Investigating the link between inner gravitational potential and star-formation quenching in CALIFA galaxies

It has been suggested that the gravitational potential can have a significant role in suppressing the star formation in the nearby galaxies. To establish observational constrains on this scenario, we investigate the connection between the dynamics, through the circular velocity curves (CVCs) as a proxy of the inner gravitational potential, and star formation quenching in 215 non-active galaxies across Hubble sequence from the Calar Alto Legacy Integral Field Area (CALIFA) survey. Our results show that galaxies with similar CVCs tend to have a certain star-formation quenching pattern. To explore these findings in more details, we construct kpc-resolved relations of the equivalent width of the H$\alpha$ ($W_{{\rm H}\alpha}$) versus the amplitude ($V_c$) and shape ($\beta= d\ln V_c/ d\ln R$) of the circular velocity at given radius. We find that the $W_{{\rm H}\alpha}-V_c$ is a declining relationship, where the retired regions of the galaxies (the ones with $W_{{\rm H}\alpha}$ values below 3 \r{A}) tend to have higher $V_c$. Differently, $W_{{\rm H}\alpha}-\beta$ is a bi-modal relationship, characterised by two peaks: concentration of the star forming regions at a positive $\beta$ (rising CVC) and another one of the retired regions with a negative $\beta$ (declining CVC). Our results show that both the amplitude of the CVC, driven by the mass of the galaxies, and the shape of the CVC, reflecting the internal structure of the galaxies, play an important role in galaxy's quenching history.


Introduction
The transition of the galaxies from blue and young to red and retired systems is referred to the process "star-formation quenching" (e.g. Strateva et al. 2001;Faber et al. 2007), which is characterised by a fast decline in the star formation rate (SFR; e.g. Corcho-Caballero et al. 2021). Several quenching mechanisms have been proposed to explain the diversity of galaxies we observe today and they can be grouped to external (outside of the galaxy) and internal (within the galaxy). Environmental quenching (Peng et al. 2010,Gunn & Gott 1972Abadi et al. 1999) is an external process. This includes rampressure stripping of the galaxies interstellar medium, strangulation (removal of the outer gaseous envelope of the galaxy falling into galaxy cluster, which will cease the star formation of the galaxy as the only available gas is hot and it cannot replenish its cold gas reservoir (Larson et al. 1980;Balogh et al. 2000) and galaxy harassment (quenching due to interactions with other members of the galaxy cluster, leading to dynamical heating Farouki & Shapiro 1981;Moore et al. 1996;Bluck et al. 2020a,b). Low-mass galaxies, in particular, can have their gas removed by tidal stripping during travel towards the centre of a galaxy cluster (e.g. Abadi et al. 1999).
In addition, a large range of internal processes are also thought to be able to quench star formation. Dark matter (DM) halo quenching is considered an internal process, related to the most extended component of the galaxies. In systems with critical DM halo mass of ∼10 12 M , the accreated gas is shocked and heated to the virial temperature of the halo, preventing star formation (Birnboim & Dekel 2003). The correlation of the stellar component with the quenching of the massive galaxies is known as mass quenching (Peng et al. 2010). Active galactic nuclei (AGN) feedback quenching is able to suppresses star formation in the galaxies (Husemann & Harrison 2018). AGN feedback can transfer radiation to the surrounding gas and suppress gas accretion (Di Matteo et al. 2005) or kinetic energy and momentum can cause expulsion of gas (Croton et al. 2006). The ejective galactic winds from supernovae or HII regions constitute another quenching feedback mechanism (stellar feedback quenching) that acts mostly in low-mass and late-type galaxies (e.g. Colling et al. 2018). Internal dynamics contributes to the quenching of the galaxies too. Secular evolution processes can form a bar structure that generates radial inflow towards the centre of the galaxies, increasing the random motion of the gas and the disc heating that stabilises the inner gaseous disc (Romeo & Fathi 2015Khoperskov et al. 2018). Additionally bars Article number, page 1 of 16 arXiv:2207.03872v1 [astro-ph.GA] 8 Jul 2022 A&A proofs: manuscript no. CVC-QS drive gas flow toward galactic centre to trigger a central starburst, which in turn can consume almost all the gas causing a quenched centre. This action leads to a periodically episodes of quenching and star formation at the centre of bars (Krumholz & Kruijssen 2015). The growth of the central spheroid (bulge) seems to play an important role in ceasing the star formation in the galaxies via stabilisation of the disk against gravitational instability, known as "morphological quenching" (Martig et al. 2009), gravitational quenching (Genzel et al. 2014) or dynamical suppression (e.g. Davis et al. 2014;Gensior et al. 2020;Gensior & Kruijssen 2021).
