Quasars at 0.2<z<0.80.2<z<0.8 whose host stellar velocity dispersions are inferred from the galaxy mass fundamental plane sit systematically above the local MBHM_{\rm BH}–σ∗\sigma_\ast relation, with offsets growing toward low σ∗\sigma_\ast, while their MBH/M∗M_{\rm BH}/M_\ast ratios match local values. Silverman, Li & Ding (2022) showed that an evolving intrinsic MBHM_{\rm BH}–σ∗\sigma_\ast relation (parametrized by α=−1.4\alpha=-1.4, β=25\beta=25) can reproduce the observations, but hypothesized instead that the apparent flattening is a projection of a non-evolving intrinsic MBHM_{\rm BH}–M∗M_\ast relation combined with quasar selection effects and ongoing host bulge growth. Deciding between these readings — determining the intrinsic intermediate-redshift MBHM_{\rm BH}–σ∗\sigma_\ast relation with virial-mass, host-size, and forward-model systematics controlled, validating the fundamental-plane assumption for quasar hosts, and establishing whether supermassive black hole mass is more fundamentally coupled to total stellar mass, bulge mass, or velocity dispersion — remains open. Post-2022 work has found negligible evolution of the MBHM_{\rm BH}–M∗M_\ast relation at z∼0.5z\sim0.5 using reverberation masses, shown that the flattened MBHM_{\rm BH}–σ∗\sigma_\ast relation persists in 4,137 direct spectroscopic measurements, and sharpened the local benchmark, but no study has yet determined the intrinsic relation with these systematics controlled in a way that settles the two readings.

Background

Supermassive black holes (SMBHs) in galaxy centers correlate tightly with several properties of their host galaxies. In the nearby universe the tightest such scaling relation links black hole mass MBHM_{\rm BH} to the stellar velocity dispersion σ∗\sigma_\ast — the line-of-sight rms of random stellar motions in a galaxy's central bulge, measured from the width of stellar absorption lines in a spectrum — with an intrinsic scatter of only ∼0.3\sim0.3 dex (a factor of about two in mass; Kormendy & Ho 2013). Whether this MBHM_{\rm BH}–σ∗\sigma_\ast relation evolves with cosmic time is a central test of black-hole–galaxy co-evolution, because the relation is widely interpreted as the fossil record of how accretion feedback regulated bulge formation.

Testing the relation beyond the local universe requires active galaxies, since dormant black holes cannot be weighed dynamically at cosmological distances. In broad-line quasars — luminous active galactic nuclei (AGN) — MBHM_{\rm BH} is usually estimated 'virially' from the width of broad emission lines together with the continuum luminosity, an empirical recipe carrying ∼0.3\sim0.3–0.50.5 dex systematic uncertainty; the higher-fidelity alternative is reverberation mapping, in which monitoring the time delay between continuum and line emission yields the broad-line-region radius and hence a more direct mass. Measuring σ∗\sigma_\ast in quasar hosts is difficult in its own right: the bright nucleus outshines the stellar absorption lines, so pre-2022 samples with direct spectroscopic σ∗\sigma_\ast were small (tens of objects; e.g. the 88 Sloan Digital Sky Survey (SDSS) Reverberation Mapping quasars of Shen et al. 2015).

