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

¹ Università di Salerno, Dipartimento di Fisica “E.R. Caianiello”, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy
² INAF – Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Napoli, Italy
³ Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, via Saragat 1, I-44122 Ferrara, Italy
⁴ INFN – Gruppo Collegato di Salerno - Sezione di Napoli, Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno, via Giovanni Paolo II, 132 - I-84084 Fisciano (SA), Italy
⁵ Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy
⁶ INAF – IASF Milano, via A. Corti 12, I-20133 Milano, Italy
⁷ INAF – OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy
⁸ Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby Rd, Portsmouth PO1 3FX, UK
⁹ Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrejón de Ardoz, 28850 Madrid, Spain
¹⁰ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany
¹¹ Università degli Studi di Trieste, Dipartimento di Fisica – Sezione di Astronomia, via Tiepolo 11, I-34143 Trieste, Italy
¹² INAF – Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy
¹³ Faculty of Physics, University Observatory, Ludwig-Maximilians-Universität München, Munich, Germany
¹⁴ Institute for Particle Physics and Astrophysics, ETH Zürich, Zürich, Switzerland

^⋆ Corresponding author.

Received: 5 March 2025
Accepted: 24 June 2025

Abstract

Aims. We investigate the fundamental plane (FP) of selected early-type member galaxies of the galaxy cluster PLCK G287.0+32.9 (z_c = 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 (f_e^⋆-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 = −0.75 ± 0.04), with an observed scatter of 0.088 dex. The f_e^⋆-HP shows no tilt relative to the theoretical plane (α = 2.0 ± 0.3 and β = −1.1 ± 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.