Astrophysical Constraints on Dark Matter

Astrophysical constraints on dark matter

Anne Green University of Nottingham

Primordial Black Holes as a dark matter candidate

Observational constraints on PBH abundance

Applying constraints to, realistic, extended mass functions arXiv:1609.01143

Astrophysical uncertainties (on microlensing constraints) arXiv1705.10818 Primordial Black Hole Astrophysical constraints on ^ dark matter

Anne Green University of Nottingham

Primordial Black Holes as a dark matter candidate

Observational constraints on PBH abundance

Applying constraints to, realistic, extended mass functions arXiv:1609.01143

Astrophysical uncertainties (on microlensing constraints) arXiv1705.10818 Primordial Black Holes as a dark matter candidate

Primordial Black Holes (PBHs) form in the early Universe (before nucleosynthesis) and are therefore non-baryonic.

PBHs evaporate (Hawking radiation), lifetime longer than the age of the Universe for M > 1015 g.

A DM candidate which (unlike WIMPs, axions, sterile neutrinos,…) isn’t a new particle (however their formation does usually require Beyond the Standard Model physics, e.g. inﬂation). LIGO has detected gravitational waves from mergers of 10 M BHs. ⇠

LIGO-Virgo, Elavsky could be formed by astrophysical processes, but such a large population possibly unexpected??

Could PBHs be the CDM?

(and potentially also the source of the GW events?? Bird et al.; Sasaki et al.) Formation

Most ‘popular’ mechanism is collapse of large (at horizon entry) density perturbations during radiation domination, forming PBHs with mass of order the horizon mass. Zeldovich & Novikov; Hawking; Carr & Hawking

For gravity to overcome pressure forces resisting collapse, size of region at maximum expansion must be larger than Jean’s length.

Simple analysis: Carr; see e.g. Harada, Yoo & Kohri and Yoo, Harada, Garriga & Kohri for reﬁnements ⇢ ⇢¯ density contrast: ⌘ ⇢¯

p 1 threshold for PBH formation: w = = c ⇠ ⇢ 3

3/2 15 t PBH mass: M w M MH 10 g H ⇠ 10 23 s ⇠ ✓ ◆ initial PBHs mass fraction (fraction of universe in regions dense enough to form PBHs):

1 (M) P (⇥(M )) d⇥(M ) ⇠ H H Zc assuming a gaussian probability distribution:

(M)=erfc c p2(M ) ✓ H ◆

σ(MH) (mass variance) typical size of ﬂuctuations PBH forming ﬂuctuations

but in fact β must be small, and hence σ ≪ δc PBH abundance

Since PBHs are matter, during radiation domination the fraction of energy in PBHs grows a . / Relationship between PBH initial mass fraction, β, and fraction of DM in form of PBHs, f: 1/2 9 M (M) 10 f ⇠ M ✓ ◆ i.e. initial mass fraction must be small, but non-negligible.

5 On CMB scales the primordial perturbations have amplitude (M ) 10 H ⇠

If the primordial perturbations are close to scale-invariant the number of PBHs formed will be completely negligible: (M) erfc(105) 105 exp (105)2 ⇠ ⇠ ⇥ ⇤ To form an interesting number of PBHs the primordial perturbations must be signiﬁcantly larger (σ(MH)~0.01) on small scales than on cosmological scales. Constraints on the primordial power spectrum

Large scale Ultracompact structure Primordial Black Holes minihalos* & the CMB

1 2 10 10 2

3 WIMP kinetic decoupling 10 10 ❙ ❙ 3 4 10 10 Allowed regions 4 5 10

10 ) ) k k Ultracompact minihalos (gamma rays, Fermi-LAT) 5 ( ( 6 10

10 R Ultracompact minihalos (reionisation, WMAP5 ⌧e) P 10 6 P 10 7 Primordial black holes 7 8 10 10 CMB, Lyman-↵, LSS and other cosmological probes 8 9 10 10 9 10 10 10

3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 10 10 10 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 1 k (Mpc ) Bringmann, Scott & Akrami

* UCMH constraints only hold if most of the DM is WIMPs. Recent studies ﬁnd they have shallower density proﬁles than previously assumed. Gosenca et al., Delos et al.

Observational constraints on PBH abundance

Microlensing

Temporary (achromatic) brightening of background star when compact object passes close to the line of sight.