Overall, the causes of star-formation quenching in galaxies are highly complex and still debated. It is not clear whether there is one dominant mechanism or a mixture of several processes that are responsible for the variety of galaxy morphologies seen today. Recent resolved studies have tried to disentangle these effects by classifying the galaxies based on their star formation activity (or its absence; Singh et al. 2013;Belfiore et al. 2016;Lacerda et al. 2018). In particular, Kalinova et al. (2021) (hereafter K21) distinguish various quenching patterns (called "quenching stages"; Fig. 1) in the spatially resolved ionised gas distribution of the galaxies, where the ratio star-forming versus quenching regions of the galaxies increases from late-type to early-type galaxy morphologies (e.g. Lacerda et al. 2018, K21). Based on Hα-equivalent-width (W Hα ) thresholds (e.g., Sánchez et al. 2014, Lacerda et al. 2020, K21 distinguished star-forming (W H α > 6 Å), mixed (3 < W H α ≤ 6 Å) and retired (W H α ≤ 3 Å) regions within the field of view of the galaxies. They proposed six quenching stages of the systems (see Fig. 1): star-forming (fully dominated by recent star-formation), quiescent-nuclearring (presence of a quiescent-ring structure in the central regions, but still dominated by star formation in the outskirts), centrally quiescent (quiescent inner region within 0.5R e of the galaxy, where R e is the effective radius of the galaxy), mixed (no clear patterns in the ionised gas distributions), nearly retired (quiescent galaxies with little star-formation regions) and fully retired (completely quiescent objects up to 2R e ).
In this study, we explore the connection between the dynamics and star-formation quenching stage of the galaxies, shedding light on the role of the dynamical suppression mechanism for their formation. To achieve this goal, we compare the circular velocities of the galaxies (tracing the total gravitational potential of the systems; Kalinova et al. 2017, hereafter K17) and the values of the W Hα (K21), serving as a star-formation/quenching marker across Hubble sequence.

Sample and Data
Our sample is derived from the 238 CALIFA (Calar Alto Legacy Integral Field Area; Sánchez et al. 2012) survey galaxies, extensively analysed in K17 and K21. It is a representative population of the CALIFA mother sample (see Fig. 1 in K17), and therefore of the nearby Universe galaxies (Walcher et al. 2014). In our analysis, we only discard few targets -the active galaxies members (15 strong AGN and 8 weak AGN galaxies) from the original sample of K17 and K21 to avoid any possible biases due to the presence of nuclear activity in the galaxies. The final sample consists of 215 non-active galaxies, spanning over six different quenching stages as defined in K21 (see Fig. 1 of K21 and Fig.  1 of this study), various morphologies (from elliptical to latetype spiral galaxies), stellar masses (from 6×10 8 M to 5×10 11 M ) and redshifts (0.005 < z < 0.03). Further details about the survey and data reduction can be found in Sánchez et al. (2012), V. Kalinova et al.: Investigating the link between inner gravitational potential and star-formation que     Quenching stages of galaxies V. Kalinova et al.: Dynamical quenching of star formation in EDGE-CALIFA galaxies       2. Left: Median curve of the CVC profiles for each quenching stage group from star-forming to fu curve of the normalised (on the asymptotic velocity) CVC profiles for each quenching stage group. The 25th percentile (below) and the 75th percentile (above) of the median distribution. To smooth the median C apply the Savitzky-Golay smoothing filter (Savitzky & Golay 1964) by adopting third degree polynomial an Sect. 3).
ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use the P (p; e.g. Kowalski 1972) to test linear relation. The Pearson's tween +1 and -1 for linear corre respectively (where 0 refers to studied data sets). The Spearm (s; e.g. Zwillinger & Kokoska relationship is monotonically i decreasing (s = 1) or ther (s = 0), independently from its 3 https://docs.scipy.org/d /scipy.stats.pearsonr.html 4 https://docs.scipy.org/d /scipy.stats.spearmanr.htm V. Kalinova et al.: Investigating the link between inner gravitational potential and star-format           ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use (p; e.g. Kowalski 1972) t linear relation. The Pear tween +1 and -1 for linea respectively (where 0 re studied data sets). The S (s; e.g. Zwillinger & Ko relationship is monotonic decreasing (s = 1) o (s = 0), independently fr V. Kalinova et al.: Investigating the link between inner gravitational potential and star-formati           ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use (p; e.g. Kowalski 1972) to linear relation. The Pears tween +1 and -1 for linear respectively (where 0 ref studied data sets). The Sp (s; e.g. Zwillinger & Kok relationship is monotonic decreasing (s = 1) or (s = 0), independently fro V. Kalinova et al.: Investigating the link between inner gravitational potential and star-formati         ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use (p; e.g. Kowalski 1972) t linear relation. The Pear tween +1 and -1 for linea respectively (where 0 ref studied data sets). The S (s; e.g. Zwillinger & Kok relationship is monotonic decreasing (s = 1) o (s = 0), independently fro V. Kalinova et al.: Investigating the link between inner gravitational potential and star-format         ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use (p; e.g. Kowalski 1972) t linear relation. The Pear tween +1 and -1 for linea respectively (where 0 re studied data sets). The S (s; e.g. Zwillinger & Ko relationship is monotoni decreasing (s = 1) o (s = 0), independently fr V. Kalinova et al.: Investigating the link between inner gravitational potential and star-forma            ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we us (p; e.g. Kowalski 1972) linear relation. The Pea tween +1 and -1 for line respectively (where 0 r studied data sets). The (s; e.g. Zwillinger & K relationship is monoton decreasing (s = 1) (s = 0), independently f 3 https://docs.scipy /scipy.stats.pearson 4 https://docs.scipy /scipy.stats.spearma V. Kalinova et al.: Investigating the link between inner gravitational potential and star-format         ies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching-CVC shape (W H↵ ), where is the derivative of V c with respect to the radius, = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c / R, giving = 1, while if the part of the curve is flat gives V c =constant, and therefore = 0 (see Sect. 4.2.4 of Leroy et al. 2008). Therefore, if the logarithmic derivative of the CVC is positive ( >0) means that the CVC is rising, and if it is negative ( <0) means that the CVC is declining, and if it is equal or close to zero ( ⇠ 0), the CVC is flat. The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use (p; e.g. Kowalski 1972) t linear relation. The Pear tween +1 and -1 for linea respectively (where 0 re studied data sets). The S (s; e.g. Zwillinger & Ko relationship is monotonic decreasing (s = 1) o (s = 0), independently fr To perform this study, we use the publicly available circular velocity curve (CVC) catalogue of K17. CVC is defined is the gravitational potential and R is the galactocentric radius, respectively. Further, the CVC is calculated through the solutions of the axisymmetric Jeans equations by applying the Jeans Axisymmetric Modelling (JAM) code of Cappellari (2008) 1 . First, K17 derive the surface brightness (SB) of the galaxies using the r-band images from Sloan Digital Sky Survey (SDSS) 2 catalogue of Data Release 12 (DR12; Alam et al. 2015) through the multi-Gaussian expansion method (MGE; Monnet et al. 1992;Emsellem et al. 1994). Given the defined SB of the galaxies, the second velocity moment of the galaxies (V rms = √ V 2 + σ 2 ) of the stellar kinematics (based on the high-resolution "V1200" dataset; Falcón-Barroso et al. 2017) is fitted, assuming constant dynamical mass-to-light ratio (γ dyn ) and constant velocity anisotropy (β z = 1 − σ 2 z /σ 2 r ) in the galactic meridional plane. In the end, the CVCs were inferred from the de-projected SB of the galaxies, scaling by the best fit value of γ dyn (see eq. 3 in K17).
The emission-line analysis for deriving the maps of the equivalent width of Hα line (W Hα ) was based on the Pipe3D pipeline (Sánchez et al. 2016c,b) calculations, assuming a Salpeter (Salpeter 1955) initial mass function (IMF). Further details about the adopted data and analysis are provided in K17 and K21.

