# Local variations of the Stellar Velocity Ellipsoid-II: the effect of the bar
in the inner regions of Auriga galaxies

Daniel Walo-Martín1,2, Francesca Pinna3, Robert J.J. Grand1,2, Isabel Pérez4,5
E-mail<EMAIL_ADDRESS>Jesús Falcón-Barroso1,2, Francesca Fragkoudi6, Marie
Martig7
1Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E-38205 La
Laguna, Tenerife, Spain
2Departamento de Astrofísica, Universidad de La Laguna, Av. del Astrofísico
Francisco Sánchez s/n, E-38206, La Laguna, Tenerife, Spain
3Max-Planck-Institut für Astronomie, Konigstuhl 17, 69117, Heidelberg, Germany
4Departamento de Física Teórica y del Cosmos, Universidad de Granada, Facultad
de Ciencias (Edificio Mecenas), E-18071, Granada, Spain
5Instituto Carlos I de Física Teórica y Computación
6Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str 1, D-85748
Garching, Germany
7Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool L3 5RF, UK

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Theoretical works have shown that off-plane motions of bars can heat stars in
the vertical direction during buckling but is not clear how do they affect the
rest of components of the Stellar Velocity Ellipsoid (SVE). We study the 2D
spatial distribution of the vertical, $\sigma_{z}$, azimuthal, $\sigma_{\phi}$
and radial, $\sigma_{r}$ velocity dispersions in the inner regions of Auriga
galaxies, a set of high-resolution magneto-hydrodynamical cosmological zoom-in
simulations, to unveil the influence of the bar on the stellar kinematics.
$\sigma_{z}$ and $\sigma_{\phi}$ maps exhibit non-axisymmetric features that
closely match the bar light distribution with low $\sigma$ regions along the
bar major axis and high values in the perpendicular direction. On the other
hand, $\sigma_{r}$ velocity dispersion maps present more axisymmetric
distributions. We show that isophotal profile differences best capture the
impact of the bar on the three SVE components providing strong correlations
with bar morphology proxies although there is no relation with individual
$\sigma$. Time evolution analysis shows that these differences are a
consequence of the bar formation and that they tightly coevolve with the
strength of the bar. We discuss the presence of different behaviours of
$\sigma_{z}$ and its connection with observations. This work helps us
understand the intrinsic $\sigma$ distribution and motivates the use of
isophotal profiles as a mean to quantify the effect of bars.

###### keywords:

galaxies: general, galaxies: evolution, galaxies: formation, galaxies:
kinematics and dynamics, galaxies: spiral

††pubyear: 2020††pagerange: Local variations of the Stellar Velocity
Ellipsoid-II: the effect of the bar in the inner regions of Auriga galaxies–B

##  1 Introduction

Since the first extensive imaging surveys it has been shown that bars are
common structure in the Local Universe that populate almost two thirds of disc
galaxies (e.g Knapen et al., 2000; Eskridge et al., 2000; Menéndez-Delmestre
et al., 2007; Aguerri et al., 2009; Gadotti, 2009; Masters et al., 2011; Buta
et al., 2015; Erwin, 2018). This fraction decreases with redshifts and at z= 1
only 20% of observed galaxies host a bar (Sheth et al., 2008; Melvin et al.,
2014), indicating that bars play a significant role in the secular evolution
of galaxies at later times (see Kormendy & Kennicutt, 2004; Athanassoula,
2013, for a review).

Theoretical studies in the last thirty years have provided valuable
information about the formation, stellar orbit distribution and influence of
bars in multiple galaxy properties (e.g. Combes & Sanders, 1981; Athanassoula,
1992a, b; Debattista & Sellwood, 2000; Athanassoula, 2003; Bureau &
Athanassoula, 2005; Martinez-Valpuesta et al., 2006; Minchev & Famaey, 2010;
Wang et al., 2012; Fragkoudi et al., 2016). Torques induced by bars can
efficiently redistribute mass and angular momentum, driving the formation of
outer rings (Buta, 1986; Buta & Combes, 1996; Rautiainen & Salo, 2000),
nuclear discs(Debattista et al., 2006; Sellwood, 2014) and inner bars (de
Lorenzo-Cáceres et al., 2012, 2019). Further, bars can also induce the
formation of vertically extended bulges, so called boxy/peanut bulges, trough
buckling instabilities (Mihos et al., 1995; Martinez-Valpuesta et al., 2006;
Fragkoudi et al., 2017).

Observational studies have investigated the morphology, kinematics and stellar
populations properties of bars in profound detail to expand our knowledge on
the role that bars play on the secular evolution of galaxies (Hawarden et al.,
1986; Martin & Friedli, 1997; Kormendy & Kennicutt, 2004; Pérez et al., 2007,
2009; Pérez & Sánchez-Blázquez, 2011; Gadotti, 2011; Sánchez-Blázquez et al.,
2011; Seidel et al., 2016; Fraser-McKelvie et al., 2019; Neumann et al.,
2020).In particular, Seidel et al. (2015) confirmed that stronger bars have
enhanced influence on inner kinematic features (e.g. stronger humps in the
velocity profiles and $\sigma$-drops) as predicted by numerical simulations
(Athanassoula, 2003; Bournaud et al., 2005; Kim et al., 2012; Kim & Stone,
2012; Łokas et al., 2014).

The information derived from external galaxies rely on projected quantities
such as the Line-of-Sight (LoS) velocity distribution from which we can infer
different parameters like the mean velocity, V, and the velocity dispersion,
$\sigma$, using both parametric (van der Marel & Franx, 1993; Kuijken &
Merrifield, 1993; Cappellari & Emsellem, 2004) and non parametric techniques
(Gebhardt et al., 2000; Fabricius et al., 2014; Falcón-Barroso & Martig,
2021). However, it is difficult to disentangle the multiple families of orbits
that coexist in a galaxy from the projected velocities alone, particularly in
the inner regions where different structures may coexist with the bar (e.g.
b/p bulge, inner bar, nuclear disc). Therefore, understanding the dynamical
state of galaxies in terms of their intrinsic components is a key factor to
comprehend observations. In other words, we require a 3D characterization of
the dynamical state of galaxies which is typically described through the
Stellar Velocity Ellipsoid (SVE) and its three velocity dispersion components,
the vertical, $\sigma_{z}$, radial, $\sigma_{r}$ and azimuthal $\sigma_{\phi}$
velocity dispersions.

