131]. The optical light responds more weakly because the old stars that dominate it sit in a more centrally concentrated profile and are tightly bound. Some of these old stars also populate the outer disk, and stripping it removes optical light at a large radius, contracting the optical size modestly, much less than the UV. The heating also thickens the survivors, but that puffing is mostly vertical and does not shrink the projected optical size further.

An individual SFG shifts to slightly lower stellar mass and a smaller size, with the UV shrinking more than the optical. The stripping that truncates the outer disk beats the heating that inflates it, and the net size falls. The effect is largest for low-mass, extended SFGs, which have the shallowest potentials and most loosely bound disks. The observational evidence for this is a size offset confined to the low-mass end, where cluster SFGs are measurably smaller than their field counterparts, with no such offset at higher mass [132]. Even so, this strong shrinkage is reached only on the rare orbits that cross the dense cluster core [129]. Net effect: within pure cluster samples, both slopes steepen and both zero points fall, with UV more than optical. Averaged across the SFG population as a whole, these shifts are negligible.

3.1.6. Ram-Pressure and Tidal Stripping

Two distinct mechanisms strip infalling cluster SFGs. Ram pressure acts hydrodynamically, as the intracluster medium sweeps the cold gas from the disk and removes the fuel for future star formation, while leaving the existing stars in place [133,134,135,136,137,138,139]. Tidal stripping acts gravitationally, as tides from massive neighbours and the cluster potential pull loosely bound stars and dark matter from the outer envelope [51,140,141,142,143,144,145].

Because cold gas (and the young stars it would form) sits preferentially in the disk outskirts, ram-pressure stripping truncates the UV-bright periphery first, leaving the older optical disk and bulge largely intact [135,137,138,139,146]. Tidal disruption, meanwhile, removes a modest fraction of the loosely bound outer stellar envelope along with the surrounding subhalo—for low-mass satellites on deep-pericentre orbits the total (dark-matter-dominated) mass loss is severe (∼80–90%, with the inner stellar cores far more resilient) [147,148]. The net result for an individual stripped SFG is a disproportionately larger reduction in UV size than in optical size.

Stripping is most efficient for low-mass galaxies, which have a smaller restoring force, and for extended ones [132,149,150]. It therefore steepens the SMR slope and lowers the zero point, more strongly in the UV. The SMR changes in the optical band share the same signs but stay weak, because ram pressure removes gas rather than the old stars that determine the optical size. These shifts become negligible in pure gas-only stripping, and appear only when the accompanying tidal forces also strip stars or the quenched outer disk fades. Net effect: the UV slope steepens and the UV zero point drops, while the optical follows weakly or not at all. The wavelength contrast is large, and the UV-side drop can persist for ∼${10}^9$ yr, set by the main-sequence lifetimes of the A and late-B stars that dominate the rest-frame UV. Few SFGs are being stripped at any epoch, and those fully stripped quench and leave the star-forming sample within ∼1 Gyr. Hence, as with high-velocity encounters, the effect on general population is small, though prominent, in cluster samples.

3.2. Quiescent Galaxies

3.2.1. Dry Major Mergers

Mergers between gas-poor, spheroid-dominated QGs are the canonical pathway to growing the most massive ellipticals [49,122,151,152,153,154,155]. For an equal-mass dry merger, a simple virial-theorem argument predicts that the remnant roughly doubles both its stellar mass and its half-light radius. The galaxy therefore climbs a merger track of unit slope, $\mathrm{d}\log R_e/\mathrm{d}\log M_{*}\approx 1$, confirmed by controlled N-body experiments [49,115].

Because both progenitors are old, the merger does not strongly redistribute UV-bright versus optical-bright stellar light: UV and optical $R_e$ grow by similar amounts. The wavelength contrast is therefore small, except for a residual UV-upturn contribution from extreme-horizontal-branch and post-AGB stars in the most massive QGs (Section 4).

During the merger, a single galaxy climbs up and to the right along its own merger track, not along the population SMR. This track runs at $\mathrm{d}\log R_e/\mathrm{d}\log M_{*}\approx 1$, steeper than the relation ($\alpha \approx 0.7$). The major-merger rate rises mildly with stellar mass, roughly $\mathcal{R}\propto M_{*}^{0.3}$ [104,156,157], and simulations place the strongest mass dependence above ∼$2\times {10}^{11}{M}_{\odot }$ [158]. Integrating this rate over $z=1\to 0$ gives of order $0.5$–1 major mergers per massive QG, enough to raise the zero point by ∼0.05–0.1 dex (Appendix A). Because the track is steeper than the relation, it also steepens the slope mildly. This steepening is only a sub-$1\sigma$ tendency, and its amplitude and even its sign depend on the assumed track slope and on the mass dependence of the rate. Over the same interval, these mergers supply only ∼15–20% of the observed QG size growth. Net effect: the zero point rises in both UV and optical, the slope steepens mildly in both bands, and the wavelength contrast is negligible.

3.2.2. Minor Mergers and Accretion

Minor mergers and smooth accretion are now widely seen as the dominant driver of QG size growth at $z\lesssim 2$ [49,113,114,115,118,119,120,122,154,159]. A galaxy accreting this material climbs a steep merger track, gaining size about twice as fast as mass ($\mathrm{d}\log R_e$/$\mathrm{d}\log M_{*}\approx 2$) [49,115]. That is twice the slope of the equal-mass dry-merger track introduced above in Section 3.2.1.

Unlike the dry-major case, the accreted satellites are typically less massive and, crucially, their stars are younger or more metal-poor than the old in-situ population [123,160]. They settle preferentially in the extended outer envelope [118]. There, their young, metal-poor stars contribute a larger share of the UV light than of the stellar mass. The QG outer envelope therefore becomes UV-brighter than the in situ core, and the half-light radius of an individual QG grows faster in the UV than in the optical.

The single galaxy moves up and to the right in the size–mass plane, with a small mass change but a large size change, and its outer envelope becomes UV-brighter than its centre. Both the accreted-to-in situ fraction and the minor-merger rate peak at high mass, above $\log M_{*}$/$M_{\odot }\simeq 10.7$ [121,123,124]. The high-mass end therefore grows most, which steepens the SMR slope and raises the zero point, with both effects larger in the UV than in the optical. Net effect: the slope steepens and the zero point rises in both bands more strongly in the UV. This large UV-biased contrast is one of the cleanest empirical fingerprints of minor mergers and accretion in the QG SMR.

