result based on CIGALE is shown on the top row and that based on the F160W band is shown on the bottom row. Note that the same disk inclination of 20° has been used for both models even though the best-fit GALFIT model on the F160W band found a larger inclination of roughly 36°. This way, we can compare more easily the impact of different bulge and disk parameters provided by the resolved stellar mass map independently of variations in inclination that will also affect the amplitude of the rotation curve. Besides, using an inclination of 36° has nearly no impact of the projected velocity field and residuals but it does affect the amplitude of the DM halo rotation curve by lowering it and therefore reducing the DM fraction. The two velocity field models shown in Fig. 8.7 and their residuals are very similar with differences of the order of a few kilometres per second, even though the contribution of the various components are different. In particular, we can note the different contribution of the bulge in the inner parts. In the case of CIGALE, the bulge is less massive than in the F160W map but peaks significantly closer to the centre, thus producing a fast rise of the rotation curve in the inner parts. Nevertheless, the total rotation curves remain close between the two models which, combined with the smoothing effect of beam smearing, produce similar velocity field models.

<figure><img src="image_81.png" /></figure>

Figure 8.7: Comparison of the best-fit MOCKING velocity field models for galaxy ID1 when using CIGALE’s stellar mass map and when using the F160W HST band. The velocity field is shown on the left-hand side and the best-fit model and corresponding residuals are represented when using the disk and bulge morphological parameters from the stellar mass map derived with CIGALE to constrain the ionised gas kinematics (top row) and when using those derived from the morphological modelling on the F160W HST band (bottom row). On the right-hand side is also shown the intrinsic rotation curve when using the morphological parameters derived from the CIGALE map (solid lines) and those derived from the F160W band (dashed lines). The total rotation curve is shown as an orange line and the bulge, disk, and DM halo components are represented in red, blue, and black, respectively. The vertical lines represent $R_{22}$, that is the radius where the maximum of the disk rotation curve is reached.
The two DM halo rotation curves are mostly similar below R<sub>22</sub> which is the position where the maximum rotation velocity of the disk is reached and where the DM fraction is typically estimated. At this radius, the model that uses the parameters from CIGALE finds a DM fraction of f<sub>DM</sub> ≈ 65% whereas the model using the parameters from the F160W band finds f<sub>DM</sub> ≈ 75% instead. Such a difference is nevertheless only mildly significant. Indeed, we can derive a rough estimate of the uncertainty of the DM fraction by loosening the bounds on the DM parameters during the kinematics modelling and see how that changes the value of f<sub>DM</sub>. When doing so using the best-fit morphological parameters derived from CIGALE’s stellar mass map, we find a new value f<sub>DM</sub> ≈ 73%. This gives an uncertainty on the DM fraction of the order of 10% which is close to the difference observed between the two best-fit models previously discussed. Furthermore, it is important to stress that this galaxy is not an optimal example to check the impact of stellar mass maps on the gas kinematics since it is nearly face-on. In practice, this means that a small change in the stellar disk inclination can have a large impact on the best-fit parameters.

## 8.2 Machine learning application with Self Organising Maps

At the beginning of the project, we had started thinking about improving the production of spatially resolved maps of physical parameters by implementing machine learning techniques. After discussing with collaborators, we converged towards an algorithm that is sufficiently simple to handle while remaining powerful enough to produce such maps by preserving the topology of the input data used for training (i.e. close input data such as fluxes or magnitudes should be mapped to close output data, i.e. stellar masses or SFR values): self-organising maps (SOMs). This work was started after the end of the first phase of the project, that is after H. Plombat had successfully produced our first results on resolved stellar mass maps during his internship. Since implementing such an algorithm requires a substantial amount of work, we supervised a second Master student to help us in this task: M. Tarrasse. The machine learning algorithm was developed by using the resolved SED library discussed in Sect. 8.1.3 as back-end with LEPHARE first, and then with CIGALE. In what follows, I will only discuss the current implementation of the SOM trained on input data from CIGALE.

Though the method has technically already produced positive results, it still remains currently in development. The test phase, (but not the training phase, see Sect. 8.2.5), were only carried out on a single galaxy (ID1) and thus are not to be taken as definite proof but rather as an indication of how the algorithm performs. We still need to assess its performance on a larger sample of galaxies, in particular in terms of speed-up compared to the classical framework discussed in Sect. 8.1, as well as better characterise its performance when dealing with noisy data.

### 8.2.1 A general overview of Self Organising Maps

A SOM is an unsupervised and competitive machine learning algorithm that can be seen as an early neural network with a single layer of neurons. It was invented by Teuvo Kohonen and described for the first time in Kohonen (1982). Its design and architecture was highly influenced by the development of biologically-influenced artificial neural networks of the 1970s and 1980s. The key feature that rendered SOMs different from contemporary early neural networks such as Perceptrons (Mcculloch & Pitts 1943; Rosenblatt 1957) was that they were designed to preserve
the topology of the input space. As previously mentioned, the SOM is an unsupervised algorithm, meaning that it learns from the input space without having to provide labelled data. In other words, the algorithm learns the structure of the data by itself during the training phase and does not rely on a “ground-truth” (labels) that it could compare to in order to improve its performance. Additionally, the SOM is also a competitive learning algorithm, meaning that, at each step during the training phase, neurons will be in competition with each other and one of them will be selected as the best (see below for more details). In essence, the SOM can be seen as a clustering algorithm that will map input data to their closest neuron. Thus, if two data points in an *N*-dimensional input space are close to each other, so will they be on the neuron layer, hence topology is preserved. Because of that feature, a very common use of SOMs is to perform a reduction of dimensionality, mapping a *N*-dimensional input space into a simpler two-dimensional output space, similarly to what could be done with principal component analysis (PCA). This is typically done to visualise the clustering of data in datasets with a large number of dimensions. Another use, which is what we are interested in here, is classification. In this case, the idea is to apply a *posteriori* label to each neuron after the training phase is completed. Afterwards, when a new data is presented to the SOM, the label corresponding to the closest neuron will be attributed to it. This application is the most common in the extragalactic literature. Examples would be to perform source (stars versus galaxies, Miller & Coe 1996) or morphological classification (Naim et al. 1997), to use multi-band photometric observations to derive photometric redshifts (e.g. Geach 2012; Way & Klose 2012; Carrasco Kind & Brunner 2014; Masters et al. 2015; Speagle & Eisenstein 2017; Davidzon et al. 2019), to infer spectroscopic properties (Jafariyazani et al. 2022), or to derive global physical properties (e.g. Hemmati et al. 2019; Davidzon et al. 2019). In our case, we extend the latter use to derive resolved maps of physical properties. In particular, I will solely focus on producing resolved stellar mass maps using SOMs since we have not tackled yet the production of SFR maps.

### 8.2.1.1 Key concepts for the SOM

A SOM contains a single layer of neurons (also called units or weights) regularly spaced onto a 2D surface. A classical layout is to consider that the neurons are laid on a flat Euclidean surface so that those located on opposite sides are far away from each other, though other types of surfaces can be used (e.g. cylindrical). The Euclidean surface is the most common and it is therefore the one that we have worked with. Let us refer to the layer of neurons (weights) as $w$. Then, the location of each neuron on the layer can be characterised by two numbers $(i, j)$ and the neuron is noted $\vec{w}_{i,j}$. The coordinates $(i, j)$ do not correspond to “physical” dimensions or dimensions in input space (e.g. fluxes, magnitudes, etc.) but to the location of the neuron on the neuron layer/map. The dimensions of the layer are ultimately free (hyper-)parameters that must be optimised (see the discussion on hyper-parameters below). Nevertheless, as a rule of thumb, we can keep in mind the few following points: (i) it is useless to have more neurons than input data because the “optimal” coverage of input space that the SOM can do is to map a single neuron to each data point, (ii) fewer neurons means faster convergence and learning phase but less precision on the inferred physical properties, and (iii) beyond a certain point, too many neurons means lower convergence without improvement in learning (i.e. we reach the irreducible error). As an indication, a $1 \times 3$ neurons' layer is shown on the top of Fig. 8.8. This dimension therefore characterises the proximity of neurons on the layer. Furthermore, each neuron acts as a vector in input space. Thus, if the input data is *N*-dimensional then so are the neurons, that is we have
$\vec{w}_{i,j} = (w_{i,j,0}, \dots, w_{i,j,N})$. In our case, the input space will typically be the pixels' fluxes or magnitudes (hence $N = 9$ bands) or colours ($N = 8$ for colours between nine successive bands), but in practice any type of data can be used (e.g. redshift, spectral features, etc.). Because neurons are located in the same space as the input data, it is possible to compute a distance between a data point and a neuron. For a given data point, the closest neuron in input space is called the best-matching unit (BMU).

As described in more details in Sect. 8.2.1.2, the goal of the SOM is to iterate through the input data one by one, find the selected data's BMU, and move the latter (in input space) towards the given data point. The BMU will be displaced in the sense that its coordinates in input space will be modified, but its location on the neurons' layer will always remain the same. To preserve the topology of the input data, not only will the data point attract its BMU towards it but it will also attract every other neuron with an amplitude that is lower for those located far away from the BMU. Let us consider the same example as in Fig. 8.8 and let us assume that the BMU is the neuron located at the position $(0, 1)$, i.e. the middle one. Then, the amplitude of the displacement will be the same for each of the two neurons on both sides since they are equidistant from the BMU (as quantified by the $(i, j)$ coordinates). Instead, if the BMU was the neuron located at position $(0, 2)$, then the neuron at position $(0, 1)$ would be more affected than the one at position $(0, 0)$ because it would be closer to the BMU on the neurons' layer. This mechanism is what allows the preservation of the topology of the input data. In practice, the function that characterises the amplitude of the displacement of a neuron is called the neighbourhood function and it is noted $\beta_{i,j}$. A typical choice that we used in our implementation is a Gaussian neighbourhood that writes

$$ \beta_{i,j} = \exp\left\{-\frac{d^2}{\sigma^2}\right\}, \qquad (8.9) $$

where $d$ is the distance between the neuron and the BMU on the neurons' layer and $\sigma$ is a free hyper-parameter called the neighbourhood radius. The distance between neurons ultimately depends upon the choice of the geometry used for the neurons' layer. In our case, we used a Euclidean surface so $d$ is simply the Euclidean distance between two neurons on the layer, that is $d = \sqrt{(i - i')^2 + (j - j')^2}$ for two neurons at positions $(i, j)$ and $(i', j')$. The neighbourhood radius characterises the strength of the neighbourhood sphere of influence. A large radius means distant neurons will be more affected than with a small radius. If $\sigma = 0$, then there is no neighbourhood and if $\sigma \to \infty$, then the same displacement is applied to all the neurons. Based on what has been said, it would not be certain that the SOM would converge to a final position after a finite number of iterations since each data point would attract the neurons towards itself, indefinitely. To counteract this effect, we must reduce the amplitude of the displacement at each iteration so that it will sufficiently converge to stop the learning phase. The function that reduces the amplitude with each iteration is called the learning rate and it is noted $\alpha$. In our case, for a maximum number of iterations $\tau$, the learning rate will linearly decrease with each iteration $t$. In other words, we have $\alpha(t) = 1 - t/\tau$ (e.g. see the evolution of the amplitude of the arrows in Fig. 8.8). Other implementations exist (e.g. exponential decline) but the linear decrease is quite common.
<figure><img src="image_82.png" /><figcaption>Figure 8.8: Illustration of the learning phase for a 1 × 3 SOM ($\sigma = 0.7$) using synthetic data. The 2-dimensional data are represented with grey points and the SOM's neurons with large symbols. For each iteration $t$, a data point is randomly chosen (highlighted in red) and the BMU is found (in red as well). Then, all the neurons are moved to their new location represented by their arrow. In this example, after nine iterations the SOM is stopped ($t = 8$). The last subplot represents which neuron would be attributed as BMU to a new data point as a function of the position in input space. At $t = 0$, the first (circle) and last (diamond) neurons do not have a large impact on each other because they are located the furthest apart on the neuron's layer (see layout on the top). A different layout and $\sigma$ would yield different results.</figcaption></figure>
### 8.2.1.2 Learning phase

Now, we turn towards the learning phase. Let us consider a $N$-dimensional input space (e.g. galaxy magnitudes or colours) with $K$ input data $\vec{x}_k$. These form a matrix $X = (\vec{x}_0, \dots, \vec{x}_K)$ where each vector $\vec{x}_k = (x_{k,0}, \dots, x_{k,N})$ corresponds to a single data (in our case a pixel). The first step before the learning phase begins is to initialise the neurons' values in input space. To avoid that one input feature<fnref n="11" /> dominates over the others during the training phase, for instance because of the range of values the data take, the input data $X$ must be normalised first as follows

$$ \tilde{X} = \frac{X - \langle X \rangle_{\vec{x}}}{\sqrt{\mathrm{Var}(X)_{\vec{x}}}}, \qquad (8.10) $$

where $\tilde{X} = (\tilde{x}_0, \dots, \tilde{x}_K)$ is the normalised data, $\langle X \rangle_{\vec{x}}$ is the mean vector taken as the average of each feature (e.g. average of fluxes in a single band), and $\sqrt{\mathrm{Var}(X)_{\vec{x}}}$ is the standard deviation taken as the standard deviation of each feature. This kind of normalisation is typical in machine learning because it allows each feature to have a null mean and a standard deviation of unity. Once the data have been normalised, the neurons' initial coordinates in input space can simply be drawn randomly from the uniform distribution $U[-1, 1]$. Because this first step is non-deterministic, each new run of the SOM will initialise the neurons to different coordinates. Thus, this will have an impact on the learning phase and on the outcome. This effect can be used to estimate the uncertainty induced by the algorithm itself by running multiple times the same SOM on the same data. To perform the learning phase, the SOM will first select a single input data $\tilde{x}_k$ and then it will compute the Euclidean distance in input space between the data point and each neuron $\vec{w}_{i,j}$ as

$$ \|\tilde{x}_k - \vec{w}_{i,j}(t)\| = \sqrt{\sum_n [\tilde{x}_{k,n} - \vec{w}_{i,j,n}(t)]^2}, \qquad (8.11) $$

where $\tilde{x}_k$ is the normalised input vector and $\vec{w}_{i,j}(t)$ is the neuron's coordinates in input space at step $t$. Afterwards, the SOM will find the BMU (closest neuron, i.e. smallest distance), it will determine the distance $d$ on the neurons' layer between the BMU and all the remaining neurons, it will compute the neighbourhood function, and it will update the learning rate. Finally, it will update the position of all the neurons using the formula

$$ \vec{w}_{i,j}(t+1) = \vec{w}_{i,j}(t) + \alpha(t)\beta_{i,j}(t) [\tilde{x}_k - \vec{w}_{i,j}(t)], \qquad (8.12) $$

where $\vec{w}_{i,j}(t+1)$ is the new position of the neuron in input space that will be used in the next iteration. Once all the neurons updated, the SOM will start the next iteration by selecting a new data and performing the same steps again. If the data points are selected randomly, the common rule is that it is done with replacement (i.e. a pixel can eventually be selected multiple times). An illustration of the learning process for a $1 \times 3$ SOM with nine synthetic input data is shown in Fig. 8.8. Each step $t$ is represented from left to right and top to bottom (from $t=0$ to $t=8$). For each step, the selected data point and the BMU are both highlighted in red and the displacement of the different neurons are shown with arrows. The last subplot (bottom right) shows the clustering prediction of the SOM for the entire input space, with each class represented with a different colour.

<footnote n="11">¹¹In machine learning jargon a feature is a single dimension in a multi-dimensional input space.</footnote>
### 8.2.2 Using the SOM for predictions

The SOM is a clustering algorithm so, once the learning phase is done, it can be used on new input data to find their BMU. Each neuron will therefore act as a potential cluster onto which new data will be mapped. Note that the mapping to a BMU after training is independent of whether the given neuron could be matched to any input data during the training phase. This means that it is possible, especially when there are many neurons or a sparsely populated input space with empty regions, that a neuron is identified as the BMU of a new data point during the prediction phase even though it was never the BMU of any of the input data during the training phase. This mapping can be used to predict unknown properties of the new data. For instance, let us assume that we trained a SOM on galaxy colours and that we also know the M/L of the galaxies in the training sample. If one of the neurons is mapped to *n* galaxies in the sample after the training phase, then we can assign to this neuron a M/L value by taking, for instance, the median M/L of the *n* galaxies. Afterwards, if we want to predict the M/L values of a new sample of galaxies for which we only know their colours, we can find for each galaxy in the sample the corresponding BMU and attribute to the galaxy the BMU's M/L value. In machine learning terminology, we say that we use the SOM to predict the M/L of the new galaxies.