Results
Our first test towards understanding the link between the inner gravitational potential of the galaxies and their quenching stage is performed in Fig. 2    galaxies become progressively more quenched, e.g., increasing in the sequence star-forming to quiescent-nuclear-ring/centrally quiescent to mixed to fully retired to nearly retired. We notice that in the central region (up to 0.4 R e ), centrally quiescent systems have similar median velocity as the quiescent-nuclear-ring galaxies, but larger velocity in the outer parts (above 0.4 R e ). This indicates that the disc of the centrally quiescent galaxies is more massive than the one of the quiescent-nuclear-ring systems. Contrary, in the outer regions (above 0.4−0.5 R e ), the nearly retired galaxies have similar disc median velocity as the fully retired systems, but larger bulge median velocity in the central parts (below 0.4−0.5 R e ). Among all quenching-stage groups, the nearly retired galaxies have the largest central mass concentration (the highest V c −peak). This might suggest that the fully retired systems (or even the rest of the quenching stage galaxies) are progenitors of the nearly retired galaxies (via re-ignition of star-formation and mass build-up by merger events), or the nearly retired class has followed a different evolutionary path from the rest of the classes. Furthermore, in the right panel of Fig. 2, the median of the CVC profiles, normalised with respect to the asymptotic velocity, show more distinguishable order: star-forming to centrally quiescent to quiescent-nuclear-ring to mixed to fully retired to nearly retired. Nevertheless, individually, the curves in a given quenching stage show a large variety.
In particular, several star-forming galaxies are characterised by declining profiles, while some retired objects have rising profiles (see also the individual CVCs in each quenching-stage group in Figs. A.1 and A.2). Figure 2 suggests that on average, there is a correlation between the CVC shape/amplitude and the quenching stage. To have a more detailed quantification of these correlations, we construct two resolved relationships using the available information from the W H α maps and CVCs.

Resolved quenching relations
The first relation, quenching-velocity (W H α − V c ), compares the W H α value and the amplitude of the circular velocity (V c ) at a given position in the field of view of the galaxy (i.e. spaxelby-spaxel). We calculate the circular velocity, V c , value at each radius of the de-projected radial map of the galaxies using the MGE_VCIRC procedure included in the JAM package of Cappellari (2008). The inclination and the MGE models of the galaxies were adopted from K17. Assuming that the galaxies are axisymmetric, we derive "maps" with constant values of V c at a given galactocentric radius. Once we obtain the V c map, we are able to construct the second relationship, quenching−CVC shape (W H α − β), where β is the derivative of V c with respect to the radius, β = d ln V c /d ln R. If the rising part of the CVC is close to a solid body rotation, then V c ∝ R, giving β = 1, while if the part of the curve is flat gives V c =constant, and therefore β = 0. If the logarithmic derivative of the CVC is negative (β<0) then the CVC is declining (e.g. Sect. 4.2.4 of Leroy et al. 2008). The statistical robustness of both resolved relationships is demonstrated in Appendix C.
Additionally, we use the Pearson's correlation coefficient 3 (ρ; e.g. Kowalski 1972) to test the strength of the monotonically linear relation. The coefficient ρ can vary between +1 and -1 for linear correlation and linear anti-correlation, respectively (where 0 refers to no correlation between the two studied data sets). The Spearman's rank correlation coefficient 4 (s; e.g. Zwillinger & Kokoska 1999) tells us whether the tested relationship is monotonically increasing (s = 1), monotonically decreasing (s = −1) or there is non-monotonic relationship (s = 0), independently from its linearity. V. Kalinova et al.: Investigating the link between inner gravitational potential and star-formation quenching in CALIFA galaxies Fig. 5. Spaxel distribution of W Hα − V c relation across properties of the host galaxies. The first and the second row panels correspond to the morphological and bar-type (A: no bar, B: bar, and AB: unsure bar) classifications of the galaxies from Walcher et al. (2014), respectively. The third row panels explore the central stellar mass surface density (Sánchez et al. 2016c,b) within 1 kiloparsec (Σ 1kpc ) of the host galaxies, following Cheung et al. (2012). The fourth row panels relates to the bulge type of the host galaxies, according to the method described in Luo et al. (2020, PB: pseudo bulge, CB: classical bulge, and E: ellipticals). The left column panels explore the statistics of the spaxels for low velocities (V c < 250 km s −1 ), while the right panels correspond to the high velocities (V c > 250 km s −1 ). The different colours and symbols of the bars distinguish star-forming (blue with black circle) and quenched (red with no symbol) regions of the galaxies. The dashed horizontal lines at Σ 1kpc =10 10 M kpc −2 are drawn to guide the eye. See more details in Sec. 3.2.