3.2.3. Adiabatic Expansion

Adiabatic expansion shapes a QG at the moment it quenches. The final starburst, together with any quasar-mode AGN activity, expels a large fraction of the central baryons from the potential [50,161]. When this loss is slow compared with the local dynamical time, the stellar orbits re-equilibrate into a less bound configuration and conserve the adiabatic invariant [162]. In this adiabatic limit $R_e$ grows in inverse proportion to the enclosed mass, and idealized models give a factor of ∼2 for a mass loss $\Delta M/M\sim 0.3$–$0.5$ [50,163]. Such large expellable gas fractions exist only during this gas-rich quenching event. Once quenched, galaxies at $z<1$ hold low molecular-gas fractions of order a few per cent [164,165]. Hence, any later gas expulsion enlarges $R_e$ only modestly.

Adiabatic expansion moves each star outward by a factor set only by the mass removed inside its orbit, regardless of its age. In an evolved QG the ongoing mass loss comes from the material that aging stars shed, which follows the stellar distribution. Therefore, expelling it stretches the galaxy nearly uniformly, and the UV and optical sizes grow by nearly the same factor, whatever the colour gradient. The one exception is the central gas expelled at the quenching event. Being more centrally concentrated than the stars, the expelled gas makes the inner regions expand more. This uneven expansion can stretch a pre-existing radial age gradient into a small difference between the UV and optical sizes. However, that difference happens while quenching (not after quenching), and its effect on the population is negligible. Adiabatic expansion therefore grows both bands by nearly the same amount, unlike the wavelength-asymmetric mergers and stripping.

The galaxy moves nearly straight up in the size–mass plane, gaining size while losing little mass. The two feedback channels act at opposite ends of the mass function. Quasar-mode episodes affect the massive compact progenitors, but they are triggered only at quenching and setting the size of newly quenched galaxies [153,161,166]. Within the established population, the ongoing expansion is driven instead by the material still expellable, dominated at late times by the slow removal of aging stars [163,167,168]. This fraction is largest in low-mass systems with shallow potentials. Adiabatic expansion therefore lifts the low-mass end of the QG SMR most. Net effect: the slope flattens, the zero point rises in both bands, and the wavelength contrast is negligible.

3.2.4. High-Velocity Encounters

For QGs in galaxy clusters, the same high-velocity encounters that affect SFGs continue to operate but without the gas-driven nuclear starbursts [47,125,127,129,130]. Repeated tidal shocks heat the stellar distribution, strip outer material, and increase the central concentration of the remnant. The observational evidence comes from the early-type dwarf populations of nearby clusters such as Virgo and Hydra I [130,169], whose structural properties are reproduced by N-body high-speed encounter simulations [129].

The wavelength response is set by the metallicity of the removed material, because a metal-poor outer shell stays UV-bright even when it is old (Section 2). Both the binding energy of the outskirts and their metallicity contrast rise steeply with mass [119]. A massive QG therefore holds a tightly bound, metal-poor, ex-situ envelope, whereas a low-mass dwarf has only a loosely bound, in situ, near-solar outskirt whose removal carries no wavelength contrast in $R_e$.

Above the pivot mass considered here, fast flybys heat that outer envelope but rarely unbind it, and most orbits strip no stars at all [129]. Its removal is instead the work of the sustained cluster tide (Section 3.2.5), leaving high-velocity encounters with no measurable imprint on the QGs we study. The effect is significant only below the SMR pivot point, a mass regime absent from Figure 2, where repeated shocks compact loosely bound dwarfs [127,130]. Net effect: above the pivot the QG slope and zero point are essentially unchanged with no UV–optical contrast, and any effect is confined to the out-of-scope low-mass regime.

3.2.5. Tidal Stripping

Tidal interactions in galaxy groups, clusters, and the central regions of host haloes can remove stars from the loosely bound outer envelopes of QGs [51,140,141,142,143,144,145,169,170]. This envelope is the diffuse, low-surface-brightness ex situ stellar halo at large radius (roughly beyond ∼$2R_e$), structurally distinct from the compact in situ core. Like high-velocity encounters, tidal stripping preferentially removes this outermost material, but the two differ in dynamical regime. The former proceeds through impulsive flyby encounters that shock-heat the stellar distribution at all radii, whereas the latter acts adiabatically through the smooth host potential and leaves the inner stellar core relatively undisturbed. That structural distinction sets which stars are removed, while the wavelength contrast developed below follows from the metallicity gradient of the removed material rather than from the impulsive or adiabatic character of the removal.

The outer envelope of a massive QG is built by past minor mergers [118,124]. As we mention in Section 3.2.2, this accreted material is more metal-poor than the in situ core at the same old age, and therefore UV-brighter [10,123,160]. Because it shines disproportionately in the UV, adding or removing it changes the UV size more than the optical. Minor mergers build this envelope in the field galaxies and raise their UV size, whereas tidal stripping removes it from cluster galaxies and shrinks their UV size more than the optical one.

A tidally stripped QG moves down and to the left, but not along the relation. Removing the outer envelope shrinks the $R_e$ faster than $M_{*}$, giving a steep track ($\mathrm{d}\log R_e/\mathrm{d}\log M_{*}>1$) well above $\alpha \approx 0.7$, and the galaxy drops below the SMR rather than sliding along it. Stripping effects are strongest at the lower-mass end of the QG range, where satellite fractions are higher and potentials shallower [144,171,172]. That end therefore drops furthest, and both slopes steepen. Because the removed envelope is metal-poor and UV-bright, its loss drops the UV $R_e$ more than the optical, and the UV slope steepens more. The per-galaxy deficit is large, with cluster QGs observed being smaller than the field [11,170], though only a minority are deeply stripped, leaving the population amplitude small. Net effect: Both slopes steepen and both zero points fall, more strongly in the UV than in the optical. This is the time-reverse of minor-merger growth, which raises the UV zero point faster than the optical.

3.2.6. Progenitor Bias

Progenitor bias is a population-level effect rather than a physical process on individual galaxies. When an SFG quenches, it joins the QG population as a newcomer. At fixed stellar mass, that newcomer is larger than the QGs which quenched earlier and have since compacted through mergers and stripping [7,10,87,173,174,175,176,177,178,179]. The relation shifts not because any galaxy grows or gains mass, but because the size distribution at fixed mass steadily gains larger members, raising the mean $R_e$ and lifting the zero point. This zero point offset survives a ∼10–30% disk-fading correction and a mild bias of quenching systems toward the compact side of the star-forming size distribution. Most galaxies that quench are of modest mass ($\log M_{*}/M_{\odot }\sim 10$–$10.7$), simply because low-mass galaxies far outnumber high-mass ones [18,180,181], even though any single massive galaxy is more likely to quench [72,89,182]. This picture applies more to the secular, non-merger quenching that dominates at these masses, whereas merger-driven quenching instead compacts the newcomer and is already captured by the merger channels above.

