© The Authors 2025

1. Introduction

It has been known for some time that the present-day distribution of light and mass of disk galaxies is often non axis-symmetric, with one side of the galaxy being more extended than the opposite side. Such asymmetries, first reported in the early work of Baldwin et al. (1980), are currently known as “lopsidedness”. The lopsided feature has been detected in galaxies in the spatial distribution of both the old stellar component observed in the near-infrared (Block et al. 1994; Rix & Zaritsky 1995) and HI gas (Richter & Sancisi 1994; Haynes et al. 1998), as well as in the large-scale kinematics of the HI gas (Swaters et al. 1999). Overall, observational studies have shown that in the Local Universe, at least 30% of disk galaxies have lopsided stellar disks, while 50% of disk galaxies have lopsided HI distributions (Zaritsky & Rix 1997; Bournaud et al. 2005). Overall, these observations suggest that lopsidedness is a common phenomenon in the disks of present-day galaxies.

Several mechanisms have been proposed to explain the origin of lopsidedness. The most commonly proposed mechanisms typically involve external processes, such as tidal encounters, dynamical friction, mergers, and asymmetric gas accretion (Beale & Davies 1969; Walker et al. 1996; Bournaud et al. 2005; Kipper et al. 2020), or internal dynamical processes within the disk, such as gravitational instabilities or instabilities in counter-rotating disks (Masset & Tagger 1997; Saha et al. 2007). Additional mechanisms also suggest that lopsidedness can originate from the response of the disk to a previously perturbed or distorted dark matter halo (Weinberg 1995; Jog 1997) or from torques resulting from an off-centered disk with respect to the dark matter halo (Noordermeer et al. 2001).

Other works have found a strong correlation between the lopsided amplitude and the internal properties of the galaxies (Conselice et al. 2000; Reichard et al. 2008; Varela-Lavin et al. 2023; Dolfi et al. 2023). Specifically, Varela-Lavin et al. (2023) characterized lopsidedness in a sample of present-day Milky Way-mass galaxies (i.e., virial mass M₂₀₀∼10^11.5−10^12.5 M_⊙), using the cosmological hydrodynamical simulations from the IllustrisTNG project (i.e., TNG50; Nelson et al. 2019). They found a strong dependence of lopsidedness on the central stellar mass density and tidal force exerted by the inner disk onto its outskirts. Strongly lopsided galaxies were also found to be typically characterized by more extended disks, lower central stellar mass density and less self-gravitating inner galactic regions than symmetric galaxies at z = 0. This is generally consistent with the observational results by Conselice et al. (2000) and Reichard et al. (2008), thereby suggesting it is the self-gravitating nature of the inner galactic regions that determines whether a galaxy can develop strong lopsidedness as a result of interactions, mergers, or torques exerted by distorted dark matter halos. In a recent work, Dolfi et al. (2023) extended the analysis of Varela-Lavin et al. (2023) by considering a larger sample of disk-like galaxies in the TNG50 simulation, which includes both centrals and satellites embedded in a range of different environments (i.e., M₂₀₀∼10^10.5−10¹⁴ M_⊙). The aim was to further study the relation between lopsidedness, local environment, and internal properties of the galaxies at z = 0. Regardless of the environmental metric used, Dolfi et al. (2023) found a lack of correlation between lopsidedness and local environment, coupled with a strong correlation between lopsidedness and the galaxy internal properties, independently of the environment. More importantly, Dolfi et al. (2023) studied the star formation histories of their lopsided and symmetric galaxy samples. They found that lopsided galaxies have a very distinct star formation history with respect to their symmetric counterparts. Lopsided galaxies have typically assembled over longer timescales with an overall constant star formation rate up to z = 0, while symmetric galaxies have typically assembled at early times between 8−6 Gyr ago with relatively short and intense burst of central star formation. These results suggest that lopsidedness in present-day disk galaxies is strongly connected to their early assembly histories.

While the previously discussed works have mainly focused on understanding the origin of lopsidedness at z∼0, recent works have detected lopsidedness in galaxies up to z∼3 (Kalita et al. 2022; Le Bail et al. 2024). Specifically, Le Bail et al. (2024) have studied the morphological and physical properties of a small sample of 22 dusty star-forming galaxies with M_*∼10¹⁰−10^11.5 M_⊙, observed with the James Webb Space Telescope (JWST) Near Infra-Red Camera between 1.5≲z≲3. They have found that a significant fraction of those galaxies (i.e., 64%) is lopsided. Therefore, the comparison between the fraction of lopsided galaxies detected at high (64%; Le Bail et al. 2024) and low (30%; Zaritsky & Rix 1997) redshift suggests that lopsidedness is a more common phenomenon at high redshift. Interestingly, Le Bail et al. (2024) classified their galaxies into three different categories based on the star formation properties of the bulge and disk components. As a result, they have found evidence of a strong correlation between lopsidedness and the internal properties of the galaxies at high redshift. Galaxies with a quenched bulge component (i.e., TypeIII galaxies), characterized by high core mass fraction, have typically lower asymmetry than galaxies with a star-forming bulge component (i.e., TypeI and TypeII galaxies), which are characterized by low core mass fraction. Since TypeIII galaxies in Le Bail et al. (2024) are at lower redshifts on average and tend to have a lower specific star formation rate than TypeI and TypeII galaxies, the authors suggested that galaxies tend to become more symmetric toward a lower redshift, due to the build up of a massive quenched bulge component that can stabilize the disk against lopsided perturbations. Overall, these results also suggest a strong correlation between lopsidedness and the internal properties of the galaxies, as similarly found in previous observational (Conselice et al. 2000; Reichard et al. 2008) and numerical (Varela-Lavin et al. 2023; Dolfi et al. 2023) studies at z = 0.

Currently, lopsidedness in simulations has been mainly investigated at z = 0 (Łokas 2022; Varela-Lavin et al. 2023; Dolfi et al. 2023). In this work, we use the IllustrisTNG¹ cosmological hydrodynamical simulations (in particular the TNG50 run) to characterize lopsidedness in the disks of galaxies up to z = 2. We study the relevant mechanisms generating lopsidedness, as well as the correlation between lopsidedness, local environment, and galaxy internal properties at different redshifts. We aim to understand whether the more widespread lopsidedness at high (rather than at low redshift) may be associated with the fact that environmental effects play a more active role in triggering lopsidedness at high redshift. In particular, we aim to understand whether the higher merger and interaction rate at high redshift, rather than low redshift (Conselice 2014) could be responsible for the high fraction of observed lopsided galaxies between 1.5≲z≲3 (Le Bail et al. 2024).

The paper is structured as follows. In Sect. 2.2, we briefly describe the simulations and the selection of our sample of disk-like galaxies. In Sect. 3, we describe the method used to quantify the global lopsidedness in our selected galaxy sample. In Sect. 4, we study the evolution of the fraction of lopsided galaxies, as well as lopsided amplitudes, as a function of redshift. We also study the correlation between lopsidedness, local environment, and galaxy internal properties as a function of redshift. Finally, we investigate the potential impact of galaxy interactions, along with gas accretion, as the mechanisms that tend to trigger lopsidedness. In Sect. 5, we present a comparison between our results and those obtained in the recent observations of Le Bail et al. (2024). Finally, in Sect. 6, we provide a summary of our results and conclusions.

2. The data

2.1. The IllustrisTNG simulations

The IllustrisTNG simulations are a suite of cosmological gravito-magnetohydrodynamical simulations that include a comprehensive physical model of galaxy formation designed to realistically trace the formation and evolution of galaxies across cosmic time (Vogelsberger et al. 2014; Genel et al. 2014; Sijacki et al. 2015; Nelson et al. 2015; Weinberger et al. 2017; Pillepich et al. 2018). The IllustrisTNG simulations were run with the moving-mesh code Arepo (Springel 2010) and have been made publicly available. A detailed description of the IllustrisTNG simulations can be found in the release paper by Nelson et al. (2019). In this work, we use the smallest cosmological volume of the simulations (i.e., TNG50: 51.7 cMpc³; Weinberger et al. 2017; Pillepich et al. 2018), which is characterized by the highest resolution (i.e., initial gas cell mass 8.5×10⁴ M_⊙). The high resolution of TNG50 allows us to better characterize and quantify asymmetries in the stellar mass density distribution of galaxies.

