(stellar) Microlensing constraints on Primordial Black Hole dark - PowerPoint PPT Presentation

(stellar) Microlensing constraints on Primordial Black Hole dark matter Anne Green University of Nottingham Theory Observations Constraints on (realistic) extended mass functions arXiv:1609.01143 Astrophysical uncertainties arXiv:1705.10818

Prelude: PBH abundance constraints on the primordial power spectrum (and hence models of inflation): Critical collapse and the PBH initial mass function: PBHs as a MACHO candidate:

Theory (stellar) Microlensing is a temporary (achromatic) brightening of background star when compact object passes close to the line of sight. [Paczynski] EROS

LMC SMC Not to scale!

Source plane Lens plane 𝛽 Observer x = D L 𝜄 𝛾 D S BH 𝐸 𝑀𝑇 𝐸 𝑀 𝐸 𝑇 [Sasaki et al.] r 2 − r 0 r − R 2 Lens equation on lens plane: E = 0 r 4 GMD L D LS Einstein radius: r = D L θ r 0 = D L β R E = D S ✓ ◆ r 1 , 2 = 1 q Image positions: r 2 0 + 4 R 2 r 0 ± E 2 ◆ 1 / 2 ✓ ◆ � 1 / 2 r ✓ ∆ ∼ R E M D S 1 − x Angular separation: = 0 . 3 mas D L 10 M � 100 kpc x

Microlensing occurs when angular resolution is too small to resolve multiple images, instead observe amplification of source: 2.5 u 2 + 2 2.0 A = 1.5 u min = 0.1 u 2 + 4 √ u lnA u min = 0.3 1.0 u min = 0.5 u min = 1 u = r 0 0.5 R E 0.0 - 2 - 1 0 1 2 τ [Sasaki et al.] at r 0 =R E A=1.34, which is usually taken as the threshold for microlensing. Duration of event: ◆ 1 / 2 ✓ ◆ 1 / 2 ✓ ◆ � 1 t = 2 R E ✓ M D S v ˆ p ≈ 4 yr x (1 − x ) 200 km s � 1 10 M � 100 kpc v n.b. this all assumes point source and lens. For sub-lunar lenses finite size of source stars reduces magnification. [Witt & Mao; Matsunaga & Yamamoto]

Di ff erential event rate assuming a delta-function lens mass function and a spherical halo with a Maxwellian velocity distribution : [Griest] Z 1 t = 32 Lu 4 − 4 R 2 d Γ  E ( x ) � T ρ ( x ) R 4 E ( x ) exp d x dˆ ˆ ˆ t 4 Mv 2 t 2 v 2 0 c c = compact object density distribution ρ ( x ) = Einstein diameter crossing time (as used by the MACHO collab., EROS ˆ t & OGLE use Einstein radius crossing time) v c = local circular speed (usually taken to be 220 km/s, ~10s% uncertainty) L = distance from observer to source (49.6 kpc for LMC) Expected number of events: Z ∞ d Γ t ✏ (ˆ t ) dˆ N exp = E t dˆ 0 E = exposure (number of stars times duration of obs.) ✏ (ˆ ˆ = e ffi ciency (prob. that an event of duration is observed) t ) t

Standard halo model cored isothermal sphere: R 2 c + R 2 0 ρ ( r ) = ρ 0 R 2 c + r 2 ρ 0 = 0 . 008 M � pc � 3 , local dark matter density R c = 5 kpc , core radius R 0 = 8 . 5 kpc , 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

Di ff erential event rate for and halo fraction f=1: M = 1 M � ( , ) t ∝ M 1 / 2 d Γ / dˆ t ∝ M − 1 ˆ d Γ (s − 2 ) dˆ t ˆ t (days) ______ = standard halo model . . . . . . = standard halo model including transverse velocity v c ∝ R 0 . 2 - - - - = Evans power law model: massive halo with rising rotation curve, v c ∝ R − 0 . 2 _ _ _ _ = Evans power law model: flattened halo with falling rotation curve, velocity anisotropy can a ff ect rate at ~10% level [De Paolis, Ingrosso & Jetzer]

Calculations of parameter constrains/exclusion limits: If no events observed: N exp < 3 at 95% confidence. If events are observed: N obs ✓ ◆ t i )d Γ Y E ✏ (ˆ t (ˆ L ( M, f ) = exp ( − N exp ) t i ; M ) dˆ i =1 ˆ where are the durations of the N obs events and other lens populations (stars t i in MW and MC) included in di ff erential event rate.

Observations MACHO Monitored 12 million stars in LMC for 5.7 years. Found 13/17 events (for selection criteria A/B, B less restrictive-picks-up exotic events) . Detection e ffi ciency 5 years A 5 years B 2 years 1 year ˆ t

Measurement of fraction of halo in compact objects, f, (assuming a delta-function mass function): selection criteria A B M/M � 0 1 MACHO f 0 1 BUT LMC-5: lens identified (using HST obs & parallax fit) as a low mass MW disc star. [MACHO] LMC-9: (criteria B) lens is a binary, allowing measurement of projected velocity, low which suggests lens is in LMC (or source is also binary). [MACHO] LMC-14: source is binary, and lens most likely to lie in LMC. [MACHO] LMC-20: (criteria B) lens identified (using Spitzer obs) as a MW thick disc star. [Kallivayalil et al.] LMC-22: (criteria B) supernova or an AGN in background galaxy. [MACHO] LMC-23: varied again, so not microlensing [EROS/OGLE]