A newcomer arrives with an extended young disk [86,87,183,184], often bearing the transitional morphology of the green valley, where quenching disks show a surplus of rings and lenses [185,186]. Because its young stars are UV-bright, this disk makes the newcomer’s UV size larger than its optical size at first. The disk then fades from the UV within ∼1 Gyr, set by the lifetimes of A and late-B main-sequence stars, but keeps shining in the optical for several Gyr. A newcomer’s UV size therefore shrinks more quickly than its optical size, and the optical zero point of the QG SMR accordingly receives a larger boost than the UV zero point. This optical-over-UV contrast is modest, and it weakens toward higher redshift. There, rapid recent quenching leaves many QGs still inside their ∼1 Gyr UV-bright window, and their surviving UV disks increase the UV zero point on par with the optical zero point.

The newcomer influx peaks at modest masses ($\log M_{*}/M_{\odot }\sim 10$–$10.7$), and their continuous addition increases the optical zero point of the QG SMR. Because they arrive preferentially near and below the pivot point, they raise the low-mass sizes more than the massive end and therefore flatten $\alpha$. The UV zero point also rises but with a smaller amplitude because each newcomer’s UV light fades within ∼1 Gyr while its optical light persists. Net effect: the optical zero point rises more than the UV, and both slopes flatten. At $z\lesssim 1$ the newcomer flux multiplies the low-mass QG number density by a factor of ∼2–3 at $\log M_{*}/M_{\odot }\approx 10$ while barely affecting the massive end [18,187]. Progenitor bias therefore reshapes the low-mass slope, while minor mergers steepen the massive end, and the observed slope is the net of the two. Progenitor bias, adiabatic expansion, and minor-merger accretion all raise the QG intercept, and their UV–optical contrasts, optical-strong, neutral, and UV-strong, are what separate the three.

4. Caveats and Degeneracies

The idealized wavelength contrasts predicted in Section 3 are subject to several observational systematics that compete with the process-driven signals in amplitude and limit the quantitative interpretation of any UV+optical SMR measurement. Below we flag, in order, the stellar-population caveats (age–metallicity degeneracy, dust attenuation, UV upturn, AGN contamination), the structural-modelling caveats (single-Sérsic vs. bulge+disk decompositions), and the measurement-systematics caveats (baseline colour gradients, PSF/resolution mismatch, surface-brightness dimming).

Age–metallicity degeneracy. Both stellar age and stellar metallicity affect the strength of the 4000 Å break and the slope of the rest-UV continuum [29,30,188]. A metal-poor old population can mimic the UV brightness of a younger metal-rich population and therefore inflate the apparent UV size of an old galaxy. The cleanest discriminant is spatially resolved spectroscopy from instruments such as MUSE, ERIS, NIRSpec IFU, and GIRMOS [189,190,191,192,193,194,195], which allows direct measurement of stellar-population gradients. In practice, this route is limited by the low surface brightness of galaxy outskirts, which is exactly where the population gradients matter most, and it is cleanest for galaxies whose outer regions stay bright enough to reach adequate signal-to-noise ratio per radial bin.

Dust attenuation. Dust reddens galaxies unevenly across their light profiles [33,53,196,197,198,199,200,201]. Because dust is centrally concentrated, it suppresses the central UV light and biases UV size measurements upward mimicking intrinsic inside-out growth [53,202]. The bias is largest for the most strongly star-forming SFGs and for wet-merger remnants. Dust content is small but not negligible in transitional and recently quenched QGs (post-starburst, K+A, fading-disk newcomers; Section 3.2.6), where residual ISM and centrally concentrated dust lanes can still bias UV size measurements during the first ∼1 Gyr after quenching. Resolved attenuation maps from JWST/MIRI and ALMA, combined with energy-balance SED fitting, are well suited to correct for dust effects. A spatially resolved multi-band UV-optical-NIR SED fit, performed pixel-by-pixel or in radial annuli, can also recover the attenuation as a function of radius and correct the inferred UV sizes. The mid-IR and sub-mm data are ideal rather than essential.

UV upturn in old populations. Hot extreme-horizontal-branch stars and post-AGB stars in the most massive, oldest QGs produce a “UV upturn” that contributes non-negligible flux at ${\lambda }_{\mathrm{rest}}\lesssim 2500$ Å [203,204,205,206,207,208,209]. The rest-frame near-UV at ∼3000 Å adopted here lies longward of the upturn and avoids most of this contamination. A residual concern is a broad-band filter that at certain redshifts still samples ${\lambda }_{\mathrm{rest}}\lesssim 2500$ Å, which can mimic extended UV light from a young population and is worst in the most massive QGs at $z\lesssim 0.5$ [206].

AGN contamination. Unresolved nuclear UV point sources from low-luminosity AGN inflate the inferred central Sérsic concentration and bias $R_e\left(\mathrm{UV}\right)$ downward relative to $R_e\left(\mathrm{opt}\right)$, mimicking the compact-nuclear-starburst signature attributed to wet major mergers (Section 3.1.3) and the post-quasar phase invoked in adiabatic expansion. Multi-wavelength AGN identification (X-ray, mid-IR, radio) is required to flag affected systems before fitting.

Single-Sérsic versus bulge+disk biases. Many of the predictions above assume that the global $R_e$ measured by a single-Sérsic fit responds to bulge growth and disk evolution together. A growing body of evidence shows that the global $R_e$ can deviate substantially from the disk $R_e$ when the bulge becomes structurally distinct [34,35,37,210]. The clearest test of the predictions in Table 1 is therefore at the level of bulge+disk decompositions, not single-band single-component fits. However, such a decomposition carries roughly twice the free parameters of a single-Sérsic fit, and the bulge and disk become degenerate at low signal-to-noise and coarse resolution. Reliable decompositions are therefore feasible only for bright, well-resolved galaxies, while fainter and smaller systems remain limited to single-Sérsic sizes.