Article number, page 5 of 16 A&A proofs: manuscript no. CVC-QS Fig. 6. Spaxel distribution of W Hα − β relation across properties of the host galaxies. The first and the second row panels correspond to the morphological and bar-type (A: no bar, B: bar, and AB: unsure bar) classifications of the galaxies from Walcher et al. (2014), respectively. The third row panels explore the central stellar mass surface density (Sánchez et al. 2016c,b) within 1 kiloparsec (Σ 1kpc ) of the host galaxies, following Cheung et al. (2012). The fourth row panels relates to the bulge type of the host galaxies, according to the method described in Luo et al. (2020, PB: pseudo bulge, CB: classical bulge, and E: ellipticals). The left column panels explore the statistics of the spaxels for the declining CVCs (β < 0), while the right panels correspond to the rising CVCs (β > 0). The different colours and symbols of the bars in the panels distinguish star-forming (blue with black circle) and quenched (red with no symbol) regions of the galaxies. The dashed horizontal lines at Σ 1kpc =10 10 M kpc −2 are drawn to guide the eye. See more details in Sec. 3.2. Article number, page 6 of 16 V. Kalinova et al.: Investigating the link between inner gravitational potential and star-formation quenching in CALIFA galaxies The resolved quenching-velocity relation, W H α − V c , which includes all values of the spaxels in each galaxy of our sample is displayed in Fig. 3 (left panel). Generally, the two quantities are moderately correlated (ρ=−0.62 and s=−0.60 ) with high significance (for both coefficients, we obtained p-value 0.01 indicating that the moderate correlation between the two quantities is not due to chance). The relation presents two slopes, as it flattens in low velocities and steepens in the high-velocities above the critical value log(V c )∼2.4 km s −1 (V c ∼ 250 km s −1 ). Overall, the retired spaxels (i.e. those with W H α < 3 Å) tend to be characterised by higher circular velocities. Furthermore, if we segregate the quenching-velocity relation into the six quenching stages (right panel of Fig. 3; see also Fig. B.1), we observe a smooth transition from star-forming to fully retired galaxies from the top-left (high W H α and low V c ) to the bottom-right (low W H α and high V c ) side of the relation. The central regions of the relation (log(W H α ) ∼ 1 Å and log(V c ) ∼ 2.3 km s −1 ) are occupied by the mixed group of galaxies, which are actually green valley galaxies on the way to be fully quenched (see Fig. 11 of K21).
In Fig. 4, the W H α and β values moderately correlate (ρ=0.49 and s=0.48), as for the previous relation, with high significance (p-value 0.01 for both tests). Contrary to the W H α −V c relation, the W H α − β relation is bi-modal and characterised by two clear peaks, where one of them is located at β ∼ −0.25 and W H α ∼ 1Å, and the second one is located at β ∼ 0.25 and W H α > 6 Å, indicating that generally star forming regions are characterised by rising rotation curves, while retired regions by declining profiles.
We also divide the W H α − β diagram across the six quenching stages (see right panel of Fig. 4 and Fig. B.2). There is a smooth transition from fully retired to star-forming galaxies from the left-bottom (low W H α and negative β) to the right-top side (high W H α and positive β) of the relation. The intermediate quenching stage spaxel distribution of the mixed galaxies is centrally located at the W H α − β plane due to the large fraction of flat CVCs that these systems possess (see Fig. 2, Fig. A.1 and Fig. A.1). In particular, the nearly retired and fully retired quenching stage galaxies are almost fully located in the quenched region with declining CVC of the W H α − β diagram (i.e. W H α < 3 Å and β < 0).

Nature of the scatter in the quenching relations
In the previous section, the W Hα − V c relation depicts a link between quenching and the amplitude of CVCs, showing in particular that a majority of spaxels with high velocity (i.e. V c ∼ 250 km s −1 ) are quenched (i.e. W Hα < 3 Å ). The W Hα − β relation, meanwhile, highlights the connection between quenching and the shapes of CVCs and exhibits a clear bi-modality: whereas star forming regions are overall characterised by rising curves, quenched regions are associated with decreasing curves.
Both quenching relations exhibit a large level of scatter, however, in the W Hα − V c plane, quenched regions can also arise at low circular velocity and there are a large number of starforming regions at high velocity. Many outlier spaxels in the W Hα − β relation represent star-forming regions with declining curves, whereas other are quenched regions with rising curves.
In this section we test whether these outliers are related to the global morphologies of the galaxies in our sample. Indeed, morphological features like bulges, bars and spiral arms are important galactic dynamical structures (see Section 4). To this end, we examine how the observed spaxels are distributed with respect to several indicators of galactic structure including Hubble type, presence of bar and bulge nature (e.g. classical or pseudo bulge, Kormendy & Kennicutt 2004; see also Figs. 5 and 6 of this work).