### 8.2.3 Optimisation of the SOM

The SOM has a few hyper-parameters that can be tuned to improve its performance for a specific task. Namely, in our implementation these are: (i) the dimensions of its neurons' layer (2D hence two hyper-parameters), (ii) the neighbourhood radius, and (iii) the maximum number of iterations. The dimensions of the SOM are important in two regards. First, the more neurons, the longer the training and the predictions will take. Second, if there are too many neurons on a too sparsely populated input space, a large number of neurons will never match any input data point and will therefore be useless for the prediction step since we will not be able to assign to them any value (e.g. stellar mass to use the same example as in Sect. 8.2.2). Hence, there is a trade off between having many neurons, thus potentially a lot of precision (because few input data will be mapped to each neuron), but with plenty of unusable neurons, or fewer neurons but with less precision when using the SOM to make predictions. Furthermore, note that it is technically possible to use SOMs with a neurons' layer with a single dimension or with more than two. For the former, it is similar to the 2D case where there is a single row or a single column in the layer. Hence, our implementation can naturally reduce to the 1D case if this is the most efficient solution. For SOMs with more than two dimensions, even though it is possible to use them, we have not tried to do so. The only caveat is that the visualisation of the SOM becomes more tedious. The neighbourhood radius is also an important hyper-parameter in the sense that even small changes in its value can have a large impact on the training phase and thus on the output. Finally, the maximum number of iterations is also an important hyper-parameter to tune. Indeed, for very large training samples, it is useless in practice to train on the entire sample since convergence will be achieved sufficiently quickly. Thus, limiting the maximum number of iterations can help in reducing the time taken to perform the training phase. Additionally, there is a risk of overfitting for large numbers of iterations, which means the neurons would become too optimised for the training sample and would not be able to properly generalise to new data.
### 8.2.4 The need for a fast implementation

Various SOM implementations are available in python, some given as stand-alone codes, others as extra-packages that can be used in combination with already existing machine learning frameworks. Codes such as `SUSI`<fnref n="12" /> or `SOMOCLU`<fnref n="13" /> have the advantage of being optimised and are usually quite complete in terms of available features but they cannot be easily combined with common machine learning frameworks. On the other hand, extra-packages such as `SKLEARN-SOM`<fnref n="14" /> are made to be used with these frameworks (in this case `SCIKIT-LEARN`<fnref n="15" />, see Pedregosa et al. 2011; Buitinck et al. 2013). During the development of the project, we decided to use `SKLEARN-SOM` as it seemed to be the simplest solution. However, it became quickly clear that this code was not optimised for large datasets. Thus, we had to either use another code or try to improve its performance. A quick inspection showed that, even though it had been written with `NUMPY`<fnref n="16" /> (Harris et al. 2020), it could be substantially optimised. Hence, after rewriting a significant portion of the code, we produced our own version that pushes its optimisation to its limits. Though the details of the optimisation are not especially important, we can nevertheless mention that it mostly consists in reducing as much as possible the number of loops in the code, maximising the use of the vectorisation capabilities of `NUMPY`, and using smarter computations with access to fewer resources whenever possible. As an indication, for a SOM with a grid of 50 × 50 neurons trained on 14 000 data points, `SKLEARN-SOM` finishes the training in roughly 2 min whereas the same result is achieved in 7 s with the optimised version. This yields a significant speedup of more than 17, which increases with the dimensions of the SOM. With this new version, a training phase that would have taken an hour before only takes four minutes to complete now. This new version of the code, dubbed `SOMPTIMISED`<fnref n="17" />, is not only an optimised version of the previous code. It also incorporates useful additional features that are explained in details and with examples in the online documentation<fnref n="18" />. First, the code now follows a modular design that allows more flexibility regarding the specific implementations that different people may want to use. The core class is labelled `SOM`, the method `SOM.fit` can be called to perform the training phase, and the method `SOM Predict` can be called to perform a prediction on new data. Additional new features that have been implemented are the possibilities to (i) set extra properties (e.g. stellar masses) to the neurons with the `SOM.set` method and to retrieve them later on with the `SOM.get` method and (ii) serialise (i.e. locally save on the computer) the `SOM` object using the `SOM.write` method and retrieve it using the `SOM.read` method. This last feature is particularly useful because it allows to locally store the python object after the training phase and to recover it later, as if the training phase had just been done. Typically, this can be used to archive previous versions that may have different parameters or handle different input data and compare them with a newer SOM version without having to perform the training phase again. Concerning the code design, there are only three modules that the `SOM` object requires: (i) the `LearningStrategy` module that defines the type of learning rate $\alpha$ to use and its parameters. Only two options are available at the moment: `LinearLearningStrategy` for a linearly decreasing function (see Sect. 8.2.1.1) and `ExponentialLearningStrategy` for an exponentially declining function.

<footnote n="12">¹²https://felixriese.github.io/susi/</footnote>

<footnote n="13">¹³https://peterwittek.github.io/somoclu/</footnote>

<footnote n="14">¹⁴https://pypi.org/project/sklearn-som/</footnote>

<footnote n="15">¹⁵https://scikit-learn.org/stable/index.html</footnote>

<footnote n="16">¹⁶https://numpy.org/</footnote>

<footnote n="17">¹⁷https://pypi.org/project/SOMptimised/</footnote>

<footnote n="18">¹⁸https://wilfriedmercier.github.io/SOMptimised/index.html</footnote>
function. (ii) The NeighbourhoodStrategy module that defines the type of neighbourhood radius $\sigma$ to use and its parameters. Two options are available as well: ConstantRadiusStrategy for a constant neighbourhood radius (see Sect. 8.2.1.1) and ExponentialRadiusStrategy for an exponentially declining radius. And (iii) the metric used to compute the distance in input space between a data point and the neurons. Three options are available: euclidianMetric() which corresponds to the classical Euclidean distance in an $N$-dimensional space (discussed in Sect. 8.2.1.2), chi2Metric() that implements a “distance” using the classical $\chi^2$ formula that takes into account the uncertainties on the input data, and chi2CigaleMetric() that implements a “distance” using the $\chi^2$ formula used in CIGALE. As discussed in Sect. 8.2.5, the last metric was included to have a SOM that maps neurons to input data in a similar fashion to how CIGALE fits its SED models to the observations.

### 8.2.5 Current implementation of the SOM

Our goal is to use the SOM to produce resolved stellar mass maps without running any SED fitting code. To do so, the procedure is as follows: (i) construct a training dataset $X = (\vec{x}_0, \dots, \vec{x}_K)$, where each vector $\vec{x}_k$ corresponds to the data of a single pixel (e.g. the fluxes in the nine HST bands), (ii) use it to train the SOM, (iii) attach to each neuron of the SOM the median CIGALE-derived stellar mass of the pixels of the training data set that are mapped to the given neuron, and finally (iv) use this trained SOM to predict the stellar mass of future pixels (e.g. new galaxies). To apply this procedure, we must define two things. First, we have to choose which data to use (e.g. fluxes, magnitudes, colours, etc.) and second, we must choose the values of the hyper-parameters of the SOM.

We tested plenty of SOM variations during the development of the algorithm. Originally, the plan was to have a single SOM that would take the fluxes (or a quantity that holds similar information, e.g. magnitudes) and the redshift of the pixels and use that information to estimate the pixels’ stellar mass values. For various reasons, this design was difficult to implement. For instance, it induces unnecessary uncertainties during the prediction phase because, in some sense, it is as if the SOM approximates the redshift of the pixels even if they are perfectly constrained (e.g. MUSE spectroscopic redshift). An early solution to this problem was to predict the M/L in the reddest band rather than the stellar mass directly since both quantities (mass and luminosity) scale equally with redshift, that is the M/L is redshift independent to first order (besides its intrinsic evolution). To improve the efficiency of the algorithm, we also decided to use colours computed as the difference between the magnitudes in successive bands, hence reducing the dimensionality of the problem by one. The argument behind their use is as follows: (i) let us imagine a galaxy but now it is located twice further away from us. Its M/L will remain the same since the galaxy has not changed, but its flux will be reduced. In practice, the change in flux is not simple to compute. Indeed, there will be a reduction of its amplitude but there will also be a shift of the galaxy’s spectrum (i.e. k-correction). So, depending on the spectrum’s shape, the flux may not necessarily be reduced<fnref n="19" />. Hence, in flux space, the galaxy will not be located at the same position because of the change in distance even though it has the same M/L. When using colours, this is not the case any more because the impact of the distance on the fluxes cancel each other out (modulo the k-correction, see below). (ii) Now, let us imagine two galaxies with the same spectrum and at the same distance from us but one of them has twice the luminosity (in the

<footnote n="19">¹⁹At the time, we had not yet thought about the impact of k-correction so the argument that follows only applied to the amplitude (see discussion below regarding the k-correction)</footnote>
reddest band) of the other. To first order, this implies that its stellar mass will be twice as
massive and thus that they have the same M/L. Hence, the galaxies will not be located at the
same position in flux space even though they share the same M/L. Once again, with colours this
is not the case any more because the same scale factor between the fluxes of the two galaxies will
apply for each band. (iii) A last argument is that it is likely that successive bands will suffer from
a roughly similar extinction, though in practice it does depend on how close the bands are to each
other, so using colours should minimise the impact of extinction as well.

Therefore, the first designs of the SOM that we tested used the pixels' colours as input data,
including or not the redshift information. However, they failed at addressing one important issue
mentioned above: the impact of the k-correction. Because galaxies located at different redshifts
will have their spectrum differently shifted (offset plus dilation), this means we will not probe the
same intrinsic wavelength range. To correct for this effect we would need to apply the k-correction
to recover the intrinsic fluxes but this would require a spectrum model which we do not have.
Incorporating the redshift among the input features and using fluxes instead of colours would not
change the problem because the k-correction would still affect the data. The solution that was
chosen is to build a library of SOMs where each SOM is optimised for a given redshift slice. For
slices with a step $\Delta z = (1+z) \times 1\%$<fnref n="20" /> in the redshift range $0.6 < z < 1.5$, corresponding to the
range of the test sample (see Table. 8.1), this amounts to optimising 46 SOMs. Furthermore, in
order to have a fair comparison, we wanted to reduce as much as possible the systematics between
the results produced by the SOM and those produced by CIGALE. Thus, we included in
SOMPTIMISED the possibility to use the same $\chi^2$ metric as in CIGALE to determine the BMU
among the neurons. The metric can be used for the training and/or for the prediction on new
data. Initially, we directly used the colours of the pixels in the 12 galaxies of the test sample as
training data set but we soon realised that this was not the optimal choice to make. Indeed, the
problem is that the SEDs of the pixels in these galaxies do not represent the variety of SED
models built by CIGALE. Thus, there was a clear risk of overfitting only a subset of CIGALE's
models. However, a clear advantage of using real data for the training phase is that they come
with a variety of S/N values that would force the SOM to adapt to the impact of noise during the
training phase. However, based on early tests, the risk of overfitting was too great and we
therefore decided to use the SED models produced by CIGALE when using its savefluxes mode as
training data set. For each redshift slice we produced a catalogue of 69 000 SED models using the
same grid of parameters as the one used when producing resolved maps with CIGALE. For each
slice we trained the SOM on a grid of hyper-parameters in order to find the combination that gives
the best result. We always considered a SOM with dimensions varying from 22 × 27 to 83 × 82
neurons and we tested a few values of neighbourhood radius between roughly $\sigma = 0.5$ and $\sigma = 3$
before settling to $\sigma = 3$ to speed-up the optimisation phase as it seemed to nearly always converge
towards this value. The dimensions of the SOM were chosen based on the number of available
training data so that there would be enough neurons to precisely map the data but not too many
to avoid having a large number of empty neurons (i.e. neurons that are mapped to no data).

8.2.6 First results and perspectives

The algorithm described in Sect. 8.2.5 has been tested so far on a few galaxies in the test sample
but only a single galaxy has been studied in details: galaxy ID1. Hence, it is difficult to
extrapolate the results without considering a larger sample of galaxies. Besides, our algorithm has

<footnote n="20">²⁰This corresponds to a constant step $\Delta\lambda/\lambda_{\text{obs}}$, where $\lambda_{\text{obs}}$ is the observed wavelength.</footnote>
technically been optimised for 46 different redshift slices but galaxy ID1 is logically located only in a single slice. Furthermore, the training dataset can also have an impact on the performance of the SOM and this will be discussed in more details in the perspectives.

<figure><img src="image_83.png" /><figcaption>Figure 8.9: Example of a resolved stellar mass map, adapted from M. Tarrasse’s Master project, using the SOM described in Sect. 8.2.5 (left-hand side) compared with the result from CIGALE (middle). The relative difference between the SOM and CIGALE is shown on the right-hand side. The algorithm was trained and optimised on the same grid of 69 000 models that CIGALE used to perform the pixel-per-pixel SED fitting. Different dimensions of SOM and values of neighbourhood radius $\sigma$ were tested to find the best optimisation before producing the map. In the end, the best hyper-parameters are a dimension of 74 × 74 neurons and $\sigma = 3$.</figcaption></figure>

As mentioned in the previous section, we trained the SOM on SED models produced by CIGALE. Once the algorithm was optimised, we attributed to each neuron in the SOM the median M/L of the input data that were matched to the neurons. We used the reddest band available to compute the M/L as we thought it would better match the galaxies’ stellar mass distribution. We then used the SOM to predict the M/L of each pixel in galaxy ID1 and then converted it to a stellar mass to reconstruct the galaxy’s stellar mass map. The result of this process is shown in Fig. 8.9. On the left-hand side is represented the stellar mass map produced by the SOM, in the middle is shown the stellar mass map produced by the pixel-per-pixel SED fitting library discussed in Sect. 8.1.3 using CIGALE, and on the right-hand side is shown the relative difference between the two maps (hereafter error) split in three different bins: small error (less than 10%), mild error (between 10% and 50%), and large error (more than 50%). Based on this example, it seems clear that, once optimised, the SOM can reproduce very accurately the results of CIGALE. The error is quite flat which means that the uncertainty induced by the SOM on the logarithm of the stellar mass is roughly constant, hence that it does not fail or perform better in specific parts of the galaxy (e.g. bulge or spiral arms). The relative error between the two maps may seem a bit high but note that an error of 50% only amounts to about 0.2 dex which is quite acceptable. Besides, these correspond to pixel-per-pixel errors but once integrated on the entire surface of the galaxy, the
two stellar masses are completely consistent with each other within 0.05 dex (i.e. less than 10%).

<figure><img src="image_84.png" /></figure>

Figure 8.10: Performance of the SOM measured with the coefficient of determination as a function of S/N, adapted from M. Tarrasse’s Master project. The algorithm was trained on noiseless input data from CIGALE’s SED models but the association of the M/L values for each neuron took the noise into account. The performance of the SOM (red dashed line) is compared with the performance for the traditional pixel-per-pixel SED fitting performed with CIGALE (blue solid line). Both were measured by evaluating the coefficient of determination (the closer to unity the better) on noisy input data. For low S/N values, hence noisy data, the SOM actually reproduces more efficiently the stellar mass maps on average whereas the opposite is true at high S/N.