Unfortunately, these kind of detailed analysis can only be performed in our
Galaxy, where thanks to missions like GAIA (Gaia Collaboration et al., 2016)
and APOGEE (Majewski et al., 2017) we have access to the full 6D information
(positions and velocities, see the works of Büdenbender et al., 2015;
Mackereth et al., 2019; Simion et al., 2021). A number of methods have been
proposed to study the SVE of external galaxies where the stellar populations
cannot be resolved either using analytic models for the disc and velocity
dispersion with photometric observations (van der Kruit & de Grijs, 1999),
spectroscopic decomposition of the line-of-sight velocity distribution
(Cappellari et al., 2007; Emsellem et al., 1999; Tempel & Tenjes, 2006; Kipper
et al., 2016; Gerssen & Shapiro Griffin, 2012), or dynamical models
(Cappellari et al., 2007; Emsellem et al., 1999; Tempel & Tenjes, 2006; Kipper
et al., 2016). Luckily, state-of-the-art hydrodynamical simulations are now
able to reproduce realistic disc galaxies with smaller bulges and extended
disc thanks to advances in resolution and sub-grid physics modules (Brooks et
al., 2011; Stinson et al., 2013; Marinacci et al., 2014a; Brooks &
Christensen, 2016), which has lead to an increasing number of studies on the
role of bars in galaxy evolution. Simulations of galaxies in the Milky-Way
mass range have particularly expanded in the last decade to understand
properties of both our Galaxy and external systems, e.g. the Auriga (Grand et
al., 2017), NIHAO-UHD (Buck et al., 2020) and FIRE (Hopkins et al., 2014)
simulations.

In this work we aim to study the local properties of the SVE in the inner
regions of Milky-Way mass galaxies. To that end we make use of the Auriga
galaxies, a set of 30 zoom-in magneto-hydrodynamical cosmological simulations
of late-type galaxies. These simulations present different morphologies and
evolutionary histories, and successfully reproduce many present day
observables such as stellar masses, sizes, rotation curves, star formation
rates (Grand et al., 2017). Thus, the Auriga sample is the ideal testbed
because it will allow us to analyze the impact of external agents and
different secular evolution processes in the SVE of high resolution disc
galaxies in a cosmological environment.

This paper is the second in a series exploring the spatial variations of the
SVE in simulated late-type galaxies. Here we focus on the connection between
the SVE and the presence of bars in the inner regions of galaxies. This paper
is organized as follows. In Sec.2 we describe the Auriga simulations, our
sample selection and the methodology used to study the SVE. In Sec.3 we
present the SVE 2D maps and profiles of Auriga galaxies. Sec. 4 explores the
correlations of the velocity dispersion differences between the major and
minor bar axis and different bar morphology proxies. In Sec.5 we analyze the
time evolution of the velocity dispersion differences and the bar formation.
We discuss our results in Sec. 6. Finally, in Sec. 7 we sum up the main
results and conclusions from this work.

##  2 Methodology

Figure 1: The panels show from left to right the surface brightness, SVE
velocity dispersions ($\sigma_{z}$, $\sigma_{r}$ and $\sigma_{\phi}$ ) and
axial ratio maps ($\sigma_{z}/\sigma_{r}$ and $\sigma_{\phi}/\sigma_{r}$) of 7
galaxies in the Auriga simulations. From top to bottom each row shows the maps
of an unbarred galaxy (AU- 3), two galaxies with weak bar (AU-21 and AU-25),
two galaxies with strong bars (AU-24 and AU-27) and two galaxies with B/P
(AU-17 and AU-23). Inner and outer circles indicate the effective radius of
the bulge and the beginning of the disc dominated region. The area covered in
this analysis is indicated (in kpc) at the bottom right of the left-most maps
of each case.

###  2.1 The Auriga simulations

The Auriga project (Grand et al., 2017) consists of 30 zoom-in high resolution
simulations of Milky-Way mass haloes taken from the largest volume Dark Matter
Only simulation of EAGLE, L100N1504 (see Schaye et al., 2015, for details).
These simulations were performed with AREPO (Springel, 2010), an N-body,
magnetohydrodynamics (MHD) code. We name each simulation by ’AU-N’ where N
ranges between 1 and 30. Simulations adopt the cosmological values taken from
Planck Collaboration et al. (2014) ($\Omega_{m}=0.307$, $\Omega_{b}=0.048$,
$\Omega_{\Lambda}=0.693$ and $H_{0}=100\ \mathrm{h}\ kms^{-1}\ Mpc^{-1}$, with
$\mathrm{h}=0.6777$). In this work, we study simulations where star and dark
matter particles have a typical mass of $m_{b}\sim 5\times 10^{4}M_{\odot}$ of
and $m_{DM}\sim 3\times 10^{5}M_{\odot}$, respectively. These values
correspond to the standard level-4 mass resolution of the Auriga galaxies (see
Table 2 in Grand et al., 2017, to find the differences between each resolution
level). The comoving softening length for star and dark matter particles is
set to $500\mathrm{h}^{-1}cpc$ before z=1, when the simulation adopts a
physical gravitational softening length of 369 pc. Groups and subhaloes were
identified by the friend-of-friends (Davis et al., 1985) and SUBFIND (Springel
et al., 2001) algorithms, respectively, while merger trees were constructed
with the LHaloTree algorithm described in Springel et al. (2005). To account
for physical processes that act below the resolution limit of the simulation
the Auriga models include an updated version of the sub-grid modules used in
Marinacci et al. (2014a, b). For a complete description of the models and the
modifications introduced we refer the reader to Grand et al. (2017).

Despite being all of the galaxies in the Milky-Way mass range and their
limited number, the Auriga sample presents different evolutionary histories
and exhibit a wide range of morphological properties. In terms of morphology,
the Hubble-Type of the galaxies ranges from Sa to Sb and individual structures
such as bars, disc breaks, pseudobulges and B/P bulges have been analyzed in
detail (Gargiulo et al., 2019; Blázquez-Calero et al., 2020; Fragkoudi et al.,
2020, 2021). In particular, Blázquez-Calero et al. (2020) extensively studied
the structural properties of the bar such as bar strength, $\rm{f_{bar}}$,
length, $\rm{L_{bar}}$ or bar to total (B/T) light fraction, based on mock
r-band images. Fragkoudi et al. (2020) analysed the chemo-dynamical and
kinematic properties of the barred galaxies while Fragkoudi et al. (2021)
studied the bar pattern speed. Table 1 contains the most relevant
morphological parameters used in this study.

Additionally, we characterize the strength of the bar by the maximum amplitude
of the m=2 Fourier mode, a common methodology used in the literature
(Athanassoula & Misiriotis, 2002; Debattista et al., 2006; Algorry et al.,
2017) . We directly obtain this parameter from the simulation as,

$A_{2}=max\frac{\sqrt{a_{2}^{2}+b_{2}^{2}}}{a_{0}}$ (1)

where $a_{2}$ and $b_{2}$ are defined as,

$a_{2}(R)=\sum_{i=0}^{N}m_{i}sin(2\theta_{i})$ (2)

$b_{2}(R)=\sum_{i=0}^{N}m_{i}cos(2\theta_{i})$ (3)

being $m_{i}$ and $\theta_{i}$ the mass and azimuthal angle of a particle i at
the the cylindrical radius R and N the total number of particles at that
radius. We use the maximum $A_{2}$ value within 10 kpc and set the limit value
of 0.3 to define the formation of a strong bar. To prevent increases in
$A_{2}$ caused by transient effects such as mergers we visually examine the
maps to verify the formation of a bar.

Auriga galaxies have evolved differently, some of them being in relative
isolation with no major mergers since z=2 while others have experienced
multiple interactions and accretion events throughout their entire lives (a
reduced number of galaxies are undergoing mergers at z=0). In this work we do
not study AU-11, AU-29 and AU-30 because they exhibit strong signs of
interactions in their z=0 surface brightness and perturbed SVE maps. We
include AU-20, a galaxy that is currently undergoing a merger at z=0 because
the infalling galaxy has not yet affected the central region. From the final
sample of 27 galaxies almost three quarters (21 of them) are barred systems.