Baseline colour gradients. Galaxies have negative colour gradients (bluer outskirts) at essentially all masses and redshifts [23,24,31,32,211,212,213], which generically produces $R_e\left(\mathrm{UV}\right)>R_e\left(\mathrm{opt}\right)$ regardless of which of the twelve processes above is at work. The predictions in Table 1 are therefore differential signals (changes in the wavelength contrast) on top of this universal baseline. The absolute UV-vs-optical size comparisons cannot be interpreted process-by-process without first subtracting it. Because this baseline is nearly a constant offset, subtracting it preserves the process-induced signals rather than erasing them, though how cleanly each is recovered depends on its amplitude relative to the residual baseline scatter. The near-neutral processes, most notably adiabatic expansion, carry the smallest differential contrast and are the hardest to recover against the residual baseline scatter. The strong-contrast channels, such as minor-merger accretion and progenitor bias, remain clearly separable above it.

PSF and resolution. The angular resolution (PSF) of both ground-based and space-based imaging is wavelength-dependent. This effect biases UV-vs-optical size differences in ways that depend on intrinsic profile shape and S/N. Although forward-modelled, PSF-matched single-Sérsic fits or PSF-convolved bulge+disk decompositions can mitigate this issue, extensive simulations are required to quantify the biases in the size measurements [10,11].

Surface-brightness limits. Although cosmological ${(1+z)}^4$ dimming is purely geometric and wavelength-independent [214], the recovered $R_e$ is not. Magnitude-limited imaging imposes limits on the surface brightness of the outer profile regions that can be traced. These surface brightness limits, especially at the outskirts, can have a negative impact on the measured sizes. Since the intrinsic rest-UV profile is typically more extended than the optical [3,10,11], UV sizes have a stronger negative bias than the optical sizes. This wavelength-dependent bias partly masks the intrinsic $R_e\left(\mathrm{UV}\right)>R_e\left(\mathrm{opt}\right)$ contrast. These biases can be quantified and corrected by the same forward-modelling injection-recovery simulations used for the PSF caveat above, run at the effective surface-brightness limit of each band.

5. Summary, Diagnostics, and Outlook

We have reviewed, process by process, the response of the galaxy size–mass relation in rest-frame UV and rest-frame optical, separately for SFGs and QGs, and set out the expected qualitative direction of each response, compiled in Table 1 and visualized in Figure 1 and Figure 2.

The central observation is that a comparison of the UV size–mass relation with the optical size–mass relation lifts several of the degeneracies that affect single-band analyses. Three cases deserve emphasis here. First, in QGs the three processes that raise the zero point the SMR—minor mergers and accretion, adiabatic expansion, and progenitor bias—predict different wavelength contrasts: minor-merger growth raises the UV intercept more than the optical, adiabatic expansion raises both nearly equally (the near-neutral case hardest to separate from the baseline colour gradient; Section 4), and progenitor bias raises the optical intercept slightly more than the UV. The relative ranking of UV and optical zero-point evolution at fixed $M_{*}$, combined with the sign of the slope change (steepening for minor mergers, flattening for the other two), therefore directly diagnoses the dominant channel2. Second, in SFGs the apparent deceleration of size growth at $z\lesssim 1$ [3,10,37] can be ascribed either to a pause in inside-out growth or to bulge-driven contraction of the global $R_e$. These two processes yield opposite UV–optical contrasts in SFG SMR evolution (bulge growth lowers the optical $R_e$ more than the UV; inside-out growth raises the UV $R_e$ at least as fast as the optical) and are therefore empirically separable in surveys with both rest-UV and rest-optical imaging. Third, environmental processes (harassment, ram-pressure and tidal stripping) leave a wavelength fingerprint biased to the UV side in SFGs but much weaker in QGs—where only tidal stripping of the UV-bright accreted envelope retains a UV bias, harassment being nearly wavelength-neutral—providing an independent test of cluster-vs-field SMR comparisons [11,215,216,217].

These considerations translate into four specific, near-term diagnostics:

• The sign of ${\alpha }_{\mathrm{UV}}-{\alpha }_{\mathrm{opt}}$ for SFGs at $z\lesssim 1$ discriminates inside-out growth from bulge-driven inside-out quenching. Bulge growth decreases the optical $R_e$ at high $M_{*}$ and flattens the optical slope while leaving the UV slope nearly unchanged, driving ${\alpha }_{\mathrm{UV}}-{\alpha }_{\mathrm{opt}}$ positive; pure inside-out growth flattens the two slopes by comparable amounts, keeping ${\alpha }_{\mathrm{UV}}-{\alpha }_{\mathrm{opt}}\approx 0$ while increasing the UV zero point. A significantly positive ${\alpha }_{\mathrm{UV}}-{\alpha }_{\mathrm{opt}}$ at fixed z therefore signals bulge-driven quenching, whereas a null slope difference accompanied by a rising UV intercept favours disk-dominated inside-out growth. Extending the joint UV+optical size–mass relation measurements from CLAUDS+HSC [10] at $0.1<z<0.9$ to higher redshifts with LSST, Euclid, Roman and JWST/NIRCam will map the cosmic epochs at which each channel dominated SFG structural evolution, and in particular when bulge growth begins to compete with disk assembly.
• Bulge growth decelerates the SFG global (single-Sérsic) zero-point evolution in the optical relative to the UV, whereas the disk zero point evolves at comparable rates in both bands, tilted only mildly UV-ward by inside-out growth. The two channels (bulge growth shrinking the optical global $R_e$; inside-out growth increasing the UV disk $R_e$) act in opposite directions on the two measurements. The UV-versus-optical split in the global single-Sérsic SMR slope can flag which channel dominates, because bulge growth flattens the optical slope while leaving the UV slope nearly unchanged, whereas inside-out growth flattens both alike. However, two-wavelength bulge–disk decompositions (e.g., [35,37]) are then needed to measure the bulge and disk zero points separately to confirm the findings of single-Sérsic modelling.
• Cluster–field SMR differences are larger in UV than in optical, with the contrast more pronounced in SFGs than in QGs. Ram-pressure stripping (SFGs only) and tidal stripping (both populations) preferentially remove UV-bright outer material—gas-rich disks in SFGs and ex situ accreted envelopes in QGs [47,129,218]. We therefore predict $|R_{e,\mathrm{UV}}^{\mathrm{cluster}}-R_{e,\mathrm{UV}}^{\mathrm{field}}|>|R_{e,\mathrm{opt}}^{\mathrm{cluster}}-R_{e,\mathrm{opt}}^{\mathrm{field}}|$ for both populations at fixed $M_{*}$, testable at $z\lesssim 1$ with existing rich-cluster and field surveys (e.g., [10,11,215]), unless preprocessing is more important than cluster-specific processes. This prediction is also testable at $z\gtrsim 1$, treated as a distinct regime, with JWST/NIRCam plus Roman, where the high-redshift (proto)cluster environments host a different member population and mass function and must be compared at fixed redshift and halo mass.
• The intrinsic scatter of the UV SMR exceeds that of the optical SMR for both SFGs and QGs. Several of the processes that show wavelength contrast in SMR evolution (mergers, stripping, in situ growth) produce stochastic, transient UV-bright outliers even when their effect on the median size–mass relation is modest. Initial evidence from CLAUDS+HSC over the COSMOS field at $0.1<z<0.9$ [10] already shows the UV scatter exceeding the optical; extending the measurement to the wider area and higher redshifts probed by LSST, Euclid, Roman and JWST/NIRCam will test whether the same wavelength asymmetry holds across cosmic time.