2.2. Sample selection

From the TNG50 simulation, we selected all galaxies with a total stellar mass between 10¹⁰ M_⊙≤M_*≤10^11.5 M_⊙ and the total number of bound stellar particles, N_tot, stars≥10⁴, at each redshift between 0<z<2². The first selection criterion was chosen to match the stellar mass range of the sample of 22 dusty star-forming galaxies studied by Le Bail et al. (2024), which were observed using the JWST Near Infra-Red Camera between 1.5<z<3. The second selection criterion was meant to ensure that each galaxy is resolved enough to quantify differences in the stellar mass density distribution, similarly to the approach taken in Varela-Lavin et al. (2023), Dolfi et al. (2023). We note that the M_*-based selection criteria used in this work differs from the star formation rate (SFR) based selection criteria used in Le Bail et al. (2024). In particular, for z>1.5, we find that our simulated galaxies are characterized by, on average, slightly lower values of the SFR and are located, on average, at lower redshift than the observed galaxies in Le Bail et al. (2024). The redshift difference is due to the different redshift intervals used in the simulations and observations, which are 1.5<z<2 and 1.5<z<3, respectively. We note here that we do not extend the redshift range up to z = 3 to match the observations from Le Bail et al. (2024) due to the low number of disk-like galaxies (<100) obtained from the TNG50 simulation, as well as to their strongly perturbed morphology, at z = 3. For the SFR difference, we checked our results when considering an additional selection criteria based on the SFR, namely SFR≳10−20 M_⊙/yr. This corresponds to the redshift-dependent SFR limit of the galaxy sample of Le Bail et al. (2024) at z = 1.5. Overall, we find that our results do not change significantly when we consider the additional SFR-based selection. For this reason, in this work, we chose to consider only the stellar mass-based selection criteria for simplicity, as well as for obtaining a more complete galaxy sample to study.

2.3. Defining galaxy sizes

For all the selected galaxies in Sect. 2.2, we define the following sizes calculated from the stellar particles alone: half-mass radius, R_h, disk size, R₉₀, and disk height, h₉₀. The stellar half-mass radius and disk size are defined as the radii enclosing 50% and 90% of the total galaxy stellar mass, respectively. These radii are calculated with respect to the position of the galaxy's particle with the minimum gravitational potential energy, which is taken as the galaxy's center. Similarly, the disk height is defined as the vertical distance above and below the disk plane enclosing 90% of the total galaxy stellar mass. Following Iza et al. (2022), we calculate the disk height along both the positive (i.e., h_90,+) and negative (i.e., h_90,−) z-directions, considering only stellar particles within the cylindrical region of radius 1 R₉₀. Then, we calculated the average value between h_90,+ and h_90,− to define the disk height, h₉₀, above and below the disk plane. The overall disk width is then defined as twice the disk height; namely, the vertical region between −h₉₀<z<+h₉₀. The calculation of all these quantities is performed after rotating the galaxy in the face-on projection, such that the galaxy disk lies on the xy-plane. We note that the calculation of the disk radii and disk height of the galaxies does not separate between bulge and disk components, and we do not use scale lengths to define the galaxy's sizes as in observations. For the objectives of this work, we are interested in defining quantities that can represent the boundaries of the disk to study its stellar mass distribution without the contamination from external sources or streams in the galactic halo.

Finally, we define the central regions of the galaxies as the spherical region enclosed within a radius of 2 kpc. We note that this definition of the central regions as a proxy of the bulge component of the galaxies is used only in Sect. 5 to perform the comparison with the observations of Le Bail et al. (2024). The observed galaxies in Le Bail et al. (2024) have an average half-light radius, calculated in the near infra-red, R_e,NIR∼2.5 kpc. For this reason, we decide to use a fixed 2 kpc radius as a proxy of the bulge region of our galaxies in Sect. 5. We note that this 2 kpc radius is also consistent with the average half-light radius of the bulges of a sample of Milky Way-like galaxies from the TNG50 simulation (Gargiulo et al. 2022). On the other hand, we defined the disk component as the cylindrical region enclosed within the radial range 2 kpc<r<R₉₀ and vertical distance above and below the disk plane −h₉₀<z<+h₉₀. Similarly, we also defined the SFR of the central component as the total SFR of all gas cells enclosed within a sphere of radius 2 kpc, while we define the SFR of the disk as the total SFR of all gas cells enclosed within a cylindrical region of radius 2 kpc<r<R₉₀ and vertical distance above and below the disk plane −h₉₀<z<+h₉₀.

2.4. Calculating the kinematic properties of the galaxies

The spin-ellipticity (i.e., λ_R−ϵ) diagram is typically used in observations to separate between fast-rotating and slow-rotating galaxies. The former are typically characterized by a disk-like structure (i.e., lenticular and spiral galaxies), while the latter are typically elliptical galaxies (Emsellem et al. 2007, 2011). For all the selected galaxies included in Sect. 2.2, we used the λ_R−ϵ diagram to select our final sample of disk-like galaxies. As described in Lagos et al. (2016), we calculated the stellar angular momentum of galaxies, λ_R, defined as:

$${\lambda }_{\mathrm{R}}=\frac{{\sum }_{j=1}^N{M}_j{R}_j{V}_{\mathrm{rot}}(R_j)}{{\sum }_j{M}_j{R}_j\sqrt{(V_{\mathrm{rot}}(R_j){)}^2+({\sigma }_{1\mathrm{D}}(R_j){)}^2}},$$(1)

where V_rot(R_j) and σ_1D(R_j) represent the galaxy rotational velocity and stellar velocity dispersion perpendicular to the mid-plane of the galactic disk calculated within the radius R_j, respectively. M_j represents the stellar mass enclosed within R_j. In Eq. (1), the sum runs over all the N radial bins. However, in this work, we consider one single bin that extends from the galaxy center to its stellar half-mass radius (i.e., R_j = 1 R_h), which we use as a proxy of the stellar half-light radius. Then, M_j represents the stellar mass enclosed within R_h, while V_rot(R_j) represents the galaxy rotational velocity at R_h. The galaxy rotation velocity is then defined as:

$$V_{\mathrm{rot}}(r)\equiv \frac{|j_{*}|}{r},$$(2)

where r=R_h and j_* is the specific angular momentum of the stellar component defined as:

$$j_{*}=\frac{{\sum }_i{m}_i(r_i-r_{\mathrm{COM}})\times (v_i-v_{\mathrm{COM}})}{{\sum }_i{m}_i}·$$(3)

Here, r_i, v_i and r_COM, v_COM are the position and velocity vectors of the i-th stellar particle and center of mass of the galaxy, respectively, while m_i is the mass of the i-th stellar particle.

The stellar velocity dispersion perpendicular to the mid-plane of the disks is calculated by considering the component of the velocity vector parallel to the total stellar angular momentum vector of the galaxy L_*. Then, the stellar velocity dispersion is defined as:

$${\sigma }_{1\mathrm{D},*}(r)=\sqrt{\frac{{\sum }_i{m}_i(\mathrm{\Delta }{v}_{i}cos{\theta }_i{)}^2}{{\sum }_i{m}_i}},$$(4)

where Δv_i is the velocity of the i-th stellar particle relative to the centre of mass of the galaxy, while cos θ_i=Δv_i·L_*/|Δv_i||L_*|.

We note that all quantities in Eqs. (1)–(4) are calculated within the inner R_h due to the fact that we are measuring λ_R within R_h. The total stellar angular momentum of the galaxy is the only quantity calculated using all stellar particles bound to the galaxy. This quantity is only used in Eq. (4) to define the angle between the stellar velocity and total angular momentum vector of the galaxy to calculate the stellar velocity dispersion perpendicular to the mid-plane of the disk.

For each of the selected galaxies included in Sect. 2.2, we also calculate the ellipticity within the inner R_h. To do this, we construct the tensor of inertia from the stellar particles; due to its symmetry, it is defined as:

$$\begin{array}{cc}\hfill I_{\mathit{xx}}(r)& =\frac{{\sum }_i{m}_i(y_i^2+z_i^2)}{{\sum }_i{m}_i},\hfill \\ \hfill I_{\mathit{yy}}(r)& =\frac{{\sum }_i{m}_i(x_i^2+z_i^2)}{{\sum }_i{m}_i},\hfill \\ \hfill I_{\mathit{zz}}(r)& =\frac{{\sum }_i{m}_i(x_i^2+y_i^2)}{{\sum }_i{m}_i},\hfill \\ \hfill I_{\mathit{xy}}(r)& =I_{\mathit{yx}}(r)=-\frac{{\sum }_i{m}_i(x_i{y}_i)}{{\sum }_i{m}_i},\hfill \\ \hfill I_{\mathit{xz}}(r)& =I_{\mathit{zx}}(r)=-\frac{{\sum }_i{m}_i(x_i{z}_i)}{{\sum }_i{m}_i},\hfill \\ \hfill I_{\mathit{yz}}(r)& =I_{\mathit{zy}}(r)=-\frac{{\sum }_i{m}_i(y_i{z}_i)}{{\sum }_i{m}_i}·\hfill \end{array}$$(5)

Then, we diagonalize the tensor of inertia to derive the corresponding eigenvalues, M₁, M₂, and M₃, which are sorted in ascending order, such as M₁<M₂<M₃, and used to define the galaxy shortest-to-longest axis ratio c/a=M₁/M₃ (Genel et al. 2015). Finally, the ellipticity of the galaxy is defined as:

$$ϵ=1-\frac{c}{a}.$$(6)

2.5. Defining the sample of disk-like galaxies

In the top panel of Fig. 1, we show the spin-ellipticity (i.e., λ_R−ϵ) diagram for all the selected galaxies included in Sect. 2.2. The red solid line shows the threshold ${\lambda }_{\mathrm{R}}=0.31\times \sqrt{ϵ}$ typically used in observations to separate between rotation-dominated and dispersion-supported galaxies within the inner stellar half-light radius at z = 0 (Emsellem et al. 2011), which we assume to be valid up to z = 2. We selected our final sample of disk-like galaxies by including all rotation-dominated (i.e., ${\lambda }_{\mathrm{R}}>0.31\times \sqrt{ϵ}$) and flattened (i.e., ϵ>0.4) galaxies with R₉₀>3 kpc at each redshift between 0<z<2 (colored points in Fig. 1). We note that the ϵ>0.4 selection criteria is an additional conservative approach to remove the roundest spheroidal-like galaxies, primarily focused on disk-like galaxies, such as spirals and lenticulars. After a visual inspection, we also removed compact galaxies with R₉₀<3 kpc that did not show evidence of a disk-like structure from the edge-on projection. These galaxies represent a minority (i.e., 17 at z = 2 and only a few at z = 0). In the bottom panel of Fig. 1, we show the number of central and satellite galaxies of our final selected sample of disk-like galaxies at each redshift. We see that the fraction of satellite galaxies increases from z = 2 up to z = 0.5 and, then, it remains overall constant up to z = 0.