In the very last phase of the project, we also started investigating the performance of the SOM in the presence of noisy data. Though current results seem particularly promising, it must be stressed that this work is still ongoing. We wondered whether it is better to train and optimise the SOM on noisy data and if so what kind of noise should we consider and in what range of S/N values. The noise can affect the SOM in two different parts: (i) during the training phase, that is when providing noisy fluxes or colours as input data so that the algorithm can learn how to cluster the neurons by taking the impact of the noise into account, and (ii) during the association of the M/L values to the neurons. The two are independent in the sense that, by construction, the SOM is a clustering algorithm. Hence, it is technically possible to train the SOM on noiseless data and afterwards use noisy data to assign a M/L to each neuron in the SOM. This is what we have done so far. The next step that shall be taken in the future is to include the noise in the input data during the training and optimisation phases so that the SOM can learn to cluster on noisy data. Thus, to estimate the performance of the algorithm on noisy data we applied the following steps: (i) we trained and optimised the SOM for each redshift slice in the same way as before, that is without adding noise to the input data, (ii) we assigned to each neuron the median M/L value of the input data that are mapped to this neuron after perturbing the input data set with a Gaussian noise (i.e. the flux in each band and therefore the colours but also the M/L). The noise level was chosen based on the typical S/N values measured in the different HST maps in the test sample, taking into account S/N variations between different bands. We varied the S/N from low values (noise-dominated) to high-values (noiseless) to see how the algorithm performs with different types of data. And (iii) we evaluated its performance on new realisations of noisy input data for different S/N values. We also performed the last step with the pixel-per-pixel SED fitting algorithm using CIGALE to compare the performance of the two techniques. The result of this first analysis is shown in Fig. 8.10 where the performance of the SOM is represented with the red dashed line and that of CIGALE with the blue solid line. In this figure, the performance is measured through the coefficient of determination $R^2$ which can be seen as one minus the ratio of
the residuals over the variance in the input data. Hence, the higher its value, the better the model can reproduce the input data. As is visible, at high S/N values CIGALE performs the best. This result is naturally expected since we are testing its performance on its own (slightly noisy) models. However, at lower S/N values (≤ 50), the SOM becomes more efficient. There are at least two explanations for this effect. First, because the SOM has much less neurons than the number of models used in CIGALE it must necessarily average its prediction which in turn smooths the output stellar mass map (e.g. see Fig. 8.9). Second, the M/L (but not the clustering for the moment) takes into account noisy fluxes or colours which will force its prediction to be closer to the true value. Beside its performance, another interesting quantity is the time it takes to produce the resolved maps. In this regard, the SOM can be quite effective because, if one uses models to train the algorithm, the optimisation can be done before the data are available. We can even imagine creating different type of SOMs, each using different SED models (e.g. IMF, SFH, etc.) and grid of parameters, and saving them locally to re-use them later on. As an indication, once the best hyper-parameters have been found, the training phase (to associate physical parameters to the neurons) is quicker than CIGALE and the prediction phase is nearly instantaneous. But once again, the training does not need to be performed each time the SOM is used since it can be saved at any time in its current state.

The previous discussion is based on quite early results and much still remains to be done. To begin with, we still need to assess whether the current implementation is robust for the 46 different redshift slices and for different types of galaxies. In particular, our results strongly depend on the grid of models that are used for the training phase and it is not yet clear whether the current implementation is well adapted for both star-forming and quiescent galaxies. We also need to assess in a much more quantitative way the impact of noise on the performance of the algorithm. Finally, we should certainly investigate much further the current design of the algorithm, in particular regarding the fact that we need to use redshift slices to solve the k-correction problem. If there is at least one thing that the current implementation has shown is that it is possible to recover resolved stellar mass maps given a redshift slice combined with fluxes or colours. Hence, we technically have all the necessary information to produce such maps using a single SOM layer. The complexity in doing so is the large number of input data that will be required to smoothly sample the input space, especially along the redshift dimension. A possible solution to this problem would be a hybrid approach where we sample the redshift dimension of the input space with a fixed number of steps and then train a single but larger SOM on this new data set by providing fluxes or colours and the redshift of the galaxies.
# Concluding remarks and perspectives

In this manuscript, I have presented the work that I have carried out throughout the three years of this Thesis regarding the impact of the environment on the dynamical properties of galaxies across cosmic time. I have shown how I was able, combining MUSE and HST data, to constrain the morphology and the kinematics of roughly 400 galaxies in the MAGIC survey. In particular, I have highlighted in Chapter 3 how valuable the MAGIC survey is to probe the impact of the environment on galaxy properties given its large homogeneity in the way precise physical, morphological, and dynamical properties are collected for a large sample of galaxies found both in the field and in environments of varying density (galaxy groups, and galaxy clusters). This has allowed me to contribute significantly to two already published papers (Abril-Melgarejo et al. 2021; Mercier et al. 2022) and to two nearly complete analyses, the first on the angular momentum in MAGIC, presented in Chapter 7, and the second on the MAGIC survey discussed in Chapter 3. Thanks to the MAGIC survey design, the powerful capabilities of MUSE, and to the precise morphological and dynamical modellings that I performed (see Chapters 4 and 5), this work has enabled me to put the most precise constraints up to date on the impact of the environment on major scaling relations at intermediate redshift. From these analyses, I have managed to highlight the environmental effect on the morphology and star formation rate of galaxies found in large structures. I have also shown that, at intermediate redshift, the environment does not have a visible impact on the galaxies kinematics (rotational velocity) but that it does seem to affect the angular momentum of the most massive galaxies found in the densest parts of their host structure.

Nevertheless, much still remains to be done. Besides, as discussed in Chapter 8, we have developed throughout the last two years a new methodology that could significantly help us to much better constrain the dynamics of intermediate redshift galaxies with MUSE and, more importantly, of high-redshift galaxies with next-generation IFS instruments such as ERIS, HARMONI, or MOSAIC that will be installed on the VLTs and ELTs. Thus, in what follows, I provide a brief account of the continuity of my research project on the short-, mid-, and long-term, in this specific order.

## Completing the analysis of the angular momentum

The shortest-term perspective that I can discuss is regarding the analysis of the angular momentum in MAGIC. As already mentioned, the analysis presented in Chapter 7 is still
preliminary. Already we can draw some early conclusions regarding the impact of the environment, stellar mass, or bulges, but there is still some work that needs to be done before a definite conclusion can be reached. In particular, one aspect that has not been discussed is the impact of the radius where the angular momentum is measured. Other radii beside $R_{22}$ may also be appropriate to probe the impact of the environment. Thus, it would be interesting to see (i) if I recover the same results when measuring the angular momentum for instance in one effective radius, or when extrapolating to infinity, and (ii) if there is some variation with environment or bulge fraction when computing the cumulative angular momentum (i.e. angular momentum versus radial distance). A second aspect that has not been discussed so far is how the population of small galaxies removed by the selection criteria compares to the Fall relation found at low stellar masses. It would be interesting to investigate whether such galaxies are found on the same relation or whether they are located below (as the Fall relation using the flat model rotation curve seems to indicate at first glance). If so, then it would be valuable to investigate whether this is physical or whether it is related to loose constraints on the galaxies' M/L, disk inclination, disk and/or bulge flux distributions, ionised gas kinematics, etc.

I will pursue these investigations and complete them in the next couple of months.

## Prospects for MAGIC, MUSCATEL, and other MUSE surveys

The MAGIC survey is now complete (Epinat et al., in prep.) and the analysis of the impact of the environment on the scaling relations, in particular the TFR and the Fall relation has already been investigated. However, as discussed in both Chapters 6 and 7, probing the impact of the environment on the dynamics of galaxies with MAGIC is entirely doable but drawing definite conclusions can be difficult for the moment. The main reason is that it seems that the differences observed between low- and high-density environments is mostly visible at high stellar masses where we lack field galaxies. Hence, it would be interesting to complement these two first studies with additional data from other MUSE surveys. The most promising one is obviously MUSE Cosmic Assembly survey Targeting Extragalactic Legacy fields (MUSCATEL) that has already been presented in Sect. 2.2.3 and that I was originally supposed to use during this Thesis. This new survey should be available soon and will provide us with deep MUSE observations with AO for four parallel fields of the HUBBLE Frontier Fields (HFF). These observations are split between four very deep (25 h), 16 deep (5 h) and 36 shallower (around 2 h) MUSE pointings spanning an area of roughly 36 arcmin². Thus, with this large spatial coverage we should expect to find a significant fraction of massive field galaxies as well as a few structures at higher redshift. Hence, this survey will be perfectly complementary to MAGIC. It will greatly extend our current sample and will allow us to probe a wider variety of environment than currently probe with MAGIC alone. Combined with other surveys, either very deep (e.g. MXDF) or shallower but spanning a larger area (e.g. HUDF and MUSE-WIDE), this should allow us to put even stronger constraints on the impact of the environment on the dynamics of galaxies at $z \sim 1$ and we might even be able to disentangle the effect of the environment from the redshift evolution of the galaxy scaling relations discussed throughout this Thesis (size-mass, MS, TFR, and Fall relations).

Beside what has been previously said, MAGIC can also be used for other purposes than probing the effect of the environment on galaxy properties. For instance, a few years back, some evidence has arisen regarding the potential existence of a few DM-poor galaxies at high redshift (e.g. Lang
et al. 2017; Genzel et al. 2017) but other authors have argued in the meantime that this may be
caused by systematics induced by the method used (e.g. Tiley et al. 2019b). However, recently
Mancera Piña et al. (2022) has found an ultra-diffuse galaxy in the local Universe whose baryonic
mass distribution (i.e. stars plus gas) is sufficient to recover the observed kinematics. They do
argue that the inclination they measure is a large source of uncertainty but that, even when
taking that into account, the galaxy’s kinematics is not compatible neither with the existence of a
cold dark matter halo around the galaxy, nor with MOND. Hence, if such a population of
DM-poor galaxies exists, then it might be a challenge to MOND, but certainly also to the current
ΛCDM paradigm of galaxy evolution. If not, then it would be particularly interesting to
understand why some galaxies do appear seemingly DM-poor. MAGIC, potentially combined
with other surveys such as MUSCATEL, can help in that matter because we do actually see in
our sample what we called in Mercier et al. (2022) baryon-dominated (i.e. DM-poor) galaxies.
However, our early investigations have led us to believe that these galaxies are not intrinsically
baryon-dominated but that they appear so because of the propagation of uncertainties in our
mass modelling. Still, I personally think that it would be enlightening to investigate these objects
on a galaxy-per-galaxy basis. Because of the homogeneity of the MAGIC sample, of its high
completeness, and because we have already acquired plenty of information regarding these
galaxies (e.g. properties of their host environment, their location on various scaling relations, their
morphology, etc.), I believe we have all the required information in MAGIC to study these
galaxies in more details. Independently of whether we find that all these objects can be accounted
for by methodological biases or systematic effects, which seems the most likely explanation at the
moment, or whether some galaxies do appear intrinsically DM-poor after all, the conclusion will
remain interesting either way. This is particularly true because, even if the former explanation is
retained (i.e. biases, systematics, etc.), we do expect such objects to be seen at higher redshift in
the intermediate-future with the development of the ELTs and their first-light instruments such
as HARMONI on the ELT and in the near-future with the JWST or ERIS.

Paving the way for next-generation IFS instruments

The methods presented in Chapters 5 and 8 of this Thesis, in particular the part regarding the
production of spatially resolved maps of physical properties, can already be applied onto MUSE
and HST data. For instance, it will be possible to use it with MUSCATEL because there is ultra
deep multi-band HST observations in the parallel fields of the HFF. However, the real objective is
to improve the method so that we are prepared for the next-generation of IFS (e.g. HARMONI,
MOSAIC) and imaging instruments such as the Multi-AO Imaging Camera for Deep
Observations (MICADO) on the future ELTs. HARMONI and MOSAIC will both observe in
the NIR with medium to high spectral resolving power and will therefore target high-redshift
galaxies. The former instrument is really optimised for high spatial resolution observations of
distant galaxies and will therefore observe one object at a time over a relatively narrow FoV. On
the other hand, the latter will have a MOS IFS mode with eight-ten IFUs that shall be deployed
over a large FoV but with a coarser spatial sampling similar to that of MUSE (i.e. about 0.2"
per spaxel). In the meantime, it will be possible to apply and further test our methodology on the
new generation of instruments such as NIRSPEC and MIRI on board of the JWST that both
host an IFS observing mode or ERIS that will be soon available on the VLT. These instruments
will provide us with integral field spectroscopy on a galaxy-per-galaxy basis in the NIR (MIR for
MIRI) with a better spatial resolution and spatial sampling that MUSE currently allows.
A first plan which is still on relatively short term is to test our current methodology on a galaxy as if it was observed with HARMONI. To do so, we will use Fourier-transform spectroscopy observations obtained with the Spectromètre Imageur à Transformée de Fourier pour l'Etude en Long et en Large de raies d'Emission (SITELLE) combined with MEGACAM and Wide-field InfraRed Camera (WIRCAM) images of M95 to simulate how this galaxy would be observed by HARMONI if it lay at high redshift. Then, we will be able to use these mock observations to assess the reliability of our current method (including the pixel-per-pixel SED fitting and the machine learning application discussed in Chapter 8). I note that it may also be doable in a slightly more distant future to perform the same tests but using outputs from simulations rather than high-resolution observations of local galaxies.

Apart from what has already been discussed, there is also the work regarding the implementation of the machine learning algorithm that needs to continue. A lot of efforts have already been put into the SOM, but there is still some work that needs to be done. We need to assess in details the advantages and drawbacks of using such a method over the more traditional pixel-per-pixel SED fitting technique, verify whether there is not a more advantageous design for the algorithm, and finally we need to test the algorithm over a large sample of galaxies, possibly from MUSCATEL. Depending on how many people work on this and how much effort is put into it, this could lead to another publication in the next one or two years. I note that this exercise will be particularly interesting because it will definitely benefit us when large surveys from next-generation instruments will become available in the medium/long-term.

Finally, I will conclude with a last prospect which is regarding the dynamical modelling of high-redshift galaxies. Indeed, with the incoming wealth of 3D spectroscopic data in the NIR, it is likely that there will be a time, perhaps in ten years from now, perhaps even before, when it will become custom to extract spatially resolved kinematics of galaxies using rest-frame optical and UV emission lines at redshifts $z \ge 6$. Actually, with NIRSPEC it is theoretically possible to reach $z \approx 12$ and, with space-based observations in the MIR such as those performed by MIRI, galaxies could be observed with relatively high spatial and spectral resolutions up to $z \approx 30$. Obviously, such observations would be limited not so much by the apparent size of the galaxies but by the scaling of the flux with redshift. Nevertheless, the point is that it will become more and more common to study the dynamics of very high redshift galaxies but, based on our current understanding of galaxy evolution and on simulations, these objects are likely to be very different from intermediate redshift galaxies. In particular, we might expect to probe a large fraction of gas-rich morphologically and kinematically disturbed galaxies. In this context, the rotationally supported disk approximation that is used in nearly all dynamical studies might begin to fail and if so it will become much harder to precisely constrain the DM fraction found in very high redshift galaxies. Hence, in my view, there should be efforts put into building dynamical models of non-axisymmetric galaxies, perhaps based on high-resolution observations of local irregular or peculiar galaxies or on outputs from simulations. The clear difficulty in doing so is that the complexity of such models will be greatly impacted by various limitations (spatial and/or spectral resolution, S/N, etc.) and by the loss of information due to integration along the LOS. Perhaps the solution will be to combine the information found in various types of data (e.g. SED, high-resolution images, data cubes, etc.) into a grand physical-morphological-dynamical model applied on spatially resolved scales, that is similarly to what is done with pixel-per-pixel SED fitting but by taking the geometry of the galaxy and its dynamics into account.
# Conclusions et perspectives

Dans ce manuscrit, j'ai présenté le travail que j'ai effectué au cours de ces trois dernières années concernant l'impact de l'environnement sur les propriétés dynamiques des galaxies au cours des dix derniers milliards d'années. J'ai montré comment j'ai été capable, en combinant des données MUSE et HST, de contraindre la morphologie et la cinématique de presque 400 galaxies dans le relevé MAGIC. En particulier, j'ai mis en avant dans le Chapitre 3 l'efficacité du relevé MAGIC pour sonder l'impact de l'environnement sur les propriétés des galaxies grâce à sa grande homogénéité dans la manière dont les propriétés physiques, morphologiques et cinématiques des galaxies sont mesurées pour un grand échantillon de galaxies à la fois dans le champs et dans des environnements de densité variable (groupes et amas de galaxies). Ceci m'a permis de contribuer fortement à deux articles déjà publiés (Abril-Melgarejo et al. 2021; Mercier et al. 2022) et à deux autres analyses quasiment terminées, la première sur le moment angulaire dans MAGIC présentée au Chapitre 7, et la seconde sur le relevé MAGIC discutées au Chapitre 3. Grâce à la conception du relevé MAGIC, aux capacités incroyables de MUSE et aux modélisations morphologiques et dynamiques précises que j'ai effectué (voir Chapitres 4 et 5), ce travail m'a permis de poser les contraintes les plus précises à ce jour sur l'impact de l'environnement sur plusieurs relations d'échelles de galaxies à redshift intermédiaire. A partir de ces analyses, j'ai pu mettre en lumière l'effet de l'environnement sur la morphologie et le taux de formation stellaire dans les structures. J'ai aussi montré que, à redshift intermédiaire, l'environnement ne semble pas affecter la cinématique des galaxies (vitesse de rotation) mais qu'il semble avoir un impact sur le moment angulaire des galaxies les plus massives que l'on trouve dans les parties les plus denses des structures.