###  2.2 SVE and central parts characterization

To analyze the SVE we study galaxies in a face-on configuration. To that end,
we first place the angular momentum vector of stars younger than 3 Gyr within
a sphere of radius 60 kpc around the center of potential parallel to the
vertical direction. These particles are mostly located in the young thin disc
and thus, their angular momentum is less affected by external agents. This
guarantees that the galaxy is properly rotated even when it is interacting
with its satellites. Then, we project galaxies along the Z axis onto a square
grid of 0.5 x 0.5 kpc pixels. The number of particles per pixel in the inner
regions is quite large and no binning is required to reach a statistically
significant number of particles compared to the analysis of the stellar disc
(Walo-Martín et al., 2021). We use a cylindrical restframe to characterize the
SVE and we calculate the vertical, $\sigma_{z}$, radial, $\sigma_{r}$, and
azimuthal, $\sigma_{\phi}$, velocity dispersions and the axial ratios
$\sigma_{z}/\sigma_{r}$ and $\sigma_{\phi}/\sigma_{r}$ in each pixel.

We aim to understand the effect of the bar on the SVE in the inner region of
galaxies, and therefore, we limit our study to the region within,
$R_{\rm{0,D}}$, the beginning of the disc-dominated region. Following Walo-
Martín et al. (2021) we characterize this parameter using the length of the
bar, $L_{\rm{bar}}$, measured by Blázquez-Calero et al. (2020) and the disc
scale length, $R_{\rm{d}}$, from Grand et al. (2017) in barred and unbarred
galaxies, respectively. On the other hand, the effect of the bar on the
stellar kinematics is less relevant in the central parts where bulges and
nuclear discs have a larger impact. Therefore, we define our region of
interest as the region between the effective radius of the bulge, $R_{\rm{B}}$
and $R_{\rm{0,D}}$. The effective radius of the bulge is taken from Blázquez-
Calero et al. (2020) for barred galaxies and Grand et al. (2017) for unbarred
systems.

Table 1: Parameters of the Auriga galaxies at z=0. The columns are:(1) Galaxy name, (2) Hubble-Type, (3) Bulge effective radius, (4) Radius where the disc starts to dominate, (5) Bar strength, (6) Bar relative size, (7) Bar light fraction, (7) $\sigma_{z}$ categories from Sec. 3.2.Galaxies with B/P bulges are indicated by ∗ $Name$ | Hubble-Type |  $R_{\rm{B}}$ (kpc) |  $R_{\rm{0,disc}}$ (kpc) | $f_{bar}$ | $L_{bar}/R_{25}$ | Bar/T | Type $\sigma_{z}$
---|---|---|---|---|---|---|---
AU-1 | SBb | 2.03 | 4.07 | 0.57 | 0.20 | 0.16 | B
AU-2 | SBc | 1.22 | 8.99 | 0.59 | 0.24 | 0.07 | B
AU-3 | Sb | 1.49 | 7.26 | 0 | 0 | 0 | -
AU-4 | Sbc | 1.74 | 3.93 | 0 | 0 | 0 | -
AU-5 | SBb | 0.77 | 4.58 | 0.50 | 0.22 | 0.06 | B
AU-6 | SBbc | 0.98 | 5.43 | 0.42 | 0.21 | 0.05 | A
AU-7 | SBb | 0.96 | 5.43 | 0.47 | 0.22 | 0.06 | A
AU-8 | Sc | 0.65 | 6.57 | 0 | 0 | 0 | -
AU-9 | SBb | 0.98 | 6.45 | 0.76 | 0.34 | 0.15 | C
AU-10 | SBa | 0.85 | 6.45 | 0.75 | 0.40 | 0.33 | C
AU-12 | SBab | 0.85 | 3.73 | 0.60 | 0.20 | 0.06 | B
AU-13∗ | SBa | 0.97 | 5.43 | 0.59 | 0.35 | 0.29 | C
AU-14 | SBb | 0.81 | 5.09 | 0.65 | 0.20 | 0.09 | B
AU-15 | Sbc | 2.22 | 5.32 | 0 | 0 | 0. | -
AU-16 | Sc | 1.83 | 9.03 | 0 | 0 | 0 | -
AU-17∗ | SBa | 1.26 | 5.43 | 0.59 | 0.34 | 19 | C
AU-18∗ | SBb | 1.11 | 6.45 | 0.61 | 0.31 | 0.12 | C
AU-19 | Sbc | 1.42 | 4.81 | 0 | 0 | 0 | -
AU-20 | SBc | 0.99 | 4.81 | 0.66 | 0.18 | 0.1 | B
AU-21 | SBb | 1.02 | 3.39 | 0.41 | 0.14 | 0.04 | A
AU-22∗ | SBa | 0.80 | 6.28 | 0.56 | 0.47 | 0.23 | C
AU-23∗ | SBbc | 1.31 | 9.5 | 0.69 | 0.38 | 0.11 | C
AU-24 | SBc | 1.10 | 5.09 | 0.52 | 0.17 | 0.04 | B
AU-25 | SBb | 1.01 | 4.41 | 0.47 | 0.21 | 0.04 | A
AU-26∗ | SBa | 1.18 | 5.43 | 0.48 | 0.30 | 0.22 | C
AU-27 | SBbc | 0.87 | 5.77 | 0.69 | 0.22 | 0.08 | B
AU-28 | SBa | 0.90 | 7.46 | 0.82 | 0.43 | 0.36 | C

##  3 The SVE of the central region

In this section we use different approaches to explore the local properties of
the SVE in the inner regions of Auriga galaxies. In Walo-Martín et al. (2021)
we focused on the SVE in the disc region of these galaxies and we showed the
importance of analyzing its spatial distribution to fully characterize the
underlying kinematic structure. In particular, we showed that similar global
values can be obtained even if the SVE profiles and 2D maps present different
behaviours. We expect that this behaviour is more pronounced in the inner
regions due to the presence of non axisymmetric structures like the bar.
Consequently, we study the SVE properties at both global and local levels.

###  3.1 2D maps

We first analyze the spatial 2D distribution of the SVE in the inner regions
of Auriga galaxies. In Fig. 1 we show the SVE 2D maps of seven galaxies
selected from our sample. From left to right the maps show the V-band surface
brightness and SVE velocity dispersions and axial ratios ($\sigma_{z}$,
$\sigma_{r}$, $\sigma_{\phi}$, $\sigma_{z}/\sigma_{r}$ and
$\sigma_{\phi}/\sigma_{r}$, respectively). From top to bottom we find galaxies
with different morphological features, e.g. an unbarred galaxy (AU-3),
galaxies with weak bars (AU- 21 and AU-25), galaxies with strong bars (AU-24
and AU-27) and galaxies with a B/P (AU-17 and AU-23). The rest of the Auriga
galaxies are shown in Appendix A. Inner and outer black circles indicate the
boundaries of the region of interest. We find that simulated galaxies present
a complex SVE spatial distribution, particularly barred galaxies, and no
general behaviour can be inferred even within each of the four categories.
Nevertheless, we note that, on average, velocity dispersions increase inwards,
except for some systems where the innermost regions have lower velocity
dispersions (see AU-4 in Fig. 7).