EROS EROS constraints on fraction of halo in compact objects, f, assuming a delta- function mass function:

log10(M/M ) EROS MACHO constraints on fraction of halo in compact objects in the 1-30 M range:

M/M MACHO

Very similar to EROS limits for M> 3 M . Ultra-faint dwarf heating

Brandt

Gravitational interactions transfer energy to stars, heating and cause the expansion of,

i) star clusters within dwarf galaxies (e.g. star cluster at centre of Eridanus II) ii) ultra-faint dwarf galaxies

increase in half-light radius with time constraint f

M/M Mass segregation in dwarf galaxies

Koushiappas & Loeb Mass segregation would lead to a deﬁcit of stars in the centre of dwarf galaxies and a ring in the projected stellar surface density proﬁle.

projected stellar mass density constraint (f=0.1, M = 30 M ) log10 f

M/M Wide binary disruption

Chaname & Gould; Yoo, Chaname & Gould; Quinn et al.; Monroy-Rodriguez & Allen

Massive compact objects perturb aﬀect the orbits of wide binaries.

Need to make assumptions about initial distribution of orbits of binaries.

dist of semi-major axes constraint (1000, 100 & 10 M v. obs) 100f

M/M Monroy-Rodriguez & Allen Cosmic Microwave Background distortions

Ricotti et al; Ali-Haϊmoud & Kamionkowski; Horowitz; Blum, Aloni & Flauger Accretion onto PBH leads to emission of X-rays which can distort the spectrum (FIRAS) and anisotropies (WMAP/Planck) of the CMB.

Signiﬁcant uncertainties in constraint due to modelling of complex astrophysical processes. � wide binaries

�� micro-lensing Planckultra-faint dwarfs

Planck ) �� (collisional ionization) �� (photoionization) ( �� ROM �� -�

��-� ��

��/�⊙

Ali-Hamoud & Kamionkowski X-ray and radio emission

Gaggero et al; Inoue & Kusenko Accretion onto PBH leads to X-ray and radio emission.

100 DM f

1 10 DM fraction

Radio constraint (2,3,5); =0.01 X-ray constraint (2,3,5); =0.01 10 2 101 102 M [M ]

But including eﬀects of gas turbulence reduces accretion onto PBH and removes X-ray bound. Hector, Hutsi & Raidal quasar microlensing

Quasar microlensing by compact objects in lens galaxy leads to variation in brightness of images in multiply lensed quasars. Chang & Refusal

α= 0.2 ± 0.05 of the mass is in compact objects with 0 . 05 M

OPTICAL rs=5 lt-day 100.0

10.0 • O

M/M 1.0

0.1

0.05 0.1 0.4 1. supernova microlensing

Compact objects affect lensing magnification distribution of type 1a SNe (most lines of sight are demagnified relative to mean, plus long-tail of high magnifications): Zumalacarregui & Seljak

magniﬁcation distribution constraint f 0.9 ↵ =0 100 ↵ =0.85 0.8 101 Maximum near 0.7 SNe lensing 80 empty beam (this work) Eridanos II =1 0.6 z (95% c.l.)

at 0.5 60 ) M 0 10 ⌦ 0.4

Magniﬁcation / µ 3 40 µ, z, ↵ tail 0.3 PBH ( / EROS ⌦

L Planck (photo) Dark Matter fraction [%]

P 0.2 ⌘ 20 ↵ 0.1 Planck (coll) 1 10 LIGO BHs 0.0 0 4 3 2 1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 MPBH [M ] µ (relative to FRW mean) M/M gravitational waves

PBH binaries can form:

i) in the early Universe (from chance proximity), Nakamura, Sasaki, Tanaka & Thorne

ii) via gravitational capture in present-day halos. Bird et al.

If PBH binaries formed in the early Universe survive to the present day then their mergers are dominant, and orders of magnitude larger than the merger rate measured by LIGO. Nakamura et al.; Ali-Haϊmoud, Kovetz & Kamionkowski

100 DM ⌦ / 1 10 PBH = ⌦

PBH 2

f 10 LIGO

EROS+MACHO 3 Eridanus II 10 Accretion - radio

DM fraction Accretion - X-ray CMB - PLANCK CMB - FIRAS 4 10 100 101 102 103 MPBH [M ] Kavanagh, Gaggero & Bertone, taking into account PBH’s dark ‘dresses’ Compilation of ~Solar mass region constraints

log10(M/M )

______. . . LMC microlensing (EROS & MACHO) - - - dwarf galaxy dynamical constraints

______- - - - - wide binary disruption (tightest) CMB constraints — — — X-ray & radio ______SNe microlensing

Doesn’t include Mediavilla et al. microlensing of quasars (no constraint on f published) or gravitational waves from mergers. microlensing of stars in M31