Fig. 1. Top panel: Spin-ellipticity (i.e., λR−ϵ) diagram of the galaxies selected from the TNG50 simulation between 0<z<2 (gray points). The colored points show our final sample of disk-like galaxies, using the selection criteria described in Sect. 2.5. The red solid line represents the threshold typically used in observations to separate between rotation-dominated and dispersion-supported galaxies within the inner stellar half-light radius at z = 0 (Emsellem et al. 2011). Bottom panel: Total number of central and satellite galaxies of our final sample of disk-like galaxies at each redshift, selected as described in Sect. 2.5.

In Fig. 2, we show the stellar mass distribution and the median value of λ_R as a function of stellar mass for our selected sample of disk-like galaxies at each redshift between 0<z<2. Overall, we see that the fraction of disk-like galaxies, as well as their degree of rotation, steeply decreases toward high stellar masses at all redshift. This is consistent with the observations that massive galaxies tend to be slow-rotating spheroidals (Emsellem et al. 2011).

Fig. 2. Stellar mass distribution (top panel) and median value of λR as a function of stellar mass (bottom panel) for our selected sample of disk-like galaxies at each redshift between 0<z<2. Shaded areas are defined by the 25th–75th interquartile range of the data in each bin.

3. Measuring lopsidedness

Similarly to the approach in Varela-Lavin et al. (2023) and Dolfi et al. (2023), we calculated the radial lopsidedness profile of our selected sample of disk-like galaxies at each redshift between 0<z<2 by performing an azimuthal Fourier decomposition of the galaxy stellar mass in the face-on projection. We measured the amplitude of the first Fourier mode, which we define here as a lopsided amplitude, in equally spaced concentric radial annuli of width 0.1 kpc and height |h_z| = 2 h₉₀ above and below the disk plane. Finally, we quantified the global lopsidedness, A₁, of our galaxies by calculating the average lopsided amplitude in the radial range of R_h<r<1.4 R₉₀.

We note that the radial range, R_h<r<1.4 R₉₀, differs from the one used in Varela-Lavin et al. (2023) and Dolfi et al. (2023) to quantify the global lopsidedness of galaxies at z = 0, namely, 0.5 R_opt<r<1.1 R_opt with R_opt being the optical radius defined as the radius where the V-band surface brightness drops down to μ_V = 26.5 mag/arcsec² at z = 0. In this work, we find that the outer radius 1.4 R₉₀ is a good proxy of the boundary of the stellar disks of galaxies at all redshift up to z = 2. On the other hand, we find that, if we use the same z = 0 definition of R_opt for all redshift, then R_opt is more extended than the boundary of galactic disks at high redshift (i.e., z≳0.5). In any case, we have studied the effect of using different radial ranges to quantify the global lopsidedness of galaxies between 0<z<2. Overall, we find that while the fraction of lopsided and symmetric galaxies varies depending on the radial range used (see also discussion in Dolfi et al. 2023 at z = 0), the general trends and results we report here are not significantly affected.

4. Results

Throughout the discussion of the results, we highlight that we are not following the evolution of a certain sample of galaxies here. Instead, we specifically aim to investigate the behavior of a defined population of disk-like galaxies at each redshift.

4.1. Lopsidedness as a function of redshift

In the left panel of Fig. 3, we show the median value of the lopsided amplitude as a function of redshift for our selected sample of disk-like galaxies between 0<z<2, divided into centrals and satellites. Overall, we see a decrease of the lopsided amplitude from z = 2 to z = 0 for both central and satellite galaxies, but this decrease is less significant for central galaxies. Additionally, we see that, while central and satellite galaxies tend to be characterized by a similar lopsided amplitude at high redshift (i.e., z>1), central galaxies have larger lopsided amplitude than satellite galaxies at low redshift (i.e., z≲1).

Fig. 3. Left panel: Median lopsided amplitude as a function of redshift for our selected sample of disk-like galaxies at each analyzed redshift between 0<z<2, divided into centrals and satellites. Shaded areas are defined by the 25th–75th interquartile range of the data at each bin. The horizontal dotted black line indicates the threshold A1 = 0.1 used to classify galaxies into lopsided (A1>0.1) and symmetric (A1<0.1) at z = 0 (Varela-Lavin et al. 2023; Dolfi et al. 2023). Right panel: Fraction of lopsided galaxies as a function of redshift for centrals and satellites.

In the right panel of Fig. 3, we show the fraction of lopsided galaxies as a function of redshift for the central and satellite galaxies, respectively. Here, we use the threshold A₁ = 0.1 to classify galaxies into lopsided (A₁>0.1) and symmetric (A₁<0.1) at all redshift, as previously done in Varela-Lavin et al. (2023) and Dolfi et al. (2023) at z = 0. At high redshift (i.e., z>1), we find similar large fractions of lopsided galaxies (i.e., ≳55%) for both centrals and satellites. This is overall consistent with the large fraction of observed lopsided galaxies (i.e., 64%) found by Le Bail et al. (2024) at high redshift (i.e., 1.5≲z≲3). For z≲1, we find that the fraction of lopsided satellite galaxies steeply decreases to ∼30−35% at z = 0. On the other hand, the fraction of lopsided central galaxies remains roughly constant at ∼55% up to z∼0.5, beyond which it drops to ∼45% at z = 0. We note that, at z∼0, we find a slightly larger number of lopsided galaxies (i.e., ∼40%) with respect to observations in the Local Universe (i.e., ∼30%; e.g., Zaritsky & Rix 1997). The reason for this discrepancy between observations and simulations can be likely attributed to the radial range used to measure the global lopsidedness of galaxies. While Zaritsky & Rix (1997) measured lopsidedness by taking the average amplitude between 1.5−2.5 disk scalelengths (i.e., ∼1−2 R_h; Łokas 2022), we measure lopsidedness reaching out to larger galactocentric radii (i.e., up to 1.4 R₉₀). For this reason, we are likely identifying a larger fraction of lopsided galaxies compared to observations, due to the fact that the strength of the lopsided perturbation typically increases toward large galactocentric radii (Varela-Lavin et al. 2023). Indeed, the estimated fraction of lopsided galaxies is sensitive to the radial range used to measure the global lopsidedness (see also the discussion in Dolfi et al. 2023).

Overall, the fact that satellite galaxies are more symmetric than central galaxies at low redshift (i.e., z≲1) may be the result of environmental effects driving more rapidly their morphological transformations to early-type disks (i.e., S0-like) (Dressler 1980; Dressler et al. 1997; Deeley et al. 2021; Montaguth et al. 2023, 2025). In fact, as previously shown in Dolfi et al. (2023) at z = 0, early-type disk galaxies typically develop lower lopsided amplitude than late-type disk galaxies. Nonetheless, the overall decrease of the lopsided amplitude from high-to-low-redshift also suggests that different mechanisms are at play at high and low redshift, or that they are more efficient at high redshift. We discuss in the mechanisms generating lopsidedness in more detail in Sect. 4.4.

4.2. Correlation between lopsidedness and local environment

We go on to study the correlation between the lopsided amplitude and the local density of the environment as a function of redshift for our selected sample of disk-like galaxies between 0<z<2. We defined the local density of the environment around each selected galaxy as:

$${\rho }_{10}=10/\left(\frac{4}{3}\pi R_{10}^3\right),$$(7)

where R₁₀ is the radius of the sphere, centered on each galaxy, enclosing the N = 10 nearest neighbors. We only consider nearest neighbors with total mass of M_tot>10⁹ M_⊙, where M_tot is computed considering all particles (i.e., dark matter+baryons) that are gravitationally bound to the galaxy. We note here that in Dolfi et al. (2023), we tested the use of different environmental metrics, as well as mass thresholds, to study the correlation between the lopsided amplitude and local environment, finding that the results did not vary significantly when using different metrics.