Unbarred galaxies present the most homogeneous picture since the three
velocity dispersions show inward increasing values in highly axisymmetric
distributions. Small scale differences are associated with stochasticity or
dynamical agents that leave only a small imprint on the SVE such as the small
spiral arms (e.g. AU-15 and AU-16 in Fig. 7). Nevertheless, we notice that
many galaxies present lower values in the innermost regions, specially in
$\sigma_{\phi}$, e.g. AU-8 and AU-4. In the case of AU-8, the $\sigma_{\phi}$
map shows a modest non axisymmetric feature that suggest the presence of a
very weak bar inside the bulge region. On the other hand, the low velocity
dispersion region of AU-4 is formed after a recent merger that has injected
gas into the galaxy. The SVE axial ratios maps are significantly different
from one galaxy to the other and we do not find a general trend.

The panels of barred galaxies show a more complex SVE distribution where the
velocity dispersion maps exhibit non-axisymmetric features that closely follow
the bar structure. These features manifest as low velocity dispersion regions
along the bar major axis within a hotter surrounding central region in the
maps of the vertical and azimuthal SVE components. In galaxies with weak bars
the low velocity dispersion region is confined to the central regions,
extending only few kilo-parsecs outside the bulge effective radius as in
AU-25. In strongly barred systems like AU-24 this feature spans over the
entire bar length. Similarly, the extent of the high velocity dispersion
region around the bar is not unique and in some cases it may cover the entire
region of study while in others it drops at intermediate radius. On the other
hand, the maps of the radial velocity dispersion are similar to those of
unbarred galaxies with larger $\sigma_{r}$ values inwards and no sign of
axisymmetric features in most of the sample. Only two barred galaxies present
an elongated high velocity dispersion feature along the bar (see the maps of
AU-2 and AU-10 in Figs. 7 -8). The axial ratio maps reveal a similar trend
where we find more oblate ellipsoids along the bar and larger axial ratios in
the rest of the inner region. These features are mostly driven by the same low
and high $\sigma_{z}$ and $\sigma_{\phi}$ features in the velocity dispersion
maps.

Figure 2: Maps and profiles of an unbarred galaxy (top left), a galaxy with a
weak bar (top right), a galaxy with a strong bar (bottom left) and a barred
galaxy with a B/P (bottom right). For each galaxy the panels show the
projected 2D maps (top), radial profiles (middle) and isophotal profiles
(bottom) of the vertical (left), radial (center) and azimuthal (right)
velocity dispersions. Inner and outer circles indicate the effective radius of
the bulge and the beginning of the disc dominated region. Symbols and
errorbars indicate the median, 16th and 84th value within each bin in the
middle and bottom panels. Red and blue colours represent the results from the
slits indicated in the maps along the bar major and minor axes. Black colour
indicates the difference between these two regions. Vertical dashed lines mark
the region of interest: position of the black circles for radial profiles and
the average surface brightness at these positions for isophotal profiles. The
area covered in this analysis is indicated (in kpc) at the bottom right of the
left-most maps of each case.

Barred galaxies hosting B/P bulges present two main differences in their
velocity dispersion maps with respect to the rest of barred galaxies. First,
the low velocity dispersion region in the $\sigma_{z}$ maps does not cover the
full extension of the bar. In fact, all six galaxies present a central high
$\sigma_{z}$ region and lower values outwards. Further, the lowest values are
found at the end of the bar e.g. AU-17. Secondly, all the B/P galaxies present
the elongated high velocity dispersion feature along the bar in the
$\sigma_{r}$ maps. Axial ratio maps are very similar to those of barred
galaxies without BP despite the differences in the velocity dispersion maps.

In summary, the SVE spatial distribution in the inner regions of barred
galaxies present non-axisymmetric features in the velocity dispersion maps
that closely match the bar location. These features present a qualitative
dependence on the bar morphology since galaxies hosting stronger bars present
more distinct regions with low $\sigma_{z}$ and $\sigma_{\phi}$ along the bar.
Interestingly, all galaxies with B/P bulges present deviations from this
general trend which emphasize the importance of this structure in the dynamics
of galaxies.

###  3.2 Profiles

We now explore the 1D profiles to better analyze the features in the SVE maps
in the previous section. To this end, we obtain the profiles along the bar
major and minor axes using slits with a width twice the effective radius of
the bulge which minimizes the bulge contribution in our analysis. Since the
bar structure is easily detectable in the surface brightness map we focus on
the isophotal profiles measured in bins of 0.5 mag/arcsec2, following Seidel
et al. (2015). The results in the following sections are centered on the
differences between these profiles in slits along the major and minor axes of
the bar. Our choice of the isophotal profiles allows to better characterize
the difference between barred and unbarred regions instead of averaging
azimuthally regions of the galaxy at the same distance from the center.
Nevertheless, we have confirmed that similar results are obtained using radial
profiles.

Fig. 2 shows the velocity dispersion maps and profiles of a galaxy without bar
(AU-19 upper left), with a weak bar (AU-21, upper right), a strong bar (AU-27,
bottom left) and a B/P (AU-17, bottom right). The panels show the projected 2D
map (top), radial profiles (middle) and isophotal profiles (bottom) of
$\sigma_{z}$ (left), $\sigma_{r}$ (center) and $\sigma_{\phi}$ (right) for
each galaxy. Symbols and errorbars indicate the median and the 16th and 84th
percentiles in each bin along the bar major (red) and minor (blue) axes. The
differences between minor and major axes (radial difference,
$\Delta_{r}\sigma$, and isophotal difference, $\Delta_{\mu}\sigma$) are shown
in black. Vertical dashed lines indicate the region of interest which
correspond to the position of the black circles in the 2D maps for radial
profiles and the average surface brightness at these positions for isophotal
profiles.

For the unbarred galaxy we randomly selected the two perpendicular directions.
Both radial and isophotal differences are negligible for the three SVE
components. This behaviour is the same for the rest of unbarred galaxies. Only
low level differences were obtained for some galaxies due to a combination of
stochasticity and the presence of spiral arms close to the disc effective
radius. This shows that there is no intrinsic difference in the inner spatial
distribution of the SVE velocity dispersions.

Barred galaxies exhibit differences as expected from the low velocity
dispersion signatures in the 2D maps. AU-21 is a weakly barred galaxy and the
radial and isophotal profiles exhibit small differences in the three velocity
dispersion components. The maps show that the characteristic low values along
the bar (particularly in the $\sigma_{\phi}$ component) are confined in the
bulge region and thus, we are not able to capture all the SVE differences
associated to the bar. On the other hand, we can appreciate that the velocity
dispersion is slightly larger in the perpendicular direction, around 10 km/s
of difference.