Same principle as MW microlensing, but sensitive to light compact objects (due to higher cadence obs.). Source stars unresolved. MM/M/M M/M 1111 11 1010-17-17 10-17 1010-7-7 10-7 10103 3 103 10101313 1013

FF WDWD F WD KK KMLML ML WBWB WB EGEG EGNSNS NS EE E 0.1000.100 0.100 mLQmLQ mLQ FIRASFIRAS FIRAS LSSLSS LSS

0.0010.001 0.001 WMAPWMAP WMAP f f f DFDF DF HSC-M31 constraint (95% C.L.)

1010-5-5 10-5 ー + one remaining candidate - - w/o one remaining candidate

1010-7-7 10-7 10101616 1016 10102626 1026 10103636 1036 10104646 10Niikura46 et al. MM/g/g M/g 1111 11

10 However analysis assumes geometric optics, however for M . 10 M wavelength of light is larger than Schwarzschild radius of lens diffraction occurs and lowers maximum magniﬁcation. Inomata et al. neutron star destruction Capture of PBHs would lead to destruction of neutron stars in high density & low velocity dispersion environments. Constraints from observation of neutron stars in globular clusters: Capela, Pshirkov & Tinyakov; Pani & Loeb

100 r = 4 102 GeVcm 3 dm ·

3 3

DM r = 2 10 GeVcm 1 dm · W 10 / PBH W 4 3 rdm = 10 GeVcm

2 10

1018 1019 1020 1021 1022 1023 1024 1025 1026 BH mass, g

But do globular clusters have a high DM density? white dwarf explosions

Transit of PBHs through white dwarf heats it, due to dynamical friction, causing it to explode. Graham, Rajendran & Varela

104

103

102 L 3 10 cm ê GeV H 1 r dark matter density in Solar neighbourhood 10-1

10-2

10-3 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 Black Hole Mass gm

H L Constraints Sasaki et al. (assuming a delta-function mass function) Requires Accretion existence of WD (X-ray) WB neutron stars OGLE in high DM Kepler density NS Eri-II Caustic regions EROS

Femtolensing Accretion (Radio) HSC UFDs Millilensing

Accretion (X-ray-II) Ignores ﬁnite size of DF Accretion disk GRBs (CMB) Katz et al. Accretion (CMB) Accretion disk (CMB-II)

10 HSC analysis assumes geometric optics, however for M . 10 M wavelength of light is larger than Schwarzschild radius of lens diffraction occurs and lowers maximum magniﬁcation Inomata et al. and also neglects ﬁnite size of sources. Bartolo et al. Constraints Bartolo et al. (assuming a delta-function mass function)

15 18 21 24 27 30 33 36 100 10 10 10 10 10 10 10 10 Kepler

WD EROS UFD -1 HSC

10 bkg

γ HSC (extrap.) 10-2

CMB 10-3

10-4

10-5 10-18 10-15 10-12 10-9 10-6 10-3 100 103 Applying constraints to, realistic, extended mass functions

Is subtle….

White Dwarf 0.100 Kepler Femtolensing EROS/MACHO

0.001 FIRAS

Can’t just compare df/dM to DM Ω / 10-5 constraints on f as a function of M EGγ PBH

(e.g. arXiv:1606.07631). Ω -7 10 WMAP3

10-9 1016 1021 1026 1031 1036

MPBH [g]

This is ‘double-counting’: for instance EROS microlensing constraints, allow f~0.2 for M~5 Msun or f~0.4 for M~10 Msun, but NOT BOTH. Extended MFs produced by inﬂation models, taking into account critical collapse, are often well approximated by a log-normal distribution: Green; Kannike et al.

2 df [log (M/M ) log (Mc/M )] (M)= exp 2 dM / ( 2 )

Constraints on the central mass, Mc, and width, σ, of log-normal MF from explicit recalculation of EROS microlensing and tightest heating of ultra-faint dwarfs limit σ

log10(MBH/M ) For realistic extended mass functions, individual constraints are smoothed out. When all constraints considered maximum allowed PBH fraction is reduced Green; Carr et al. monochromatic log-normal (ﬁxed width) log10 f monochromatic lognormal, σ=2 0.0 0.0

Planck

K M -0.5 WD NS -0.5 WB FL EROS -1.0 -1.0

Eri II

Evaporation PBH PBH f -1.5 f -1.5 10 10 log log

-2.0 HSC -2.0

Seg I

-2.5 -2.5

-3.0 -3.0 -15 -10 -5 0 -15 -10 -5 0

log10(Mc /M⊙) log10(Mc /M⊙) M log 10 M ✓ ◆

Carr et al. Astrophysical uncertainties (on microlensing constraints)

Evans power law halo models: self-consistent halo models, which allow for non-ﬂat rotation curves.

