Context. Giant radio galaxies (GRGs, or colloquially 'giants') are the Universe's largest structures generated by individual galaxies. They comprise synchrotron-radiating AGN ejecta and attain cosmological (Mpc-scale) lengths. However, the main mechanisms that drive their exceptional growth remain poorly understood. Aims. To deduce the main mechanisms that drive a phenomenon, it is usually instructive to study extreme examples. If there exist host galaxy characteristics that are an important cause for GRG growth, then the hosts of the largest GRGs are likely to possess them. Similarly, if there exist particular large-scale environments that are highly conducive to GRG growth, then the largest GRGs are likely to reside in them. For these reasons, we aim to perform a case study of the largest GRG available. Methods. We reprocessed the LOFAR Two-metre Sky Survey (LoTSS) DR2 by subtracting compact sources and performing multiscale CLEAN deconvolution at 60 and 90 resolution. The resulting images constitute the most sensitive survey yet for radio galaxy lobes, whose di use nature and steep synchrotron spectra have allowed them to evade previous detection attempts at higher resolution and shorter wavelengths. We visually searched these images for GRGs. Results. We discover Alcyoneus, a low-excitation radio galaxy with a projected proper length l p = 4.99 ± 0.04 Mpc. Its jets and lobes are all four detected at very high signi cance, and the SDSS-based identi cation of the host, at spectroscopic redshift z spec = 0.24674 ± 6•10 −5 , is unambiguous. The total luminosity density at ν = 144 MHz is L ν = 8±1•10 25 W Hz −1 , which is below-average, though near-median (percentile 45 ± 3%), for GRGs. The host is an elliptical galaxy with a stellar mass M = 2.4 ± 0.4 • 10 11 M and a supermassive black hole mass M • = 4 ± 2 • 10 8 M , both of which tend towards the lower end of their respective GRG distributions (percentiles 25±9% and 23±11%). The host resides in a lament of the Cosmic Web. Through a new Bayesian model for radio galaxy lobes in three dimensions, we estimate the pressures in the Mpc 3 -scale northern and southern lobe to be P min,1 = 4.8 ± 0.3 • 10 −16 Pa and P min,2 = 4.9±0.6•10 −16 Pa, respectively. The corresponding magnetic eld strengths are B min,1 = 46±1 pT and B min,2 = 46±3 pT. Conclusions. We have discovered what is in projection the largest known structure made by a single galaxy -a GRG with a projected proper length l p = 4.99 ± 0.04 Mpc. The true proper length is at least l min = 5.04 ± 0.05 Mpc. Beyond geometry, Alcyoneus and its host are suspiciously ordinary: the total low-frequency luminosity density, stellar mass and supermassive black hole mass are all lower than, though similar to, those of the medial GRG. Thus, very massive galaxies or central black holes are not necessary to grow large giants, and, if the observed state is representative of the source over its lifetime, neither is high radio power. A lowdensity environment remains a possible explanation. The source resides in a lament of the C...

Context. Many massive galaxies launch jets from the accretion disk of their central black hole, but only ∼10 3 instances are known in which the associated outflows form giant radio galaxies (GRGs, or giants): luminous structures of megaparsec extent that consist of atomic nuclei, relativistic electrons, and magnetic fields. Large samples are imperative to understanding the enigmatic growth of giants, and recent systematic searches in homogeneous surveys constitute a promising development. For the first time, it is possible to perform meaningful precision statistics with GRG lengths, but a framework to do so is missing. Aims. We measured the intrinsic GRG length distribution by combining a novel statistical framework with a LOFAR Two-metre Sky Survey (LoTSS) DR2 sample of freshly discovered giants. In turn, this allowed us to answer an array of questions on giants. For example, we can now assess how rare a 5 Mpc giant is compared to one of 1 Mpc, and how much larger -given a projected lengththe corresponding intrinsic length is expected to be. Notably, we can now also infer the GRG number density in the Local Universe. Methods. We assumed the intrinsic GRG length distribution to be Paretian (i.e. of power-law form) with tail index ξ, and predicted the observed distribution by modelling projection and selection effects. To infer ξ, we also systematically searched the LoTSS DR2 for hitherto unknown giants and compiled the largest catalogue of giants to date. Results. We show that if intrinsic GRG lengths are Pareto distributed with index ξ, then projected GRG lengths are also Pareto distributed with index ξ. Selection effects induce curvature in the observed projected GRG length distribution: angular length selection flattens it towards the lower end, while surface brightness selection steepens it towards the higher end. We explicitly derived a GRG's posterior over intrinsic lengths given its projected length, laying bare the ξ dependence. We also discovered 2050 giants within the LoTSS DR2; our sample more than doubles the known population. Spectacular discoveries include the largest, second-largest, and fourth-largest GRG known (l p = 5.1 Mpc, l p = 5.0 Mpc, and l p = 4.8 Mpc), the largest GRG known hosted by a spiral galaxy (l p = 2.5 Mpc), and the largest secure GRG known beyond redshift 1 (l p = 3.9 Mpc). We increase the number of known giants whose angular length exceeds that of the Moon from 10 to 23; among the discoveries is the angularly largest known radio galaxy in the Northern Sky, which is also the angularly largest known GRG (ϕ = 2°). Combining theory and data, we determined that intrinsic GRG lengths are well described by a Pareto distribution, and measured the index ξ = −3.5 ± 0.5. This implies that, given its projected length, a GRG's intrinsic length is expected to be just 15% larger. Finally, we determined the comoving number density of giants in the Local Universe to be n GRG = 5 ± 2 100 Mpc −3 . Conclusions. We developed a practical mathematical framework that elucidates the statistics of giant ...