AND Distribution of timescales is narrower than expected for lenses in MW halo (assuming standard halo model). [Green & Jedamzik] ✏ d Γ dˆ t ˆ t X durations of observed events _____ best fit distribution assuming standard halo model + delta-function mass function - - - - best fit gaussian di ff erential event rate

Limits on halo fraction for from 1 < M/M � < 30 MACHO null search for long (> 150 day) duration events: f M/M � MACHO

EROS Monitored 67 million stars in LMC and SMC for 6.7 years. Use bright stars in sparse fields (to avoid complications due to ‘blending’-contribution to baseline flux from unresolved neighbouring star) . 1 SMC event (also seen by MACHO collab.) consistent with expectations for self-lensing (SMC is aligned along line of sight) . [Gra ff & Gardiner] Earlier candidate events eliminated: 7 varied again and 3 identified as supernovae. Constraints on fraction of halo in compact objects, f, (DF MF): f log 10 ( M/M � ) EROS

OGLE OGLE-II and III monitored 41 million stars in LMC and SMC for 12 years. Total of 8 events. All but 1 (SMC-02) consistent (number/duration/lensed star location/ detailed modelling of light curve including parallax) with lens being a star in the MW or MCs. SMC-02: Light curve shows parallax e ff ect and additional Spitzer observations find deviation from single lens model [Dong et al.] . Consistent with lens being a ~10 Solar mass BH binary in MW halo ( no light observed from lens). best-fit binary microlensing fit standard microlensing fit also including parallax OGLE-2005-SMC-001 OGLE-2005-SMC-001 15 15 OGLE I 3 OGLE I OGLE V OGLE V CTIO I CTIO I 15.5 15.5 AUCKLAND AUCKLAND 2.5 CTIO V CTIO V SPITZER SPITZER 16 16 2 16.5 16.5 1.5 17 17 3560 3580 3600 3620 3640 3560 3580 3600 3620 3640 -0.06 -0.06 -0.04 -0.04 Residuals -0.02 -0.02 0 0 0.02 0.02 0.04 0.04 0.06 0.06 3560 3580 3600 3620 3640 3560 3580 3600 3620 3640

Constraints on fraction of halo in compact objects, f, (assuming a delta-function mass function): f halo fraction from BH binary candidate event log 10 ( M/M � ) OGLE

M31 with Subaru Hyper Suprime-Cam Same principle as MW microlensing, but sensitive to lighter compact objects (due to higher cadence obs.). Source stars unresolved. M PBH [ M � ] Ignores 10 � 15 10 � 10 10 � 5 10 0 finite size of Kepler GRBs Femto EROS/MACHO CMB [Katz et al.] 10 � 1 f= Ω PBH / Ω DM 10 � 2 BH Evaporation 10 � 3 10 � 4 10 � 5 10 15 10 20 10 25 10 30 10 35 [Niikura et al.] M PBH [g] Finite size of source stars and effects of wave optics (Schwarzschild radius of BH comparable to wavelength of light) leads to reduction in maximum magnification for M . 10 � 11 M � M . 10 � 7 M � and respectively. [Witt & Mao; Gould; Nakamura]

OGLE Galactic bulge Observed events consistent with expectations from stars, except for 6 ultra-short (0.1-0.3) day events: 10 2 OGLE data Galactic bulge/disk models OGLE data (2622 events) Number of events per bin Number of events per bin PBH MS M PBH = 10 � 3 M � 10 2 f PBH = 1 BD 10 1 besft-fit PBH model 10 1 ( M PBH = 9 . 5 × 10 � 6 M � , f PBH = 0 . 026) WD 10 0 10 0 NS 10 − 1 10 0 10 1 10 2 10 − 1 10 0 ML LC timescale: t E [days] ML LC timescale: t E [days] Niikura et al.

Exclusion limit Allowed region assuming no PBH lensing observed assuming 6 ultra-short events are due to PBHs M PBH [ M � ] M PBH [ M � ] 10 � 8 10 � 7 10 � 6 10 � 5 10 � 4 10 � 10 10 � 6 10 � 2 10 2 10 0 10 0 OGLE alone Kepler Caustic CMB f PBH = Ω PBH / Ω DM f PBH = Ω PBH / Ω DM EROS/MACHO 10 � 1 HSC 10 � 1 OGLE+HSC 10 � 2 OGLE excl. region (95% CL) HSC excl. region 10 � 2 10 � 3 allowed region (95% CL) 10 � 4 10 � 3 10 20 10 25 10 30 10 35 10 26 10 27 10 28 10 29 M PBH [g] M PBH [g]

Constraints on (realistic) extended mass functions Applying constraints calculated assuming a DF MF to extended MFs is subtle. Can’t just compare df/dM to constraints on f as a function of M. White Dwarf Kepler 0.100 Femtolensing EROS/MACHO 0.001 FIRAS Ω PBH /Ω DM 10 - 5 EGγ 10 - 7 WMAP3 10 - 9 10 16 10 21 10 26 10 31 10 36 M PBH [ g ]

Constraints on (realistic) extended mass functions Applying constraints calculated assuming a DF MF to extended MFs is subtle. Can’t just compare df/dM to constraints on f as a function of M. Beware double counting. e.g. EROS microlensing constraints allow f~0.1 for M~M sun or f~0.5 for M~10 M sun, but NOT BOTH . f log 10 ( M/M � )