Traditionally used in microlensing studies since there are analytic expressions for velocity distribution. Rotation curve

1 vc (km s )

r (kpc) ______standard halo (SH) — — — top: power law halo B (massive halo, rising rotation curve) bottom: power law halo C (light halo falling rotation curve) ……….. envelope of MW rotation curve data [Bhattacharjee et al.] Microlensing diﬀerential event rate (f=1 M= 1 M , and perfect detection eﬃciency)

d dt

Einstein diameter crossing time (days)

Microlensing: ______standard halo (SH) — — — power law halos B and C - - - - - SH local circular speed, 200 & 240 km/s Constraints on halo fraction for delta-function MF: f

log10(M/M ) ______standard halo (SH) — — — power law halos C and B 3 ………. SH local density, 0.005 and 0.015 M pc - - - - - SH local circular speed, 200 & 240 km/s Brandt dwarf galaxy constraints ______

Clustering of PBHs would also aﬀect microlensing (and other) constraints. Garcia-Bellido & Clesse Constraints on width of log-normal MF with f=1

microlensing constraints exclude bottom left

CMB and dynamical constraints exclude top right

log10(Mc/M )

______standard halo (SH) ______Brandt dwarf galaxy constraints — — — power law halos C and B _____ Horowitz CMB contraint ………. SH local density, 0.005 and 0.015

- - - - - SH local circular speed, 200 & 240 km/s

See also Calcino, Garcia-Bellido & Davis. Are PBHs a viable dark matter candidate? multi-Solar mass Primordial Black Holes making up all of the DM appears to be excluded (possible caveat: clustering).

However there are, hard to probe, open windows for very light PBHs:

15 18 21 24 27 30 33 36 100 10 10 10 10 10 10 10 10 Kepler

WD EROS UFD -1 HSC

10 bkg

γ HSC (extrap.) 10-2

CMB 10-3

10-4

10-5 10-18 10-15 10-12 10-9 10-6 10-3 100 103

Bartolo et al. X-ray pulsars (in particular Small Magellanic Cloud X1) are a good candidate for detecting microlensing by light compact objects (large photon energy & counts to reduce wave effects, small geometric size comp to Einstein radius so ﬁnite size effects small, distant so large optical depth, steady ﬂux): Bai & Orlofsky

M M PBH ( ⊙) 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10 AstroSat RXTE(10 d)

d) 1 (100 ( 100 /Lynx

d Athena ) 0.100 PBH f

Evaporation

0.010 BH LOFT(300 d) Subaru

0.001 1016 1017 1018 1019 1020 1021 1022 1023 1024

MPBH (g) If PBHs form from collapse of large curvature perturbations, then these perturbations act as 2nd order source of (non-gaussian) gravitational waves, which could be detected by LISA. Saito & Yokoyama; Cai, Pi & Sasaki; Bartolo et al.

PBH mass

10-7 10-8 10-9 10-10 10-11 10-12 10-13 10-14 10-15 10-6

GW energy 10-7 density 10-8

10-9

10-10

10-11 ��

10-12

10-13

10-14

10-15 10-5 10-4 10-3 10-2 10-1 GW frequency Bartolo et al.

delta-function and gaussian primordial curvature power spectrum which give fPBH = 1. Summary

Primordial Black Holes can form in the early Universe, for instance from the collapse of large density perturbations during radiation domination.

There are numerous constraints on the abundance of PBHs from gravitational lensing, their dynamical eﬀects, accretion and other astrophysical processes.

(realistic) extended mass functions are more tightly constrained than delta-function MF (which is usually assumed when calculating constraints).

(1 100) M PBHs making up all of the DM appears to be excluded, but there is still 10 an open mass window at M< 10 M which can be partly probed by LISA.

Short summary

Are Primordial Black Holes a viable dark matter candidate?