Following, K21 we adopt the Hubble type classifications for the galaxies in our sample from Walcher et al. (2014). As discussed in K21 (see their Fig. 7), late-type spiral galaxies (Sb−Sdm) mostly make up the star-forming class while early-type spirals (Sab−Sbc) predominantly make up the quiescent-nuclear-ring and centrally quiescent galaxies. The mixed quenching stage includes a large variety of morphologies (covering early-type spirals, lenticulars and ellipticals). The nearly and fully retired galaxies consist only of early-type galaxies (both lenticulars and ellipticals).
According to the bar-type statistics of K21, the quiescentnuclear-ring group contains the largest number of barred galaxies, followed by centrally quiescent, mixed and star-forming quenching stages. The nearly-and fully-retired stages, on the other hand, are mostly constituted by unbarred galaxies (see their Fig.8). It is notable that the quiescent-nuclear-ring, centrally quiescent and mixed quenches stages consist of early-type spiral galaxies (both lenticulars and discy ellipticals). These systems are thus dominated by secular evolution/dynamical features (e.g. spiral arms and bars; K21), and we expect these galaxies to be the largest contributor to the scatter in the two trends revealed in this work.
Additionally, we use the central stellar mass surface density of the galaxies within 1 kpc, Σ 1kpc . This parameter has been used as a proxy for both the mass and compactness of the central region (bulge/spheroid) of the galaxies, scaled on the same physical size (Cheung et al. 2012;Luo et al. 2020 and references therein).
For each galaxy we define Σ 1 kpc ≡ is the stellar mass of the galaxy integrated within the 1 kpc ellipsoid (obtained by de-projecting the radial map on the plane of the galaxy) and R 1kpc ≡ 1 kpc (e.g. Cheung et al. 2012). For each galaxy, the stellar mass in a given spaxel is determined from the stellar mass surface density map (Σ * , included in the PIPE3D dataset; Sánchez et al. 2016c,b) multiplied by the area of the spaxel. Using Σ 1kpc , we adopt the method of Luo et al. (2020) to classify the central stellar concentration of the galaxies as either "classical bulges" (CB) or "pseudo bulges" (PB). (Elliptical galaxies are omitted from the bulge classification analysis and are designated with the label "E" in the bulge statistics of Figs. 5 and 6.) Following Luo et al. (2020), we use the residual ∆Σ 1kpc from their measured relation between Σ 1kpc and global stellar mass (M * , calculated integrating across the entire stellar mass map; equation 2 of Luo et al. 2020) to classify galaxies as classical bulge hosts (or elliptical galaxies) when log(∆Σ 1kpc ) > 0 and as pseudo bulge hosts when log(∆Σ 1kpc ) < 0 (see Luo et al. (2020); their Fig. 7). As a basic classification of the outliers around the general trend in W H α vs. V c (Fig. 5), we define a critical velocity log(V c )∼2.4 km s −1 (V c ∼ 250 km s −1 ) where the trend changes from relatively flat (in the low V c region) to steep (in the high V c region). Across this critical value, galaxies also separate broadly into star-forming and quenched groups. Most of the spaxels in low V c region (< 250 km s −1 ) belong to disc galaxies (S0−Sdm) without a dominant bar type (barred, unbarred or unsure) whereas most of the spaxels in the high V c (> 250 km s −1 ) zone, on the other hand, come from early-type and unbarred galaxies (E1−Sab).
The critical velocity also tends to mark a separation in the behaviour of Σ 1kpc . For spaxels in the high V c region (e.g. mainly above 10 10 M kpc −2 ), Σ 1kpc is also high independent of whether Article number, page 7 of 16 A&A proofs: manuscript no. CVC-QS the region is star-forming or quenched. This indicates that regions with high V c amplitudes are also those with high central stellar mass density. In the low-V c regime (e.g. mainly below 10 10 M kpc −2 ), in contrast, only quenched regions have high Σ 1kpc whereas the star-forming spaxels exhibit a broad bi-modal distribution. The broadness of the Σ 1kpc distribution for this latter subset of galaxies reveals a large variety of central stellar concentrations, and the bi-modality tends to correlate with bulge classification, in the sense that star-forming regions can be arise in galaxies that host both classical bulges (with high Σ 1kpc ) and pseudo bulges (with low Σ 1kpc ). Quenched regions are almost exclusively hosts of classical bulges.