The diagnostic predictions here are qualitative and directional rather than numerical, but they are testable in exactly this differential form. These diagnostics (the sign of ${\alpha }_{\mathrm{UV}}-{\alpha }_{\mathrm{opt}}$, the cluster–field UV-versus-optical size contrast, and the excess UV scatter) can each be measured with joint UV+optical imaging. The per-process prescriptions of Appendix A are equally portable and can be incorporated into semi-analytic and semi-empirical models of galaxy evolution to turn these qualitative directions into quantitative, model-specific predictions.

The unifying message is that the joint UV+optical size–mass relation is no longer a niche measurement: with CLAUDS+HSC, LSST, Euclid, and Roman now or soon delivering rest-frame UV and optical imaging out to $z\sim 2$ for samples of ${10}^6$–${10}^7$ galaxies, two-rest-frame-wavelength SMR analysis can become a standard observable. The addition of rest-frame near-IR sizes from Euclid NISP and Roman WFI will extend this to a three-wavelength view in which the near-IR anchors the bulk-mass radius even more robustly than the optical. Confronting it with the process-by-process predictions of Table 1 should allow the community to identify which structural-evolution channels dominate at which mass and redshift, and to use the residuals as a direct test of the sub-grid feedback prescriptions in cosmological simulations.

Appendix A.1. Forward Monte Carlo Framework

To determine the relative strength of the impact of the processes considered in Section 3 on the SMR, we build a common forward model of the SMR and apply each process to it. We first seed a mock population on a power-law baseline,

$$\log R_e=\log R_0+\alpha (\log M_{*}-\log M_0)+\mathcal{N}(0,\sigma ),$$

(A1)

where $R_e$ is the half-light radius, $M_{*}$ the stellar mass, $\alpha$ the slope, and $\sigma$ the intrinsic (Gaussian) scatter in dex. The normalization $\log R_0$ is the characteristic size, defined at the fiducial mass $M_0=5\times {10}^{10}{M}_{\odot }$ ($\log M_0=10.7$, the knee of the Davidzon et al. [18] mass function), and it is at $M_0$ that we report every zero-point shift. The SFGs follow a single slope $\alpha =0.22$ with $\log R_0=0.74$ and $\sigma =0.20$ dex. The QGs follow a continuous broken power law whose slope changes from $\alpha =0.70$ to $\alpha \approx 0.15$ at the pivot mass $\log M_{\mathrm{p}}\approx 10.4$, with $\log R_0=0.55$ and $\sigma =0.16$ dex [3,8,10]. Stellar masses are drawn by inverse-cumulative sampling of the Davidzon et al. [18] double-Schechter functions.

Each process then displaces every affected galaxy along its own size–mass track,

$$\Delta \log R_e=s\Delta \log M_{*},s\equiv {\displaystyle \frac{\mathrm{d}\log R_e}{\mathrm{d}\log M_{*}}},$$

(A2)

where s is the track slope and $\Delta \log M_{*}$ the change in stellar mass. Writing $\mu \left(M_{*}\right)$ for the mean change in $\log M_{*}$ of galaxies that start at mass $M_{*}$, each galaxy ends at a mass $\log M_{*}+\mu$ and a size $\log R_e+s\mu$. Differentiating the new size with respect to the new mass gives a refitted slope ${\alpha }_{\mathrm{new}}=(\alpha +s^{\prime }\mu +s{\mu }^{\prime })/(1+{\mu }^{\prime })$. For the zero point, a galaxy that ends at $M_0$ began at $\log M_0-\mu$ and has therefore gained $(s-\alpha )\mu$ in size relative to the original relation. Hence, to first order,

$$\Delta \alpha \simeq {\displaystyle \frac{s^{\prime }\mu +(s-\alpha ){\mu }^{\prime }}{1+{\mu }^{\prime }}},\Delta \log R_0\simeq (s-\alpha )\mu \left(M_0\right),$$

(A3)

where $\Delta \alpha$ is the change in slope, $\Delta \log R_0$ the change in the characteristic size at $M_0$, and ${\mu }^{\prime }\equiv \mathrm{d}\mu /\mathrm{d}\log M_{*}$ and $s^{\prime }\equiv \mathrm{d}s/\mathrm{d}\log M_{*}$ are the mass gradients of the growth and of the track slope. The adopted track slopes follow the virial arguments of Naab et al. [49] and Bezanson et al. [152]. For a track whose slope does not vary with mass ($s^{\prime }=0$), the slope change reduces to $(s-\alpha ){\mu }^{\prime }/(1+{\mu }^{\prime })$. We adopt a single representative slope near the fiducial mass for every channel except wet major mergers, where the dissipative compaction makes s explicitly mass-dependent (Appendix A.3). Equation (A3) therefore serves as an order-of-magnitude cross-check on the Monte Carlo rather than an exact result.

Equation (A3) serves as an analytic cross-check of the full Monte Carlo, in which discrete channels (mergers, stripping) are Poisson-sampled over $z=1\to 0$, smooth channels (in situ growth, stellar mass return) are applied continuously by integrating their rates over the same interval, and $\mu \left(M_{*}\right)$ is built from the measured event rate, per-event mass change and mass-dependent affected fraction [70,103,104]. We assign each galaxy a rest-UV (3000 Å) and a rest-optical (5000 Å) half-light radius from its stellar light in one of two ways. The first applies an evolving colour-gradient law that sets how much larger the UV size is than the optical [3]. The second treats the galaxy as its separate stellar components (bulge, disk, envelope) and gives each an age-dependent brightness in each band from single-population fading curves (piecewise power-law fits to the luminosity-per-unit-mass of Bruzual and Charlot [25] and FSPS [222]). Thus, young components are UV-bright and old ones are UV-faint. Then we measure $R_e$ in each band. Either way, we fit the relation in Equation (A1) separately to the optical size and UV size.