Néanmoins, beaucoup de travail reste encore à effectuer. Qui plus est, comme discuté au Chapitre 8, nous avons développé au cours des deux dernières années une nouvelle méthodologie qui pourrait nous aider grandement à mieux contraindre la dynamique des galaxies à redshift intermédiaire avec MUSE et, de manière encore plus importante, des galaxies à grand redshift avec la prochaine génération de spectrographes à intégrale de champs tels que ERIS, HARMONI ou MOSAIC qui seront installés sur les très grands télescopes actuels et sur les télescopes extrêmement grands du futur. Ainsi, dans ce qui suit, je discute rapidement de la continuité de mon projet de recherche à la fois sur le court, moyen et long termes.

## Finaliser l'analyse du moment angulaire

La perspective sur le plus court terme possible que je peux discuter concerne l'analyse du moment angulaire dans MAGIC. Comme précédemment discuté, l'analyse présentée au Chapitre 7 est encore préliminaire. Il est possible de tirer quelques premières conclusions concernant l'impact de
l'environnement, de la masse stellaire, ou encore des bulbes, mais il reste encore du travail avant de pouvoir conclure définitivement. En particulier, un aspect qui n'a pas été discuté est l'impact du rayon où le moment angulaire est mesuré. Des rayons autres que $R_{22}$ pourraient aussi être considérés pour sonder l'impact de l'environnement. Ainsi, il serait intéressant de tester (i) si je retrouve les mêmes résultats quand je mesure le moment angulaire par exemple à un rayon effectif, ou quand j'extrapole à l'infini, et (ii) s'il y a des variations avec l'environnement ou la fraction de masse dans le bulbe quand je calcule le moment angulaire cumulatif (c'est-à-dire le moment angulaire en fonction de la distance radiale). Un autre aspect qui n'a pas été discuté pour le moment est où se situent les petites galaxies enlevées par la sélection par rapport à la relation de Fall trouvée à faibles masses stellaires. Il serait intéressant de voir si de telles galaxies se trouvent sur la même relation ou si elles sont situées en-dessous (comme le relation de Fall obtenue en utilisant les courbes de rotation issues du modèle « flat » semble indiquer). Si c'est le cas, alors cela mériterait d'étudier si l'effet est physique ou si c'est lié à des contraintes faibles sur le rapport masse-lumière des galaxies, sur leurs distributions de flux dans le disque ou le bulbe, sur la cinématique du gaz ionisé, etc.

Je compte continuer ces investigations et arriver à des conclusions définitives dans les mois qui suivent.

## Perspectives concernant MAGIC, MUSCATEL et d'autres relevés MUSE

Le survey MAGIC est à présent terminé (Epinat et al., in prep.) et l'analyse de l'impact de l'environnement sur les relations d'échelles, en particulier les relations de Tully-Fisher et de Fall, a été effectuée. Cependant, comme discuté dans les Chapitres 6 et 7, sonder l'impact de l'environnement sur la dynamique des galaxies avec MAGIC est entièrement faisable, mais conclure définitivement peut être difficile pour le moment. La raison principale est qu'il semble que les différences observées entre les galaxies dans les environnements à basse et à haute densité soient majoritairement visibles à grande masse stellaire où nous manquons de galaxies de champs. Ainsi, il serait intéressant de compléter ces deux premières études avec des données supplémentaires issues d'autres relevés MUSE. Le plus prometteur est évidement MUSCATEL qui a déjà été présenté en Sect. 2.2.3 et dont j'étais censé utiliser les données pendant cette Thèse. Ce nouveau relevé devrait être disponible bientôt et fournira des observations MUSE profondes avec optique adaptative pour quatre champs parallèles du HFF. Ces observations sont séparées entre quatre champs MUSE très profonds (25 h), 16 profonds (5 h), et 36 moins profonds (environ 2 h) qui couvriront au total une surface de 36 arcmin². Ainsi, avec cette grande couverture spatiale on devrait s'attendre à trouver une fraction significative de galaxies de champs massives ainsi que quelques structures à plus haut redshift. De fait, ce relevé sera parfaitement complémentaire de MAGIC. Il permettra d'étendre grandement notre échantillon actuel et nous permettra de sonder une plus grande variété d'environnements que ce que MAGIC peut faire seul actuellement. Combiné avec d'autres relevées MUSE, soit très profonds (ex : MXDF), soit un peu moins mais couvrant une plus grande surface (ex : HUDF et MUSE-WIDE), cela devrait nous permettre de poser des contraintes encore plus fortes sur l'impact de l'environnement sur la dynamique des galaxies à $z \sim 1$, et l'on peut même imaginer séparer l'effet de l'environnement d'une potentielle évolution en redshift des relations d'échelle discuté au cours de cette Thèse (relation masse-taille, séquence principale, relation Tully-Fisher, relation de Fall).
Au-delà ce qui a été dit, MAGIC peut aussi être utilisé pour d'autres objectifs que sonder l'impact de l'environnement sur les propriétés des galaxies. Par exemple, quelques années en arrière, certaines études ont semblé montrer l'existence potentielle de quelques galaxies pauvres en matière noire à grand redshift (e.g. Lang et al. 2017; Genzel et al. 2017) mais d'autres auteurs ont montré par la suite que cela pourrait être dû à des effets systématiques induits par la méthode utilisée (e.g. Tiley et al. 2019b). Cependant, Mancera Piña et al. (2022) ont trouvé récemment une galaxie ultra diffuse dans l'Univers local dont la distribution de matière baryonique (c'est-à-dire étoiles plus gaz) est suffisante pour reconstruire la cinématique observée. Ils discutent bien du fait que l'inclinaison de la galaxie qu'ils mesurent est une grande source d'incertitude, néanmoins, ils affirment que, même en prenant cela en compte, la cinématique n'est compatible ni avec un halo de matière noire froide, ni avec MOND. Ainsi, si une telle population de galaxies pauvres en matière noire existe, alors cela pourrait être un challenge pour le paradigme actuel ΛCDM concernant l'évolution des galaxies et, si ce n'est pas le cas, alors il serait particulièrement intéressant de comprendre pourquoi certaines galaxies semblent apparaître pauvre en matière noire. MAGIC, potentiellement combiné avec d'autres relevés MUSE comme MUSCATEL, peuvent aider parce que nous observons en effet dans notre échantillon ce que nous appelions dans Mercier et al. (2022) des galaxies dominées par les baryons (c'est-à-dire pauvres en matière noire). Cependant, nos premières investigations nous ont amené à conclure que ces galaxies ne sont pas intrinsèquement dominées par les baryons mais plutôt qu'elles apparaissent ainsi à cause de la propagation d'incertitudes jusque dans notre modélisation de masse. Et pourtant, je pense personnellement qu'il serait intéressant d'étudier au cas par cas ces objets. Grâce à la grand homogénéité de notre échantillon MAGIC, de sa haute complétude, et parce que nous avons acquis de nombreuses informations sur ces galaxies (ex : propriété de leur environnement, leur localisation dans plusieurs relations d'échelle, leur morphologie, etc.), je pense que nous avons en notre possession toutes les informations nécessaires pour étudier ces galaxies plus en détail. Indépendamment du résultat (effet systématique pour toutes les galaxies ou certaines d'entre elles sont intrinsèquement pauvres en matière noire), la conclusion sera dans tous les cas enrichissante. Cela est particulièrement vrai, même si la première explication s'avère la bonne (ex : biais systématiques), parce que l'on s'attend à ce que de tels objets soient détectés à haut redshift dans le futur à moyen terme avec le développement des télescopes extrêmement grands et leur instruments de première lumière comme HARMONI sur l'ELT et dans le future proche avec le JWST ou encore ERIS.

## Préparer le chemin pour les spectrographes à intégrale de champs de prochaine génération

Les méthodes présentées aux Chapitres 5 et 8 de cette Thèse, en particulier concernant la production de cartes spatialement résolues de propriétés physiques, peuvent déjà être appliquées sur des données MUSE et HST. Par exemple, il sera possible de l'utiliser avec MUSCATEL car il y a des observations HST multi-bandes ultra profondes dans les champs parallèles du HFF. Cependant, le véritable objectif est d'améliorer la méthode de telle sorte qu'elle soit disponible pour les prochaines générations de spectrographes à intégrale de champs (ex : HARMONI, MOSAIC) et d'imageurs haute résolution tel que MICADO sur les futures télescopes extrêmement grands. HARMONI et MOSAIC observeront tous les deux dans le proche infrarouge avec des résolutions spectrales moyennes et hautes et pourront de fait cibler des
galaxies à haut redshift. Le premier des deux instruments est vraiment optimisé pour des observations à haute résolution spatiale de galaxies distantes et observera donc un objet à la fois sur un champs de vue relativement petit. A l'inverse, le second instrument aura un mode spectroscopie à intégrale de champs multi-objets composé de huit à dix unités d'intégrale de champs (IFU) qui seront déployées sur un grand champs de vue mais avec une moins bonne résolution spatiale, similaire à celle de MUSE (c'est-à-dire environ 0.2" par spaxel). En attendant, il sera possible d'appliquer et de tester notre méthodologie sur les instruments de nouvelle génération que sont NIRSPEC et MIRI sur le JWST, qui tout les deux ont un mode spectroscopie à intégrale de champs, ou encore ERIS qui sera bientôt disponible sur le VLT. Ces instruments nous fourniront des observations dans le proche infrarouge (infrarouge moyen pour MIRI) avec une meilleure résolution spatiale que ce que MUSE offre actuellement.

A relativement court terme, je testerai notre méthodologie sur une galaxie comme si elle était observée avec HARMONI. Pour ce faire, nous allons utiliser des observations de spectroscopie à transformée de Fourier de M95 obtenues avec SITELLE combinées avec des images issues de MEGACAM et WIRCAM pour simuler comment cette galaxie nous apparaîtrait si elle était située à grand redshift et vue par HARMONI. Ensuite, nous serons en mesure d'utiliser ces observations synthétiques pour estimer l'efficacité de notre méthode (incluant la technique d'ajustement pixel par pixel et son application d'apprentissage machine discutés au Chapitre 8). Je précise qu'il serait aussi possible dans une avenir légèrement plus lointain d'effectuer ce même type de tests sur des données issues de simulations plutôt que sur des observations à haute résolution de galaxies dans l'Univers local.

Au-delà de ce qui a déjà été dit, il reste encore du travail concernant l'implémentation de l'algorithme d'apprentissage machine qui doit être terminée. Beaucoup d'efforts ont déjà été fournis dans le SOM, mais il reste encore du travail. Nous devons déterminer en détails les avantages et les inconvénients d'une telle méthode par rapport à des techniques plus classiques d'ajustement pixel par pixel, vérifier qu'il n'existe pas une manière plus efficace d'utiliser l'algorithme et enfin le tester sur un grand échantillon de galaxies, potentiellement avec MUSCATEL. En fonction de combien de personnes travailleront dessus et de l'effort général qui sera fourni, cela pourrait mener à une autre publication dans les deux prochaines années à venir. Cet exercice sera particulièrement intéressant car ce qu'il en ressortira sera d'autant plus utile pour les grands relevés à venir issus des futurs instruments sur le moyen/long terme.

Finalement, je conclurai avec une dernière perspective concernant la modélisation dynamique des galaxies à grand redshift. En effet, avec l'avènement prochain de nombreuses données 3D dans le proche infrarouge, il est plus que probable qu'il adviendra une époque, peut-être dans une dizaine d'années, où il sera coutume d'extraire la cinématique spatialement résolue de galaxies à l'aide de raies d'émission dans l'ultra-violet ou le visible (au repos) à des redshifts $z \ge 6$. En principe, avec NIRSPEC il est déjà théoriquement possible d'atteindre des redshifts $z \approx 12$ et, avec des observations spatiales dans l'infrarouge moyen comme celles fournies par MIRI, des galaxies pourraient être observées avec une relativement haut résolution spatiale jusqu'à $z \approx 30$.

Évidemment, de telles observations seraient limitées non pas vraiment par la taille apparente des galaxies mais plutôt par l'évolution du flux avec le redshift. Néanmoins, l'argument reste qu'il sera de plus en plus commun d'étudier la dynamique de galaxies à très grand redshift mais, basé sur notre connaissance actuelle de leur évolution et sur les résultats de simulations, ces objets sont censés être très différents de ceux à redshift intermédiaire. En particulier, on peut s'attendre à observer une fraction importante d'objets riches en gaz et perturbés morphologiquement et cinématiquement. Dans ce contexte, l'approximation de disques supportés par leur propre
rotation qui est utilisée dans la plupart des études pourrait ne plus être valide, auquel cas il
deviendrait d'autant plus difficile de contraindre correctement la fraction de matière noire à grand
redshift. Ainsi, de mon point de vue, il devrait y avoir des efforts mis dans la conception de
modèles dynamiques de galaxies non-axisymétriques, peut-être en se basant sur des observations
de galaxies locales irrégulières ou particulières ou via des résultats de simulations. La difficulté
évidente dans cet effort est que la complexité de tels modèles sera fortement impactée par la perte
d'information due à l'intégration le long de la ligne de visée. Peut-être que la solution serait de
combiner les informations contenues dans différents types de données (ex : distribution d'énergie
spectrale, images haute résolution, cubes de données, etc.) dans un super modèle
physico-morphologico-dynamique qui serait appliqué sur des échelles résolues, dans la veine de ce
qui est actuellement fait avec l'ajustement de distribution d'énergie spectrale pixel par pixel mais
en prenant en compte la géométrie de la galaxie et sa dynamique.
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Observatories, instruments, and
surveys

2MASS The Two Micron All Sky Survey is a NIR photometric survey carried out between 1997 and 2001 that observed nearly completely the celestial sphere. 53

3D-MPE An early German IFS instrument that integrated an image slicer with a long-slit and that was deployed at the WHT, at the observatory of Calar Alto in Andalusia, Spain, and at the 2.2m MPG-ESO telescope in La Silla, Chile. 52

AEGIS All-Wavelength Extended Groth Strip International Survey is a multi-wavelength ground- and space-based survey that studies the evolution of galaxies at $z \sim 1$. 53

AIP The Leibniz Institute for Astrophysics Potsdam is an astrophysics research institute located in Potsdam, Germany. 54

ALMA Atacama Large Millimeter/submillimeter Array is an international observatory comprised of 66 radio telescopes observing at millimetre and sub-millimetre wavelengths and located in the Chajnantor plateau, Chile. 7

AMAZE Assessing the Mass-Abundance redshift[-Z] Evolution is an ESO large program spectroscopic survey obtained with SINFONI whose goal was to study the mass-metallicity relation in the range $3 < z < 5$. 52

ATLAS<sup>3D</sup> A multi-wavelength survey from the radio to the optical of 260 early type galaxies whose stellar and ionised gas kinematics has been derived from SAURON observations. 52

BLUEMUSE A future IFS instrument for the VLT similar to MUSE but optimised to observe in the blue part of the spectrum. 59

CALIFA Calar Alto Legacy Integral Field Area Survey is a large IFS survey of galaxies in the local Universe carried out at the Calar Alto 3.5m telescope. 44

CANDELS The Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey is a HST imaging survey observing galaxies at high redshift. 92

CFHT Canada-France-Hawaii Telescope is a Canadian-French-Hawaiian observatory and a 3.5m telescope located on mount Mauna Kea in Hawaii. 33, 49–51
**COSMOS** Cosmic Evolution Survey, a two square degree photometric survey from X-ray to radio bands designed to probe the evolution of galaxies as a function of cosmic time. The version of the catalogue is usually appended with the year of its release, that is COSMOS2015 for the old version and COSMOS2020 for the new one. 6, 53, 58, 61, 63, 65, 68, 73, 75, 84

**CRAL** The Centre de Recherche Astrophysique de Lyon is an astrophysics research institute located in Lyon, France. 54

**ELTs** Extremely Large Telescopes are the next-generation 30–40 m class telescopes that should be available around the years 2030–2040. 49, 58, 59, 237, 239

**ELT** Extremely Large Telescope, the next-generation extremely large European (ESO) telescope with a 39 m diameter telescope located at Cerro Armazones, Chile, and that should be launched around the year 2025. 59, 239, 243

**ERIS** The Enhanced Resolution Imager and Spectrograph is a new instrument installed on the UT4 platform of the VLT that has an integral field spectroscopy mode from the J to the K bands. 59, 60, 237, 239, 241, 243, 244