The other two barred galaxies (a strongly barred galaxy and a barred galaxy
with B/P bulge) show a more complex scenario with significantly different
velocity dispersion profiles. The plots of AU-27 reveal that all velocity
dispersion profiles along the bar major axis present a smooth decreasing
tendency with lower values than along the bar minor axis, leading to positive
differences. The average behaviour is the same for radial (middle panels) and
isophotal (bottom panels) profiles. Interestingly, we find that the vertical
and azimuthal velocity dispersions have more similarities with each other,
with a peak within the regions of interest along the minor axis, than with the
radial component. In addition, we find that isophotal profiles better capture
the influence of the bar in $\sigma_{r}$ compared to radial profiles where the
differences are significantly lower.

The plots of AU-17 quantitatively show the behaviour previously described in
Sec. 3.1 where the velocity dispersion profile along the bar major axis
presents a minimum close to the bar ends (both in the radial and isophotal
profiles). In other words, the spatial behaviour of the velocity dispersions
cannot be described with simply decreasing component, and in contrast to
AU-27, the profiles along the bar minor axis present a smooth decreasing
profile. We also find that isophotal differences are on average larger than
radial differences and are always positive ($\sigma_{r}$ radial differences
are negative in the inner regions due to high velocity dispersion features
along the bar major axis).

Figure 3: The panels show the maximum isophotal difference of the vertical
(top), radial (middle) and azimuthal (bottom) velocity dispersions as a
function of the bar strength (left), relative size (center) and light fraction
(right). Each panel shows the results for the full sample where each symbol
represent a galaxy. Blue and red colours represent unbarred and barred
galaxies, respectively. Filled red stars represent galaxies with a B/P.
Numbers within each panel indicate the Spearman correlation coefficient and
p-value of the barred sample. Figure 4: Same as Fig. 3, color coding each
galaxy according to the $\sigma_{z}$ classification from Sec. 3.2

The rest of barred galaxies in our sample present a combination of the
features shown in these three examples which difficults the development of a
complete classification of the SVE inner spatial distribution. Nevertheless,
the results of such classification would be hard to test observationally
except for the vertical velocity dispersion, which can be directly obtained
form the line-of-sight kinematics in face on galaxies. This component is
particularly important since it measures the vertical heating and is related
to the thickening of disks. Thus, we divide galaxies into three categories
depending on the properties of their $\sigma_{z}$ profiles and differences.

Type-A: the imprint of the bar is limited to the innermost region within the
bulge and the differences in the region of interest are negligible, e.g.
AU-21.

Type-B: the vertical velocity dispersion along the bar major axis smoothly
decreases. Along the bar minor axis $\sigma_{z}$ increases until it peaks and
then decreases to values similar to those along the bar major axis.

Type-C: the profiles along the bar present a multi component behaviour with a
clear minimum close to the bar ends e.g. AU-17.

The results of this classification are shown in Table 1. We find that there
are 4, 8 and 9 galaxies in each of these categories, and interestingly all B/P
galaxies belong to the type-C one. The fact that not all type-C galaxies
contain a B/P bulge suggests that these profiles are either obtained through
two different mechanisms for galaxies with and without B/P bulge or that there
is only one mechanisms that takes place in all the B/P galaxies in our sample.

The profiles of the SVE axial ratios of the barred sample do not exhibit the
same differences as the individual dispersions and present a much homogeneous
picture (not shown). This is in agreement with our previous results on the
radial distribution of $\sigma_{z}/\sigma_{r}$ in the disc region where
similar profiles were obtained with different velocity dispersion behaviours
(Walo-Martín et al., 2021).

To summarize, these results manifest that even though galaxies share some
similitudes in the 2D maps, e.g. lower velocity dispersions along the bar
major axis, a more in depth analysis is required to fully understand the
behaviour of the SVE in the inner regions. In fact, these features can be very
varied, from almost no imprint within the bar region to different combinations
of increasing and decreasing profiles along the direction of the bar major and
minor axes.

##  4 Correlation with bar properties

In the previous section we found that stronger bars exhibit more clear
differences between the velocity dispersion along the bar major and minor axes
than smaller bars. Therefore, we now aim to quantify the connection between
the bar morphology and the velocity dispersion features. To this end, we first
choose a metric that allow us quantify the differences seen in Fig. 2. In Sec.
3.2 we presented the radial and isophotal SVE profiles along the bar minor and
major axes for a limited number of galaxies and we showed that while both
radial and isophotal profiles have a similar behaviour, the latter provided
larger differences. We also showed that in some cases, differences from radial
profiles take negative values, particularly in the $\sigma_{r}$ component. For
these reasons, we will use the maximum isophotal difference,
$\Delta_{\mu,max}\sigma$, i.e. the maximum difference between the isophotal
profiles from the slits along the bar major and minor axes of the bar (see
black lines in the bottom panels of Fig. 2). This parameter allows us to
better quantify the differences between the direction along and perpendicular
to the bar. We have further ensured that similar results, although less
significant, are obtained if we choose a different parameter (e.g. mean or
median difference). Table 2 contains the maximum differences from radial and
isophotal profiles for the three velocity dispersions for all galaxies in our
sample.

In terms of the average velocity dispersion in the region of interest, we find
that both barred and unbarred galaxies have similar $\sigma_{r}$ and
$\sigma_{\phi}$. In terms of the vertical component, $\sigma_{z}$, barred
galaxies show on average larger values. This is somewhat expected since Grand
et al. (2016) showed that in Auriga galaxies bars can effectively heat stellar
particles in the disc along the vertical direction, especially bars that
experience buckling, in agreement with previous studies (Sellwood & Carlberg,
1984; Minchev & Quillen, 2006; Saha et al., 2010; Yurin & Springel, 2015).
However, since not all galaxies have the same stellar mass and the number of
unbarred galaxies in our sample is significantly lower we did not further
explore the differences in the absolute velocity dispersion values between
these two types of galaxies.

Figure 5: Average inner $\sigma_{z}/\sigma_{r}$ (top) and
$\sigma_{\phi}/\sigma_{r}$ (bottom) values along the bar minor axis
(perpendicular direction ) as a function of the average values along the major
axis (parallel direction). Blue squares represent unbarred galaxies and barred
galaxies are indicated as circles that are color coded according to the bar
strength. The black dashed lines represent the one to one correlation.

In Fig. 3 we explore how the differences of the three velocity dispersions
depend on different bar properties. From top to bottom the panels show the
maximum isophotal differences of the vertical, radial and azimuthal velocity
dispersions, as a function of the strength of the bar, $f_{bar}$ (left), the
relative size of the bar, $L_{bar}/R_{25}$ (center) and the bar light
fraction, Bar/L (right). Barred galaxies are represented with red circles and
B/P with red stars. We use, unbarred galaxies (blue squares), as a control
sample by setting the "bar properties" equal to zero. Numbers within each
panel indicate the Spearman correlation coefficient and p-value of the barred
sample (red symbols).

Figure 6: From top to bottom the panels show the time evolution of the
vertical, radial and azimuthal velocity dispersion maximum isophotal
differences, and the bar strength of AU-12 (left), AU-2 (center) and AU-23
(right). The vertical dashed lines denote the formation of a strong bar with
bar strength $A_{2}$ larger than the 0.3 threshold marked by the horizontal
line in the bottom panels.