In Fig. 4, we show that independently of the redshift considered, the median lopsided amplitude does not strongly depend on the local environmental density of the galaxies. We find that the results are similar when we are considering the central and satellite galaxies separately. Overall, this suggests that whether a galaxy develops strong lopsidedness does not primarily depend on the local environment (i.e., number of nearest neighbors) at any redshift considered. This is consistent with the mild or lack of correlation between lopsidedness and local environment observed at z = 0 both in simulations (Dolfi et al. 2023; Fontirroig et al. 2025) and observations (Wilcots 2010), as well as in high-redshift observations (Le Bail et al. 2024).

Fig. 4. Median lopsided amplitude as a function of the local density of the environment between 0<z<2 for our selected sample of disk-like galaxies. The local density of the environment is calculated as described in Sect. 4.2, considering the ten nearest neighbors with total mass Mtot>109 M⊙. Shaded areas are defined by the 25th–75th interquartile range of the data in each bin.

4.3. Correlation between lopsidedness and internal galaxy properties

Previous theoretical and observational studies have found that lopsidedness strongly correlates with the central stellar mass density and disk size of galaxies at z∼0 (Reichard et al. 2008; Varela-Lavin et al. 2023).

In top panels of Fig. 5, we study the correlation between lopsidedness and the internal properties of galaxies as a function of redshift for our selected sample of disk-like galaxies between 0<z<2. From the left to the right panel of Fig. 5, we show the median lopsided amplitude as a function of the central stellar mass density (i.e., μ_*³), stellar half-mass radius (R_h), and disk size (R₉₀), respectively. We note that here we define the central stellar mass density within R_h as previously done in the observational results from Reichard et al. (2008). We find a strong correlation between the median lopsided amplitude and the different internal properties of galaxies at all redshift considered. Galaxies characterized by (on average) a lower central stellar mass density, larger half-mass radii, and larger R₉₀ values tend to be more lopsided than galaxies with higher central stellar mass density, smaller half-mass radii, and smaller R₉₀ values. We note that we have also studied the correlation between lopsidedness and galaxy size (i.e., R₉₀) for fixed concentration values (i.e., c^*=R₉₀/R₅₀). We have continued to observe a strong correlation between lopsidedness and galaxy size, suggesting that the correlation is not influenced by variations in bulge properties. However, we also see two additional trends. First of all, the median lopsided amplitude tends to increase from low-to-high redshift at a fixed μ_*, R_h, and R₉₀ values. Furthermore, the median lopsided amplitude increases more rapidly with increasing R_h and R₉₀ at high redshift, rather than at low redshift (i.e., steeper A₁ slope for R_h and R₉₀ at high redshift). This means that at high redshift, lopsidedness is more sensitive to variations in the galaxy's radii. Overall, these results suggest that the development of the lopsided perturbation is strongly connected to the internal properties of the galaxies at all redshift, consistent with the results from previous works at z = 0 (Reichard et al. 2008; Varela-Lavin et al. 2023; Dolfi et al. 2023). In a recent work, Fontirroig et al. (2025) trained a Random Forest algorithm on a large sample (∼8000) of disk-like galaxies selected from the IllustrisTNG simulations between 0<z<0.5. The algorithm was trained only considering the galaxy internal properties as features, meaning that no information about the environment of the galaxies was provided. They showed that their algorithm can perform an accurate and rapid classification of lopsided galaxies. This is also in agreement with our current results shown in Figs. 4 and 5, and confirms that the internal properties of the galaxies play a primary role in determining whether a galaxy can develop strong lopsidedness as a result of external perturbations.

Fig. 5. Top: Median lopsided amplitude as a function of the central stellar mass density (μ*), stellar half-mass radius (Rh) and disk size (R90) for our selected sample of disk-like galaxies between 0<z<2. Bottom: Median μ*, Rh and R90 as a function of redshift for our selected sample of disk-like galaxies between 0<z<2. Shaded areas are defined by the 25th–75th interquartile range of the data in each bin.

In the bottom panels of Fig. 5, we study the evolution of the median μ_*, R_h, and R₉₀ as a function of redshift for our selected sample of disk-like galaxies between 0<z<2. We see that at high redshift (e.g., z>1), galaxies have larger central stellar mass density and smaller sizes than at lower redshift. This means that, if mechanisms triggering lopsidedness are the same as a function of redshift, low-redshift galaxies would be more susceptible at developing strong lopsidedness than high-redshift ones, due to the observed strong correlation between lopsidedness and galaxy internal properties. The fact that the median lopsided lopsided amplitude, as well as fraction of lopsided galaxies, decrease from high-to-low redshift (see Fig. 3). This suggests that mechanisms triggering lopsidedness are different or more efficient at high (rather than low) redshift.

We also studied the correlation between lopsidedness and the galaxy stellar mass. At all redshift considered, we only find a mild decrease of the median lopsided amplitude with increasing stellar mass. This is consistent with the decrease of A₁ toward large stellar masses seen in observations at z = 0 (Reichard et al. 2008), as previously shown in Dolfi et al. (2023). We note that we find similar results when we consider central and satellite galaxies separately.

4.4. Origin of lopsidedness at high and low redshift

In the sections above, we explain that while the internal properties of galaxies always play an important role for the development of lopsidedness, this perturbation seems to be significantly more common and stronger at high (rather than low) redshift. It is thus interesting to explore whether different mechanisms are at play or whether the same mechanisms driving lopsidedness at z∼0 are simply more efficient at high redshift.

In this section, we study the relative importance as a function of redshift of different mechanisms that can potentially trigger lopsidedness, specifically focusing on close tidal interactions and gas accretion. We recall in particular that we are selecting a new galaxy sample at each redshift, using the same selection criteria described in Sect. 2.5.

4.4.1. Galaxy interactions

To study the role of tidal interactions as triggers of lopsidedness, we quantify the number of nearest neighbors within 0.5×R₂₀₀ during the last 3 Gyr of the galaxy history, where R₂₀₀ is the virial radius of the galaxy. Here, we only consider nearest neighbors with stellar mass-ratio⁴ >1:10. This mass threshold was chosen to include both major (mass-ratio 1:1−1:4) and minor (mass-ratio 1:4−1:10) mergers, which are expected to be massive enough to influence the morphology of the galaxy (Bournaud et al. 2004; Gómez et al. 2017). We only include the results for the central galaxies in our sample and not for the satellites. This is because, by definition, satellite galaxies have at least one massive neighbor that they are interacting with; namely, the central galaxy of the group or cluster to which they belong.

In Fig. 6, we show the number of lopsided and symmetric galaxies with one or more massive neighbors (stellar mass-ratio <1:10) within 0.5×R₂₀₀ during the last 3 Gyr of the galaxy history, normalized by the total number of lopsided and symmetric galaxies, respectively, at the five different specific redshift z = 2, 1.5, 1, 0.5 and 0. Here, we note that we are looking at different galaxies as a function of redshift, since we are selecting our samples of disk-like galaxies independently at each redshift (see Sect. 2.2). We see that the fraction of lopsided galaxies with one or more massive neighbors remains overall constant between ∼35%−40% between 0<z<2, with the exception of z = 1 where this fraction becomes close to ∼50%. This means that close tidal interactions are similarly common at high and low redshift, and that they can be a trigger of lopsided perturbations in, on average, ∼40% of lopsided galaxies between 0<z<2. However, the fact that on average, ∼60% of lopsided galaxies do not have any nearby massive neighbors within the last 3 Gyr means that tidal interactions are not the only mechanism at play in the origin of lopsidedness. Other mechanisms must also be triggering lopsidedness at all redshift between 0<z<2.

Fig. 6. Number of lopsided and symmetric galaxies with one or more massive neighbors (stellar mass-ratio >1:10) within 0.5×R200 during the last 3 Gyr of the galaxy history, normalized by the total number of lopsided and symmetric galaxies, respectively, at the five different specific redshift z = 2, 1.5, 1, 0.5, 0. Here, we only focus on the central galaxies in our sample.

For the symmetric galaxies, we see that the fraction of galaxies with one or more massive neighbors slightly increases from ∼15% at z = 2 up to ∼35% at z = 0. This means that, at high redshift, these galaxies have not experienced significant interactions that could have triggered lopsidedness. At low redshift, the role of tidal interactions is similar for both lopsided and symmetric galaxies, which means that the particular internal properties of the symmetric galaxies (i.e., high central stellar mass density and small disk size) cause them to be less prone to develop lopsidedness with respect to lopsided galaxies (see Sect. 4.3). Overall these results show that, while external interactions can be a trigger of lopsidedness at all redshift up to z = 2, the internal properties of the galaxies play an important role in determining whether a lopsided perturbation will rise as a result of such an interaction, as previously discussed in Sects. 4.2 and 4.3.