Two channels need their own displacement rule. When a central mass fraction f is removed, the system re-equilibrates to a larger size $R_f$ from its initial size $R_i$. In the slow (adiabatic) limit,

$${\displaystyle \frac{R_f}{R_i}}={\displaystyle \frac{1}{1-f}},$$

(A4)

and in the fast (impulsive) limit $R_f/R_i=(1-f)/(1-2f)$, which diverges as $f\to \frac{1}{2}$ [162,163]. A fast tidal encounter is instead governed by the ratio of the injected energy $\Delta E$ to the target’s binding energy $\left|E\right|$,

$${\displaystyle \frac{\Delta E}{\left|E\right|}}\sim {\left({\displaystyle \frac{M_{\mathrm{pert}}}{m}}\right)}^2{\left({\displaystyle \frac{r_h}{b}}\right)}^4{\left({\displaystyle \frac{{\sigma }_v}{V}}\right)}^2\propto {\displaystyle \frac{r_h^3}{m}},$$

(A5)

for a perturber of mass $M_{\mathrm{pert}}$ passing a target galaxy of mass m, half-mass radius $r_h$, and internal velocity dispersion ${\sigma }_v$, at impact parameter b and relative speed V, using the virial scaling ${\sigma }_v^2\propto m/r_h$ [47,125,223,224]. The final proportionality $r_h^3/m\propto 1/\overline{\rho }$, with $\overline{\rho }$ the mean density, shows that diffuse low-mass systems are the most susceptible.

These are deliberately simplified forward models meant to check the sign and rough amplitude of each imprint, validated against the sign conventions of Table 1, not precise quantitative predictions. All toy UV profiles are attenuation-free. Therefore, the quoted UV–optical contrasts are intrinsic stellar-population contrasts and exclude dust gradients.

Appendix A.2. Adopted Tracks, Affected Fractions, and Wavelength Splits

Table A1 lists, for each process, the adopted track slope or displacement rule, the affected mass range and mean log-mass growth $\mu \left(M_0\right)$ at the fiducial mass, the assumption that splits the rest-UV from the rest-optical size, and the recovered imprint. The signs reproduce Table 1 in every case.

Appendix A.3. Additional Notes on Individual Processes

We provide additional notes only for the processes whose toy-model amplitude estimates (Table A1) depend on a specific adopted parameter or formula that needs justification. The remaining channels (in situ growth, bulge growth, dry major mergers, and ram-pressure stripping) follow straightforward track slopes and affected fractions there.

Wet major mergers. The collisionless remnant grows along the dissipationless track $s=1$ [49,75]. In a gas-rich merger the cold gas dissipates, sinks to the centre, and forms a compact nuclear starburst, and the remnant ends up smaller than the dissipationless expectation [225,226,227]. We capture this with a per-event displacement $\Delta \log R_e=\log (1+{\mu }_{\mathrm{r}})-\log (1+f_{\mathrm{gas}}/f_0)$, where ${\mu }_{\mathrm{r}}$ is the merger mass ratio, for which we adopt a mean of $0.5$ and hence a per-event growth ${\mu }_{\mathrm{ev}}=0.18$ dex, and $f_{\mathrm{gas}}$ the cold-gas fraction at coalescence [107]. The compaction scale $f_0=0.27$ is adopted, within the range $0.25$–$0.30$ that the dissipation models motivate [225,226,227]. The effective slope $s_{\mathrm{eff}}=1-\log (1+f_{\mathrm{gas}}/f_0)/\log (1+{\mu }_{\mathrm{r}})$ falls with gas fraction and turns negative for gas-rich low-mass mergers. Because $f_{\mathrm{gas}}$ declines with stellar mass, $s_{\mathrm{eff}}$ rises with mass, and this is the one channel for which we propagate a mass-dependent track slope, that is $s^{\prime }\ne 0$ in Equation (A3). During the starburst the young light is centrally concentrated, giving a transient per-event UV size drop of $\approx -0.25$ dex relative to the optical [102]. The merger rate meanwhile rises weakly with mass as $M_{*}^{0.3}$ [104], giving ${\mu }^{\prime }>0$, and the two gradients enter $\Delta \alpha$ with opposite signs and largely cancel. Massive galaxies are moreover gas-poor, hence the population-level imprint stays small, with $\Delta \alpha \approx 0$ and $|\Delta \log R_0|\lesssim 0.12$ dex.

Table A1. Toy-model ingredients and recovered imprints for the twelve processes. The track slope is $s\equiv \mathrm{d}\log R_e/\mathrm{d}\log M_{*}$, and $\mu \left(M_0\right)$ is the mean log-mass growth (dex) at the fiducial mass $M_0=5\times {10}^{10}{M}_{\odot }$. The recovered $\Delta \alpha$ and $\Delta \log R_0$ (dex) are over $z=1\to 0$, optical unless noted, with the amplitude grade of Table 1 in parentheses.