In the W H α − β diagram, we see that the majority of the spaxels again separate into two main areas: a star-forming− rising CVC zone (W H α > 6 Å and β > 0) and a quenched−declining CVC region (W H α < 3 Å and β < 0). In other words, the CVCs of the star-forming galaxies are generally slowly rising (β > 0), while the CVCs of the quenched systems have a faster inner rise, followed by either a flattened part or a slowly declining part (with β < 0; outside a central peak caused by a central mass concentration). The star-forming −rising CVC zone is largely populated by spaxels of late-type spirals (Sb−Sdm), both barred and unbarred, while the quenched−declining CVC region is populated by the spaxels of early-type galaxies (E1−Sb), mostly without bars. This behavior generally agrees with the idea that galactic regions with high stellar mass concentrations (bulges or spheroids) are quenched (e.g. Martig et al. 2009;Gensior et al. 2020).
There are significant outliers from these general trends, however, with some star-forming galaxies exhibiting declining CVCs (W H α > 6 Å and β < 0) and some quenched galaxies exhibit rising CVCs (W H α < 3 Å and β > 0). (We note, though, that these outliers represent only a minority of the sample.) Inspection of the individual CVCs in these cases (and across all quenching stages; see Figs. A.1 and A.2), indicates that this quenched subset (of nearly and fully retired class galaxies) are lacking peaks in their CVCs while the subset of star-formers have prominent peaks in their CVCs (e.g., prominent bulges). The latter are found to arise predominantly in classical bulge hosts. (In this same zone, quenched regions are also dominated by ellipticals and classical bulges.) Note, though, that for both star-forming and quenched regions, rising CVCs are found in galaxies with both classical and pseudo bulges.
Another characteristic of these two outlier regions is that the median of the spaxel distributions is shifted towards early-type disc galaxies (S0−Sbc), both with and without bars (Fig. 6). The Σ 1kpc spaxel distribution also tends to be tighter compared to the broad distribution characteristic of the star-forming−rising CVC zone.

Concluding remarks
In this paper, we explore the link between the inner gravitational potential of 215 (E−Sdm) non-active CALIFA galaxies through their circular velocity curves (CVC) (calculated from stellar dynamics within 1.5 R e ), and the star-formation quenching parameter given by the value of the equivalent width of the H α (W H α ). Our main findings can be summarised as follows: (i) galaxies with certain CVC shape and amplitude tend to be associated with a specific quenching-stage group; (ii) there is a moderate correlation between the amplitude of the velocity of the spaxels (V c ) and their quenching-proxy values (W H α ), where the relationship steepens at higher amplitudes above log V c ∼ 2.4 km s −1 (V c ∼ 250 km s −1 ); (iii) the relation between the shape of the CVC (β) and W H α is moderate and bi-modal, showing that the quenching regions of the galaxies (where W H α is below 3 Å) are overall characterised by high V c and negative β (declining CVC), and the star-forming regions (where W H α is above 6 Å) are overall characterised by low V c and positive β (rising CVC); (iv) the outlier spaxels of the W H α − β relation is largely coming from early-type disc galaxies (S0−Sbc) and follows the opposite trend of the main spaxel distribution, described above; (v) the spaxels of the nearly and fully retired class galaxies almost fully occupy the quenched−declining CVC region of the W H α − β relation (where W H α < 3 Å and β < 0).
These findings indicate an important link of the gravitational potential to the present day star-formation quenching stages of the galaxies. The quenching spaxels lay in regions of the galaxies with a large gravitational potential due to a mass concentration (e.g. bulge/spheroid, bar and rings). Star formation quenching regulated by gravitational potential is generally referred to "dynamical suppression" (or "morphological quenching" Martig et al. 2009;Genzel et al. 2014;Gensior et al. 2020;Gensior & Kruijssen 2021). Our results support such a scenario, as we find that a proxy for quenching (e.g. W H α ) appears correlated with quantities that trace features of the galactic potential, namely circular velocity curve amplitude (V c ) and shape (β).
A number of other studies have argued that reductions in star formation efficiency (SFE) are responsible for quenching observed in galaxies globally and/or locally (see, e.g. Colombo et al. 2018Colombo et al. , 2020Ellison et al. 2020). In particular, a link between reduced SFE, morphology, increased circular velocity and rotation curve shear (expressed through the Oort's constant A = 0.5V c /R(1 − β)) has been observed by Colombo et al. (2018) within a sample of galaxies with largely similar molecular gas mass surface densities. That study supports the idea that high shear contributes to the stabilisation of gas discs, preventing the fragmentation in the molecular gas (e.g. Toomre 1964;Romeo & Mogotsi 2017) and reducing the rate of star formation (e.g. Martig et al. 2009;Meidt et al. 2013;Davis et al. 2014;Meidt et al. 2018).