4.4.2. Net accretion rate

In this section, we explore the potential role of gas accretion as trigger of lopsidedness. For this purpose, we calculate net accretion rates of our samples of disk-like galaxies selected at the five different specific redshift z = 2, 1.5, 1, 0.5 and 0. The net accretion rates are computed only considering a 1 Gyr time period by following the procedure described in Iza et al. (2022):

$${\dot{M}}_{\mathrm{net}}(i)=\frac{M_{\mathrm{gas}}(i)-M_{\mathrm{gas}}(i-1)+M_{*}}{t(i)-t(i-1)},$$(8)

where M_gas(i) is the total gas mass in the disk at snapshot (i), while M_* is the total stellar mass born in the disk in the corresponding time interval [t(i)−t(i−1)] considered for this analysis. We recall here that the disk is defined as the cylindrical region enclosed within the radius 2 kpc<r<R₉₀ and vertical distance from the disk plane −h₉₀<z<+h₉₀, as described in Sect. 2.3.

In Fig. 7, from the left to the right panel, we show the net accretion rate, the galaxy specific star formation rate and the specific star formation rate within the disk as a function of redshift. As previously in Fig. 6, we divide our galaxies into lopsided and symmetric, using the threshold A₁ = 0.1 as described in Sect. 4.1.

Fig. 7. Left panel: Net accretion rate onto the galactic disk within the last 1 Gyr for our selected sample of disk-like galaxies at the five different specific redshift z = 2, 1.5, 1, 0.5, and 0. Middle panel: Total galaxy star formation rate. Right panel: Star formation rate within the galactic disk, where the disk is defined as described in Sect. 2.3. We divide our galaxies into lopsided and symmetric, using the threshold A1 = 0.1. Shaded areas are defined by the 25th–75th interquartile range of the data at each redshift.

In the left panel of Fig. 7, we see that lopsided galaxies are characterized by larger net accretion rate than symmetric galaxies at all redshift. In particular, for lopsided galaxies, we see that the net accretion rate decreases from z = 2 to z = 0, indicating that these galaxies experience more significant gas accretion at high- (i.e., z>1) than low redshift. On the contrary, symmetric galaxies are mainly characterized by negative net accretion rate, meaning either that they experienced more significant outflow events that removed the gas from the disk, or that they consumed more gas than have accreted during the last ∼10 Gyr of evolution.

In the middle and right panel of Fig. 7, we also see that lopsided galaxies are characterized by, on average, larger specific star formation rate than symmetric ones at all redshift both globally and within the disk region. This suggests that the gas accreted onto the disk is being more efficiently converted into stars through subsequent star formation in lopsided galaxies. We note that we find similar results when we consider central and satellite galaxies, separately.

Overall, the results in Fig. 7 show that gas accretion with subsequent star formation is clearly more prominent for lopsidedness at all redshift between 0<z<2. The fact that the net accretion rate decreases from high to low redshift also suggests that the role of gas accretion behind lopsidedness is more significant at high (rather than low) redshift, consistent with the decreasing fraction of lopsided galaxies toward low redshift, as shown in Fig. 3. In a previous work, Łokas (2022) also suggested that asymmetric gas accretion followed by star formation is a probable mechanism for the origin of lopsidedness at z = 0, based on the finding that lopsided galaxies are typically characterized by larger gas fraction, larger star formation rate, lower metallicity, and bluer color than symmetric galaxies, using the IllustrisTNG simulations. The connection between lopsidedness and star formation rates has also been recently detected in high-redshift disk galaxies (i.e., 1<z<4) observed with JWST (Ganapathy et al. 2025). In another work, Bournaud et al. (2005) simulated an ideal case where the accretion of gas onto the galaxy occurs along an off-centered filament or along several filaments at once. They found that this asymmetric gas accretion can produce strong lopsided amplitudes. However, they did not try to study more realistic gas accretion scenarios predicted by cosmological models. We note that, here, we have not studied whether the gas is being accreted asymmetrically nor its angular momentum. The latter is also expected to be an important parameter due to the recent finding that lopsided galaxies tend to live in high-spin halos (Varela-Lavin et al. 2023), which can lead to the formation of their more extended disks (Grand et al. 2017). We will perform this analysis in details in a follow-up work to test the scenario of lopsidedness being generated as a result of asymmetric gas accretion with subsequent star formation in realistic cosmological models.

5. Comparison with observations

In a recent work, Le Bail et al. (2024) investigated the properties of a small sample of 22 dusty star-forming galaxies observed with the JWST Near Infra-Red Camera between 1.5<z<3. They found that 64% of their galaxies are lopsided. Specifically, they find that the sub-sample of strongly lopsided galaxies tend to be characterized by a star-forming core with low core mass fraction compared to the symmetric sub-sample, which is typically characterized by a quenched bulge with high core mass fraction (see Fig. 14 in Le Bail et al. 2024). Based on these results, Le Bail et al. (2024) suggest that the build-up of a quenched massive bulge component can have the effect of stabilizing the disk against the lopsided perturbation, consistent with the results from previous works at z = 0 that demonstrated that the low central stellar mass densities are key for the onset of lopsided perturbations (Reichard et al. 2008; Varela-Lavin et al. 2023; Dolfi et al. 2023). In this section, we propose performing a comparison with the observations from Le Bail et al. (2024) by classifying our selected sample of simulated disk-like galaxies into different types based on their star formation activity, similarly to Le Bail et al. (2024). We recall that we are selecting a new galaxy sample at each redshift, using the same selection criteria described in Sect. 2.5.

5.1. Classification: Star-formation rate of cores and disks

First, we calculated the star formation rate (SFR) of the central and disk components of each of our selected disk-like galaxies between 0<z<2, as described in Sect. 2.3. Then, we calculate the specific star-formation rate (sSFR) of the central and disk components of our galaxies by normalizing the SFR of each component by its total stellar mass.

In Fig. 8, we show the distribution of the sSFR of the central and disk components of our selected sample of disk-like galaxies at z = 0 and z = 1.5, as an example. At each redshift, we separate the distribution of the sSFR of both components in tertiles (dashed lines in Fig. 8) and we select the four galaxy sub-samples lying on the outside of the tertile contours (colored points in Fig. 8). We define the following four categories of galaxies as:

• TypeI: sub-sample of galaxies characterized by both a star-forming core and a star-forming disk (SF bulge+SF disk, hereafter);

• TypeII: sub-sample of galaxies characterized by a star-forming core and a quenched disk (SF bulge+Q disk, hereafter);

• TypeIII: sub-sample of galaxies characterized by a quenched core and a star-forming disk (Q bulge+SF disk, hereafter);

• TypeIV: sub-sample of galaxies characterized by both a quenched core and a quenched disk (Q bulge+Q disk, hereafter).

We note that the first three categories (i.e., TypeI, TypeII, and TypeIII) refer to those defined in Le Bail et al. (2024), while the fourth category (i.e., TypeIV) is only defined in this work and is not included in Le Bail et al. (2024). Furthermore, we note that our galaxy classification into TypeI, TypeII, and TypeIII differs from that of Le Bail et al. (2024). In this work, we use the instantaneous star formation rate of the gas cells to calculate the total SFR of each galaxy component. Le Bail et al. (2024) estimated the SFR, stellar mass and magnitudes of each galaxy component by fitting its observed SED. Then, they separated each galaxy component into quiescent and star-forming based on its location on the UVJ color diagram. Creating mock observations to reproduce the galaxy classification from Le Bail et al. (2024) is outside the scope of this paper. Nonetheless, the selection criteria applied in this work allowed us to carry out a first and reasonable qualitative comparison with the observational results of Le Bail et al. (2024). For this reason, in this work, the term “quenched” does not necessarily mean that the central regions or disks are completely quiescent, as they can still be mildly star-forming according to our classification. We applied the classification shown in Fig. 8 to all our sample of disk-like galaxies selected at each redshift between 0<z<2.

Fig. 8. Distribution of the specific star-formation rate (sSFR) of the central regions and disks of our selected sample of disk-like galaxies at z = 0 (top panel) and z = 1.5 (bottom panel). The sSFR of the different components of the galaxies are calculated as described in Sect. 5.1. We divide the sSFR of the central regions and disks in tertiles (dashed lines) and we select the four galaxy sub-samples lying on the outside of the tertile contours (coloured points). These four galaxy sub-samples represent the sub-samples of star-forming cores (disks) and quenched cores (disks) defined in Sect. 5.1, similarly to Le Bail et al. (2024).

5.2. Star-forming main sequence

In Fig. 9, we show the sSFR-stellar mass relation of the central and disk components of our selected sample of disk-like galaxies between 0<z<2. We divide our galaxies into the four galaxy types defined in Sect. 5.1. Small symbols represent the sSFR and stellar mass of the central and disk components of individual galaxies, while large symbols represent the average sSFR and stellar mass of the central and disk components for the galaxies belonging to each type. Solid lines represent the redshift-dependent parametrization of the star-forming main-sequence (SFMS) relation defined in Eq. (5) of Ilbert et al. (2015). Here, and throughout the rest of the paper, we will separate our galaxies in two redshift intervals: 1.5<z<2 (i.e., high redshift) and 0<z<1 (i.e., low redshift), where the high-redshift interval partly overlaps with the redshift range of the observations from Le Bail et al. (2024). This figure shows that the mean values of the sSFR of the star-forming central and disk components of TypeI, TypeII, and TypeIII galaxies are consistent with the SFMS of the corresponding high- and low-redshift intervals. On the other hand, we see that the quenched core and disk components of TypeII, TypeIII, and TypeIV galaxies all lie below the SFMS of the corresponding high- and low-redshift intervals. These results are in agreement with our classification of the galaxies into star-forming (quenched) cores and/or disks, described in Sect. 5.1. We note however that, in the low-redshift interval, the quenched disks of TypeII galaxies have, on average, higher sSFR than the other quenched components.