Yes, but….

probably not planetary—multi-Solar mass PBHs

need BSM physics to form them (AFAIK…) deviations from simple scenario: i) non-gaussianity

Since PBHs are formed from rare large density fluctuations, changes in the shape of the tail of the probability distribution (i.e. non-gaussianity) can significantly affect the PBH abundance. Bullock & Primack; Ivanov;… Byrnes, Copeland & Green;…

Franciolini, Kehagias, Matarrese & Riotto use a path integral formalism to derive an exact expression for the PBH abundance. However it involves all of the smoothed N-point connected correlation functions… ii) critical collapse

Choptuik; Evans & Coleman; Niemeyer & Jedamzik

BH mass depends on size of M = kM ( ) ﬂuctuation it forms from: H c

log10(MBH/MH)

Musco, Miller & Polnarev

using numerical simulations (with appropriate initial conditions) ﬁnd k=4.02, γ=0.357

log(δ-δc) Get PBHs with range of masses produced even if they all form at the same time i.e. we don’t expect the PBH MF to be a delta-function iii) clustering

Is potentially important (aﬀects merger rate and also many of the abundance constraints), but hard to calculate. Is currently a somewhat open issue.

PBHs don’t form in clusters Ali-Haϊmoud (previous work Chisholm extrapolated an expression for the normalized correlation beyond its range of validity).

Clustering depends on shape of power spectrum, small if PBHs form from spike in power spectrum. Desjacques & Riotto; Ballesteros, Serpico & Taoso Inﬂation models with (potentially) large perturbations on small scales i) Monotonically increasing power spectrum

1 2 2 e.g. running-mass inﬂation Stewart V ()=V + m () 0 2

potential primordial power spectrum

Leach, Grivell, Liddle ii) models with a feature in the power spectrum

From models with two fields or two periods of inflation or features in the potential. e.g. hybrid inflation with a mild waterfall transition

potential primordial power spectrum

0.01

" ! 4

k 10 ! Ζ P

10! 6

10! 8

10! 10 !70 !60 !50 !40 !30 !20 !10 0

N k

Buchmuller Clesse & Garcia-Bellido n.b. In single ﬁeld models need to violate slow roll and hence standard expressions for amplitude of ﬂuctuations aren’t valid. Kannike et al.; Germani & Prokopec; Motohashi & Hu; Ballesteros & Taoso

Can get large ﬂuctuations by over-shooting a local minimum, but need to ﬁne-tune potential: Ballesteros & Taoso; Hertzberg & Yamada

When slow-roll violated need to take into account quantum diﬀusion. Biagetti et al. Ezquiaga & Garcia-Bellido;

potential primordial power spectrum 1.2

1.0

) 10-2 0

( 0.8 U 0.6 ) 0.7380 k -5 )/ 10 ( ζ

ϕ 0.7375 ( 0.4 P 0.7370 U 0.7365 0.2 10-8 0.88 0.90 0.92 0.94 0.96 0.98 1.00

0.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.001 10.000 105 109 1013 1017 ϕ end of -1 CMB/LSS k [Mpc ] inﬂation Hertzberg & Yamada Other mechanisms Collapse of cosmic string loops Hawking; Polnarev & Zemboricz;

Cosmic strings are 1d topological defects formed during symmetry breaking phase transition.

String intercommute producing loops.

Small probability that loop will get into conﬁguration where all dimensions lie within Schwarzschild radius (and hence collapse to from a PBH with mass of order the horizon mass at that time).

Probability is time independent, therefore PBHs have extended mass spectrum. Bubble collisions Hawking

1st order phase transitions occur via the nucleation of bubbles.

PBHs can form when bubbles collide (but bubble formation rate must be ﬁne tuned).

PBH mass is of order horizon mass at phase transition:

GUT scale: ~103 g

electroweak scale: ~1028 g

QCD scale: ~1032 g PBH formation during an early (pre nucleosynthesis) period of matter domination During matter domination PBHs can form from smaller fluctuations (no pressure to resist collapse) in this case fluctuations must be sufficiently spherically symmetric Yu, Khlopov & Polnarev; Harada et al. and (M) 0.0565(+1.5?) ⇡ The required increase in the amplitude of the perturbations is reduced Georg, Sengör & Watson; Georg & Watson; Carr, Tenkanen & Vaskonen; Cole & Byrnes:

ℛ(k) Primordial required amplitude for curvature 10-2 standard cosmological perturbation history power spectrum

10-5

-28 t1=10 s -24 t1=10 s varying duration -21 t1=10 s -19 t1=10 s of matter -14 t1=10 s 10-8 t =10-9s domination 1 t =10-5s amplitude on 1 Radiation domination, δc =0.42 CMB scales

kMpc-1 108 1010 1012 1014 1016 1018

Cole & Byrnes k Carr, Kuhnel & Sandstad method:

Divide relevant mass range into bins, I, II, III etc.