The panels show that there is a clear positive correlation between the
velocity dispersion differences and the bar properties, i.e. galaxies with
bars that present larger bar strength, relative size and light fraction
exhibit larger differences in the three velocity dispersions. Unbarred
galaxies present very small differences as seen from their homogeneous 2D maps
(see top maps in Fig. 1). The average difference of unbarred galaxies is about
10 km/s. This value could be considered a lower limit to assess whether the
velocity dispersion differences of a barred galaxy are significant or not and
therefore, they would be considered significant above this limit. For example,
in the left panels, we find that the SVE differences of barred galaxies
without B/P bulge and with $f_{bar}$ around to 0.4 are close to this limit. In
other words, the trend would not extend to lower values if we study galaxies
with bar strength below 0.4.

These differences present on average, significant and strong correlations with
the bar parameters, showing an average Spearman coefficient around 0.7 and
p-value below 0.01% for the three velocity dispersions. We notice that the
range of velocity dispersion differences is not the same for the three
components being $\sigma_{\phi}$ the component that presents the largest
values with a maximum of 103 km/s while $\sigma_{z}$ and $\sigma_{r}$ maximum
values are around 80 and 75 km/s, respectively.

B/P galaxies exhibit intermediate differences that closely follow the general
trend of the barred sample although there are two outliers in the left and
middle panels. The first one is AU-26, which is the only system that presents
larger differences than what would be appropriate for a galaxy with $f_{bar}$
of 0.5. This discrepancy is likely caused by a complex bar formation with
significant changes in the $A_{2}$ parameter between 4 and 2 Gyrs ago, and the
subsequent buckling instabilities which are present until present time. AU-26
is undergoing a secondary bucking phase (as seen by (Fragkoudi et al., 2020),
and Athanassoula (2005) and Martinez-Valpuesta et al. (2006) in other
simulations). The second outlier is AU-22 a galaxy with a weak B/P that
exhibits lower velocity dispersions than the general trend of barred galaxies
in the $\Delta_{\mu,max}\sigma$-$L_{bar}/R_{25}$ panels. Remarkably, both
galaxies follow the general trend if we use the Bar/T parameter to
characterize the bar.

The 2D maps show that the most evident imprint of bars in the SVE spatial
distribution are low velocity dispersion regions along the bar major axis.
Therefore, it could be argued whether the trends seen in Fig. 3 between the
velocity dispersions and bar morphology proxies are genuine or rather, a
connection between the velocity dispersion along the bar major axis ultimately
drives these results. We repeated our analysis, calculating the average
velocity dispersion in the slits along the bar major and minor axes, and
studied their correlation with the strength of the bar. The results, (not
shown), revealed only weak correlations with average 111To compute these
averages we consider the correlation coefficients and p-value of each
direction, i.e. along the bar major and minor axes, and the three velocity
dispersions,$\sigma_{z}$, $\sigma_{r}$ and $\sigma_{\phi}$ Spearman
correlation coefficient of 0.2 and an average p-value of 40%, thus, the null
hypothesis cannot be discarded. These results pose a scenario where the
morphology of the bar determines the extent of the velocity dispersion
differences but does not affect the average value along the major and minor
axes. We have further explored if these differences depend on other global
properties of the galaxy, such as galaxy size or halo mass but the results
(not shown) revealed that bar related parameters are the only ones that are
correlated with the SVE differences.

Notwithstanding, we note that the parameters we use to characterize the
morphology of the bar is an additional source of uncertainty. For example, in
Fig.3 we use bar strengths measurements based on the ellipticity of the bar
from Blázquez-Calero et al. (2020) which is only one of the multiple
possibilities to characterize it. Alternatives would have been the use of
radial to tangential forces, m=2 Fourier amplitudes or 2D kinematics.
Fortunately, it has been shown that all these bar strength proxies are well
correlated and we do not expect significant differences by choosing one over
the others (Seidel et al., 2015; Díaz-García et al., 2016).

We have further explored if there is any connection between the $\sigma_{z}$
categories from Sec. 3.2 and the position in these panels. Fig.4 shows the
same results colour-coding each galaxy by the category they belong to (red,
green and blue for type-A, B and C, respectively). The left panels show that
type-A galaxies have the lowest values of bar strength and velocity dispersion
isophotal difference. Galaxies in the other two categories show larger values
in both quantities and are distributed in a very similar way. Conversely, the
middle and right panels show a more interesting picture where the three
categories occupy different regions. In particular, we find that there are two
values that divides type-C galaxies from the rest of the barred sample at
$L_{bar}/R_{25}$ equal to 0.3 and Bar/T equal to 0.1. The type-A and type-B
are reasonably divided in terms of their velocity dispersion differences.
These results are particularly important because the classification was solely
based on the isophotal profiles of the vertical velocity dispersion and did
not take into account any morphological input.

The SVE axial ratio maps presented a much homogeneous picture than the three
velocity dispersions maps with more oblate ellipsoids in the bar region than
its surroundings. In Fig. 5 we plot the average value of
$\sigma_{z}/\sigma_{r}$ (top) and $\sigma_{\phi}/\sigma_{r}$ (bottom) along
the bar minor axis ($\perp$) as a function of the average value along major
axis ($\parallel$). Blue squares represent unbarred galaxies, barred galaxies
are coloured according to the strength of the bar and the black dashed lines
indicate the 1 to 1 correlation. As expected from the 2D maps, unbarred
galaxies fall in the 1 to 1 correlation while the majority of barred galaxies
lie above it. Following the previous results, most of the galaxies with low
bar strength values present similar values to unbarred galaxies with almost
the same SVE ratios along the two perpendicular directions. These galaxies
belong to the type-A category. Surprisingly, there is no apparent correlation
with the strength of the bar and the location of barred galaxies in these
diagrams where they describe a well defined parallel sequence to the 1 to 1
correlation. Other bar morphology proxies and the type-B/C classification
provide similar results. The average difference of the SVE axial ratios of
barred galaxies between the minor and major axes is 0.08.

In summary, our results show that bars do not affect the global properties of
the SVE (velocity dispersions and axial ratios) but they determine the
differences measured between the regions along the bar major and minor axes.

##  5 SVE time evolution

The results of the previous section showed a clear connection between the
velocity dispersion differences and the bar properties at z=0. We now aim to
understand what is the physical origin of these differences and how the
formation and evolution of the bar affects the SVE. To that end, we study how
the SVE evolve with time from z=2 to z=0, applying the same methodology
described in Sec.2 at different snapshots. In particular, we fixed the z=0
region of interest to study the SVE differences at higher redshifts. The
average time step between snapshots is 0.16 Gyrs.

Fig. 6 shows from top to bottom the temporal evolution of the maximum
isophotal difference of the vertical, $\sigma_{z}$, radial, $\sigma_{r}$ and
azimuthal, $\sigma_{\phi}$, velocity dispersions, and the amplitude of the m=2
Fourier mode, $A_{2}$, as a function of the look back time. From left to right
we find the results of AU-12, (type-B), AU-2, (type-B), and AU-23, (type-C
with B/P). Vertical dashed lines indicate the time when we consider that the
bar becomes strong i.e. $A_{2}$ > 0.3 and there is a clear bar in the surface
brightness maps. This time does not always coincide with the bar formation
since some bars have initial lower $A_{2}$ values and grow stronger over time
(e.g. AU-12 where the bar formation began 5 Gyrs ago and it has steadily
increased its strength) while others present large values since they are
formed. Short-lived high $A_{2}$ values at high look back times are associated
with mergers and satellite interactions.