Fig. 9. Specific star-formation rate (sSFR)-stellar mass relation of the central regions and disk components of our selected sample of disk-like galaxies between 0<z<2, divided into the four galaxy types defined in Fig. 8 and in the two redshift intervals: 1.5<z<2 (i.e., high redshift) and 0<z<1 (i.e., low redshift). Small symbols represent the sSFR and stellar mass of the central regions and disk components of individual galaxies, while large symbols represent the mean values of the sSFR and stellar mass of the central regions and disk components for the galaxies belonging to each type. Error bars represent the standard error of the mean. The light-to-dark green solid lines represent the redshift-dependent parametrization of the star-forming main-sequence (SFMS) relation defined in Eq. (5) of Ilbert et al. (2015).

5.3. Lopsidedness

In Fig. 10, we show the correlation between lopsidedness and the internal properties of our selected sample of disk-like galaxies between 0<z<2. We focus here on the correlation between lopsidedness and disk size (R₉₀), fraction of stellar mass within the inner 2 kpc (M_*,≤2 kpc/M_*; core mass fraction, hereafter) and central stellar mass density (μ_*) within the inner R_h.

Fig. 10. Lopsided amplitude as a function of the disk size (R90), fraction of stellar mass within the central regions (M*(≤2 kpc)/M*), and central stellar mass density (μ*), respectively, for our selected sample of disk-like galaxies between 0<z<2, divided into the four galaxy types defined in Fig. 8 and in the two redshift intervals: 1.5<z<2 (i.e., high redshift) and 0<z<1 (i.e., low redshift) shown from left to right. Small symbols represent individual galaxies, while large symbols represent the mean distribution of each galaxy type. Error bars represent the bootstrap standard error, calculated from the standard deviation of the means of 500 bootstrap samples obtained by sampling with replacement our original dataset.

For the high-redshift interval, we find a clear correlation between the lopsided amplitude and central stellar mass density. Indeed, the top-right panel shows A₁ decreasing with increasing μ_* from TypeI to TypeIV galaxies. TypeII and TypeIII galaxies, which are characterized by similar central stellar mass density, show similar lopsided amplitude. The core mass fraction (top-middle panel) and disk size (top-left panel) also decrease from TypeI to TypeIV galaxies. However, the TypeII and TypeIII galaxies, which have similar A₁, differ in these quantities such that TypeII galaxies have smaller disk size and larger core mass fraction compared to the other galaxy types. Overall, these results confirm the finding that the central stellar mass density is one of the most important parameters for determining the strength of the lopsided amplitude at high redshift, with the disk size and core mass fraction playing a secondary role.

For the low-redshift interval, we see that TypeI and TypeIII galaxies have, on average, larger lopsided amplitude than TypeII and TypeIV galaxies. However, while we see that the central stellar mass density and the strength of the lopsided amplitude show a very strong correlation, we see that the disk size and core mass fraction are also strongly correlated to A₁. In fact, we see that TypeIII galaxies have, on average, larger lopsided amplitude than TypeII galaxies for similar (or slightly larger) central stellar mass density, but TypeIII galaxies tend to be characterized by lower core mass fraction and larger disk size than TypeII galaxies. In particular, TypeIII galaxies have similarly extended star-forming disks to TypeI galaxies. This suggests that lopsidedness in TypeI and TypeIII galaxies could be connected to the properties of their star-forming disk component, indicating that this perturbation may be associated with gas accretion at low redshift, as previously discussed in Sect. 4.4.

We note that our results in Fig. 10 show some differences with respect to the observational results from Le Bail et al. (2024). Specifically, Le Bail et al. (2024) found that TypeIII galaxies have, on average, the highest core mass fraction and lowest lopsided amplitude (see their Fig. 14). Additionally, Le Bail et al. (2024) also found that TypeIII galaxies have lower overall sSFR and are located at lower redshift than TypeI and TypeII galaxies (see their Fig. 12). For this reason, Le Bail et al. (2024) suggest that the TypeIII galaxy population is more evolved compared to the TypeI and TypeII galaxy populations and, thus, had more time to build up a massive quenched bulge component that can make the galaxy less susceptible to develop strong lopsidedness. On the contrary, in this work, we find that TypeIII galaxies at high redshift have low core mass fraction, more similarly to TypeI galaxies, and can show large lopsided amplitude (see top-middle panel in Fig. 10). This discrepancy between the observational results and our models may be due to several reasons. Firstly, it could be a result of the different methods adopted in this work to define the central and disk components of galaxies, as well as to measure the sSFR of each galactic component, with respect to the observations of Le Bail et al. (2024). This can lead to differences in the estimation of the internal properties of galaxies (e.g., stellar mass, central mass density) between observations and simulations. Secondly, it could be due to the different methods used to divide our galaxies into the four galaxy types with respect to Le Bail et al. (2024) (see Sect. 5.1), as well as to the different methods used to classify galaxies into lopsided and symmetric in the observations of Le Bail et al. (2024) and our simulations. In fact, Le Bail et al. (2024) quantified lopsidedness by rotating the galaxy image by 180° and subtracting it from the original image, while we quantify lopsidedness via Fourier analysis as described in Sect. 3. Finally, it could also be a result of the different redshift interval covered by our simulated galaxy models (i.e., 1.5<z<2) and observed galaxies from Le Bail et al. (2024) (i.e., 1.5<z<3) at high redshift.

Despite all the previously mentioned differences between observations and simulations, the overall result that the presence of a massive and/or dense central component can suppress the development of strong lopsidedness by stabilizing the disk is consistent both with the observations and simulations at high redshift.

5.4. Origin of lopsidedness: galaxy interactions vs. gas accretion

In the left panel of Fig. 11, we show the number of galaxies of a given type and with one or more massive neighbors, normalized by the total number of galaxies of all types and with one or more massive neighbors at the five different specific redshift z = 2, 1.5, 1, 0.5, 0. Similarly to Fig. 6, we focus here on the central galaxies in our sample, and we consider only massive neighbors with stellar mass-ratio <1:10 within 0.5×R₂₀₀ during the last 3 Gyr of the galaxy history. Similarly to Fig. 7, in the right panel of Fig. 11, we show the net accretion rate within the last 1 Gyr onto the galactic disk for the five different specific redshift z = 2, 1.5, 1, 0.5, 0. We consider here all the central galaxies (i.e., with number of nearest neighbors ≥0) in our sample.

Fig. 11. Left panel: Number of galaxies of a given type and with one or more massive neighbors (stellar mass-ratio >1:10) within 0.5×R200 during the last 3 Gyr of the galaxy history, normalized by the total number of galaxies of all types and with one or more massive neighbors at the five different specific redshift z = 2, 1.5, 1, 0.5 and 0. We only consider here the central galaxies in our sample. Right panel: Mean net accretion rate within the last 1 Gyr onto the galactic disk of our selected sample of galaxies at the five different specific redshift z = 2, 1.5, 1, 0.5, 0. Here, we consider all the central galaxies in our sample (i.e., with number of nearest neighbors ≥0). Shaded areas are defined by the 25th–75th interquartile range of the data at each redshift. We divide our galaxies into the four different types, as described in Sect. 5.1.

We see that the fraction of TypeI galaxies with one or more massive neighbors remains typically high, and it only slightly decreases from ∼60% at z = 2 to ∼40% at z = 0. Since this fraction is also the highest compared to the other galaxy types, it indicates that close tidal interactions are typically more significant in TypeI galaxies and, thus, more often behind the triggering of their lopsidedness at all redshift. In fact, in Fig. 10, we see that TypeI galaxies typically show the strongest lopsided amplitude, as well as the lowest central stellar mass density and largest disk size, with respect to the other galaxy types both at high and low redshift. However, as previously discussed in Sect. 4.4, close tidal interactions cannot be the only mechanism at play triggering lopsidedness. In fact, in the right panel of Fig. 11, we see that TypeI galaxies are always characterized by positive net accretion rate, slightly decreasing toward low redshift. Specifically, at high redshift (i.e., z>1), we find that TypeI galaxies have experienced significant recent gas accretion with respect to the other galaxy types. This is generally consistent with the results from Le Bail et al. (2024), where they found that their TypeI galaxies are characterized by clumpy and heterogeneous star-forming disks, suggesting a recent gas-rich major merger. Overall, these results suggest that both close tidal interactions and recent gas accretion can play a fairly relevant role in the origin of lopsidedness in TypeI galaxies at all redshift between 0<z<2.

