Carlo Rovelli

Planck Stars and Fast Radio Burts

Carlo Rovelli

with Aurélien Barrau, Hal Haggard, Francesca Vidotto

Planck stars Black hole fireworks: quantum-gravity effects outside Francesca Vidotto, C.R. the horizon spark black to white hole tunneling Int.J.Mod.Phys. D23 (2014) 2026 Hal Haggard, C.R. arXiv:1401.6562 arXiv:1407.0989

Planck star phenomenology Fast Radio Bursts and White Hole Signals Aurelien Barrau, C.R. Aurélien Barrau, Francesca Vidotto, C.R. Phys.Lett. B739 (2014) 405 Phys.Rev. D90 (2014) 12, 127503 arXiv:1404.5821 arXiv:1409.4031 4

Fast Radio Bursts

The FRB 121102 event Figureseen 2. by theCharacteristic Arecibo observatory plots of FRB 121102. In each panel the data were smoothed in time and frequency by a factor of 30 and 10, respectively. The top panel is a dynamic spectrum of the discov- ery observation showing the 0.7s during which FRB 121102 swept across the frequency band. The signal is seen to become signifi- Duration:cantly ~ milliseconds dimmer towards the lower part of the band, and some arti- facts due to RFI are also visible. The two white curves show theex- −2 −3 Frequency:pected 1.3 sweep GHz for a ν dispersed signal at a DM = 557.4 pc cm . The lower left panel shows the dedispersed pulse profile averaged across the bandpass. The lower right panel compares the on-pulse Observedspectrum at: Parkes, (black) Arecibo with an off-pulse spectrum (light gray), andfor reference a curve showing the fitted spectral index (α =10)isalso Origin: Likelyoverplotted extragalactic (medium gray). The on-pulse spectrum was calculated by extracting the frequency channels in the dedispersed datacor- responding to the peak in the pulse profile. The off-pulse spectrum Estimatedis theemitted extracted power: frequency 1038 channels erg for a time bin manually chosen to be far from the pulse. Physical source: unknown. time, and pulsar broadening were all fitted. The Gaus- sian pulse width (FWHM) is 3.0 ± 0.5 ms, and we found an upper limit of τd < 1.5 ms at 1.4 GHz. The residual DM smearing within a frequency channel is 0.5msand 0.9 ms at the top and bottom of the band, respectively. The best-fit value was α = 11 but could be as low as α =7.Thefitforα is highly covariant with the Gaussian amplitude. Every PALFA observation yields many single-pulse events that are not associated with astrophysical sig- Figure 1. Gain and spectral index maps for the ALFA receiver. nals. A well-understood source of events is false positives Figure a): Contour plot of the ALFA power pattern calculated from Gaussian noise. These events are generally isolated from the model described in Section 3 at ν =1375MHz.The (i.e. no corresponding event in neighboring trial DMs), contour levels are −13, −10, −6, −3(dashed),−2, and −1dB have low S/Ns, and narrow temporal widths. RFI can (top panel). The bottom inset shows slices in azimuth for each beam, and each slice passes through the peak gain for its respec- also generate a large number of events, some of which tive beam. Beam 1 is in the upper right, and the beam number- mimic the properties of astrophysical signals. Nonethe- ing proceeds clockwise. Beam 4 is, therefore, in the lower left. less, these can be distinguished from astrophysical pulses Figure b): Map of the apparent instrumental spectral index due in a number of ways. For example, RFI may peak in S/N to frequency-dependent gain variations of ALFA. The spectral in- at DM = 0 pc cm−3, whereas astrophysical pulses peak dexes were calculated at the center frequencies of each subband. − Only pixels with gain > 0.5 K Jy−1 were used in the calculation. at a DM > 0pccm 3. Although both impulsive RFI The rising edge of the first sidelobe can impart a positive appar- and an astrophysical pulse may span a wide range of ent spectral index with a magnitude that is consistent with the measured spectral index of FRB 121102. trial DMs, the RFI will likely show no clear correlation of S/N with trial DM, while the astrophysical pulse will of parameters. The model assumes a Gaussian pulse have a fairly symmetric reduction in S/N for trial DMs profile convolved with a one-sided exponential scatter- just below and above the peak value. RFI may be seen ing tail. The amplitude of the Gaussian is scaled with simultaneously in multiple, non-adjacent beams, while a a spectral index (S(ν) ∝ να), and the temporal loca- bright, astrophysical signal may only be seen in only one tion of the pulse was modeled as an absolute arrival time beam or multiple, adjacent beams. FRB 121102 exhib- plus dispersive delay. For the least-squares fitting the ited all of the characteristics expected for a broadband, DM was held constant, and the spectral index of τd was dispersed pulse, and therefore clearly stood out from all fixed to be −4.4. The Gaussian FWHM pulse width, other candidate events that appeared in the pipeline out- the spectral index, Gaussian amplitude, absolute arrival put for large DMs. Main message of this talk:

Consider the possibility that this (or similar) signals be of quantum gravity origin. -33 What is the relation between quantum gravity (lPlanck ~ 10 cm) and radio waves (λ ~ 1 cm) ??

t Hubble 1060 tPlanck ⇠

t Hubble l .02cm ⇠ t Pl ⇠ r Planck We see large amount of matter falling into astrophysical black holes

The black hole (here Cygnus X-!) pulls gas of the star orbiting around it. The gas heats up and emits X rays (yellow) as it falls into the black hole. What happens to all matter falling into a black hole?

? GR predict a “singularity”, but this only means that the theory goes wrong: it disregards quantum phenomena. ?

Black hole in Eddington-Finkelstein coordinates

m ds2 = r2d!2 +2dv dr F (r, t)du2. =0 =2 2m r r F (r, t)=1 r r What happens to all matter falling into a black hole? An input from quantum cosmology

a˙ 2 8⇡G ⇢ = ⇢ 1 a2 3 ⇢ ✓ Pl ◆

A strong repulsive force when matter reaches the Planck density Several other similar inputs

• LQC bounce • Maximal curvature

• Maximal acceleration [Vidotto, CR: PRL 111 (2013) 091303] • Electrons in atoms • …

Reasonable hypothesis:

Quantum mechanics prevents the formation of the singularity by developing an effective strong repulsion when matter reaches Planck density. Planck density does not mean Planck size !

Example: if a star collapses (M ~ M☉ ), Planck density is reached at 10-12 cm, which is 1020 times the Planck length!

Planck Planck black star length star hole

10-33 10-12 105 1013 cm 1 R 1 R = m 3 R =2m R m ⇠

1 There is a relevant intermediate scale between the M 3 L LP Schwarzschild radius LS and the Planck scale LP ⇠ M ✓ P ◆ ?

(Static) Planck star ?

? t m =0 =2 r r r

Static black hole with Planck star

Exemple (out of many possible) 2mr2 F (r)=1 r3 +2↵2m

r = rin r =2m r [Hayward] Dynamics of a Planck star, I

Example (out of many possible)

2m(t)r2 F (r)=1 r3 +2↵(t)2m

r = rin r =2m r

Important: matter-energy can escape from the trapped region, independently form Hawking radiation. What happens to all matter falling into a black hole?

? ? ? What happened at the big bang?

2 2 a˙ 8⇡G ⇢ LQG a˙ 8⇡G ⇢ = ⇢ 1 2 2 = ⇢ 1 a 3 ⇢Pl a 3 ⇢ ✓ ◆ ✓ Pl ◆

Effective repulsive force - elastic bounce Dynamics of a Planck star, I What is the physics here?

Non-evaporating Evaporating black hole black hole Planck star Physical problem:

- How can matter and energy come out from a quantum region?

Key idea:

- Neglect Hawking radiation in the ﬁrst approximation. - Energy conservation at inﬁnity → elastic bounce. - GR is time reversal invariant! Use this. time

Matter falls into a trapped region. Classically (but not in reality) matters goes into a singularity: Black hole time Matter emerges from a trapped region. Classically (but not in reality) matters emerges from a singularity: White hole!

But there are no white holes!

Matter falls into a trapped region. Classically (but not in reality) matters goes into a singularity: Black hole Black holes:

Weinberg’s “Gravitation and cosmology” (1972):

“There is no Schwarzschild singularity [black hole] in the gravitational ﬁeld of any known object of the universe”

“The Schwarzschild singularity does not seem to have much relevance for the world.” Black holes:

Are you sure, Steven? Black holes:

Cygnus-X1. Today there is no decent alternative to its interpretation as black hole.

Astronomers currently estimates that there are 10 millions black holes just in our galaxy. White holes:

Wald’s “General Relativity” (1984):

“Regions III and IV of the extended Schwarzschild solution [white holes] are probably unphysical.”