**ESA** The European Space Agency is a European organisation begun in 1975 that is dedicated to the exploration of space. 33

**ESO** European Southern Observatory is an intergovernmental European research organisation specialised in ground-based astronomy in the southern hemisphere. 32, 52–54, 58, 59, 61

**ETH** Eidgenössische Technische Hochschule Zürich is a University located in Zürich, Switzerland. 54

**FLAMES** Fibre Large Array Multi-Element Spectrograph is a fiber facility installed on the UT2 platform of the VLT. It hosts two different spectrographs: GIRAFFE and UVES. 52, 53

**GALACSI** The AO facility associated to MUSE at the VLT. 55, 61

**GALEX** An American UV telescope launched in 2003 and designed to study star formation in the Universe. 34

**GASP** GAs Stripping Phenomena in galaxies with MUSE is a MUSE survey designed to study gas stripping in local galaxies. 6

**GIRAFFE** A medium-high resolution spectrograph in the visible fed by FLAMES. It can either offer simultaneous spectroscopy for 132 targets in its MEDUSA mode, or perform IFU spectroscopy for simultaneously 15 targets, or perform a single IFU observation on a larger FoV. 52, 53, 58, 59

**GOODS** Great Observatories Origins Deep Survey is a multi-wavelength ground- and space-based survey that studies the evolution of galaxies over a large FoV combining optical, IR, and X-ray data. 53

**GAIA** A space mission that performs precise astrometry and proper motion measurement of mainly stars. 89
**HARMONI** High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph will be one of the first-light instrument on the future ELT that will perform diffraction-limited integral field spectroscopy across the optical and the IR. 59, 237, 239-241, 243, 244

**HDFS** The Hubble Deep Field South is a deep HST multi-wavelength observation of a portion of the sky visible in the southern hemisphere that followed the Hubble Deep Field observations. 5

**HDF** The Hubble Deep Field is the first deep HST multi-wavelength observation performed in the Northern Hemisphere. 33

**HST-ACS** Hubble Space Telescope Advanced Camera for Surveys is a third generation camera installed on the HST since 2002 that can observe from the UV to the IR. 207

**HST-WFC3** Hubble Space Telescope Wide Field Camera 3 is a fourth generation camera installed on the HST since 2009 designed for its large FoV and high sensitivity that can observe from the UV to the IR. 207

**HST** Hubble Space Telescope is a 2.4 m diameter telescope operating from UV to near-IR with spectroscopy and imaging capabilities. 6, 33, 34, 64, 67, 92, 105, 173, 199, 201, 202, 207, 208, 211-214, 222, 231, 234, 237, 239, 241, 243

**HUDF** The Hubble Ultra Deep Field survey is a deep HST survey that targeted a low density region of the sky first with the ACS instrument and later with WFC3 in the infrared. 33, 201, 238, 242

**IAG** The Institute for Astrophysics and Geophysics is an astrophysics and geophysics research institute located in Göttingen, Germany. 54

**IMAGES** Intermediate MAss Galaxy Evolution Sequence, an ESO large program spectroscopic survey performed with FLAMES-GIRAFFE multi-IFU instrument designed to probe the dynamics of 63 intermediate redshift ($0.4 < z < 0.75$) galaxies. 53

**IRAP** The Institut de Recherche en Astrophysique et Planétologie is an astrophysics research institute located in Toulouse, France. 54

**IRIS** The Infrared Imaging Spectrograph is a future first-light IFS instrument on the TMT that will perform diffraction-limited observations in the NIR. 59

**JWST** James Webb Space Telescope is a new-generation space-based telescope observing the in IR with a primary mirror of 6.5 m launched in 2021 and operational since mid-2022. 59, 239, 243, 244

**KMOS3D** K-band Multi Object Spectrograph 3D, an integral field spectroscopic survey performed with KMOS on more than 600 galaxies in the redshift range $0.7 < z < 2.7$. The goal was to build a large sample of galaxies with spatially resolved kinematics at intermediate to high redshift using the high capabilities of KMOS. 58, 137

**KMOS** K-band Multi Object Spectrograph is a second-generation near-infrared spectrograph mounted on the UT1 platform of the VLT that can perform integral field spectroscopy on 24 targets at the same time. 53, 56, 58, 59
KROSS KMOS Redshift One Survey, an ESO GTO KMOS survey targeting 795 star forming galaxies around redshift $z \sim 0.9$. 58, 137

KECK telescopes Two 10 m-class American telescopes located on Mauna Kea in Hawaii and operated by Caltech and NASA. 52, 53

LAM Laboratoire d'Astrophysique de Marseille is a French astrophysics research institute located in Marseille. 131

LSD Lyman-break galaxies Stellar populations and Dynamics, a spectroscopic survey performed with SINFONI whose goal was to have integral field spectroscopic observations for a complete sample of Lyman-break galaxies at $z \sim 3$. 52

MAGPI Middle Ages Galaxy Properties with Integral Field Spectroscopy is a MUSE survey dedicated to studying resolved properties and kinematics of 60 central galaxies and roughly 100 satellites in dense environments at $z \approx 0.3$ where the impact of the environment on galaxies is supposed to be the most prominent. 7

MASSIV Mass Assemby Survey with SINFONI in VVDS is a spectroscopic survey performed with SINFONI in the VIMOS VLT Deep Survey whose goal was to probe the kinematics and chemical content of a representative sample of galaxies at $1 \le z \le 2$. 44, 52

MAUVE MUSE and ALMA Unveiling the Virgo Environment is a future MUSE survey that will observe massive disk galaxies in the Virgo cluster in combination with ALMA to study the link between environment and star-formation. 7

MEGAFLOW MusE GAs FLOw and Wind survey is a MUSE survey that observes galaxy-quasar pairs to study gas inflows and outflows in the range $0.4 < z < 1.4$. 6

MICADO Multi-AO Imaging Camera for Deep Observations will be one of the first-light instrument on the future ELT that will take images at a diffraction-limited spatial resolution in the NIR. 239, 243

MIRI The JWST Mid-Infrared Instrument is a spectrograph mounted on the JWST and observing in the MIR between roughly 5 µm and 28 µm. 59, 239, 240, 244

MOSAIC A future MOS and IFS instrument that will be located on the ELT. 59, 237, 239, 241, 243

MUDF The MUSE Ultra Deep Field is a deep survey of 200 h on-source integration time composed of two largely overlapping MUSE fields that targeted various structures along with two bright quasars. 7

MUSE-HDFS The MUSE Hubble Deep Field South is a deep 27 h integration time early MUSE survey that targeted the HDFS region of the sky. 5, 58, 64, 65, 68, 71

MUSE-HUDF The MUSE Hubble Ultra Deep Field survey is a deep MUSE survey that targeted ten fields in the HUDF area. 5, 56, 58, 65, 68, 71, 201, 221

MUSE-WIDE A complementary survey to MUSE-HUDF that performed shallow observations of 1 h exposure but over a large FoV comprised of 24 different MUSE fields. 5, 58, 238, 242
MUSE-FAINT MUSE-FAINT is a MUSE survey of ultra-faint dwarf galaxies designed to put constraints on DM properties. 45

MUSE Multi-Unit Spectroscopic Explorer is an IFS spectrograph with large FoV of one arc minute square. It is a second generation instrument mounted on the UT4 platform on the VLT. 5–7, 9, 43, 47, 49, 53–61, 63–65, 67–69, 71–73, 83–85, 105, 109, 110, 119, 122, 125, 128, 131, 134, 137, 138, 173, 199, 201, 207, 231, 237–239, 241–244, 272

MXDF The MUSE Extremely Deep Field is an ultra deep MUSE survey of 140 h of integration time located in the HUDF field. 6, 46, 56, 58, 201, 238, 242

MANGA Mapping Nearby Galaxies at APO is an ongoing survey that uses 17 different fibre-based IFUs to observe local galaxies that are part of the SDSS. 58

MEGACAM A wide-field optical imager installed on the CFHT which is complementary to WIRCAM. 240, 244

NASA The National Aeronautics and Space Administration is an American agency founded in 1958 dedicated to space exploration. 33

NIRSPEC The JWST Near-Infrared Spectrograph is a spectrograph mounted on the JWST and observing in the NIR between roughly 0.9 µm and 5 µm. 59, 239, 240, 244

OASIS Optically Adaptive System for Imaging Spectroscopy, the second IFS instrument developed in Lyon under the supervision of Roland Bacon that was installed on the CFHT. 50–52

OHP Observatory of Haute-Provence is a French observatory located in Provence, France. 50

ORELSE Observations of Redshift Evolution in Large-Scale Environments is a photometric and spectroscopic survey of galaxies located in 20 clusters in the redshift range 0.6 < z < 1.3 designed to probe the impact of the environment on galaxy properties over a large range of environments. 48, 137

OSIRIS OH-Suppressing Infra-Red Imaging Spectrograph is an IFS instrument that observes in the infrared and is mounted on one of the Keck telescopes. 53

SAMI Sydney-AAO Multi-object Integral field spectrograph is a MOS and IFS instrument that combines 13 deployable fibre bundles. It is also the name of a large IFS survey of galaxies in the nearby Universe. 44, 59

SAURON Spectroscopic Areal Unit for Research on Optical Nebulae was an early IFS instrument built in collaboration between the observatories of Lyon, Leiden, and Durham with a large FoV designed to study the dynamics of galaxies. 51, 52

SAURON project A survey of 72 spiral, lenticular, and elliptical galaxies observed with SAURON. 51

SDSS Deep Imaging Multi-Object Spectrograph is a MOS slit spectrograph mounted on the Keck II telescope. 48
**SDSS** The Sloan Digital Sky Survey is a large FoV deep spectroscopic multi-band survey observing galaxies in the local Universe. 37, 41, 53, 58, 92, 94

**SINFONI** A second-generation IFS instrument located on the UT4 platform at the VLT that observed in the infrared. 52, 54, 60

**SINS** Spectroscopic Imaging survey in the Near-infrared with SINFONI is a survey performed with SINFONI aimed at studying the dynamics of galaxies at high redshift ($z \ge 2$). 52

**SITELLE** Spectromètre Imageur à Transformée de Fourier pour l'Etude en Long et en Large de raies d'Emission is an imaging Fourier-transform spectrograph installed at the CFHT. 240, 244

**SPARCS** Spitzer Photometry and Accurate Rotation Curves is a sample of nearby galaxies with high-quality rotation curves and SPITZER photometry. 44

**SPITZER** An American infrared telescope launched in 2003. 6, 34

**TIGER** Traitement Intégral des Galaxies par l'Etude de leurs Raies, the first IFS prototype imagined by Georges Courtès in 1982, developed between the observatories of Lyon and Marseille, and installed on the CFHT and then on the WHT. 49–52

**TMT** The Thirty Meter Telescope is a future American extremely large telescope that will host a 30 m primary mirror. 59

**VIMOS** VISIBLE Multi-Object Spectrograph was a large FoV imager, multi-object spectrograph, and IFU instrument previously mounted on the UT3 platform of the VLT. 52, 65

**VLTs** Very Large Telescopes, a generation of 8 – 10 m class telescopes. 34, 59, 237

**VLT** Very Large Telescope, a 8 m ground-based large telescope operated by ESO on Cerro Paranal in Chile. 52, 55, 58, 59, 61, 239, 244

**VUDS** The VIMOS Ultra-Deep Survey is a large spectroscopic survey of roughly 10 000 galaxies at $2 \le z \le 6$ carried out with VIMOS. 84

**WHT** William Herschel Telescope is a 4.2 m telescope located at the Roque de los Muchachos Observatory in La Palma, Spain. 51, 52

**WIRCAM** The Wide-field InfraRed Camera is a NIR imager installed at the CFHT which is complementary to MEGACAM. 240, 244

**zC-SINF** zCOSMOS-SINFONI is a spectroscopic survey that combined SINFONI observations with the zCOSMOS information to study a sample of massive star forming galaxies around $z \approx 2$. 52

**zCOSMOS** A 600 h ESO Large Program carried out with the VIMOS spectrograph to provide a spectroscopic follow-up to roughly 20 000 galaxies between redshift 0.2 and 3. The sample with about 20 000 galaxies is sometimes referred as the 20K sample whereas the earlier version with only 10 000 galaxies is named 10K sample. 61, 65, 84, 85
*Gemini* The Gemini Observatory is composed of two 8 m-class telescopes, one located at Cerro Pachón in Chile and the other at Mauna Kea in Hawaii, thus covering almost the entire sky.
52

*MPE* A German research institute in astrophysics located in Garching. 52
Glossary

ACDM A Cold Dark Matter is the current cosmological model that assumes an homogeneous and isotropic Universe with a dark matter component and a cosmological constant that acts as dark energy and is responsible for the current accelerated expansion of the Universe. 23–25, 28, 44–46, 239, 243

CIGALE Code Investigating GALaxy Emission, a SED fitting code initially written in Fortran but rewritten in Python that can fit SEDs from the X-ray/UV to the radio bands. 73, 75, 78, 80–83, 134, 201, 203, 206, 208–211, 214–217, 222–224, 231–235

EAGLE The Evolution and Assembly of GaLaxies and their Environments project is a large cosmological hydrodynamical simulation from the Virgo consortium. 35

FAST A SED fitting code written in IDL that can work in combination with the photometric redshift code EAZY. 73, 74, 78, 80, 82

FIRE Feedback In Realistic Environments is a suite of high-resolution cosmological designed to probe star formation and feedback processes on spatially resolved scales. 35

GALPAK$^{3D}$ Galaxy parameters and kinematics is a full 3D kinematics fitting code based on a parametric disk model and MCMC sampling that directly fits the data cubes to extract the intrinsic kinematics of a galaxy. 130, 131, 201

GALFIT A widely used 2D fitting software optimised to fit the surface brightness distribution of galaxies from single-band photometry. 72, 87, 132, 133, 201, 222

HFF The HUBBLE Frontier Fields are ultra deep HST and SPITZER observations that focus on six massive high-magnification lensing clusters as well as six blank parallel fields. This survey provides the deepest HST observation ever taken. 6, 238, 239, 242, 243

ILLUSTRIS TNG The Next Generation Illustris Simulations is a new version of ILLUSTRIS that covers boxes of different volume and spatial resolution. 35

ILLUSTRIS A large cosmological hydrodynamical simulation designed to study the evolution of galaxies. 35

LEPHARE A SED fitting/photometric redshift code written in Fortran used in particular for the COSMOS catalogue. 73, 75, 201, 203, 206, 208–211, 214–217, 224
**MAGIC** The MUSE-gAlaxy Groups In Cosmos is a deep MUSE survey targeting already known groups in the COSMOS field and designed to probe the impact of the environment on galaxy properties. 6, 9, 41–43, 57, 58, 61, 63–65, 68, 69, 71, 72, 77, 83, 84, 87, 97, 101, 105, 109, 110, 119, 122, 130–134, 137, 138, 173, 237–239, 241–243

**MARZ** Manual and Automatic Redshifting Software is a template-matching automatic and manual redshifting software. 65

**MUSCATEL** The MUSE Cosmic Assembly survey Targeting Extragalactic Legacy fields is a MUSE survey targeting the four HFF fields with deep 25 h observations surrounded by a mosaic of shallower MUSE fields. 6, 238–240, 242–244

**MAGPHYS** Multi-wavelength Analysis of Galaxy Physical Properties is SED fitting code written in Fortran77 that can model SEDs from the UV to the IR. 201

**MILLENIUM** The first large semi-analytical cosmological simulation from the Virgo consortium. 35

**MULTIDRIZZLE** A program used on HST observations that perform various tasks to combine dithered images. 213

**PROSPECTOR** A SED fitting code that performs Bayesian inference on highly flexible models and that can combine photometric and spectroscopic data. 209

**SEXTRACTOR** Source-Extractor is a program that automatically creates catalogues of objects from images. 211

**ZAP** Zurich Atmosphere Purge is a sky subtraction software for integral field spectroscopy data that uses PCA analysis, wavelength segmentation and masking. 65

**³D BAROLO** 3D-Based Analysis of Rotating Object via Line Observation is a full 3D fitting code that uses the tilted-ring formalism to model and fit an input cube. 130

**STARLIGHT** A stellar spectral synthesis code that estimates stellar physical parameters from spectra. 209

**absolute magnitude** Noted $M$, it is defined as the apparent magnitude if the object was seen at 10 pc from us. Depending on the definitions it can also be corrected of the extinction and the k-correction. In essence, it is a proxy for the intrinsic luminosity of the object. 74