Check integral of MF in bin I is less than weakest limit on f in this bin.

Check integral of MG in bins I+II is less than weakest limit on f in these bins. And so on… 1.0 f 0.8 dwarf WB EROS heating 0.6 Eridanus II

0.4 EROS

0.2

I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI 0.0 1 5 10 50 100 500 1000 M/M M/M ⊙11 This underestimates the strength of the constraints.

If the integral of the MF exceeds weakest limit on f it’s deﬁnitely excluded, but some which don’t are also excluded. (Conversely all MFs which don’t exceed tightest limit are deﬁnitely allowed, but some which do are also allowed.) Method for applying delta-function constraints to extended mass functions: Carr, Raidal, Tenkanen, Vaskonen& Veermae:

If (as is usually case) diﬀerent mass PBHs contribute independently to constraint can write observable, A, as:

A[ ]=A0 + dM (M)K1(M) Z K1(M) encodes the underlying physics (& also depends on astrophysical parameters).

And if fmax(M) is the maximum allowed PBH fraction for a delta-function MF can show: (M) dM 1 f (M)  Z max

Bellomo, Bernal, Raccanelli & Verde:

Introduced concept of equivalent mass, Meq, of extended mass function (and cautioned about EMF extending beyond validity of constraint). PBH fraction (considering all constraints) maximised by a MF which is a sum of delta-functions Lehmann, Profumo & Yant:

robust constraints all constraints

100 100 WB SNe WB K M K M

FL FL

1 1 WD 10 EriII 10 EriII EROS SegI EROS SegI DM DM ⌦ ⌦ / /

evap evap

PBH PBH NS

CMB CMB ⌦ ⌦ 10 2 10 2 HSC HSC

Set AB Set ABC¯ 3 3 10 10 17 14 11 8 5 2 1 17 14 11 8 5 2 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 MPBH [M ] MPBH [M ]

fmax =2.0 fmax =0.4 iii) phase transitions

Reduction in the equation of state parameter (w=p/ρ) at phase transitions decreases the threshold for PBH formation δc and enhance the abundance of PBHs formed on this scale. (Horizon mass at QCD phase transition is of order a solar mass.) Jedamzik

Using new lattice calculations of QCD phase transition Byrnes et al. transition ﬁnd a 2 order of magnitude enhancement in β (but still need a mechanism for amplifying the primordial perturbations):

10-4

10-5

10-6 f

10-7

10-8

0.001 0.010 0.100 1 10 100 1000

M/M⊙ Byrnes et al. A microlensing event occurs when a compact object passes through the microlensing ‘tube’ which has radius uT RE:

uT RE x L

RE = Einstein radius: GMx(1 x)L 1/2 R (x)=2 E c2  M = lens mass L = distance from observer to source (50 kpc for LMC & SMC) x = distance of MACHO from observer (in units of L)

uT = minimum impact parameter for which amplification is above threshold (amplification threshold usually taken as 1.34 which corresponds to uT = 1) Differential event rate, assuming a delta-function lens mass function and a spherical halo with a Maxwellian velocity distribution (and neglecting the transverse velocity of the microlensing tube): Griest

d 32Lu4 1 4R2 (x) = T ⇢(x) R4 (x)exp E dx dtˆ tˆ4Mv2 E tˆ2v2 c Z0  c

⇢ ( x ) = compact object density distribution

tˆ = Einstein diameter crossing time (as used by the MACHO collab., EROS & OGLE use Einstein radius crossing time)

vc = local circular speed (usually taken to be 220 km/s, ~10% uncertainty)

Expected number of events: 1 d Nexp = E ✏(tˆ)dtˆ dtˆ Z0 E = exposure (number of stars times duration of obs.)

✏ ( tˆ ) = eﬃciency (prob. that an event of duration tˆ is observed) Standard halo model cored isothermal sphere: 2 2 Rc + R0 ⇢(r)=⇢0 2 2 Rc + r

3 ⇢0 =0 . 008 M pc , local dark matter density

Rc =5kpc, core radius

R0 =8.5kpc , Solar radius

‘Backgrounds’

i) variable stars, supernovae in background galaxies cuts/ﬁts developed to eliminate them (but some events only rejected years later, after star’s brightness varied a 2nd time!)

ii) lensing by stars in MW or Magellanic Clouds themselves (‘self-lensing’) model and include in event rate calculation