Silverman, Li & Ding (2022) sidestepped this spectroscopic bottleneck for 584 SDSS quasars at 0.2<z<0.80.2<z<0.8 whose hosts were characterized from Hyper Suprime-Cam imaging (a wide-field camera on the Subaru Telescope): they inferred σ∗\sigma_\ast from each host's stellar mass M∗M_\ast and effective (half-light) radius via the galaxy mass fundamental plane — the tight empirical relation among M∗M_\ast, size, and σ∗\sigma_\ast obeyed by inactive galaxies — under the assumption that quasar hosts obey the same plane. The quasars lie elevated above the local MBHM_{\rm BH}–σ∗\sigma_\ast relation, with offsets growing toward lower σ∗\sigma_\ast (an apparent flattening), while their MBH/M∗M_{\rm BH}/M_\ast ratios remain consistent with local values (Li et al. 2021). Forward modeling — simulating the observed population from an assumed intrinsic relation plus the quasar selection function, the probability that a quasar of given properties enters the flux-limited sample — admits two readings: either the intrinsic MBHM_{\rm BH}–σ∗\sigma_\ast relation genuinely evolves with redshift (their best-fitting evolutionary parametrization is log⁡MBH=4.58×(1+z)α×log⁡σ∗p−2.31+β×log⁡(1+z)\log M_{\rm BH}=4.58\times(1+z)^{\alpha}\times\log\sigma_\ast^{p}-2.31+\beta\times\log(1+z) with α=−1.4\alpha=-1.4 and β=25\beta=25, where σ∗p\sigma_\ast^{p} is the fundamental-plane-predicted velocity dispersion), or the intrinsic relation does not evolve and the flattening is a projection of a non-evolving MBHM_{\rm BH}–M∗M_\ast relation combined with selection effects and the fact that high-redshift bulges are still growing. Shen et al. (2015) had earlier concluded from direct measurements that selection effects could account for practically all of the observed flattening (no evolution); the 2022 forward modeling found selection insufficient. The same study reports 'migration' of the most massive, quiescent, or compact hosts onto the local relation, consistent with bulges still assembling at these epochs. Independently, Shankar et al. (2016) argued from residual analysis and selection-bias modeling that locally σ∗\sigma_\ast — not M∗M_\ast — is the more fundamental quantity, and that the local MBHM_{\rm BH}–M∗M_\ast relation is largely a projection of the MBHM_{\rm BH}–σ∗\sigma_\ast and σ∗\sigma_\ast–M∗M_\ast relations. Which host property SMBH mass is fundamentally coupled to is therefore contested both locally and at intermediate redshift.

Problem Statement

Determine the intrinsic MBHM_{\rm BH}–σ∗\sigma_\ast relation of broad-line quasars at intermediate redshift (0.2≲z≲0.80.2\lesssim z\lesssim 0.8) with measurement and systematic uncertainties controlled, and decide whether the observed offset and flattening relative to the local relation reflects genuine redshift evolution of the intrinsic MBHM_{\rm BH}–σ∗\sigma_\ast relation, or is instead a projection of a non-evolving intrinsic MBHM_{\rm BH}–M∗M_\ast relation combined with quasar selection effects and ongoing host bulge growth.

The following related questions serve this single objective: (a) validate, or correct, the assumption underlying fundamental-plane inference of σ∗\sigma_\ast — that quasar hosts follow the same stellar-mass fundamental plane as inactive galaxies at comparable epochs — across host types (star-forming versus quiescent, disk-dominated versus bulge-dominated) and redshift; (b) quantify how the fitted intrinsic slope and normalization depend on the assumed virial black-hole-mass uncertainties (e.g. 0.3 versus 0.4 dex) and on the adopted host size–mass relations; (c) propagate the input-distribution choices of the forward modeling into the inferred intrinsic relations — the stellar mass function (the space density of galaxies as a function of stellar mass), host-size distributions, the Eddington-ratio distribution (accretion luminosities in units of the Eddington luminosity at which radiation pressure balances gravity), bolometric corrections (the factors converting a luminosity measured in one photometric band into the total radiated luminosity), and selection functions — so that the robustness of the conclusions is ascertained; (d) assess whether the proposed evolving parametrization (α=−1.4\alpha=-1.4, β=25\beta=25) — offered by the source only as an exercise, without claiming an exact form of the intrinsic high-redshift relation — is unique and physically well founded; and (e) establish whether MBHM_{\rm BH} is more fundamentally coupled to the host's velocity dispersion, its total stellar mass, or its bulge mass (the candidate couplings named in the source paper).

Of these, parts (a)–(c) are required to complete the task and part (e) is the fundamental-coupling phrasing of the same objective rather than an additional deliverable, while part (d) is an optional interpretive check on the source's illustrative parametrization that is not required to resolve the task.

An answer addresses the problem when the intrinsic intermediate-redshift MBHM_{\rm BH}–σ∗\sigma_\ast relation is measured by analysis that carefully considers measurement uncertainties and all underlying systematic effects — including the assumed level of virial black-hole-mass uncertainty, the adopted host size–mass relations, the validity of the fundamental-plane premise for quasar hosts, and the forward-modeling inputs and quasar selection function — and the resulting evidence is brought to bear on whether the observed offsets and flattening reflect genuine redshift evolution of the intrinsic MBHM_{\rm BH}–σ∗\sigma_\ast relation or a non-evolving intrinsic MBHM_{\rm BH}–M∗M_\ast relation combined with selection effects and ongoing host bulge growth.

References

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