On the other hand, the fraction of TypeII galaxies with one or more massive neighbors remains overall constant at ∼20% as a function of redshift, with only a mild decrease by ∼10% between z = 0.5 and z = 0. This behavior is similar to that of TypeIII galaxies, although there are only ∼10% of TypeIII with one or more massive neighbors at all redshift between 0<z<2. Furthermore, we see that while TypeIII galaxies show an overall positive and constant net accretion rate at all redshift, TypeII galaxies show a decreasing net accretion rate toward low redshift. Overall, these results suggest that, while close tidal interactions are typically more significant in triggering lopsidedness in TypeII galaxies than TypeIII ones, recent gas accretion is more often behind the triggering of lopsidedness in TypeIII galaxies. At high redshift (i.e., z>1), the effect of these two different mechanisms can produce the overall similar lopsided amplitude observed in TypeII and typeIII galaxies in Fig. 10. At low redshift, the decreasing role of close tidal interactions in TypeII galaxies is consistent with the overall decreasing fraction of lopsided galaxies toward low redshift (see e.g., Fig. 3), thus producing an overall lower lopsided amplitude in TypeII galaxies than TypeIII galaxies, as shown in Fig. 10. Although we do not show it here, we note that we found similar behaviors of the net accretion rate for the different galaxy types when we consider only central galaxies with no massive neighbors within 0.5×R₂₀₀ at any chosen redshift. This suggests that a recent gas accretion event onto the disks of our galaxies is not only associated with the accretion of gas from nearby massive satellite galaxies through gas-rich minor or major mergers (i.e., with stellar mass-ratio >1:10), but also to the continuous accretion of cold gas from the intergalactic medium, from along filamentary streams or from smaller gas-rich dwarf galaxies (see, e.g., Sancisi et al. 2008 for a review). The study of the source of gas accretion onto our galaxies, as well as its potential for producing lopsidedness, will be left to a follow-up work. The fact that the source of lopsidedness may be different in TypeII and TypeIII galaxies could also explain the different star formation properties of the bulge and disk components of these galaxies. Lopsidedness in TypeIII galaxies is mainly produced through recent gas accretion, thus affecting the star-forming properties of their disk component. On the other hand, lopsidedness in TypeII galaxies is mainly produced through close tidal interactions. These interactions are expected to trigger starbursts within the inner galactic regions and increase the bulge mass (Bekki & Couch 2011), in addition to triggering lopsidedness. Therefore, the result of this would be a lopsided disk galaxy (i.e., S0-like) with younger stellar populations in the bulge than in the disk due to the induced central starbursts. This outside-in quenching scenario and lopsidedness generated by repetitive tidal interactions in TypeII galaxies seem consistent with the observational results from Kalita et al. (2022), who observed highly star-forming bulges embedded in quiescent lopsided stellar disks in three group galaxies at z∼3. Alternatively, different AGN feedback processes may be active in TypeII and TypeIII galaxies and affecting the structural properties of the galaxies in a different way; namely, by suppressing the central star formation or the star formation in the outer regions through the ejection of the central gas or by heating up the halo gas, respectively (Irodotou et al. 2022). It remains to be explored what mechanisms are primarily driving the quenching of the disk and central regions in TypeII galaxies with respect to TypeIII galaxies at high redshift.

Finally, for TypeIV galaxies, we see that the fraction of galaxies with one or more massive neighbors increases toward low redshift. Nonetheless, TypeIV galaxies are always characterized by low asymmetry at all redshift. This is likely the result of the specific internal properties of these galaxies (i.e., high central stellar mass density, high core mass fraction, and small disk size), which makes them less prone to develop strong lopsidedness due to external interactions. Furthermore, we see that TypeIV galaxies are characterized by negative net accretion rate at all redshift, ruling out recent gas accretion as trigger of lopsidedness in these galaxies. In fact, we find that TypeIV galaxies are the most symmetric galaxies both at high and low redshift, as shown in Fig. 10. We note that the lack of recent gas accretion does not exclude that TypeIV galaxies may have either experienced earlier accretion of gas, which was rapidly converted into stars, or gas-poor mergers that contributed to the rapid mass assembly of these galaxies (i.e., high core mass fraction and high central stellar mass density), specifically at high redshift.

6. Summary and conclusions

In this work, we have characterized the lopsidedness nature seen in the mass distribution of the stellar component of our samples of disk-like galaxies selected from the TNG50 simulation at specific redshift between 0<z<2. We highlight the fact that our method consisted of selecting a new galaxy sample at each redshift, using the selection criteria described in Sect. 2.2. Similarly to Varela-Lavin et al. (2023) and Dolfi et al. (2023), we quantified lopsidedness via Fourier analysis by decomposing the face-on projected galaxy stellar mass distribution in concentric radial annuli and measuring the amplitude of the first Fourier m = 1 mode in each radial bin within R_h<r<1.4 R₉₀ and |h_z|<2 h₉₀, as described in Sect. 3. We then quantified the fraction of lopsided galaxies as a function of redshift and we studied the correlation between lopsidedness and both the local environmental density and galaxy internal properties as a function of redshift. We summarize our main results below.

• We find that the median lopsided amplitude, as well as the fraction of lopsided galaxies, decreases from high to low redshift. Additionally, we find that, while both central and satellite galaxies show similar lopsided amplitude at high redshift (i.e., z>1), central galaxies tend to be more lopsided than satellite ones at low redshift (i.e., z≲1). Overall, this suggests that mechanisms triggering lopsidedness are different or more efficient at high redshift, rather than at low redshift. In particular, environmental interactions could be playing a more active role in triggering lopsidedness at high redshift.

• We find that independently of the redshift considered, the lopsided amplitude does not strongly depend on the local environmental density of the galaxies, consistent with the mild or lack of correlation observed between lopsidedness and the local environment at z = 0 (Wilcots 2010; Dolfi et al. 2023).

• We found a strong correlation between the median lopsided amplitude and the internal properties of the galaxies at all redshifts considered up to z = 2. This suggests that independently of the mechanisms producing lopsidedness, galaxies with low central stellar mass density, large stellar half-mass radius, and large disk size are more prone to develop strong lopsidedness. This finding is consistent with previous results at z = 0 (Reichard et al. 2008; Varela-Lavin et al. 2023; Dolfi et al. 2023).

In Sects. 4.4.1 and 4.4.2, we give the details of our study of different mechanisms triggering lopsidedness based on their relative importance as a function of redshift. Specifically, we focus on close tidal interactions and gas accretion to explore whether different mechanisms are at play at high and low redshift or whether the same mechanisms driving lopsidedness at z∼0 are simply more efficient at high redshift. We summarize our main findings below.

• As a proxy of the probability of close tidal interactions, we quantified the number of nearest neighbors with a stellar mass-ratio of >1:10, located within half the virial radius of the galaxy (i.e., 0.5×R₂₀₀) during the last 3 Gyr of the galaxy history. In Sect. 4.4.1, we can see that the fraction of lopsided galaxies with at least one massive neighbors remains overall constant at ∼40% as a function of redshift, suggesting that close tidal interactions are similarly efficient at triggering lopsidedness at high and low redshift. However, we also find that, on average, ∼60% of lopsided galaxies do not have any massive neighbors, suggesting that other mechanisms must be at play with respect to the origin of lopsidedness at all redshifts.

• We calculated the net accretion rate onto the galactic disk within the last 1 Gyr for our selected sample of disk-like galaxies, as described in Sect. 4.4.2. We find that lopsided galaxies are always characterized by positive and larger net accretion rate compared to symmetric galaxies, which mainly show a negative net accretion rate. In particular, lopsided galaxies show significant net accretion rate at high redshift, decreasing toward z = 0. This suggests that gas accretion can play a significant role in triggering lopsidedness at all redshifts, specifically at high redshift (i.e., z>1).

In summary, we find that multiple mechanisms (i.e., gas accretion and close tidal interaction) are playing a more active and efficient role in triggering lopsidedness at high redshift (i.e., z≳1). This would mean that the seemingly more widespread lopsidedness detected with observations at high redshift (Le Bail et al. 2024) can be associated with the more efficient conditions existing at high redshift. In a follow-up work, we aim to study the distribution of lopsidedness within the large-scale environment as a function of redshift to determine how the connectivity of galaxies to filaments can influence the distinct net accretion rate, as well as the star formation histories of galaxies up to z = 0 (see Figs. 13 and 14 in Dolfi et al. 2023). This will also allow us to understand the origin of lopsidedness through asymmetric gas accretion with subsequent star formation in realistic cosmological models.

In Sect. 5, we present a more detailed comparison with the observations of Le Bail et al. (2024) by classifying our galaxies into four galaxy types depending on the star formation rate of the central and disk components, similarly to Le Bail et al. (2024). The galaxy classification is described in details in Sect. 5.1. We summarize our main results below.

• We find that at high redshift (i.e., z>1), the lopsided amplitude decreases with increasing central stellar mass density from TypeI to typeII and TypeIII to TypeIV galaxies. This suggests that the central stellar mass density is one of the key parameters responsible for determining the strength of the lopsided perturbations at high redshift. This is consistent with the more efficient role of external interactions in perturbing the galaxies at high redshift, where the strength of the perturbation also depends on the specific internal galaxy properties, as shown by the TypeI galaxies included in Figs. 10 and 11. On the other hand, at low redshift, we find that the disk size of the galaxies plays a more relevant role for lopsidedness. This is consistent with the finding that TypeI and TypeIII galaxies with star-forming disks are the most lopsided, as shown in Fig. 10.