“There is no reason to believe that the initial conﬁguration of any region of our universe corresponds to these initial conditions, so there is no reason to believe that any region of our universe corresponds to the fully extended Schwarzschild solution.” White holes:

Are you sure, Bob? General relativity predicts an extraordinary number of processes and objects, which at ﬁrst nobody believed (including Einstein):

- Black holes - Expansion of the universe - Gravitational waves - White holes - …. White Holes can be out there in the sky time But can an that solves the classical vacuum vacuum that the solves classical where there is where there noand nomater and a (in a single spacetime Einstein equations quantum effect?) later whitelater hole Seems impossible… Seems stay together? stay earlier blackhole region region Take for simplicity an ingoing and the outgoing null shell

This0 is what we want: = r

- Region I: ﬂat (inside the shell) u - Region II: Schwarzschildt = 0 (outside theτ shell) - Region III: Quantumv region R - a well =inside the horizon! E = r r (seems impossible…) Two ideas to solve the problem!

I: The Crossed Fingers

II: The Crystal Ball r = 0 r = a r = R τ t = 0 v u is in diagram the whitea Kruskal hole seemsimpossible,because A white hole before , not the whiteafter hole! after ablackhole II: The Fingers Crossed 0 = r Full metric: join the pieces u t = 0 τ Spherical symmetry: v R 2 2 2 2 2 a = ds = F (u, v)dudv + r (u, v)(d✓ + sin ✓d ) = r r

vI uI Region I (Flat): F (u ,v )=1,r(u ,v )= . I I I I I 2 Bounded by: vI < 0. 3 32m r r r F (u, v)= e 2m 1 e 2m = uv. Region II (Schwarzschild): r 2m ⇣1 u⌘I uI 4m Matching: rI (uI ,vI )=r(u, v) u(uI )= 1+ e . vo 4m 3 ⇣ ⌘ 32m rq Region III (Quantum): F (uq,vq)= e 2m ,rq = vq uq. rq

The metric is determined by the constants: m, vo,,✏ The metric is determined by four constants: m, vo,,✏

- m is the mass of the collapsing shell.

1 m 3 - ✏ is the radius where quantum effect start on the shell ✏ l . ⇠ m3 P ✓ P ◆ - is the minimal distance from the horizon where the theory is entirely classical 0

- v o is the key parameter: it determines the= external time the full process takes r u

t = 0 τ v R a What determines the last two constants? = = r

r An argument from above:

The ﬁrewall argument (Almheiri, Marolf, Polchinski, Sully) shows that “something” unusual must happen before the Page time (half of the Hawking evaporation time).

Therefore the hole lifetime must be shorter or of the order of ~ m^3. An argument from below: For something quantum to happen of the validity of the semiclassical approximation must fail.

The classical theory is reliable as long as we are in a “small action” regime (typically in quantum gravity: high curvature). How small? Small effects can pile up (typical example tunnelling: a small probability per unit of time gives a probable effect on a long time.)

Therefore there two possible quantum effects:

(i) when Curvature ~ (LP)-2

(ii) when Curvature X (time) ~ (LP)-1 “Classicality parameter” q = l ⌧. P R Look for its max in the radius, and, at the max, for the time it gets to unity. A (long) straightforward calculation give the max radius at 7 R = 2m 6

m2 And the time ⌧ =2c . lP

c m m Which in turn give v e 2lp , . o ⇠ ⇠ 3

The metric is entirely determined by the mass. The lifetime of the hole is huge

m2 ⌧ =2c . lP

How can this be compatible with a bounce, which is short ? r = 0 I: Ball The Crystal r = a r = R τ t = 0 v u that: in the Schwarzschildshows metric A simple straightforward calculation at ﬁxed ⌧

becomes arbitrary large ! arbitrary becomes ⌧ t = = r R 1 R a , 2 when R m ✓ R a

a approaches 2 m ln R a 2 2 m m ◆ . 2 , m t

I: The Crystal Ball in Scharzschild-like coordinates:

The external proper time can be

I ⌧ II arbitrary long. 2 While the internal proper time is arbitrarily short.

2m a 2m m a ⌧ = 1 R a 2m ln .

R R R 2m

= r ✓ ◆ =0 =2 = r r r r t = R a r ⌧ r = 0 r internal = a Time dilation r = R ⇠ τ t m = 0 ⇠ v u 1 ms “A black hole is a short cut to the future” ⌧ external ⇠ m 2 ⇠ 10 9 years Can all this be observable? [Vidotto CR, 2014. Barrau CR, 2014]

Final stage of the evaporation can be at a radius larger than LPlanck !

