Testing a Galactic Lensing Hypothesis with the Prompt Emission of GRB 221009A

Even at modest amplification, the optical depth to gravitational lensing through the Galaxy is $<10^{-5}$. However, the large apparent isotropic-equivalent energy of GRB 221009A coupled with a path through low Galactic latitude suggests that the conditional probability that this particular GRB was lensed is greater than the very low a priori expectation. With the extreme brightness of the prompt emission, this Galactic lensing hypothesis can be constrained by autocorrelation analysis of Fermi photons on 0.1-1000 ms timescales. In relating lensing mass, magnification, and autocorrelation timescale, I show that a lensed-induced autocorrelation signature by stellar lenses falls below the minimal variability timescale (MVT) expected from a black hole central engine. However, lensing by Galactic dark matter MACHOs ($M_l>10-1000\,M_\odot$) could be confirmed with this approach. Regardless, at a peak $\gamma$-ray photon rate of $>30$ ms$^{-1}$, GRB 221009A represents a prime opportunity to measure the smallest MVTs of GRBs.


INTRODUCTION
At a redshift of z GRB = 0.1505 (de Ugarte Postigo et al. 2022;Castro-Tirado et al. 2022), GRB 221009A (Dichiara et al. 2022;Veres et al. 2022) is among the lowest redshift long-soft GRBs known.With an isotropic-equivalent energy of E γ (iso) > 3 × 10 54 erg (Frederiks et al. 2022) (the lower limit arising from detector effects) GRB 221009A was one of the most (if not the most) intrinsically luminous GRBs ever observed (cf.Cenko et al. 2011).Observed in the direction of a low Galactic latitude (b = 4.32 • , l = 52.96• ), we expect larger stellar mass and dark matter columns than other well-observed GRBs far off the Galactic plane.This, coupled with (near) record-setting E γ (iso) (and likely the lowest V /V max of any GRB) suggests that the apparent prompt emission brightness could have been amplified by gravitational lensing.
Gravitational lensing of GRBs has been considered and purportedly observing previously in a cosmological context (Paczyński 1986;Loeb & Perna 1998;Hirose et al. 2006;Ji et al. 2018;Paynter et al. 2021;Lin et al. 2022;Veres et al. 2021).Here we (primarily) consider lensing by a Galactic object.In additional to amplification, lensing by a point mass forms two images of the source on the sky with an arrival time delay between each image (Ji et al. 2018): with β = θ s /θ E , M l (z l ) the mass (redshift) of the lens, and θ s the angular misalignment of the lens and the GRB.
The Einstein radius is with D s (D l ) as the angular diameter distance to the GRB (lens).On a cosmological scale, lensing by galaxies or clusters give rise to days-to-months delays between events or by seconds (Hirose et al. 2006) if the lens mass is an intermediate black hole or large dark matter halo (M l = 10 4 −10 6 M ⊙ ).Instead, as is our focus here, with M l = 30 M ⊙ in the Galaxy at D l = 8 kpc then θ E = 5.5 milliarcsec and ∆t = 1.2 ms at β = 1.These short timescales can only be probed with bright, fast changing GRBs.The total magnification A(β) = (β 2 + 2)/(β β 2 + 4) establishes an implicit anti-correlation between ∆t and A.
. The total magnification of GRB 221009A as a function of correlation timescale ∆t for a range of Galactic lens masses at D l = 8 kpc.Lines of constant θs are shown in dotted red, labeled in units of milliarcsec.The grey region corresponds to excluded times less than the redshifted timescale associated with a black hole central engine.The vertical lines correspond to the 10% and 50% minimal variability timescale of Fermi events in the 300-1000 keV range from Golkhou et al. (2015) (GRL), corrected to the redshift of GRB 221009A using the median redshift (zGBL = 1.72) of the GRL sample.The horizontal lines labelled at right show the approximate optical depth to massive compact dark matter objects in the Galaxy.
The lensing time delay could give rise to an observable signature, a significant peak in the autocorrelation function of the γ-ray light curve (Ji et al. 2018), detectable only if the minimal variability timescale (MVT) (MacLachlan et al. 2013) is less than ∆t.As the observed variability in a GRB directly corresponds to the activity of the central engine in the internal shock model (cf.Piran 2004), the MVT lower limit should be set by the minimum possible timescale for variability in the central engine.Assuming emission from the innermost stable orbit (ISCO) of a non-rotating 5 M ⊙ black hole, the MVT of GRB 221009A must be greater than (1 + z GRB )R ISCO /c = 0.17 ms.
