We present the discovery of nine quasars at z ∼ 6 identified in the Sloan Digital Sky Survey (SDSS) imaging data. This completes our survey of z ∼ 6 quasars in the SDSS footprint. Our final sample consists of 52 quasars at 5.7 < z ≤ 6.4, including 29 quasars with z AB ≤ 20 mag selected from 11,240 deg 2 of the SDSS single-epoch imaging survey (the main survey), 10 quasars with 20 ≤ z AB ≤ 20.5 selected from 4223 deg 2 of the SDSS overlap regions (regions with two or more imaging scans), and 13 quasars down to z AB ≈ 22 mag from the 277 deg 2 in Stripe 82. They span a wide luminosity range of −29.0 ≤ M 1450 ≤ −24.5. This well-defined sample is used to derive the quasar luminosity function (QLF) at z ∼ 6. After combining our SDSS sample with two faint (M 1450 ≥ −23 mag) quasars from the literature, we obtain the parameters for a double power-law fit to the QLF. The bright-end slope β of the QLF is well constrained to be β = −2.8 ± 0.2. Due to the small number of low-luminosity quasars, the faint-end slope α and the characteristic magnitude M * 1450 are less well constrained, with α = −1.90The spatial density of luminous quasars, parametrized as ρ(M 1450 < −26, z) = ρ(z = 6) 10 k(z−6) , drops rapidly from z ∼ 5 to 6, with k = −0.72 ± 0.11. Based on our fitted QLF and assuming an IGM clumping factor of C = 3, we find that the observed quasar population cannot provide enough photons to ionize the z ∼ 6 IGM at ∼ 90% confidence. Quasars may still provide a significant fraction of the required photons, although much larger samples of faint quasars are needed for more stringent constraints on the quasar contribution to reionization.

This paper addresses the question of whether there are 4D Lorentz invariant unitary quantum field theories with scale invariance but not conformal invariance. An important loophole in the arguments of Luty-Polchinski-Rattazzi and DymarskyKomargodski-Schwimmer-Theisen is that trace of the energy-momentum tensor T could be a generalized free field. In this paper we rule out this possibility. The key ingredient is the observation that a unitary theory with scale but not conformal invariance necessarily has a non-vanishing anomaly for global scale transformations. We show that this anomaly cannot be reproduced if T is a generalized free field unless the theory also contains a dimension-2 scalar operator. In the special case where such an operator is present it can be used to redefine ("improve") the energy-momentum tensor, and we show that there is at least one energy-momentum tensor that is not a generalized free field. In addition, we emphasize that, in general, large momentum limits of correlation functions cannot be understood from the leading terms of the coordinate space OPE. This invalidates a recent argument by Farnsworth-Luty-Prilepina (FLP). Despite the invalidity of the general argument of FLP, some of the techniques turn out to be useful in the present context.

Gauge-gravity duality is arguably our best hope for understanding quantum gravity. Considerable progress has been made in relating scattering amplitudes in certain gravity theories to those in gauge theories — a correspondence dubbed the double copy. Recently, double copies have also been realized in a classical setting, as maps between exact solutions of gauge theories and gravity. We present here a novel map between a certain class of real, exact solutions of Einstein’s equations and self-dual solutions of the flat-space vacuum Maxwell equations. This map, which we call the Newman-Penrose map, is well-defined even for non-vacuum, non-stationary spacetimes, providing a systematic framework for exploring gravity solutions in the context of the double copy that have not been previously studied in this setting. To illustrate this, we present here the Newman- Penrose map for the Schwarzschild and Kerr black holes, and Kinnersley’s photon rocket.