The mass of a primordial black hole 3 tH 14 r = l 10 cm 348⇡ t P ⇠ exploding no, for a long lifetime, r P

t In the case of a short lifetime, r H 1cm ⇠ t ⇠ we can get to ~ 1 cm r P

The ratio of cosmological time to Planck time provides a large multiplicative factor that can make quantum gravity effects observable. Detectable? Eburst = hc/(2rf ) 3.9 GeV Already detected? ⇡

~10 MeV

energy spectrum of photons ] -1 eGamma 4 From ~200 light years 10 Entries 318043 [GeV dN dE Mean 0.09656 103 RMS 0.3454 Short gamma-ray burts 102

10 Isotropic

10-1 One event per day

10-2

10-2 10-1 1 10 E [GeV] 4

Detectable? Already detected?

Figure 2. Characteristic plots of FRB 121102. In each panel the data were smoothed in time and frequency by a factor of 30 and 10, respectively. The top panel is a dynamic spectrum of the discov- ery observation showing the 0.7s during which FRB 121102 swept Duration:across the frequency ~ milliseconds band. The signal is seen to become signifi- cantly dimmer towards the lower part of the band, and some arti- facts due to RFI are also visible. The two white curves show theex- Frequency:pected sweep for 1.3 a ν− GHz2 dispersed signal at a DM = 557.4 pc cm−3. The lower left panel shows the dedispersed pulse profile averaged Observedacross the bandpass. at: Parkes, The lower Arecibo right panel compares the on-pulse spectrum (black) with an off-pulse spectrum (light gray), andfor reference a curve showing the fitted spectral index (α =10)isalso Origin:overplotted Likely (medium extragalactic gray). The on-pulse spectrum was calculated by extracting the frequency channels in the dedispersed datacor- responding to the peak in the pulse profile.38 The off-pulse spectrum Estimatedis the extracted emitted frequency power: channels for 10 a time erg bin manually chosen to be far from the pulse. Physical source: unknown. time, and pulsar broadening were all fitted. The Gaus- sian pulse width (FWHM) is 3.0 ± 0.5 ms, and we found an upper limit of τd < 1.5 ms at 1.4 GHz. The residual DM smearing within a frequency channel is 0.5msand 0.9 ms at the top and bottom of the band, respectively. The best-fit value was α = 11 but could be as low as α =7.Thefitforα is highly covariant with the Gaussian amplitude. Every PALFA observation yields many single-pulse events that are not associated with astrophysical sig- Figure 1. Gain and spectral index maps for the ALFA receiver. nals. A well-understood source of events is false positives Figure a): Contour plot of the ALFA power pattern calculated from Gaussian noise. These events are generally isolated from the model described in Section 3 at ν =1375MHz.The (i.e. no corresponding event in neighboring trial DMs), contour levels are −13, −10, −6, −3(dashed),−2, and −1dB have low S/Ns, and narrow temporal widths. RFI can (top panel). The bottom inset shows slices in azimuth for each beam, and each slice passes through the peak gain for its respec- also generate a large number of events, some of which tive beam. Beam 1 is in the upper right, and the beam number- mimic the properties of astrophysical signals. Nonethe- ing proceeds clockwise. Beam 4 is, therefore, in the lower left. less, these can be distinguished from astrophysical pulses Figure b): Map of the apparent instrumental spectral index due in a number of ways. For example, RFI may peak in S/N to frequency-dependent gain variations of ALFA. The spectral in- at DM = 0 pc cm−3, whereas astrophysical pulses peak dexes were calculated at the center frequencies of each subband. − Only pixels with gain > 0.5 K Jy−1 were used in the calculation. at a DM > 0pccm 3. Although both impulsive RFI The rising edge of the first sidelobe can impart a positive appar- and an astrophysical pulse may span a wide range of ent spectral index with a magnitude that is consistent with the measured spectral index of FRB 121102. trial DMs, the RFI will likely show no clear correlation of S/N with trial DM, while the astrophysical pulse will of parameters. The model assumes a Gaussian pulse have a fairly symmetric reduction in S/N for trial DMs profile convolved with a one-sided exponential scatter- just below and above the peak value. RFI may be seen ing tail. The amplitude of the Gaussian is scaled with simultaneously in multiple, non-adjacent beams, while a a spectral index (S(ν) ∝ να), and the temporal loca- bright, astrophysical signal may only be seen in only one tion of the pulse was modeled as an absolute arrival time beam or multiple, adjacent beams. FRB 121102 exhib- plus dispersive delay. For the least-squares fitting the ited all of the characteristics expected for a broadband, DM was held constant, and the spectral index of τd was dispersed pulse, and therefore clearly stood out from all fixed to be −4.4. The Gaussian FWHM pulse width, other candidate events that appeared in the pipeline out- the spectral index, Gaussian amplitude, absolute arrival put for large DMs. Covariant loop quantum gravity. Full definition.