Besides shear due to differential rotation (inferred from the CVCs), other kinds of shear may be present due to local gas flows (e.g. Meidt et al. 2018), associated either with spirals arms (Dobbs & Baba 2014;Meidt et al. 2013) or bars (Athanassoula 1992;Sormani et al. 2015). According to a recent analytical model, local and global (orbital) motions in the galactic potential can lead to a dynamical suppression of star formation on the scales of star-forming giant molecular clouds (Meidt et al. 2018(Meidt et al. , 2020Liu et al. 2021). In this "bottleneck" model, galactic orbital motions compete with gas self-gravity on cloud scales. These motions contribute to the velocity dispersion within clouds, thus increasing their stable mass (see, e.g. Hughes et al. 2013 for an observational example of this effect) and introducing environmental variations in gas dynamical state that can influence its ability to collapse and form stars Leroy et al. 2017).
Whether the reductions in the SFE predicted in some environments by this type of model lead to reductions in the observed rate of star formation depends on the amount of fuel available for star formation. Based on the results of this work, the galaxies most likely to show signs of dynamical suppression are the nearly and fully retired class galaxies in our sample, where the spheroidal component is prominent (e.g. quencheddeclining CVC area of the W H α − β relation, Figs. 4 and 6) and gas fractions are low (see also Martig et al. 2013;Gensior & Kruijssen 2021). For other systems, dynamical suppression may be less efficient due to the amount and organization of the gas reservoir. This could explain why several of the galaxies in our sample with peaks in their CVCs (indicating a large central mass concentration or bulge) are fully star-forming and why some of the galaxies that are quenched (or are approaching quenching) possess rising curves in our sample. In the first case (i.e. spaxels with W H α > 6 Å and β < 0), it is noteworthy that galaxies with star-forming classical bulges are less massive and less concentrated than quenched systems with classical bulges (Yu et al. 2022), possibly indicating that they have a larger gas reservoir (according to the observed increase in gas fraction with decreasing stellar mass; Saintonge et al. 2017). Central star formation enhancements in classical bulges (e.g. Luo et al. 2020) could also be induced by bars (Athanassoula 1992;Wang et al. 2012Wang et al. , 2020 and spirals (Kim & Kim 2014;Yu et al. 2022). In the second case (i.e. spaxels with W H α < 3 Å and β > 0), it may be relevant that anaemic spirals and S0s are disk galaxies that can have rising CVC and no star-formation (Kormendy & Bender 2012). Therefore, environmental quenching mechanisms may preferentially remove the gas from galaxies with rising CVCs, stopping their star-formation. Spirals could also fade away as the gas fraction decreases (Yu et al. 2021) and becomes S0s (Kormendy & Bender 2012).
In summary, our results show that the largest number of spaxels come from the two extreme quenching stages: starforming and fully retired classes (corresponding to late-type spirals and elliptical galaxies, respectively) for which V c and β spaxel distribution coincide with the dynamical suppression hypothesis: the non-quenching or quenching of the galaxies depends on the absence or presence of bulge/spheroid, respectively (which corresponds to low and high velocities in W H α − V c relation or rising and declining CVC in W H α − β relation, respectively). On the other hand, the scatter analysis indicates that the presence of classical bulge is not the only necessary condition. It seems that galaxies need to further have: higher central density (approximately above 10 10 M kpc −2 ), no bar, and early-type morphologies (meaning no tight and prominent spiral arms).
Future work will explore the cold gas content of the studied systems (Kalinova et al., in prep.) and will give a complementary perspective on the dynamics and mechanisms responsible for the formation of the various quenching stages of the nearby galaxies.  Number of overlapping spaxels  All data (50% and 95% contour) Single realisation (50% and 95% contour) The red lines indicate the spaxels of randomly selected 60 galaxies from the original data set (where 10 galaxies from each of the six quenching stages are chosen) for 100 single realisations. The inner and the outer contours represent the relationships constructed with 50% and 95% of the spaxel distribution, respectively, of the full sample of 215 galaxies (black contours) and each sub-sample of 60 galaxies (red contours). Both relationships overall show stability regarding the studied sample across quenching stage.