The bottom left panel shows that the bar strength of AU-12 has steadily grown
with time since the bar started to form 5 Gyrs ago and became strong 0.7 Gyrs
ago. Interestingly, the upper panels show that the velocity dispersion
isophotal differences follow a similar trend, increasing since the bar
formation until present time. The vertical and azimuthal components present a
time evolution that closely follows the evolution of $A_{2}$ while the the
radial velocity dispersion shows a much weaker evolution.

Middle and right panels present a different formation scenario for AU-2 and
AU-23 where both galaxies form a strong bar in a short period of time that
causes an equally fast increase of the three velocity dispersion differences.
Surprisingly, while there are fluctuations in small time scales we find that
the differences of both galaxies present similar values in long periods of
time, although a small decreasing trend can be appreciated mainly in the
$\sigma_{\phi}$ component. We also find that the radial velocity dispersions
present the lowest differences at all times. We notice that the bar of AU-23
experiences a buckling phase after the bar forms that causes the formation of
a B/P bulge 2.5 Gyrs ago but no clear signatures of this process can be seen
in the evolution of the velocity dispersion isophotal differences. In some
situations low level differences can be associate with the formation of the BP
bulge but the changes are similar to those experienced by galaxies without BP.

The rest of galaxies in our sample follow similar trends as these three
examples. In galaxies where the bar experiences a more complex formation (e.g.
substantial changes in bar strength during the formation or buckling of the
bar) we find that the velocity dispersion differences tightly follow the
$A_{2}$ evolution. In terms of the $\sigma_{z}$ classification, type-A
galaxies present the weakest evolutions of the velocity dispersion isophotal
differences and bar strength while type-B and type-C galaxies present similar
behaviours. Lastly, the isophotal differences of the unbarred galaxies sample
exhibit low-value flat curves with randomly distributed high values associated
with mergers.

These plots confirm that the z=0 correlation between the bar properties and
the velocity dispersion differences is caused by the bar since its formation
induces the differences in the velocity dispersion. Further, they reveal that
these differences remain fairly constant since the formation of the bar.
Surprisingly, our results indicate that there is no apparent connection
between the velocity dispersion differences and the B/P formation.

##  6 Discussion

Previous sections show that bars affect the kinematics of stars in the inner
regions of galaxies inducing non axisymmetric features in the z=0 velocity
dispersion maps that coincide with the bar light distribution. Further, we
find that velocity dispersion differences between the minor and major bar axes
are larger for galaxies with stronger bars. We now explore whether there is an
observational counterpart to our findings and if other simulations provide
similar results.

Thanks to the development of Integral Field Units we are able to fully study
the 2D kinematics of galaxies. From the first works from the SAURON survey
(Cappellari et al., 2007) to the latest results from ManGA (Graham et al.,
2018) and SAMI (van de Sande et al., 2021a, b) our understanding of the
kinematic properties of galaxies is steadily expanding. However, many of these
works have focused on global properties galaxies and the number of dedicated
works to study in detail the differences caused by the bar in late-type
systems is scarce. Seidel et al. (2015) is one of the first works to study
both the radial and isophotal velocity dispersion profiles and 2D maps of 16
galaxies from the BALROG sample. These galaxies are not face-on and thus it is
difficult to compare the line-of-sight velocity dispersion from BALROG with
intrinsic velocity dispersions from AURIGA. Nevertheless, the profiles
presented in their Appendix C show that many galaxies are compatible with
type-A and type-C classifications although the 2D features are not so clearly
seen in the maps.

The velocity dispersion signatures from type-B galaxies have been observed in
the velocity dispersion maps of inner bars in double barred galaxies. de
Lorenzo-Cáceres et al. (2008); de Lorenzo-Cáceres et al. (2012, 2019) have
shown that these systems present low velocity dispersions along the inner-bar
major axis and larger values with a peak in the perpendicular direction,
features that have been coined as $\sigma$-hollows (de Lorenzo-Cáceres et al.,
2008) and $\sigma$ humps (Du et al., 2016), respectively. de Lorenzo-Cáceres
et al. (2008) proposed that these features were caused by a contrast between
the cold orbits of stars from the inner bar and the high velocity dispersion
from the bulge component. N-body simulations have shown that these features
can also be detected in the presence of B/P bulge formed by the buckling of
the outer bar (Du et al., 2016). On the other hand, by applying Jeans
equations to a bar + disc system Du et al. (2017) have shown that there is no
need for a hot component to obtain a $\sigma$ hollow+hump pattern in the
vertical velocity dispersion since these features appear naturally as a
dynamical response of stars to the presence of a massive vertically thin bar.

Bars in Auriga galaxies extend far beyond the bulge effective radius and the
contribution of the bulge hot orbits to our velocity dispersion features
should be minimal. Nevertheless, we repeated our analysis applying a vertical
cut to stellar particles beyond two bulge effective radius from the plane of
the disc which also discards halo stars. The results (not shown) provide
velocity dispersion maps with the same features and similar correlations
between the minor to major velocity dispersion differences and bar morphology
proxies, thus discarding this possibility.

From the point of view of simulations, the number of works exploring the
kinematics of galaxies in a cosmological environment have increased in the
last years. Nevertheless, they have focused mostly on global kinematic
properties and do not explore in detail the questions proposed here (e.g.
Lagos et al., 2017, 2018; Pillepich et al., 2018; Schulze et al., 2018; Lagos
et al., 2020; Walo-Martín et al., 2020; Frankel et al., 2022). Interestingly,
Pillepich et al. (2018) presented the analysis of the kinematics of galaxies
from IllustrisTNG50 (a high resolution cosmological volume simulation) and
showed that the velocity dispersion maps of face-on disc galaxies (see Fig. 4
in their work) present similar low velocity dispersion features along the bar
major axis to those presented in this work. This opens the possibility to
study the role of the bar in the SVE in a different environments with an
independent and larger dataset.

Our results, motivate more theoretical studies aimed to understand how the
kinematics and morphology of bars coevolve to form the $\sigma$-hollows at z=0
and how they are connected to the evolution of the rest of the components. In
addition, our search of the literature indicate that more integral-field
observations of low inclination single barred galaxies are required to
understand how common these features are in our Universe. Furthermore, these
works would provide observational constrains that will allow to determine
wether the intrinsic kinematic properties of simulated galaxies are well
reproduced by currently used models.

We have shown that the velocity dispersion differences of the three components
are well stablished since the formation of the bar but is not clear how do
different velocity dispersion distributions are formed. To answer these
questions we could explore if there is any signature at z=0 that could help
distinguish the formation mechanism and evolution of the bar by studying the
SVE of different stellar populations. Preliminary results show that on average
young(old) stellar populations have colder(hotter) kinematics in agreement
with a secular evolution scenario. However, we did not find any general
behaviour in the velocity dispersion maps. Young and old populations present
complex spatial variations that deviate from the patterns obtained for all the
particles to a greater or lesser extent for each galaxy depending on the
individual dynamical evolution and the star formation history in the bar
region. To fully characterize the SVE for different populations in the inner
regions exceeds the scope of this paper and we will focus on this topic in
future works.