n0 l 2 L N hl State space = L [SU(2) /SU(2) ] (h ) n H 3 l

Kinematics Operators: where spin network (nodes, links)

Transition amplitudes W (hl)=N dhvf (hf ) A(hvf ) hf = hvf C C SU(2) v Z f v Y Y Y 8⇡~G =1 Dynamics

j (j+1) j 1 Vertex amplitude A(h )= dg0 (2j + 1) D (h )D (g g ) vf e mn vf jmjn e e0 SL(2,C) j Z Yf X

e v f All you need is here

spinfoam (vertices, edges, faces) Quantum transition

j,k 2 2 (j+ j) (j+ j) A(j, k)= dg dh dh+ e e SL(2C) SU2 SU2 Z Z Z j+j X k3 k3 n n Trj+ [e Y †gY ]Trj [e Y †gY ] v

j,-k A(j, k) 1 ⇠ ! relation j-k relation m-time Physical picture and main messages:

- Collapsing matter bounces. In a very short proper time (or order m).

- Incoming phase: black hole, Outgoing phase: white hole

- No singularity. 2 trapped regions. Information preserved.

- Time dilation. From the outside, long proper time, order m2. (For a stellar black hole, m is microseconds, m2 is billions of years). 4 “A black hole is a bouncing star seen in super-slow motion”.

- Observable? Primordial black holes

- An LQG calculation may be doable.

t Hubble 1060 tPlanck ⇠

E = hc/(2r ) 3.9 GeV burst f ⇡ Short gamma-ray burts ~10 MeV Isotropic One event per day FigureThe 2. FRBCharacteristic 121102 event plots of FRB 121102. In each panel the data were smoothed in time and frequency by a factor of 30 and 10, respectively.seen by the The Arecibo top panelobservatory is a dynamic spectrum of the discov- ery observation showing the 0.7s during which FRB 121102 swept across the frequency band. The signal is seen to become signifi- cantly dimmer towards the lower part of the band, and some arti- facts due to RFI are also visible. The two white curves show theex- pected sweep for a ν−2 dispersed signal at a DM = 557.4 pc cm−3. The lower left panel shows the dedispersed pulse profile averaged across the bandpass. The lower right panel compares the on-pulse spectrum (black) with an off-pulse spectrum (light gray), andfor reference a curve showing the fitted spectral index (α =10)isalso overplotted (medium gray). The on-pulse spectrum was calculated by extracting the frequency channels in the dedispersed datacor- responding to the peak in the pulse profile. The off-pulse spectrum is the extracted frequency channels for a time bin manually chosen to be far from the pulse.

