Is there a tilt in the fundamental (hyper)plane?

Aims. We investigate the fundamental plane (FP) of selected early-type member galaxies of the galaxy cluster PLCK G287.0+32.9 (zc = 0.3833), also exploring 4D hyperplane extensions. Methods. We measured early-type galaxy structural parameters and photometry from Hubble Space Telescope observations. We used high-quality spectroscopic data from the Multi Unit Spectroscopic Explorer to measure the galaxy central stellar velocity distribution and stellar population properties. With these data, we constructed the FP through a robust fitting procedure and analyzed its tilt and scatter. We then introduced two hyperplane extensions, one including the stellar mass (M∗-HP) and the other the stellar-over-total mass fraction (fe∗-HP), and compared their coefficients and scatter to those of the FP. Results. The FP of PLCK G287.0+32.9 is found to have best-fit parameter values consistent with those in the literature (α = 1.2 ± 0.1 and β = a0.75 ± 0.04), with an observed scatter of 0.088 dex. The fe∗-HP shows no tilt relative to the theoretical plane (α = 2.0 ± 0.3 and β = a1.1 ± 0.1), with an observed scatter of 0.036 dex, and the M∗-HP has an even tighter relation, with an observed scatter of only 0.036. Conclusions. Our findings support the idea that the FP is a lower-dimensional projection of a more complex hyperplane and confirm that the variations in the dark matter content contribute significantly to the tilt of the FP. Future studies incorporating larger samples of galaxies and additional physical parameters may further refine our understanding of the FP and its higher-dimensional extensions.