Single Parameter Galaxy Classification: The Principal Curve through the Multi-dimensional Space of Galaxy Properties

We propose to describe the variety of galaxies from the Sloan Digital Sky Survey by using only one affine parameter. To this aim, we construct the principal curve (P-curve) passing through the spine of the data point cloud, considering the eigenspace derived from Principal Component Analysis (PCA) of morphological, physical, and photometric galaxy properties. Thus, galaxies can be labeled, ranked, and classified by a single arc-length value of the curve, measured at the unique closest projection of the data points on the P-curve. We find that the P-curve has a "W" letter shape with three turning points, defining four branches that represent distinct galaxy populations. This behavior is controlled mainly by two properties, namely u - r and star formation rate (from blue young at low arc length to red old at high arc length), while most other properties correlate well with these two. We further present the variations of several important galaxy properties as a function of arc length. Luminosity functions vary from steep Schechter fits at low arc length to double power law and ending in lognormal fits at high arc length. Galaxy clustering shows increasing autocorrelation power at large scales as arc length increases. Cross correlation of galaxies with different arc lengths shows that the probability of two galaxies belonging to the same halo decreases as their distance in arc length increases. PCA analysis allows us to find peculiar galaxy populations located apart from the main cloud of data points, such as small red galaxies dominated by a disk, of relatively high stellar mass-to-light ratio and surface mass density. On the other hand, the P-curve helped us understand the average trends, encoding 75% of the available information in the data. The P-curve allows not only dimensionality reduction but also provides supporting evidence for the following relevant physical models and scenarios in extragalactic astronomy: (1) The hierarchical merging scenario in the formation of a selected group of red massive galaxies. These galaxies present a lognormal r-band luminosity function, which might arise from multiplicative processes involved in this scenario. (2) A connection between the onset of active galactic nucleus activity and star formation quenching as mentioned in Martin et al., which appears in green galaxies transitioning from blue to red populations.