time, and pulsar broadening were all fitted. The Gaus- sian pulse width (FWHM) is 3.0 ± 0.5 ms, and we found an upper limit of τd < 1.5 ms at 1.4 GHz. The residual DM smearing within a frequency channel is 0.5msand 0.9 ms at the top and bottom of the band, respectively. The best-fit value was α = 11 but could be as low as α =7.Thefitforα is highly covariant with the Gaussian amplitude. Every PALFA observation yields many single-pulse events that are not associated with astrophysical sig- Figure 1. Gain and spectral index maps for the ALFA receiver. nals. A well-understood source of events is false positives Figure a): Contour plot of the ALFA power pattern calculated from Gaussian noise. These events are generally isolated from the model described in Section 3 at ν =1375MHz.The (i.e. no corresponding event in neighboring trial DMs), contour levels are −13, −10, −6, −3(dashed),−2, and −1dB have low S/Ns, and narrow temporal widths. RFI can (top panel). The bottom inset shows slices in azimuth for each beam, and each slice passes through the peak gain for its respec- also generate a large number of events, some of which tive beam. Beam 1 is in the upper right, and the beam number- mimic the properties of astrophysical signals. Nonethe- ing proceeds clockwise. Beam 4 is, therefore, in the lower left. less, these can be distinguished from astrophysical pulses Figure b): Map of the apparent instrumental spectral index due in a number of ways. For example, RFI may peak in S/N to frequency-dependent gain variations of ALFA. The spectral in- at DM = 0 pc cm−3, whereas astrophysical pulses peak dexes were calculated at the center frequencies of each subband. − Only pixels with gain > 0.5 K Jy−1 were used in the calculation. at a DM > 0pccm 3. Although both impulsive RFI The rising edge of the first sidelobe can impart a positive appar- and an astrophysical pulse may span a wide range of ent spectral index with a magnitude that is consistent with the measured spectral index of FRB 121102. trial DMs, the RFI will likely show no clear correlation of S/N with trial DM, while the astrophysical pulse will of parameters. The model assumes a Gaussian pulse have a fairly symmetric reduction in S/N for trial DMs profile convolved with a one-sided exponential scatter- just below and above the peak value. RFI may be seen ing tail. The amplitude of the Gaussian is scaled with simultaneously in multiple, non-adjacent beams, while a a spectral index (S(ν) ∝ να), and the temporal loca- bright, astrophysical signal may only be seen in only one tion of the pulse was modeled as an absolute arrival time beam or multiple, adjacent beams. FRB 121102 exhib- plus dispersive delay. For the least-squares fitting the ited all of the characteristics expected for a broadband, DM was held constant, and the spectral index of τd was dispersed pulse, and therefore clearly stood out from all fixed to be −4.4. The Gaussian FWHM pulse width, other candidate events that appeared in the pipeline out- the spectral index, Gaussian amplitude, absolute arrival put for large DMs. 4

The FRB 121102 event seen by the Arecibo observatory

time, and pulsar broadening were all fitted. The Gaus- sian pulse width (FWHM) is 3.0 ± 0.5 ms, and we found an upper limit of τd < 1.5 ms at 1.4 GHz. The residual DM smearing within a frequency channel is 0.5msand 0.9 ms at the top and bottom of the band,? respectively. The best-fit value was α = 11 but could be as low as α =7.Thefitforα is highly covariant with the Gaussian amplitude. Every PALFA observation yields many single-pulse events that are not associated with astrophysical sig- Figure 1. Gain and spectral index maps for the ALFA receiver. nals. A well-understood source of events is false positives Figure a): Contour plot of the ALFA power pattern calculated from Gaussian noise. These events are generally isolated from the model described in Section 3 at ν =1375MHz.The (i.e. no corresponding event in neighboring trial DMs), contour levels are −13, −10, −6, −3(dashed),−2, and −1dB have low S/Ns, and narrow temporal widths. RFI can (top panel). The bottom inset shows slices in azimuth for each beam, and each slice passes through the peak gain for its respec- also generate a large number of events, some of which tive beam. Beam 1 is in the upper right, and the beam number- mimic the properties of astrophysical signals. Nonethe- ing proceeds clockwise. Beam 4 is, therefore, in the lower left. less, these can be distinguished from astrophysical pulses Figure b): Map of the apparent instrumental spectral index due in a number of ways. For example, RFI may peak in S/N to frequency-dependent gain variations of ALFA. The spectral in- at DM = 0 pc cm−3, whereas astrophysical pulses peak dexes were calculated at the center frequencies of each subband. − Only pixels with gain > 0.5 K Jy−1 were used in the calculation. at a DM > 0pccm 3. Although both impulsive RFI The rising edge of the first sidelobe can impart a positive appar- and an astrophysical pulse may span a wide range of ent spectral index with a magnitude that is consistent with the measured spectral index of FRB 121102. trial DMs, the RFI will likely show no clear correlation of S/N with trial DM, while the astrophysical pulse will of parameters. The model assumes a Gaussian pulse have a fairly symmetric reduction in S/N for trial DMs profile convolved with a one-sided exponential scatter- just below and above the peak value. RFI may be seen ing tail. The amplitude of the Gaussian is scaled with simultaneously in multiple, non-adjacent beams, while a a spectral index (S(ν) ∝ να), and the temporal loca- bright, astrophysical signal may only be seen in only one tion of the pulse was modeled as an absolute arrival time beam or multiple, adjacent beams. FRB 121102 exhib- plus dispersive delay. For the least-squares fitting the ited all of the characteristics expected for a broadband, DM was held constant, and the spectral index of τd was dispersed pulse, and therefore clearly stood out from all fixed to be −4.4. The Gaussian FWHM pulse width, other candidate events that appeared in the pipeline out- the spectral index, Gaussian amplitude, absolute arrival put for large DMs.