Figure 1 demonstrates the phase space for lensing-induced autocorrelation detectability for GRB 221009A.Larger lensing masses, such as from MACHOs, could give rise to a detectable signal corresponding to large amplitude magnification.If the MVT of GRB 221009A is at the 10th percentile of the GRL sample, then any autocorrelation lensing signature detected from a M l < 10M ⊙ would correspond to less than an factor of 2 in magnification amplitude.
There are potentially confounding matters to consider.First, as the ratio of the flux from the primary to the counter image location increases with decreasing A, small magnification events will also require larger signal-to-noise to be detected.Second, finite source effects limit the largest possible A. The largest radius at which internal shocks can dissipate energy before deceleration, is R γ ∼ 4 × 10 13 (Γ max /100)2 (T 90 /10 s) cm ≈ 2.5 × 10 15 cm with Γ max ≈ 440 (following from Lazzati et al. 1999).At an angular diameter distance D s = 546.9Mpc the source size during the prompt emission was R γ /D s ≈ 3 µarcsec.Figure 1 shows that finite-source effects are not relevant at ∆t ∼ > (1 + z GRB )R ISCO /c.If we instead consider a lensing mass at cosmological distances, say D s /2, then finite-source effects limit the largest amplitude to A ∼ < 30 for all ∆t with M l < 100M ⊙ .
It is reasonable to ask what the probability that this event was lensed and, relatedly, what the chance that any GRB has been lensed with ∼ms time delays from a Galactic lens.At a fixed mass density and profile of dark matter in the Galaxy, the number density of primordial black holes (Bird et al. 2016) and MACHOs (Alcock et al. 2001;Muñoz et al. 2016;Blaineau et al. 2022) decreases proportionally to the lens mass.The optical depth to lensing through the Galaxy can be calculated assuming a DM profile.I find τ DM = 4.8 × 10 −6 β 2 in the direction of GRB 221009A following from the calculation of Griest (1991).As a function of amplification this is τ DM (A) ≈ 10 −5 × [A/(A 2 − 1)1/2 − 1].Thus, the detectability phase space autocorrelation (say A ∼ [1.3 − 30]) occurs with an optical depth from 5 × 10 −6 to 5 × 10 −9 .In a cosmological context, Ji et al. (2018) found an optical depth of τ DM,eg ≈ 0.15 to the entire GRB-observable volume if all dark matter resides in ∼ 30M ⊙ MACHOs.Given that GRB 221009A originated from low redshift, the calculation for fast radio bursts (FRBs) is more relevant: Bird et al. (2016) found τ ≈ 0.02 to z = 0.5.Scaling by the ratio of the comoving volumes between z = 0.5 and z GRB gives an approximate extragalactic optical depth to this GRB of τ DM,eg ∼ 0.0007.Though the a priori expectation of lensing to any single event is evidently very low, the extreme apparent luminosity and low Galactic latitude clearly raises the conditional probability that GRB 221009A itself was lensed.I do not attempt to calculate this conditional probability.
The Fermi GBM saw 0.375 × 10 6 photons per second during the peak of the prompt phase (Lesage et al. 2022), corresponding to an average of 1 photon per 3 µs.As the MVT of GRBs is seen to decrease with increasing energy (Golkhou et al. 2015), focus on the higher energy photons should give access to the largest phase space for lensing autocorrelation discovery.It is possible that intrinsic processes give rise to apparent peaks in the autocorrelation function.However two observations would be the telltale confirmation of lensing for GRB 221009A.First, the autocorrelation timescale should be the same at all energies.Second, over the very long duration of the γ-ray event (> 300 s; Lesage et al. 2022), as the emission transitions from internal to external shocks, the MVT at high energy should increase; so long as MVT(t) < ∆t, the measured autocorrelation peak should remain the same in the lensing scenario 1 .Dead time, pile up, and saturation effects will make an autocorrelation analysis on 0.1-100 ms timescales challenging (C.Guidorzi, priv.comm.) but hopefully this note motivates such an effort.I thank S. Bradley Cenko, Christiano Guidorzi, Raffaella Margutti, Liang Dai, Wenbin Lu, and Guy Nir for helpful conversations and comments.