##  7 Conclusions

We have analyzed a set of 27 high resolution cosmological zoom-in simulations
of Milky-Way mass late-type galaxies from the Auriga sample to study the
spatial distribution of the Stellar Velocity Ellipsoid (SVE). In particular,
we studied the impact of the bar on the SVE properties by analyzing the 2D
maps of the three velocity dispersions and their ratios (e.g. $\sigma_{z}$ ,
$\sigma_{r}$, $\sigma_{\phi}$, $\sigma_{z}$/$\sigma_{r}$ and
$\sigma_{\phi}$/$\sigma_{r}$.)

The 2D velocity dispersion maps show a variety of features in the different
components that depend on the presence of structures such as bars or B/P
bulges. The most common feature is a low velocity dispersion region along the
bar in the $\sigma_{z}$ and $\sigma_{\phi}$ maps which is more evident in
galaxies with strong bars. Conversely, SVE axial ratio maps are very similar
with no strong dependence on the bar properties.

A more in depth analysis of the isophotal profiles exhibit different
behaviours along the bar major and minor axes. We classified galaxies into
three categories attending to the vertical velocity dispersion profiles which
could be confirmed observationally. In particular, galaxies can exhibit
similar profiles along the two direction with almost no differences (type-A),
smooth decreasing profile along the bar and increasing profile with a peak in
the perpendicular direction (type-B) and a multi component decreasing profile
along the bar major axis with a minimum close to the bar ends (type-C).

The minor-major velocity dispersion differences obtained from the isophotal
profiles are well correlated with structural properties for the three SVE
components. In other words, the velocity dispersions differences are more
significant in barred galaxies with larger bar strength, relative size and
light fraction. On the contrary, the correlation between the velocity
dispersions and the bar properties is significantly weaker.In terms of the SVE
axial ratios, the bar regions present more oblate ellipsoids but there is no
apparent connection with the bar properties.

The time evolution of the SVE and the bar strength confirm that the velocity
dispersion differences seen at z=0 are caused by the bar formation. In bars
that slowly grow over time the velocity dispersion differences follow an
equally steady evolution, while strong bars that are formed in short time
intervals present a rapid increase in the differences that remain very stable
over large periods of time.

Future works using cosmological volume simulations such as IlustrisTNG50
(Pillepich et al., 2018) will help us understand how do these findings adapt
to galaxies in different environments and with different masses.

## Acknowledgements

D. W-M, F. P and J. F-B acknowledge support through the RAVET project by the
grant PID2019-107427GB-C32 from the Spanish Ministry of Science, Innovation
and Universities (MCIU), and through the IAC project TRACES which is partially
supported through the state budget and the regional budget of the Consejería
de Economía, Industria, Comercio y Conocimiento of the Canary Islands
Autonomous Community. RG acknowledges financial support from the Spanish
Ministry of Science and Innovation (MICINN) through the Spanish State Research
Agency, under the Severo Ochoa Program 2020-2023 (CEX2019-000920-S)

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A 2D SVE maps

Figure 7: Continuation of Fig.1 for the rest of Auriga galaxies. Figure 8:
Continuation of Fig.1 for the rest of Auriga galaxies. Figure 9: Continuation
of Fig.1 for the rest of Auriga galaxies.

##  Appendix B SVE isophotal and radial differences values

Table 2: Maximum radial and isophotal differences of Auriga galaxies at z=0. The columns are:(1) Galaxy name, (2)-(4) Maximum radial difference of the vertical, radial and azimuthal velocity dispersion, (5)-(7) Maximum isophotal difference of the vertical, radial and azimuthal velocity dispersion $Name$ |  $\Delta_{r,max}\sigma_{z}$ [km/s] |  $\Delta_{r,max}\sigma_{r}$ [km/s] |  $\Delta_{r,max}\sigma_{\phi}$ [km/s] |  $\Delta_{\mu,max}\sigma_{z}$ [km/s] |  $\Delta_{\mu,max}\sigma_{r}$ [km/s] |  $\Delta_{\mu,max}\sigma_{\phi}$ [km/s]
---|---|---|---|---|---|---
AU1 | 19.2 | 14.7 | 19.9 | 14.8 | 11.0 | 20.6
AU-2 | 13.7 | 13.8 | 26.2 | 28.9 | 22.7 | 36.7
AU-3 | 6.4 | 11.0 | 12.2 | 10.6 | 11.6 | 12.8
AU-4 | 8.6 | 12.8 | 11.6 | 5.4 | 9.8 | 6.5
AU-5 | 26.8 | 12.7 | 58.9 | 42.8 | 26.5 | 64.1
AU-6 | 8.9 | 5.7 | 19.5 | 12.0 | 14.0 | 22.9
AU-7 | 17.2 | 10.8 | 27.3 | 18.6 | 15.1 | 31.0
AU-8 | 5.1 | 13.5 | 9.3 | 4.9 | 8.2 | 6.2
AU-9 | 38.1 | 14.1 | 63.2 | 68.1 | 51.6 | 80.1
AU-10 | 36.9 | 24.9 | 45.8 | 46.9 | 40.6 | 66.6
AU-12 | 25.9 | 12.0 | 44.0 | 28.0 | 20.7 | 39.1
AU-13∗ | 24.5 | 25.1 | 42.4 | 50.4 | 37.6 | 58.4
AU-14 | 28.0 | 14.2 | 45.7 | 46.0 | 36..3 | 59.0
AU-15 | 3.8 | 4.3 | 4.0 | 1.2 | 1.5 | 1.4
AU-16 | 7.0 | 11.2 | 5.7 | 5.9 | 7.5 | 6.5
AU-17∗ | 19.3 | 19.4 | 18.4 | 50.6 | 39.9 | 59.5
AU-18∗ | 8.3 | 14.7 | 18.2 | 33.3 | 22.9 | 30.7
AU-19 | 6.3 | 8.9 | 11.0 | 3.3 | 2.3 | 7.4
AU-20 | 19.8 | 11.7 | 30.2 | 25.4 | 23.7 | 37.2
AU-21 | 9.8 | 6.5 | 14.2 | 9.0 | 8.7 | 13.8
AU-22∗ | 26.7 | 29.3 | 30.1 | 43.2 | 34.7 | 54.0
AU-23∗ | 16.3 | 18.9 | 35.1 | 50.0 | 36.0 | 52.8
AU-24 | 33.6 | 12.8 | 50.5 | 36.7 | 31.6 | 44.6
AU-25 | 11.1 | 14.4 | 15.9 | 7.7 | 8.8 | 12.9
AU-26∗ | 24.4 | 17.1 | 64.8 | 62.3 | 49.5 | 78.0
AU-27 | 31.2 | 9.4 | 53.5 | 51.1 | 39.0 | 64.2
AU-28 | 31.6 | 38.3 | 59.0 | 80.4 | 75.7 | 103.7