Process	Track s/Rule	Affected Range, $\mathit{\mu }$	Wavelength Split	Recovered Imprint
Star-forming galaxies
In situ SF, inside-out growth	$s=0.30$	whole pop., $\mu =0.35$	evolving colour gradient, UV zero point ∼+0.01 over opt	$\Delta \alpha =-0.04$, $\Delta \log R_0=+0.04$ (significant)
Bulge growth, inside-out quenching	light-weighted shrink at fixed $M_{*}$, $R_{\mathrm{b}}/R_{\mathrm{d}}=0.2$	$\log M_{*}>10$ SFGs	disk/bulge $L/M$: 1.5 (opt), 20 (UV)	$\Delta \alpha =-0.04$, $\Delta \log R_0=-0.03$, UV $\approx 0$ (modest)
Wet major mergers	$s=1$ less compaction $\log (1+f_{\mathrm{gas}}/f_0)$, $f_0=0.27$	$0.5$–$1.3$ events, ${\mu }_{\mathrm{ev}}=0.18$ (per event)	per-event UV compaction $-0.25$ (burst)	$\Delta \alpha \approx 0$, $|\Delta \log R_0|\lesssim -0.12$ (modest)
Minor mergers, accretion	$s=1$ ($0.5$–2)	ex situ $\mu \left(M_0\right)=0.03$	shared track, UV∼opt	$\Delta \alpha =+0.03$, $\Delta \log R_0=+0.03$ (modest)
High-velocity encounters	$s=2$ (outer-disk stripping)	$f_{\mathrm{pop}}\approx 0.004$, $\log M_{*}<10$	UV outer disk first, per-event $-0.10$	$\Delta \alpha ,\Delta \log R_0\lesssim {10}^{-3}$ (negligible)
Ram-pressure, tidal stripping	UV near-vertical $s_{\mathrm{UV}}\approx 17$, $s_{\mathrm{opt}}\approx 0$	$f_{\mathrm{eff}}\approx 0.04$ field	UV traces $<300$ Myr, per-event $-0.22$	field negligible, cluster-only $\Delta \log R_0^{\mathrm{UV}}=-0.09$
Quiescent galaxies
Dry major mergers	$s=1$	$\sim 0.8$ events, $\mu \left(M_0\right)=0.18$	UV≈opt (added stars old)	$\Delta \alpha =+0.03$ ($+0.02$–$0.08$), $\Delta \log R_0=+0.08$ (modest)
Minor mergers, accretion	$s=2.3$	$\mu \left(M_0\right)=0.014$	track steeper in UV $s_{\mathrm{UV}}\approx 3.7$	$\Delta \alpha =+0.05$, $\Delta \log R_0=+0.03$, UV>opt $+0.02$ (significant)
Adiabatic expansion	$R_f/R_i=1/(1-f)$, $s\approx -\eta (1-f_{\mathrm{DM}})$	stellar return $f\approx 0.11$, $\eta =0.5$	UV≈opt (homologous)	$\Delta \alpha =-0.006$, $\Delta \log R_0=+0.02$ (negligible)
High-velocity encounters	$s=0.7$ ($0.4$–$1.1$)	$f_{\mathrm{aff}}\lesssim 0.008$	UV≈opt (uniformly old)	$\Delta \alpha ,\Delta \log R_0\approx 0$, $s\approx \alpha$ (negligible)
Tidal stripping	$s_{\mathrm{opt}}\approx 1.7$, $s_{\mathrm{UV}}\approx 2.0$ (truncating an $n=4$ profile)	$⟨\Delta \log M_{*}⟩=-0.07$, $f_{\mathrm{aff}}=f_{\mathrm{sat}}\times f_{\mathrm{strip}}$	metal-poor UV envelope removed	$\Delta \alpha =+0.01$, $\Delta \log R_0=-0.005$, UV>opt (negligible)
Progenitor bias	mixture $f_{\mathrm{new}}\left(M\right)\Delta \left(M\right)$, not a track	$f_{\mathrm{new}}=0.67/0.44/0.23$	UV fades $<1$ Gyr, adopted $w\approx 0.7$	$\Delta \alpha =-0.12$ (flatten), $\Delta \log R_0=+0.02$ at $M_0$, UV<opt (significant)

Table A1. Toy-model ingredients and recovered imprints for the twelve processes. The track slope is $s\equiv \mathrm{d}\log R_e/\mathrm{d}\log M_{*}$, and $\mu \left(M_0\right)$ is the mean log-mass growth (dex) at the fiducial mass $M_0=5\times {10}^{10}{M}_{\odot }$. The recovered $\Delta \alpha$ and $\Delta \log R_0$ (dex) are over $z=1\to 0$, optical unless noted, with the amplitude grade of Table 1 in parentheses.

Process	Track s/Rule	Affected Range, $\mathit{\mu }$	Wavelength Split	Recovered Imprint
Star-forming galaxies
In situ SF, inside-out growth	$s=0.30$	whole pop., $\mu =0.35$	evolving colour gradient, UV zero point ∼+0.01 over opt	$\Delta \alpha =-0.04$, $\Delta \log R_0=+0.04$ (significant)
Bulge growth, inside-out quenching	light-weighted shrink at fixed $M_{*}$, $R_{\mathrm{b}}/R_{\mathrm{d}}=0.2$	$\log M_{*}>10$ SFGs	disk/bulge $L/M$: 1.5 (opt), 20 (UV)	$\Delta \alpha =-0.04$, $\Delta \log R_0=-0.03$, UV $\approx 0$ (modest)
Wet major mergers	$s=1$ less compaction $\log (1+f_{\mathrm{gas}}/f_0)$, $f_0=0.27$	$0.5$–$1.3$ events, ${\mu }_{\mathrm{ev}}=0.18$ (per event)	per-event UV compaction $-0.25$ (burst)	$\Delta \alpha \approx 0$, $|\Delta \log R_0|\lesssim -0.12$ (modest)
Minor mergers, accretion	$s=1$ ($0.5$–2)	ex situ $\mu \left(M_0\right)=0.03$	shared track, UV∼opt	$\Delta \alpha =+0.03$, $\Delta \log R_0=+0.03$ (modest)
High-velocity encounters	$s=2$ (outer-disk stripping)	$f_{\mathrm{pop}}\approx 0.004$, $\log M_{*}<10$	UV outer disk first, per-event $-0.10$	$\Delta \alpha ,\Delta \log R_0\lesssim {10}^{-3}$ (negligible)
Ram-pressure, tidal stripping	UV near-vertical $s_{\mathrm{UV}}\approx 17$, $s_{\mathrm{opt}}\approx 0$	$f_{\mathrm{eff}}\approx 0.04$ field	UV traces $<300$ Myr, per-event $-0.22$	field negligible, cluster-only $\Delta \log R_0^{\mathrm{UV}}=-0.09$
Quiescent galaxies
Dry major mergers	$s=1$	$\sim 0.8$ events, $\mu \left(M_0\right)=0.18$	UV≈opt (added stars old)	$\Delta \alpha =+0.03$ ($+0.02$–$0.08$), $\Delta \log R_0=+0.08$ (modest)
Minor mergers, accretion	$s=2.3$	$\mu \left(M_0\right)=0.014$	track steeper in UV $s_{\mathrm{UV}}\approx 3.7$	$\Delta \alpha =+0.05$, $\Delta \log R_0=+0.03$, UV>opt $+0.02$ (significant)
Adiabatic expansion	$R_f/R_i=1/(1-f)$, $s\approx -\eta (1-f_{\mathrm{DM}})$	stellar return $f\approx 0.11$, $\eta =0.5$	UV≈opt (homologous)	$\Delta \alpha =-0.006$, $\Delta \log R_0=+0.02$ (negligible)
High-velocity encounters	$s=0.7$ ($0.4$–$1.1$)	$f_{\mathrm{aff}}\lesssim 0.008$	UV≈opt (uniformly old)	$\Delta \alpha ,\Delta \log R_0\approx 0$, $s\approx \alpha$ (negligible)
Tidal stripping	$s_{\mathrm{opt}}\approx 1.7$, $s_{\mathrm{UV}}\approx 2.0$ (truncating an $n=4$ profile)	$⟨\Delta \log M_{*}⟩=-0.07$, $f_{\mathrm{aff}}=f_{\mathrm{sat}}\times f_{\mathrm{strip}}$	metal-poor UV envelope removed	$\Delta \alpha =+0.01$, $\Delta \log R_0=-0.005$, UV>opt (negligible)
Progenitor bias	mixture $f_{\mathrm{new}}\left(M\right)\Delta \left(M\right)$, not a track	$f_{\mathrm{new}}=0.67/0.44/0.23$	UV fades $<1$ Gyr, adopted $w\approx 0.7$	$\Delta \alpha =-0.12$ (flatten), $\Delta \log R_0=+0.02$ at $M_0$, UV<opt (significant)