• We find that the results given in Fig. 10 slightly differ from the observational results of Le Bail et al. (2024) at high redshift. Specifically, Le Bail et al. (2024) found that TypeIII galaxies are the most symmetric when they exhibit the highest core mass fraction in their quenched bulge. On the other hand, we find that TypeIII galaxies have similar lopsidedness, but a lower core mass fraction as compared to TypeII galaxies. In Sect. 5.3, we note several reasons that could produce this discrepancy between observations and simulations at high redshift. However, despite these differences, the general result that the presence of a centrally massive and dense bulge component can suppress the development of strong lopsidedness is consistent both in terms of the simulations and observations.

• We find that close tidal interactions are typically more effective in triggering lopsidedness in TypeI galaxies than in TypeII, TypeIII, and TypeIV galaxies at all redshift. Furthermore, we find that TypeI galaxies experience significant recent gas accretion at high redshift, which could be a key factor behind their strong lopsidedness. This is consistent with the observations from Le Bail et al. (2024), where the authors found that TypeI galaxies have clumpy and heterogeneous star-forming disks that could be signature of a recent gas-rich major merger.

• In Fig. 11, we find that the source of lopsidedness may be different in TypeII and TypeIII galaxies, thus producing the different star formation properties of the bulge and disk component in these galaxies. Lopsidedness in TypeIII galaxies is mainly produced through recent gas accretion, while lopsidedness in TypeII galaxies is mainly produced through close tidal interactions. These interactions can trigger central starbursts and increase the bulge mass (Bekki & Couch 2011), in addition to triggering lopsidedness. Therefore, a lopsided disk galaxy is created with younger stellar populations in the bulge than in the disk. This outside-in quenching scenario and lopsidedness generated by repetitive tidal interactions in TypeII galaxies would be consistent with the observational results from Kalita et al. (2022), where highly star-forming bulges embedded in quiescent lopsided stellar disks were reported in three group galaxies at z∼3.

Data availability

The data used in this work come from the IllustrisTNG simulations, which are publicly available at: https://www.tng-project.org/data/ and described in Nelson et al. (2019).

Acknowledgments

We thank the anonymous referee for their constructive comments and suggestions that helped to improve this paper. We also thank Dr. Gissel Dayana Pardo Montaguth for providing the A₁₈₀ values for a test sample of galaxies from the TNG50 simulation to study the differences with the A₁ values measured in this work. The authors gratefully acknowledge support by the ANID BASAL project FB210003. A.M. acknowledges support from the FONDECYT Regular grant 1212046. F.A.G. acknowledges support from the FONDECYT Regular grant 1211370. A.M. and F.A.G. gratefully acknowledge funding from the Max Planck Society through a “PartnerGroup” grant as well as from the HORIZON-MSCA-2021-SE-01 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement number 101086388. P.B.T. acknowledges partial funding from FONDECYT 1240465 and the LACEGAL Network (Horizon 2030).

References

All Figures

Fig. 1. Top panel: Spin-ellipticity (i.e., λR−ϵ) diagram of the galaxies selected from the TNG50 simulation between 0<z<2 (gray points). The colored points show our final sample of disk-like galaxies, using the selection criteria described in Sect. 2.5. The red solid line represents the threshold typically used in observations to separate between rotation-dominated and dispersion-supported galaxies within the inner stellar half-light radius at z = 0 (Emsellem et al. 2011). Bottom panel: Total number of central and satellite galaxies of our final sample of disk-like galaxies at each redshift, selected as described in Sect. 2.5.

In the text

	Fig. 2. Stellar mass distribution (top panel) and median value of λR as a function of stellar mass (bottom panel) for our selected sample of disk-like galaxies at each redshift between 0<z<2. Shaded areas are defined by the 25th–75th interquartile range of the data in each bin.
In the text

Fig. 3. Left panel: Median lopsided amplitude as a function of redshift for our selected sample of disk-like galaxies at each analyzed redshift between 0<z<2, divided into centrals and satellites. Shaded areas are defined by the 25th–75th interquartile range of the data at each bin. The horizontal dotted black line indicates the threshold A1 = 0.1 used to classify galaxies into lopsided (A1>0.1) and symmetric (A1<0.1) at z = 0 (Varela-Lavin et al. 2023; Dolfi et al. 2023). Right panel: Fraction of lopsided galaxies as a function of redshift for centrals and satellites.

In the text

	Fig. 4. Median lopsided amplitude as a function of the local density of the environment between 0<z<2 for our selected sample of disk-like galaxies. The local density of the environment is calculated as described in Sect. 4.2, considering the ten nearest neighbors with total mass Mtot>109 M⊙. Shaded areas are defined by the 25th–75th interquartile range of the data in each bin.
In the text

Fig. 5. Top: Median lopsided amplitude as a function of the central stellar mass density (μ*), stellar half-mass radius (Rh) and disk size (R90) for our selected sample of disk-like galaxies between 0<z<2. Bottom: Median μ*, Rh and R90 as a function of redshift for our selected sample of disk-like galaxies between 0<z<2. Shaded areas are defined by the 25th–75th interquartile range of the data in each bin.

In the text

	Fig. 6. Number of lopsided and symmetric galaxies with one or more massive neighbors (stellar mass-ratio >1:10) within 0.5×R200 during the last 3 Gyr of the galaxy history, normalized by the total number of lopsided and symmetric galaxies, respectively, at the five different specific redshift z = 2, 1.5, 1, 0.5, 0. Here, we only focus on the central galaxies in our sample.
In the text

Fig. 7. Left panel: Net accretion rate onto the galactic disk within the last 1 Gyr for our selected sample of disk-like galaxies at the five different specific redshift z = 2, 1.5, 1, 0.5, and 0. Middle panel: Total galaxy star formation rate. Right panel: Star formation rate within the galactic disk, where the disk is defined as described in Sect. 2.3. We divide our galaxies into lopsided and symmetric, using the threshold A1 = 0.1. Shaded areas are defined by the 25th–75th interquartile range of the data at each redshift.

In the text

Fig. 8. Distribution of the specific star-formation rate (sSFR) of the central regions and disks of our selected sample of disk-like galaxies at z = 0 (top panel) and z = 1.5 (bottom panel). The sSFR of the different components of the galaxies are calculated as described in Sect. 5.1. We divide the sSFR of the central regions and disks in tertiles (dashed lines) and we select the four galaxy sub-samples lying on the outside of the tertile contours (coloured points). These four galaxy sub-samples represent the sub-samples of star-forming cores (disks) and quenched cores (disks) defined in Sect. 5.1, similarly to Le Bail et al. (2024).

In the text

Fig. 9. Specific star-formation rate (sSFR)-stellar mass relation of the central regions and disk components of our selected sample of disk-like galaxies between 0<z<2, divided into the four galaxy types defined in Fig. 8 and in the two redshift intervals: 1.5<z<2 (i.e., high redshift) and 0<z<1 (i.e., low redshift). Small symbols represent the sSFR and stellar mass of the central regions and disk components of individual galaxies, while large symbols represent the mean values of the sSFR and stellar mass of the central regions and disk components for the galaxies belonging to each type. Error bars represent the standard error of the mean. The light-to-dark green solid lines represent the redshift-dependent parametrization of the star-forming main-sequence (SFMS) relation defined in Eq. (5) of Ilbert et al. (2015).

In the text

Fig. 10. Lopsided amplitude as a function of the disk size (R90), fraction of stellar mass within the central regions (M*(≤2 kpc)/M*), and central stellar mass density (μ*), respectively, for our selected sample of disk-like galaxies between 0<z<2, divided into the four galaxy types defined in Fig. 8 and in the two redshift intervals: 1.5<z<2 (i.e., high redshift) and 0<z<1 (i.e., low redshift) shown from left to right. Small symbols represent individual galaxies, while large symbols represent the mean distribution of each galaxy type. Error bars represent the bootstrap standard error, calculated from the standard deviation of the means of 500 bootstrap samples obtained by sampling with replacement our original dataset.

In the text

Fig. 11. Left panel: Number of galaxies of a given type and with one or more massive neighbors (stellar mass-ratio >1:10) within 0.5×R200 during the last 3 Gyr of the galaxy history, normalized by the total number of galaxies of all types and with one or more massive neighbors at the five different specific redshift z = 2, 1.5, 1, 0.5 and 0. We only consider here the central galaxies in our sample. Right panel: Mean net accretion rate within the last 1 Gyr onto the galactic disk of our selected sample of galaxies at the five different specific redshift z = 2, 1.5, 1, 0.5, 0. Here, we consider all the central galaxies in our sample (i.e., with number of nearest neighbors ≥0). Shaded areas are defined by the 25th–75th interquartile range of the data at each redshift. We divide our galaxies into the four different types, as described in Sect. 5.1.

In the text