**AGN** An active galactic nucleus is a luminous compact region in the central parts of a galaxy that emits all across the wavelength range from radio to gamma rays and is supposedly associated to a supermassive black-hole surrounded by an accretion disk. 35, 38, 46, 53, 56

**angular diameter distance** A distance in cosmology used to derive the physical extension of an object onto the sky. 27, 28

**AO** Adaptive Optics, a technology used in modern ground-based telescopes to actively compensate for atmospheric distortion by deforming the mirror. 6, 50, 52, 53, 55, 59, 60, 65, 238
**apparent magnitude** Noted *m*, it is defined as -2.5 times the decimal logarithm of the observed flux. 74

**asymmetric drift** The tendency of a test particle to rotate slower than the circular velocity it should have because of non-streaming motions. The asymmetric drift is usually associated to an inward radial pressure gradient that compensates parts of the inward gravitational force and therefore lead to a reduction of the rotation velocity. 109, 112, 113

**attenuation curve** The curve that describes the strength of the dust attenuation as a function of wavelength. 208, 210

**B/T** Bulge-to-disk flux ratio, the ratio between the integrated flux of the bulge component and that of the sum of all the components in a galaxy and in a given aperture. 97, 99, 101, 201

**Balmer line** A set of hydrogen lines corresponding to an energy transition from the second level to a higher one (absorption) or vice-versa (emission). 57, 65

**beam smearing** An instrumental effect that affects kinematics maps extracted from data cubes by convolving nearby pixels with different flux and line-of-sight velocity. Its effect is particularly relevant for the velocity dispersion whose value is increased where the flux and the velocity gradient are the highest. 39, 69, 119, 129–131, 223

**bias frame** An image generated instantaneously that only contains undesired signal from the electronics independently of the exposure. 65

**BMU** The best-matching unit is the closest neuron in a SOM to a given data point. 226, 228, 229, 232

**Boltzmann's equation** An equation of conservation of mass along a flow in phase space derived by Boltzmann in 1872. 25–27

**Burkert** A core DM profile first proposed by Andreas Burkert to fit DM haloes of dwarf spiral galaxies. 118

**CCD** Charge-Coupled Devices are a type of light detector that converts incoming photons into electrons that can then be read and stored on a device. 54

**CDM** Cold Dark Matter, the most massive type of DM which is currently part of the standard cosmological model. Its high mass means it must have a low thermal velocity and thus it is "cold". 44

**CGM** The gas located beyond the extent of a galaxy stellar disk but still gravitationally bound that is thought to be source of accreted cold gas. 41

**circular velocity** Rotation velocity of a test particle in equilibrium embedded in the gravitational potential generated by a mass distribution so that its centrifugal acceleration compensates the inward force generated by the mass distribution. 112, 114

**CMB** The cosmic microwave background is an electromagnetic signal coming from the youngest detectable age of the Universe, the epoch of recombination, when protons and electrons recombined into neutral hydrogen. 24, 44
**colour excess** A measurement of the excess of colour, that is reduction in flux, between two bands because of their differential dust extinction. 74

**comoving distance** The distance to an object that is measured with respect to a reference distance that stretches with the expansion of the Universe. Thus, this distance remains constant with cosmic time if the object has no proper motion. 24, 27

**core** A family of DM profiles that exhibit a core in their central parts. 46, 118–120

**COSMOS-WALL** A large scale filamentary structure located at $z \approx 0.73$ in the COSMOS field. 61, 65

**critical density** The density that is required to compensate the expansion of the Universe when there is no cosmological constant. It corresponds to the density of such a Universe without curvature. 25

**cusp-core problem** A current problem in extragalactic astrophysics where DM-only simulations predict cuspy DM profiles whereas observations favour cored profiles instead. 46, 118, 201

**cuspy** A family of DM profiles that exhibit a cusp in their central parts. 46, 118, 119

**dark energy** A component of the Universe to which is attributed the acceleration of its expansion. 25

**dark frame** An image taken for a given exposure time in complete darkness that contains dark current signal plus electronics signal. The bias frame is usually removed from the dark frame to produce a new dark frame that only contains the dark current signal. 65

**de Vaucouleurs profile** A surface brightness profile first proposed by Gérard de Vaucouleurs in 1948. 36, 89, 99, 115

**Dekel-Zhao** Another parametric DM profile of the family of the Zhao profiles with fewer parameters and first proposed by Jonathan Freundlich. 118

**DM** Dark Matter, an hypothetical transparent component visible in galaxies and galaxy clusters through its impact on galaxies dynamics via gravity. 6, 23, 33, 35, 37–40, 42–46, 87, 110, 114, 118–120, 133, 134, 138, 139, 174, 199, 201, 214, 222–224, 238–240

**double exponential disk** A thick disk model represented by the product of two exponential laws, one along the radial direction in the plane of the disk, the other along the vertical direction. 94–96, 113, 114, 133

**Einasto** A parametric cuspy/core model defined by Jaan Einasto to describe stellar systems that has been used to model DM haloes. 118, 119

**exponential disk** A disk model whose surface brightness follows an exponential law, usually combined with a Dirac delta to produce a razor-thin disk for morphological and mass models. 89, 94, 95, 97, 99, 113
**extinction** The drop in flux, or equivalently the increase in magnitude, due to the absorption of a part of the emitted light by dust. There are two different type of extinctions: (i) the intrinsic extinction that affects the rest-frame light due to the dust embedded in the observed galaxy and (ii) the Galactic extinction that affects the observed/redshifted light because of the dust located in the Milky Way. 74

**Faber-Jackson relation** A scaling relation observed for elliptical galaxies between their luminosity or stellar mass and their velocity dispersion. 36

**Fabry-Pérot spectroscopy** A type of spectroscopic observation where the spectrum of an object is taken by varying the distance between two semi-reflective blades. 48

**FIR** Far-infrared, a sub-portion of the IR part of the spectrum that goes beyond 25 µm. 201, 210

**flat-field** Correcting pixel-per-pixel sensitivity variations so that a uniform input signal produces a uniform image on the detector. 65

**FLRW** Friedmann-Lemaître-Robertson-Walker metric is the exact solution of Einstein's field equations for a homogeneous, isotropic, and expanding Universe. 24-27

**flux calibration** Processing step that transforms an input signal in ADUs or electrons into a physical units. 65

**FoF** Friends of Friends are a type of grouping algorithm that associate objects to the same group. To objects belong to the same group if they are separated by a distance which is less than the linking length, a free parameter chosen beforehand. 83-85, 139

**FoV** Field of View, the area over which an observation is made. 6, 9, 49-56, 58-61, 63-65, 71, 84, 85, 213, 239

**Freeman's disk** Another name for the razor-thin exponential disk. Named after Kenneth Charles Freeman after his derivation of the expression of the circular velocity required to compensate the gravitational force generated by this mass distribution. 114, 115, 118

**FTS** A type of spectroscopic observation where the spectrum of an object is taken by varying the distance between the two arms of a Michelson interferometer. The resulting observation is the Fourier transform of the incoming spectrum. 48

**fundamental plane** A multivariate scaling relation that links the luminosity, velocity dispersion and size of elliptical galaxies by placing them on a plane. 36

**FWHM** Full Width at Half Maximum, a measure of the width of a distribution where its value equals half its maximum. It is used for a PSF to describe the quality of an image. 51, 53, 55, 61, 71, 72, 122, 125, 131, 132, 211

**galaxy harassment** The effect of gravitational interactions with multiple galaxies in a clusters across a few fly-bys. 38, 41

**geodesic** The path of shortest distance in an arbitrary geometry. For instance, in a Euclidian space it corresponds to a straight line whereas on a sphere it corresponds to a great circle. 26
**GLAO** Ground Layer Adaptive Optics, a type of AO technology that uses a single deformable mirror and whose correction only applies for atmospheric distortion close to the ground. 61

**GR** General Relativity is a geometrical theory of gravity developed by Albert Einstein in 1915 that expands upon Special Relativity. 24, 25, 30

**grating** An optical element composed of a multitude of equally spaced ridges or slits that act as a dispersive system in spectrographs. 48, 59

**green valley** A population of scarce galaxies that lie between the blue and red sequences, probably in a transition from an active to a passive state. 77

**grism** An optical element made of the combination of a prism and a dispersion grating used in modern spectrographs. 48, 52

**GTO** Guaranteed Time Observation, a fixed amount of guaranteed observing hours provided by ESO to a consortium that has provided software or hardware solutions. 5, 6, 57, 58, 61

**Hernquist's model** A spherical mass distribution derived by Lars Hernquist that is close to a de Vaucouleurs profile once projected onto the sky. 114, 115, 133

**Hubble modified** A cored DM profile whose surface brightness approximates the Hubble-Reynolds law for elliptical galaxies. 119

**ICM** The intra-cluster medium is a medium mainly composed of hot X-ray emitting gas located between galaxies in galaxy clusters. 41, 44

**IDL** Interactive Data Language is a dynamic proprietary language mainly used for data analysis and data visualisation. 131

**IFS** Integral Field Spectrograph, an instrument that combines spectroscopy and photometry by splitting the FoV in multiple parts that are fed to spectrographs to recover one spectrum per spatial element, thus producing a data cube. 49–53, 55–60, 237, 239

**IFU** Integral Field Unit, an instrument that performs integral field spectroscopy. Some IFS instruments combine multiple IFUs together to observe a large FoV with good spatial resolution. 53, 54, 58–60, 239, 244

**IMF** Initial mass function, the mass distribution of stars originating from a single star formation event when entering the stellar Main Sequence. It describes how likely it is to get a star of a given mass on the stellar Main Sequence as a result of star formation. 73, 75, 80, 82, 208–210, 214, 235

**intrinsic axis ratio** The intrinsic ratio between the scale height and the scale length of a thick disk. This ratio is different from the observed axis ratio on the plane of the sky since the latter is impacted by projection effects. 94, 117, 119

**IR** Infrared, the part of the spectrum located between the visible and the submillimeter that roughly spans from 8000 Å (or similarly 0.8 µm) to 300 µm. 6, 34, 47, 52, 53, 58, 60, 200, 201, 206–208
ISM The gaseous and dusty medium located in a galaxy in-between stars. 35, 41, 210

isochrone A line of constant age in a stellar evolution diagram. 209

isophotal fitting A technique which consists in fitting circular or elliptical isophotes on an image of a galaxy to measure key properties such as its flux, size, or axis ratio. 101

Jeans' equations A set of four equations derived by Jeans in 1915 from Boltzmann's equation which describe the dynamics of collisionless particles whose mass is conserved along a flow in phase space. 11, 25–28

k-correction A correction that needs to be applied to the flux measured in a given band of redshifted galaxies to recover their intrinsic flux in the band. Indeed, redshifted galaxies see a reduction of their flux and a scaling of the wavelength of their emitted light because of cosmological redshift. Thus, the flux measured in a given band does corresponds to the flux emitted in a narrower band located at shorter wavelengths. To recover the intrinsic flux in the observed band, a correction must be applied that assumes a galaxy's spectrum. 74, 231, 232

Kuzmin's disk A theoretical razor-thin disk model with an analytical potential derived by Grigori Kuzmin in 1956. 88, 114

LBGs Lyman-break galaxies are galaxies selected from photometric observations in multiple UV and optical filters given that their signal significantly drops below the Lyman limit at 912 Å in rest-frame, since below this limit the radiation is sufficiently energetic to be almost completely absorbed by the neutral gas content of the galaxies. 52

long-slit spectroscopy A type of spectroscopic observation where the spectrum of an object is taken along a slit, thus giving 1D spatial plus spectral information at the same time. 48

LOS line-of-sight. 6, 24, 27, 32, 83, 125, 126, 210, 240

LSF Line Spread Function, the response of a spectrograph to a source whose spectrum is Dirac delta distribution. 51, 69, 71, 125–131

luminosity distance A distance in cosmology that appears in the expression relating the luminosity and the flux of an object. 27, 28

M/L Mass-to-light ratio defined as the ratio between the mass and the luminosity (or flux depending on definitions) in a given aperture. The mass-to-light ratio can be assumed constant throughout a galaxy or varying. 115, 134, 199, 200, 202, 214, 222, 229, 231–235, 238

MACHOs Massive astrophysical compact halo objects are DM candidates that would be composed of baryonic but low-luminosity matter such as black holes, neutron stars, white dwarfs, or brown dwarfs. 45

MIR Mid-infrared, a sub-portion of the IR part of the spectrum that spans roughly from 3 µm to 25 µm. 201, 239, 240
**Moffat profile** A PSF profile used to better characterise the outer parts, first derived by Anthony F. J. Moffat in 1969. 72, 134, 211

**MOND** Modified Newtonian dynamics is a theory of modified gravity proposed by Mordehai Milgrom to account for the missing mass problem in galaxies without relying on DM. 45, 239

**MOS** Multi-Object Spectrographs, an instrument that perform spectroscopy on multiple targets at the same time, usually using a bundle of fibres whose position can be placed on the FoV. 34, 52, 53, 59, 239

**MS** The Galaxies main sequence is a scaling relation that relates the galaxies' stellar mass to their star formation rate, galaxies with larger stellar masses are also found to be more star-forming. Quenched galaxies are found below the MS and starburst galaxies are found above. 5, 6, 34, 37, 38, 42, 57, 73, 77, 78, 80–83, 138, 238

**MZR** The mass-metallicity relation is a scaling relation observed at low- and high-redshift between stellar mass and gas metallicity. 33, 34

**NFW** Navarro-Frenk-White profile, a cuspy DM profile predicted from numerical simulations and extensively used in the literature. 46, 114, 118, 119, 133, 134, 139

**NIR** Near-infrared, a sub-portion of the IR part of the spectrum that goes up to roughly 3 µm. 53, 58, 59, 201, 239, 240

**OB** Observing Block, a unit observation made of multiple combined single observations with fixed integration time. 6, 64, 65

**PA** Position angle, the angle between the major axis and a reference direction (usually the north). There are two types of PA: (i) the morphological PA which measures the direction of the morphological major axis and (ii) the kinematics PA which measures the direction of the velocity field major axis. 133

**PCA** Principal component analysis is a statistical technique that allows to define a new orthogonal basis where the first vector shares the most variance of the input data, the second vector the second most variance, and so on. 225

**photometric redshift** Redshift measured from a SED obtained from multi-band photometric observation, usually through Bayesian inference. 65

**proper distance** The distance to an object that is measured with respect to a reference distance that does not scale with the expansion of the Universe. Thus, this distance is varies when the Universe is expanding. 24, 27

**pseudo isothermal sphere** A cored extension of the isothermal sphere model that avoid that singularity at the centre and that can be used to model DM haloes. 118–120

**PSF** Point Spread Function, the response of an optical system to a point source. 53, 55, 61, 65, 69, 71, 72, 127–129, 131, 134, 211
**QSOs** Quasi-stellar objects are particularly luminous AGNs powered by the accretion of material onto a supermassive black hole. 33

**ram-pressure stripping** The removal of cold gas found in galaxies because of the pressure exerted by the ICM. 6, 38, 41

**red sequence** A population of red, passive and mostly elliptical galaxies that is seen in colour-magnitude diagrams and in the MS. 77, 78

**S/N** Signal to Noise ratio, defined as the ratio between the true signal and the noise in an image or in a spectrum. It is an indicator of data quality. 105, 125, 130, 133, 201, 213, 222, 232, 234, 235, 240

**scale height** Scale parameter for the thickness profile of thick disks which can be either a constant or vary with the radius in the plane of the disk. 94, 117

**sech law** A thickness profile derived in the approximation of an isothermal sheet represented by a sech function. 25, 28, 94, 95, 113

**SED** Spectral Energy Distribution, the wavelength or frequency variation of the flux or energy of a source for multiple broad-band filters. 29, 33, 73-75, 78, 80, 139, 199-203, 206-217, 222, 224, 231-235, 240

**SFH** Star formation history, the evolution of the SFR across cosmic time. 73, 75, 77, 78, 80, 82, 208-210, 214, 216, 235

**SFR** Star Formation Rate, a measure of the total mass of new stars being born in a given amount of time. 33, 34, 37-42, 47, 65, 69, 73-75, 77, 78, 80-83, 138, 200, 201, 206, 209, 211, 213, 216, 224, 225

**SHMR** The stellar-to-halo mass relation is a scaling relation that links on a statistical level the stellar mass of the galaxies with the mass of their host dark matter halo. 119, 120

