Photon ring autocorrelations

Shahar Hadar, Michael D. Johnson, Alexandru Lupsasca, and George N. Wong

Phys. Rev. D 103, 104038 – Published 17 May 2021

Abstract

In the presence of a black hole, light sources connect to observers along multiple paths. As a result, observed brightness fluctuations must be correlated across different times and positions in black hole images. Photons that execute multiple orbits around the black hole appear near a critical curve in the observer sky, giving rise to the photon ring. In this paper, a novel observable is proposed: the two-point correlation function of intensity fluctuations on the photon ring. This correlation function is analytically computed for a Kerr black hole surrounded by stochastic equatorial emission, with source statistics motivated by simulations of a turbulent accretion flow. It is shown that this two-point function exhibits a universal, self-similar structure consisting of multiple peaks of identical shape: while the profile of each peak encodes statistical properties of fluctuations in the source, the locations and heights of the peaks are determined purely by the black hole parameters. Measuring these peaks would demonstrate the existence of the photon ring without resolving its thickness, and would provide estimates of black hole mass and spin. With regular monitoring over sufficiently long timescales, this measurement could be possible via interferometric imaging with modest improvements to the Event Horizon Telescope.

Received 18 October 2020
Accepted 13 April 2021

DOI:https://doi.org/10.1103/PhysRevD.103.104038

Physics Subject Headings (PhySH)

Classical black holes Fluids & classical fields in curved spacetime General relativity Radio, microwave, & sub-mm astronomy

Accretion disk & black-hole plasma Astronomical black holes

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Shahar Hadar ^1,*, Michael D. Johnson ^2,†, Alexandru Lupsasca ^3,‡, and George N. Wong ^4,§

¹Center for the Fundamental Laws of Nature, Harvard University, Cambridge, Massachusetts 02138, USA
²Center for Astrophysics | Harvard & Smithsonian, Cambridge, Massachusetts 02138, USA
³Princeton Gravity Initiative, Princeton University, Princeton, New Jersey 08544, USA
⁴Department of Physics, University of Illinois, Urbana, Illinois 61801, USA

^*shaharhadar@g.harvard.edu
^†mjohnson@cfa.harvard.edu
^‡lupsasca@princeton.edu
^§gnwong2@illinois.edu

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Vol. 103, Iss. 10 — 15 May 2021

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Images

Figure 1
Universal structure in the autocorrelation function $C (T, Φ)$ of intensity fluctuations in the photon ring, as a function of the separation in time $T$ and azimuthal angle $Φ$ around the ring. This particular plot corresponds to polar observations of random fluctuations in an equatorial disk surrounding a Kerr BH with spin $a / M = 94 %$ [Eq. (5.1)]; the same structure holds to leading order in small observer inclination [Eq. (7.18)] and for all spin. Strong lensing by the BH enables a single source to connect to a given observer along multiple paths (Fig. 2, left), giving rise to correlations within the photon ring (Fig. 2, right). These correlations display a universal structure governed by the critical exponents $γ$ , $δ$ , and $τ$ [Eqs. (2.8)] that govern BH lensing. Two light rays that are emitted from the same source and circumnavigate the BH $k$ and $k^{'}$ times, respectively, before reaching the observer contribute to a peak in the autocorrelation $C (T, Φ)$ labeled by $m = k - k^{'}$ . Here, we display the $| m | = 0$ , 1, 2, 3 peaks in red, blue, green and purple, respectively. The dashed grey lines (with $Φ = \pm 180 °$ identified) connect peaks with neighboring values of $m$ . All the peaks share an identical profile that depends on the source statistics. In particular, the peak width is set by the correlation length of fluctuations in the source; for this plot, we used the correlation lengths $ℓ_{t}$ and $ℓ_{ϕ}$ [Eq. (5.2)] inferred from the GRMHD-simulated accretion flow in Fig. 3. On the other hand, the locations and relative heights of the peaks are fixed by the BH parameters, with each successive peak an echo of its predecessor, suppressed by $e^{- γ}$ and translated by $(τ, δ)$ (colored arrows). Observations of this correlation structure in the photon ring of a BH could therefore provide a measurement of its critical exponents $τ$ and $δ$ , which would in turn produce estimates of its mass and spin.
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Figure 2
Left: Kerr black hole with spin $a / M = 94 %$ , surrounded by a geometrically and optically thin equatorial accretion disk terminating at the innermost stable circular orbit (ISCO) radius $r_{ms}$ . Right: image of the disk seen by a far observer at an inclination $θ_{o} = 17 °$ , assuming a simple stationary and axisymmetric source profile [22]. Light rays that execute multiple orbits around the black hole intersect the emission region multiple times, accruing additional intensity at every crossing (orange dots on the left) according to Eq. (2.12), and resulting in a brightness enhancement near the critical curve: the photon ring. The polar coordinates (2.10), illustrated on the right image, are generically offset from the ring centroid. Due to the strong lensing in the photon shell, light rays shot back from different times and positions on the photon ring can connect to the same spacetime event in the bulk (blue and green dots on the right image, corresponding to blue and green rays in the bulk). As a result, black hole images display autocorrelations in the photon ring.
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Figure 3
Normalized correlation function $S$ of fluctuations in 230 GHz synchrotron emissivity computed in the midplane of a numerical GRMHD simulation. The configuration consists of a magnetically arrested disk accreting onto a Kerr BH of spin $a / M \approx 94 %$ . Top left: correlation function in $T$ assuming $\hat{ϕ} = Δ r = 0$ . Top right: azimuthal correlation in $\hat{ϕ}$ for $T = Δ r = 0$ . Both panels in the top row are evaluated at $\bar{r} = {\tilde{r}}_{0} \approx 2.5 M$ . Bottom row: radial correlation in $Δ r$ evaluated at two different radii $\bar{r} = {\tilde{r}}_{0}$ and $\bar{r} = 4 M$ . The emissivity is set to zero behind the event horizon $r = r_{+}$ , so correlations vanish for $Δ r \geq 2 (\bar{r} - r_{+})$ . This occurs in the shaded region in the bottom left panel. For a unit-height Gaussian, the standard deviation width ( $1 σ$ ) is achieved at height $e^{- 1 / 2} \approx 60.65 %$ . The correlation lengths at $\bar{r} = {\tilde{r}}_{0}$ are $ℓ_{t} \approx 3.0 M$ , $ℓ_{ϕ} \approx 4.3 °$ , and $ℓ_{r} \approx 0.4 M$ (while $ℓ_{r} \approx 0.3 M$ at $\bar{r} = 4 M$ ).
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Figure 4
Universal structure in the time autocorrelation function $C_{1 D} (T)$ [Eq. (3.3)], with $T$ the time separation, for polar observations of random fluctuations in an equatorial disk surrounding a Kerr BH with spin $a / M = 94 %$ . Universal aspects of $C_{1 D} (T)$ are governed by the critical exponents $γ$ and $τ$ [Eqs. (2.8)]. The autocorrelation function consists of a sum (displayed with orange shading) of self-similar peaks localized around $T = m τ_{0}$ for every integer $m$ , differing only by an overall demagnification factor $e^{- | m | γ_{0}}$ . Here, we show the $| m | = 0$ , 1, 2 peaks in red, blue, and green, respectively. The profile of each peak, which is nonuniversal, depends on statistical properties of the flow and is taken here to correspond to our toy model (5.2). This structure may be obtained by integrating over $Φ$ in Fig. 1.
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