High-velocity encounters. From Equation (A5) the susceptibility scales as $r_h^3/m$, which favours low-mass extended disks. We therefore taper the effect above $\log M_{*}/M_{\odot }=10$ with the factor $\frac{1}{2}[1-\tanh ((\log M_{*}-10)/0.15)]$, following the decline of stripping efficiency toward higher mass reported by Boselli and Gavazzi [128] and Boselli et al. [149]. Summing the tidal-shock energy from each encounter along a realistic cluster orbit gives a cumulative $\Delta E/\left|E\right|$ well below unity for a median orbit, consistent with the N-body result that more than three quarters of orbits produce no stellar mass loss even for the loosely bound early-type dwarfs of Virgo-like clusters [129]. For QGs the removal scale is the King tidal radius $r_t\simeq R_{\mathrm{peri}}{[m/3M(<R_{\mathrm{peri}})]}^{1/3}$ [228], where $R_{\mathrm{peri}}$ is the pericentric distance and $M(<R_{\mathrm{peri}})$ the host mass enclosed within it. Massive QGs above the pivot sit in deeper potentials than those dwarfs; hence, $r_t$ remains well outside their stellar body on a typical orbit and a single encounter strips at most a few per cent of their stars. The QG track slope $s\approx 0.7$ nearly matches the relation slope above the pivot, which pushes the imprint toward zero independently of the population dilution.

Minor mergers and accretion. For dry spheroids the accreted low-density satellites are deposited at a large radius and grow in size as $R_e\propto M_{*}^2$ ($s\approx 2$ in the energy argument for accreted material with negligible internal binding; [49]), and N-body experiments of 1:5 mass ratio mergers give $s\approx 2.3$ once the satellites retain their dark-matter haloes [115]. We adopt $s=2.3$ for QGs. This dry-spheroid scaling does not carry over to gas-rich disk hosts, where accreted gas can sink to the centre and ongoing in situ star formation dilutes the size growth. For SFGs we therefore adopt a lower, low-confidence $s\approx 1$, bracketed by the dry-spheroid limit ($s=2$) and a strong central-star-formation offset ($s=0.5$; [111]).

Tidal stripping of quiescent galaxies. We strip a galaxy by truncating an $n=4$ Sérsic profile at radius $R_{\mathrm{cut}}=x{R}_e$ and solving $L(<R_{\mathrm{new}})=\frac{1}{2}L(<R_{\mathrm{cut}})$ from the curve of growth $L(<x)\propto \gamma (2n,b_n{x}^{1/n})$, where $\gamma$ is the lower incomplete gamma function and $b_n$ the Sérsic constant [229]. Averaged over retained-mass fractions of $0.70$–$0.95$ this gives a track slope $s\approx 1.7$ for $n=4$, and $\approx 1.4$ for $n=2.5$. The UV light is more extended ($R_{\mathrm{UV}}\approx 1.15R_{\mathrm{opt}}$) and hence, the same physical cut removes a larger UV fraction and the per-galaxy contrast is $\mathrm{d}\log R_e^{\mathrm{UV}}/\mathrm{d}\log R_e^{\mathrm{opt}}\approx 1.2$ [124]. The affected fraction $f_{\mathrm{sat}}{f}_{\mathrm{strip}}$, the product of the satellite fraction and the stripped-satellite fraction, dilutes this to $\Delta \log R_0\approx -0.005$ dex at the population level.

Adiabatic expansion of quiescent galaxies. Equation (A4) gives expansion factors $R_f/R_i=1.4$–$2.0$ for a removed fraction $f=0.3$–$0.5$, the range set by the gas-rich quenching event [50,163]. The impulsive form unbinds the system at $f=0.5$. For already-quiescent galaxies at $z<1$ the only steady expellable reservoir is stellar-evolution mass return, $f\approx 0.11$ at the fiducial mass [230], driving a near-vertical effective track $s\approx -\eta (1-f_{\mathrm{DM}})$ with expulsion efficiency $\eta \lesssim 1$ and dark-matter fraction $f_{\mathrm{DM}}\approx 0.15$. This returns $\Delta \log R_0\approx +0.02$ dex over $z=1\to 0$, at or below the measurement yardstick.

Progenitor bias. This is the one channel modelled as a change in the population mixture rather than a displacement of individual galaxies. At each mass a fraction $f_{\mathrm{new}}\left(M\right)=1-1/g\left(M\right)$ of today’s QGs quenched since $z=1$, where $g\left(M\right)=n_Q(M,0)/n_Q(M,1)$ is the quiescent number-density growth. We adopt $g\approx 3/1.8/1.3$ at $\log M_{*}/M_{\odot }=10.0/10.7/11.3$ [187,231], which gives $f_{\mathrm{new}}=0.67/0.44/0.23$. These newcomers enter at their quench-epoch star-forming size, larger at fixed mass than the incumbent quiescent galaxies. The relation therefore shifts by $\Delta \log R_0\left(M\right)=f_{\mathrm{new}}\left(M\right)\Delta \left(M\right)$, with $\Delta \left(M\right)$ the mean newcomer size offset. Because $f_{\mathrm{new}}$ falls with mass, the low-mass end is lifted most and the slope flattens. Each newcomer stays UV-bright only while its disk survives, for under 1 Gyr after quenching, and fades to a mild excess afterward. Averaged over arrival times this gives a mean UV weight $w\approx 0.7$ and a weak UV-minus-optical signal below $0.01$ dex whose sign is not robust. The result is a slope change $\Delta \alpha =-0.12$ with a zero-point lift $\Delta \log R_0=+0.02$ dex at the fiducial mass.