**sky subtraction** Processing step that removes from input data signal coming from foreground and background unresolved sources. 65

**SMF** The stellar mass function is a measure of the comoving stellar mass density as a function of stellar mass. It characterises how many galaxies with a given mass there are. 41

**SOM** A self-organising map, also called Kohonen map, is an unsupervised machine learning algorithm originally created to reduce the number of dimensions in a multi-dimensional dataset. 224-226, 228-235, 240, 244

**spaxel** Spatial pixel, the name given for a pixel in a data cube for a given spectral slice. A spaxel can only vary in space, not in wavelength. 51, 53, 55, 59, 69, 71, 122-126, 128-130, 133, 201, 239, 244

**spexel** Spectral pixel, the name given for a pixel in a data cube along the spectral dimension, that is at a fixed spatial position. 51, 55, 67, 69, 124, 126
**SSP** Single Stellar population, a cluster of stars with various ages all born as a result of the same star formation event. By extension, it designates the spectrum of the whole cluster as a function of time. To model a SSP, one must use a model for stellar evolution and assume given IMF and metallicity. 73, 75, 208–210, 214, 216

**starvation** The cessation of cold gas accretion from the CGM. 41

**strangulation** A synonym of starvation. 41

**Sérsic profile** A general parametric profile used to describe the surface brightness distribution of disk-like and elliptical galaxies, derived by José Luis Sérsic. 88, 89, 97, 99, 102, 113, 201

**TFR** Tully-Fisher relation, a scaling relation between stellar mass or luminosity and velocity first described by Richard Brent Tully and James Richard Fisher in 1977. 6, 33–35, 37, 39, 43, 80, 137, 138, 238

**thermal evaporation** The removal of the cold gas in a galaxy through heating when interacting with the hot ICM. 38, 41

**throughput** The efficiency of an optical system to let light pass through. It characterises the loss of incoming flux from the beginning of the optics to the detector at the end. 51, 206, 207

**total-to-selective extinction ratio $R_V$**, the ratio of the total extinction in the V band over the difference between the extinctions in the B and V bands. 74, 75

**UV** Ultraviolet, the part of the spectrum located between the X-ray part and the visible that roughly spans from 100 Å to 3000 Å. 33, 34, 58, 200, 201, 210, 240

**WCS** World coordinate system, the physical coordinates attached to some observations to locate them on the sky. 211

**WDM** Warm Dark Matter, an type of DM that has a theoretical mass below that of CDM but above that of neutrinos. 44

**wide-field mode** The largest of the two observing modes for MUSE with a 1' × 1' FoV and pixel scale of 0.2". 61

**Zhao** A parametric DM profile first suggested by HongSheng Zhao. 118
First author or significant contributions

**Epinat et al. (in prep.)**

**MAGIC:** MUSE-gAlaxy Groups In Cosmos - I. A survey of galaxies in groups at intermediate redshift

Paper describing the *MAGIC* survey. It includes the observing strategy, an evaluation of its completeness, a description of the spectroscopic redshift determination, the structure determination, and a thorough discussion of the morpho-physical properties of the galaxies as a function of environment.

**Contribution:** Morphological and kinematics modellings of the galaxies, measurements of their key properties ([OII] flux, bulge-to-total flux ratio, size, DM fraction, velocity dispersion, etc.), pre-analysis of the morpho-dynamics dependence of galaxies with environment

**Keywords:** Galaxy morpho-dynamics, large spectroscopic survey, galaxy groups/clusters, structure characterisation, red sequence, HST, MUSE, MAGIC

**Mercier et al. (in prep., 2023)**

**Impact of the environment on the angular momentum of $z \sim 0.25 - 1.5$ galaxies in the MAGIC survey**

Analysis of the stellar angular momentum as a function of environment, bulge fraction, and stellar mass for intermediate redshift galaxies in the MAGIC survey. This paper includes a new method to derive the galaxies' angular momentum that uses high-resolution HST images and robust kinematics mass models derived from MUSE data. This analysis discusses mainly the impact of the environment on the Fall relation and how this relates to the fraction of mass found in galactic bulges. We show that massive galaxies are depleted in angular momentum by $\sim 20\%$ with respect to the field and that it correlates with high-density environments.

**Contribution:** Nearly everything. Including for the major parts: the development of the new method to measure angular momentum, the sample selection, assessment of the method performance, analysis, and writing of the entire paper.

**Keywords:** Galaxy dynamics, angular momentum, Fall relation, galaxy groups/clusters, bulges, HST, MUSE, MAGIC

**Mercier et al. (2022)**

**Scaling relations of $z \sim 0.25 - 1.5$ galaxies in various environments from the morpho-kinematics analysis of the MAGIC sample**

A self-consistent joint analysis of the size-mass relation, MS relation, and TFR for galaxies in the field and in environments of various density. The galaxies' morphology is obtained from high-resolution HST images and the ionised gas kinematics from deep MUSE observations of intermediate redshift galaxies in the MAGIC survey using a mass modelling approach. Our self-consistent analysis and robust sample selection allow us to put the strongest constraints up to date on the impact of the environment on these scaling relation at $z \sim 0.7$. We find that there is a visible impact of massive structures on the size and SFR of galaxies at fixed stellar mass but that their dynamics is unaffected.

**Contribution:** Nearly everything. Including for the major parts: the morpho-kinematics modelling, sample selection, full analysis, and redaction of the entire paper. Excluding: data reduction.

**Keywords:** Galaxy morpho-kinematics modelling, mass models, ionised gas, size-mass relation, main sequence relation, Tully-Fisher relation, galaxy groups/clusters, quenching, baryon contraction, dark matter fraction, 3D spectroscopy, HST, MUSE, MAGIC

**Bouché et al. (2022)**

**The MUSE Extremely Deep Field: Evidence for SFR-induced cores in dark-matter dominated galaxies at $z \approx 1$**

A dynamical analysis of nine $z \approx 1$ emission line galaxies in the ultra deep MXDF survey using a mass modelling approach. The high S/N of these galaxies combined with our robust 3D modelling approach allowed to constrain the shape of DM haloes at intermediate redshift. We find that DM cores are preferred over cuspy profiles and that they correlate with higher SFR, showing an interplay between feedback processes and the formation of cored DM profiles in intermediate redshift galaxies.

**Contribution:** Morphological modelling used in the mass models and in the kinematics modelling of the ionised gas

**Keywords:** Galaxy dynamics, mass modelling, ionised gas, dark matter, cusp-core problem, 3D spectroscopy, HST, MUSE, MXDF

**Abril-Melgarejo et al. (2021)**

**The Tully-Fisher relation in dense groups at $z \sim 0.7$ in the MAGIC survey**

First analysis with the MAGIC survey of the TFR as a function of environment. Our MUSE-derived galaxies' kinematics is compared to that of other galaxies in shallower groups. An effect of the environment is found, consistent with baryon contraction and/or quenching.

**Contribution:** Morphological characterisation and sample selection for the analysis of the TFR as a function of environment.

**Keywords:** Galaxy morphology & 2D kinematics, 3D spectroscopy, MUSE, HST, TFR, large spectroscopic survey, ionised gas

**Richard et al. (2021)**

**An atlas of MUSE observations towards twelve massive lensing clusters**

A large and deep spectroscopic survey of 12 massive lensing clusters observed with MUSE providing spatially resolved observations of galaxies in clusters and magnified background sources.

**Contribution:** Led the Toulouse redshift determination team.

**Keywords:** Galaxy clusters, strong lensing, 3D spectroscopy, MUSE, large spectroscopic survey, redshift measurement
## Minor contributions

**Langan et al. (in prep., 2023)**
MusE GAs FLOw and Wind (MEGAFLOW) IX. The impact of gas flows on the relations between the mass, star formation rate and metallicity of galaxies
A joint analysis of the MS relation, MZR, and FMR for galaxies with and without gas inflows/outflows at intermediate redshift. Current results suggest a link between (i) the SFR and inflow event and (ii) the metal content and outflow events, visible in the three scaling relations.

**Keywords:** Gas flows, MgII absorption, main sequence relation, mas-metallicity relation, fundamental metallicity relation, MUSE, X-SHOOTER

**Bouché et al. (2021)**
The MUSE Hubble Ultra Deep Field Survey. XVI. The angular momentum of low-mass star-forming galaxies: A cautionary tale and insights from TNG50
A joint analysis of the angular momentum of the ionised gas for galaxies in the MUSE-HUDF survey and in the TNG50 simulation. The key result is the discovery of a correlation between the scatter in the relation and the dynamical states of galaxies.

**Keywords:** Galaxy dynamics, angular momentum, Fall relation, ionised gas, MUSE, TNG50

## Proceedings

**Bouché et al. (2021)**
Cored dark-matter profiles in $z \approx 1$ star forming galaxies
Presentation of the analysis of the shape of the dark matter haloes for nine galaxies in the MXDF survey.

**Keywords:** Galaxy dynamics, mass modelling, dark matter, core-cusp problem, ionised gas, MUSE, MXDF

**Mercier et al. (2021)**
Morpho-kinematics of galaxies in various environments at $z \sim 0.2-1.5$
Presentation of the first results of the analysis of the impact of the environment on the dynamics of galaxies at intermediate redshift in the MAGIC survey.

**Keywords:** Galaxy dynamics, mass modelling, Tully-Fisher relation, ionised gas, MUSE, MAGIC
# Appendices
<figure><img src="image_85.png" /></figure>
# Appendix A

## List of major MUSE surveys

Below are listed<fnref n="1" /> the main GTO and non-GTO MUSE surveys that are related to the study of galaxy evolution.

### A.1 Main GTO MUSE surveys

Several important GTO surveys have been carried out thanks to MUSE. On the topic of galaxy evolution the main ones are:

(i) MUSE-HDFS described in *Bacon et al. (2015)*, the first deep observation (27 h) carried out with MUSE over a portion of the Hubble Deep Field South (HDFS) and whose goal was to assess the performance of the instrument for long integration times during its commissioning. Its most significant scientific results were the first ionised gas kinematics analysis of intermediate redshift galaxies that especially probed low-mass galaxies (*Contini et al. 2016*) and the discovery of extended Lyα haloes at high redshift (*Wisotzki et al. 2016*).

(ii) MUSE-WIDE (Herenz et al. 2017; Urrutia et al. 2019), a shallow survey (1 h integration time) targeting a large area made of 100 different MUSE pointings following the release of the MUSE-HDFS. It is complementary to the MUSE-HUDF survey.

(iii) MUSE-HUDF (Bacon et al. 2017), a deep and ultra-deep MUSE survey following the release of the MUSE-HDFS that was complementary to MUSE-WIDE. Its approach was to observe first a mosaic of nine MUSE fields (hereafter MOSAIC) with a total integration time of roughly 10 h over an area of almost 10 square arcminutes and then a tenth field (hereafter UDF-10) that partially overlap the MOSAIC with an ultra-deep integration time of 31 h. Significant contributions concerning intermediate and/or high redshift galaxies were the first spatially resolved stellar kinematics modelling at $0.2 \le z \le 0.8$ (Guérou et al. 2017), the study of galactic winds at $z \approx 1.3$ (Finley et al. 2017), the measurement of the evolution of the fraction of galaxy mergers up to redshift $z \approx 6$ (Ventou et al. 2017), constraining the low-mass end of the galaxy MS relation (Boogaard et al. 2018), exploring the physical origin

<footnote n="1">¹The list is not intended to be exhaustive but rather to give a census of the important surveys that have impacted the field.</footnote>
of Mg II emission seen in some star-forming galaxies (Feltre et al. 2018), or studying the
angular momentum of the ionised gas in galaxies as a function of their redshift and
dynamical state (Bouché et al. 2021).

(iv) MusE GAs FLOw and Wind survey (MEGAFLOW) that observed 79 Mg II absorbers located along the LOS of 22 selected quasars to study galactic super-winds, inflows, and outflows in intermediate redshift galaxies (Schroetter et al. 2016, 2019, 2021; Bouché et al. 2016; Zabl et al. 2019, 2021).

(v) MXDF (Bacon et al. 2021), an extremely deep survey located within the MOSAIC and that partially overlaps the UDF-10 field and uses AO. Its design that involves a rotation of the FoV of a few degrees between each OB allows extremely deep observations of 140 h in an inner circle of 1' in diameter and deep observations of around 10 h in the outer parts. Among important results we can cite the detection of Lyα emission that traces the cosmic web at redshift 3 ≲ z ≲ 4.5 (Bacon et al. 2021), constraining the DM profile inner slope of z ≈ 1 star-forming galaxies (Bouché et al. 2022), or the serendipitous detection of a large Mg II emission nebula that spans the intra-group medium of a galaxy group located at z ≈ 1.3 (Leclercq et al. 2022).

(vi) MAGIC (Epinat et al., in prep.), a deep survey of 17 MUSE fields that was specifically designed to study the impact of the environment on the dynamics of galaxies at z ~ 0.7 by targeting already known structures in the COSMOS field as well as foreground and background sources (see Chapter 3 for more details). Current publications concern the first detection of ram-pressure stripping at z ~ 0.7 (Boselli et al. 2019), the serendipitous detection of a unique large ionised gas structure in a galaxy group (Epinat et al. 2018), and the study of the impact of the environment on the TFR, MS relation, and size-mass relation at intermediate redshift (Abril-Melgarejo et al. 2021; Mercier et al. 2022).

(vii) MUSCATEL (PIs: Lutz Wisotzki, Roland Bacon, and Thierry Contini), an ongoing deep MUSE survey with AO targeting four parallel fields in the HFF survey (Lotz et al. 2017). The design is composed of one deep 25 h observation per parallel field surrounded by four 5 h FoVs and nine shallower FoVs of roughly 2 h that map the entire four parallel fields observed by the HST and SPITZER. The fact that the survey targets parallel fields allows to minimise the effect of lensing while still having access to the deepest HST observations ever taken combined with deep IR SPITZER data. Expected science cases are the study of the physical mechanisms at the origin of high-redshift Lyα haloes and their dependence on environment, as well as building a statistically significant large sample of intermediate redshift galaxies to study their spatially resolved dynamics.

## A.2 Main non-GTO MUSE surveys

Similarly, important non-GTO surveys that probe the evolution of galaxies have also been carried
out with MUSE, including:

(i) GAs Stripping Phenomena in galaxies with MUSE (GASP) described in Poggianti et al. (2017), a survey aimed at studying with MUSE gas stripping phenomena by observing galaxies in the local Universe. It has led to a significant series of publications on that topic,
starting with Poggianti et al. (2017) and currently continuing with Luber et al. (2022) and
Deb et al. (2022).

(ii) Middle Ages Galaxy Properties with Integral Field Spectroscopy (MAGPI) described in Foster et al. (2021), an ongoing survey targeting 60 central galaxies and around 100 satellites located in galaxy clusters at $z \approx 0.3$. Based on predictions from simulations, this survey was specifically designed to maximise the chance to probe the impact of the environment on galaxy properties.

(iii) MUSE Ultra Deep Field (MUDF) described in Lusso et al. (2019) and Fossati et al. (2019), an extremely deep survey of 200 h on-source integration time made of two largely overlapping MUSE field. The survey targeted a region of the sky with multiple structures and two bright quasars that allowed the detection of seven galaxy groups at $0.5 < z < 1.5$, four of which showed the presence of cold gas in the intra-group medium through Mg II absorption.

(iv) MUSE and ALMA Unveiling the Virgo Environment (MAUVE)<fnref n="2" />, a future survey that will combine MUSE and Atacama Large Millimeter/submillimeter Array (ALMA) observations of 40 massive disk galaxies in the Virgo cluster. The survey is designed to study the link between the cold gas found in galaxies, star formation, and the galaxies' environment.

<footnote n="2">²https://astronomyaustralia.org.au/blog/news/australian-based-astronomers-to-take-a-deep-dive-into-the-cosmos-with-time-awarded-on-one-of-esos-most-powerful-instruments/</footnote>
# Appendix B

## MAGIC field-of-views

Here are shown the MUSE FoVs for the 15 targeted groups. Even though these figures illustrate the MAGIC survey and are discussed in various parts of Sect. 3, I moved them to their own appendix to save space in the main part of the manuscript.
<figure><img src="image_86.png" /><figcaption>Figure B.1: HST image of the MUSE observation CGr23. The main MUSE FoV is represented as a square with dimensions roughly equal to 1' × 1', with north and east pointing up and left, respectively. The galaxies observed in the FoV are colour-coded according to their MUSE spectroscopic redshift. Field galaxies are shown with crosses whereas each structure is shown with its own marker (see the top left part of the main plot). Field galaxies with low confidence flag (CONFID) are shown with smaller markers (by definition there are no such galaxies in the structures). On the top is shown the redshift histogram for field galaxies, segregated between low (black dashed line) and high (black solid line) CONFID value. The structures are shown with vertical lines with the same colour as that of the galaxies in the main plot and below each line is shown the corresponding symbol used in the main plot.</figcaption></figure>
<figure><img src="image_87.png" /><figcaption>Figure B.2: HST image of the MUSE observation CGR26. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_88.png" /><figcaption>Figure B.3: HST image of the MUSE observation CGR28. See Fig. B.1 for the legend. One group is not shown in the histogram because its redshift is beyond the plot limits but its symbol and its redshift are indicated below.</figcaption></figure>
<figure><img src="image_89.png" /></figure>

Figure B.4: HST image of the MUSE observation CGR30. See Fig. B.1 for the legend. Three groups are not shown in the histogram because their redshift is beyond the plot limits, but their symbol and their redshift are indicated below. The reason why a large number of galaxies is found beyond the FoV limits is that these galaxies were observed in an early snapshot at the result of which the FoV was rotated by roughly 30° and its centre was moved to the south west.
<figure><img src="image_90.png" /><figcaption>Figure B.5: HST image of the MUSE observation CGR32. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_91.png" /><figcaption>Figure B.6: HST image of the MUSE observation CGR34. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_92.png" /><figcaption>Figure B.7: HST image of the MUSE observation CGR35. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_93.png" /><figcaption>Figure B.8: HST image of the MUSE observation CGR51. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_94.png" /><figcaption>Figure B.9: HST image of the MUSE observation CGR61. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_95.png" /><figcaption>Figure B.10: HST image of the MUSE observation CGR79. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_96.png" /><figcaption>Figure B.11: HST image of the MUSE observations CGR84-M1 and CGR84-M2. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_97.png" /><figcaption>Figure B.12: HST image of the MUSE observation CGR87. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_98.png" /><figcaption>Figure B.13: HST image of the MUSE observation CGR114. See Fig. B.1 for the legend.</figcaption></figure>
<figure><img src="image_99.png" /><figcaption>Figure B.14: HST image of the MUSE observation CGR172. See Fig. B.1 for the legend.</figcaption></figure>
# Appendix C

## Derivation of the Jeans' equations and their applications

In this section, we provide a complete derivation of the theoretical thickness profile one can get for a disk galaxy when making the isothermal sheet approximation. This derivation was first proposed by Spitzer (1942) and Camm (1950) as an analytical solution for a self-gravitating distribution of stars. Here, the term "isothermal sheet" refers to the assumption that the velocity dispersion and thus the temperature do not vary with z, where z represents the vertical coordinate with respect to the x-y plane of symmetry of the disk galaxy. The reason to provide the derivation in this appendix was twofold: (i) to give a quick access for any reader having a look at Sect. 4.1.2.2 who might be interested in the derivation of the sech law and (ii) to have a concise and hopefully clear version of the demonstration that uses more modern notations for some parts.

### C.1 From Boltzmann's to Jeans' equations

First, let us start with the derivation of the Jeans' equations since they are the basis for both the asymmetric drift correction discussed in Sect. 5.2.2 and the theoretical thickness profile for the isothermal sheet in Sect. 4.1.2.2. The starting point of the derivation is to consider that stars are collisionless particles that can be described by their distribution function $f(t, \vec{x}, \vec{v})$ and that their mass is conserved along a flow in phase space, that is $df/dt = 0$. This is the so-called Boltzmann's equation:

$$ \frac{\partial f}{\partial t} + \frac{\vec{\partial} \cdot \vec{\partial} f}{\partial \vec{x}} + \frac{\vec{\partial} \cdot \vec{\nabla} f}{\partial \vec{v}} = 0, \qquad (C.1) $$

where we have made the following definitions

$$ \vec{\nabla} = \sum_{i} \frac{\partial f}{\partial x_{i}} \hat{e}_{i} \qquad \vec{\nabla}_{\vec{v}} = \sum_{i} \frac{\partial f}{\partial v_{i}} \hat{e}_{i} \qquad (C.2) $$

$$ \frac{d\vec{x}}{dt} = \sum_{i} \frac{dx_{i}}{dt} \hat{e}_{i} \qquad \frac{d\vec{v}}{dt} = \sum_{i} \frac{dv_{i}}{dt} \hat{e}_{i}. \qquad (C.3) $$
The first line corresponds to the gradient operators in position and velocity space, respectively, and the second line represents the vectors of the time derivative of the position and velocity vector coordinates, respectively. We note that, depending on the coordinate system used, these two vectors may not correspond to the velocity and acceleration vectors since they do not take into account the time derivative of the basis vectors. In other terms, based on the coordinate system used, the following equalities may not hold: $d\vec{x}/dt = d\vec{x}/dt$ and $d\vec{v}/dt = d\vec{v}/dt$.

## C.2 Jeans' equations in Cartesian coordinates

In Eq. C.1, we see the acceleration which, in Cartesian coordinates, matches the gradient of the gravitational potential $\Phi$, and we also have $\vec{v} = d\vec{x}/dt$. Thus, Boltzmann's equation becomes

$$ \frac{\partial f}{\partial t} + \vec{v} \cdot \vec{\nabla} f - \vec{\nabla} \Phi \cdot \vec{\nabla} \vec{v} f = 0. \qquad (C.4) $$

Let us remind a few important properties of the distribution function. To start with, depending on the convention used, the integral of $f$ in phase space yields either the total number of stars or the total stellar mass at time $t$. In what follows, we will use the latter convention. Thus, it follows that the integral of $f$ in velocity space yields the stellar mass density $\rho_M$, that is

$$ \rho_M = \int_{\vec{v} \in \mathbb{R}^3} d^3 \vec{v} f. \qquad (C.5) $$

Another useful property is that for all $i, j$ the integral of $v_j \partial f / \partial v_i$ in velocity space always yields 0 if $i \neq j$ and $-\rho_M$ otherwise since we have

$$ \begin{aligned} \int_{\vec{v} \in \mathbb{R}^3} d^3 \vec{v} v_j \frac{\partial f}{\partial v_i} &= \underbrace{[v_j f]^{+\infty}}_{0} - \underbrace{\int_{\vec{v} \in \mathbb{R}^3} d^3 \vec{v} \frac{\partial v_j}{\partial v_i} f}_{\delta_{ij}} && (C.6) \\ &= -\delta_{ij} \rho_M, \end{aligned} $$

where we have used the constraint that the distribution function should rapidly vanish at infinity. To compute the first Jeans' equations, we also need to define the average velocity vector $\langle \vec{v} \rangle$ at each position in space. Here, the average must be understood as the mean velocity of all the stars given their distribution function $f$, thus for any component $\langle v_i \rangle$ we have

$$ \langle v_i \rangle = \frac{1}{\rho_M} \int_{\vec{v} \in \mathbb{R}^3} d^3 \vec{v} v_i f. \qquad (C.7) $$

The first Jeans' equations can be derived by integrating Eq. C.4 in velocity space:

$$ \frac{\partial \rho_M}{\partial t} + \vec{\nabla} \cdot (\rho_M \langle \vec{v} \rangle) = 0, \qquad (C.8) $$

where we have used that $\vec{\nabla} \cdot (f \vec{v}) = \vec{\nabla} f \cdot \vec{v}$, with $\vec{\nabla}$ the divergence operator. The second Jeans' equations (which is actually a set of three equations) can be obtained by multiplying Eq. C.4 by $\vec{v}$ and integrating in velocity space:
$$ \frac{\partial(\rho_M \langle \vec{v} \rangle)}{\partial t} + \vec{\nabla} \cdot (\rho_M \langle \vec{v} \otimes \vec{v} \rangle) + \rho_M \vec{\nabla} \Phi = 0, \qquad (C.9) $$

where we have used the fact that $\partial(f\vec{v})/\partial t = \vec{v} \partial f/\partial t$ and where $\otimes$ denotes the tensor product operation. Alternatively, the second Jeans' equations can be rewritten by developing the first term in Eq. C.9, using Eq. C.8 and defining the stress tensor $\overline{\sigma^2}$ as

$$ \begin{aligned} \overline{\sigma^2} &= \langle (\vec{v} - \langle \vec{v} \rangle) \otimes (\vec{v} - \langle \vec{v} \rangle) \rangle \\ &= \langle \vec{v} \otimes \vec{v} \rangle - \langle \vec{v} \rangle \otimes \langle \vec{v} \rangle. \end{aligned} \qquad (C.10) $$

The alternate form for the second Jeans' equations therefore writes

$$ \rho_M \frac{\partial \langle \vec{v} \rangle}{\partial t} + \rho_M (\langle \vec{v} \rangle \cdot \vec{\nabla}) \langle \vec{v} \rangle + \vec{\nabla} \cdot (\rho_M \overline{\sigma^2}) + \rho_M \vec{\nabla} \Phi = 0. \qquad (C.11) $$

## C.3 Jeans' equations in cylindrical coordinates

Now we turn to the Jeans' equations written in cylindrical coordinates since this coordinate system is more appropriate for rotating axisymmetric disks in galaxies. To do so, we need to write Boltzmann's equation in cylindrical coordinates this time. Using Eq. 5.4 for the expression of the acceleration and equating with the gradient of the gravitational potential, we get

$$ \frac{\partial v_R}{\partial t} = \frac{v_\theta^2}{R} - \frac{\partial \Phi}{\partial R}, \qquad (C.12) $$

$$ \frac{\partial v_{\theta}}{\partial t} = -\frac{1}{R} \left( \frac{v_R v_{\theta}}{R} + \frac{\partial \Phi}{\partial \theta} \right), \qquad (C.13) $$

$$ \frac{\partial v_z}{\partial t} = -\frac{\partial \Phi}{\partial z}, \qquad (C.14) $$

which we can plug into Eq. C.1 to retrieve its expression in cylindrical coordinates

$$ \begin{aligned} & \frac{\partial f}{\partial t} + v_R \frac{\partial f}{\partial R} + \frac{v_\theta}{R} \frac{\partial f}{\partial \theta} + v_z \frac{\partial f}{\partial z} + \left( \frac{v_\theta^2}{R} - \frac{\partial \Phi}{\partial R} \right) \frac{\partial f}{\partial v_R} \\ & - \frac{1}{R} \left( v_R v_\theta + \frac{\partial \Phi}{\partial \theta} \right) \frac{\partial f}{\partial v_\theta} - \frac{\partial \Phi}{\partial z} \frac{\partial f}{\partial v_z} = 0. \end{aligned} \qquad (C.15) $$

For an axisymmetric disk, Eq. C.15 can be further simplified by considering that the mass density $\rho_M$ and therefore the distribution function $f$ and the gravitational potential $\Phi$ must be independent of the azimuthal angle $\theta$. Then, integrating Eq. C.15 in velocity space yields the first Jeans' equations in cylindrical coordinates for an axisymmetric disk:

$$ \frac{\partial \rho_M}{\partial t} + \frac{1}{R} \frac{\partial (\rho_M R \langle v_R \rangle)}{\partial R} + \frac{\partial (\rho_M \langle v_z \rangle)}{\partial z} = 0. \qquad (C.16) $$
The three other Jeans' equations can be derived by integrating in velocity space the product of Eq. C.15 with each of the three velocity vector components and using the same tricks as in Sect. C.2

$$ \frac{\partial(\rho_M \langle v_R \rangle)}{\partial t} + \frac{\partial(\rho_M \langle v_R^2 \rangle)}{\partial R} + \frac{\partial(\rho_M \langle v_R v_z \rangle)}{\partial z} + \rho_M \left[ \frac{\langle v_R^2 \rangle - \langle v_z^2 \rangle}{R} + \frac{\partial \Phi}{\partial R} \right] = 0, \quad (C.17) $$

$$ \frac{\partial(\rho_M \langle v_\theta \rangle)}{\partial t} + \frac{\partial(\rho_M \langle v_R v_\theta \rangle)}{\partial R} + \frac{\partial(\rho_M \langle v_\theta v_z \rangle)}{\partial z} + \frac{2\rho_M}{R} \langle v_R v_\theta \rangle = 0, \quad (C.18) $$

$$ \frac{\partial(\rho_M \langle v_z \rangle)}{\partial t} + \frac{\partial(\rho_M \langle v_R v_z \rangle)}{\partial R} + \frac{\partial(\rho_M \langle v_z^2 \rangle)}{\partial z} + \rho_M \left[ \frac{\langle v_R v_z \rangle}{R} + \frac{\partial \Phi}{\partial z} \right] = 0. \quad (C.19) $$

## C.4 Isothermal sheet thickness profile

A common vertical profile used for disk galaxies is the isothermal sheet profile. This profile can be derived by considering a sheet at a given position $R$ in steady state with a constant temperature and therefore velocity dispersion. The vertical support of this sheet is described by Eq. C.19 after getting rid of derivatives with respect to $t$ and $R$ and by assuming a diagonal stress tensor (i.e. $\langle v_R v_z \rangle = 0$):

$$ \frac{d\rho_M}{dz} = - \frac{\rho_M}{\langle v_z^2 \rangle} \frac{\partial \Phi}{\partial z}. \quad (C.20) $$

This equation can be solved using Poisson's equation. Similarly to Jeans' equations, the vertical gradient will mostly contribute to the dynamical support of the sheet and therefore Poisson's equation can be written as

$$ \frac{\partial^2 \Phi}{\partial z^2} = -4\pi G \rho_M. \quad (C.21) $$

The differential equation to solve in order to find the vertical profile therefore writes

$$ \frac{d^2 \log \rho_M}{dz^2} = \frac{4\pi G \rho_M}{\langle v_z^2 \rangle}. \quad (C.22) $$

Considering the boundary condition $\rho_M(z \to \infty) = 0$, the solution of Eq. C.22 is given by (Spitzer 1942; Camm 1950)

$$ \rho_M(R, z) = \rho_{M,0}(R) \operatorname{sech}^2(z/z_0(R)), \quad (C.23) $$

where $z_0(R) = \sqrt{\langle v_z^2 \rangle/(2\pi G \rho_{M,0}(R))}$ with $\rho_{M,0}(R) = \rho_M(R, z=0)$ the mass density in the plane of the disk. This solution is usually referred to as the sech law.
# Appendix D

## Impact of convolution on the variance

In this appendix, I derive the impact of the convolution of a signal $S$ by a kernel $K$ on its variance map. This result was used in Sect. 8.1.2.2 to produce resolved map using pixel-per-pixel SED fitting. We consider a signal that is represented by a random variable $S(x_0, y_0)$ at position $(x_0, y_0)$ on an image. By definition, we can estimate its variance as $Var_S = \sum_i (S_i - \bar{S})^2/n$, where $S_i$ is a single realisation of the random variable, $n$ is the number of realisations, and $\bar{S}$ is its average value. We could also use $n-1$ instead of $n$ to have an unbiased estimator but the demonstration would remain the same. Now, we consider a new signal $S'$ that has been convolved with a kernel $K$, hence

$$ 
\begin{aligned}
S'(x_0, y_0) &= [S \otimes K](x_0, y_0) \\
&= \sum_{x,y} S(x, y) K(x_0 - x, y_0 - y).
\end{aligned} 
\quad (D.1) 
$$

Its variance can be estimated in the same way. If we develop the terms we have

$$ Var_{S'}(x_0, y_0) = \frac{1}{n} \sum_i \left[ \sum_{x,y} S_i(x, y) K(x_0 - x, y_0 - y) - \frac{1}{n} \sum_j \sum_{x,y} S_j(x, y) K(x_0 - x, y_0 - y) \right]^2 . \quad (D.2) $$

Inverting the two sums in the last term and combining the sum along $x$ and $y$ for the two terms, we get

$$ Var_{S'}(x_0, y_0) = \frac{1}{n} \sum_i \left[ \sum_{x,y} K(x_0 - x, y_0 - y) [S_i(x, y) - \bar{S}(x, y)] \right]^2 . \quad (D.3) $$

The square can be written as the product of a sum along $x$ and $y$ with a sum along $x'$ and $y'$, in which case we can rewrite it as a first sum when $x = x'$ and $y = y'$ and a second sum when $x \neq x'$ and/or $y \neq y'$, that is