PBHs as dark matter: CMB bounds and GW radiation. Yacine Ali-Haïmoud Johns Hopkins University

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1 PBHs as dark matter: CMB bounds and GW radiation Yacine Ali-Haïmoud Johns Hopkins University Universitat de Barcelona, 18 January 2017

2 micro-lensing Planck (strong feedback) wide binaries Planck (no feedback) ultra-faint dwarfs ( ) ROM - - Ali-Haïmoud & Kamionkowski arxiv: /

3 Note: the CMB also constrains lowermass PBHs (will not discuss these here) Luminosity is produced by Hawking radiation L ~ 1/M 2 BH EGB Femtolensing Planck (grams) Poulin, Lesgourgues & Serpico 2016 m BH (g) Clark et al. 2016

4 Physics underlying CMB bounds for the M & M mass range PBHs accrete baryons a fraction of the accreted mass is re-radiated a fraction of this luminosity is deposited into the plasma some is deposited as heat => CMB spectral distortions some leads to extra ionizations => change the recombination history and visibility function => affects CMB temperature and polarization anisotropies

5 Adopted philosophy (i) Strive to derive the minimum physically plausible luminosity to get the most conservative (and robust) bounds (ii) Accretion physics is complex. Bounds (even if conservative) are robust at the order-of-magnitude level only

6 Accretion model overview Steady-state spherical accretion model. Difficult to estimate angular momentum, which typically leads to a disk and greater luminosity => conservative hypothesis Build on Shapiro (1973), adding Compton drag and cooling Account for relative velocity of PBHs wrt baryons, à la Bondi & Hoyle: replace cs by (cs 2 + vrel 2 ) 1/2 throughout, and average the luminosity over the Gaussian distribution of vrel Consider two limiting cases for feedback: (i) no feedback: xe = <xe> deep inside the Bondi radius (ii) strong feedback: xe = 1 beyond the Bondi radius.

7 Luminosity Luminosity is due to free-free radiation near the horizon. Integrated emissivity (ergs/s/cm 3 ): j = n 2 e c T T J (T ) Accretion rate: Ṁ =4 r 2 b v 4 r 2 S cm p n e => Luminosity is quadratic in accretion rate: L 4 r 3 Sj T S m p c 2 J (T S) Ṁc2 L Edd Ṁc 2

8 Accretion rate Ṁ = 4 1 (GM) 2 c 3 s (Bondi 1952) If Compton cooling time << Bondi time, isothermal accretion: c 2 s =(1+x e ) T 1 m p = 1 4 e3/ Otherwise, adiabatic accretion: c 2 s = 5 3 (1 + x e) T 1 m p (3/5) 3/2 0.12

9 Accretion rate T 2/3 d dt 2/3 = cool (T cmb T ), cool 8x1 e T cab 3m e c(1 + x 1 e ) We compute (t B cool ) numerically and provide an analytic fit

10 10 4 M 10 2 M t B γ cool 1 M no feedback strong feedback isothermal adiabatic z Difference with Ricotti, Ostriker & Mack 2008 (ROM): ROM assume the isothermal limit

11 If Compton drag time << Bondi time, gravity counteracted by Compton drag instead of pressure dv dt = GM 1 r 2 dp dr dragv, drag 4 3 x e T cmb m p c v r 2 GM drag ) Ṁ 4 1r 2 v =4 1 GM drag We use the analytic approximation of Ricotti (2007) for (t B drag )

12 Accretion rate 10 4 M ṁ Ṁc2 L Edd 10 2 M - - no feedback 1 M strong feedback z

13 Temperature profile T ~10 11 K ~10 9 K m e c 2 / 1/r 2/3 ionized background neutral background ~10 4 K / 1/r collisional ionization T 1 = T cmb r S derived in YAH & MK x 1 e 2 8 r ion r ion r B (Shapiro 1973) r

14 Effect of Compton cooling T m e c 2 ionized background neutral background ~10 4 K collisional ionization T 1 = T cmb r S r B /( cool t B ) 2/3 r

15 T T S m e c 2 / 1/r 2/3 strong feedback / 1/r no feedback T ion T 1 = T cmb r S inner adiabatic ionization outer, constant xe 1+x 1 e 2 8 r ion r ion min[1, 2/3 ]r B r

16 Temperature near the horizon Compton cooling collisional ionizations 1 M ( ) 10 2 M 10 4 M no feedback strong feedback z

17 Radiative efficiency - 1 M L = Ṁc2 ϵ M 10 4 M - no feedback strong feedback ROM assume /ṁ =0.011

18 Luminosity M < >/ M 1 M no feedback strong feedback

19 Relative velocities Baryons and dark matter have large-scale relative motions (see e.g. Tseliakhovich & Hirata 2010) before recombination: - baryons tightly coupled to photons, acoustic oscillations - dark matter perturbations grow vrel 30 km/s (~Mach 5 at recombination) after recombination: baryons become cold like DM. vrel 1/a In addition, small-scale motions due to non-linear clustering. We do not account for those.

20 Relative velocities L / Ṁ 2 / 1 (c 2 s + v 2 rel )3 (in the simple Bondi case) hli / 1 (c 2 s + vrel 2 )3 1 c 3 shv 2 rel i3/2, hv2 reli c 2 s hli L(v rel = 0) cs hv 2 rel i1/ at recombination ROM have vrel 4 km/s. cs at recombination.

21 Energy deposition into the plasma Energy injected as ~flat photon spectrum up to ~0.2 to 6 MeV. Dominant deposition mechanism: Compton scattering Boltzmann equation for photon number density per energy bin: a 2 d dt (a2 N E ) 1 E d inj E(E)N E @E E(E) n H ch Ei rate of energy loss by Compton scat. d inj de inj E max = 1 E max f pbh dm M pbh hli

22 h Ei/ TE Thomson limit ΔE << E Ultra-relativistic limit σ << σt /

23 We approximate h Ei 0.1 T E Z dep = n H c deh EiN E After some algebra, arrive at the very simple ODE a 7 d dt (a7 dep ) 0.1 n H c T ( inj dep ). Integral forms of this equation (with incorrect scalings) were previously given in Cirelli, Iocco & Panci (2009), Natarajan & Schwarz (2010), Giesen, Lesgourgues, Audren & Ali-Haïmoud (2012), Poulin, Serpico & Lesgourgues (2015)

24 10 2 M ρ inefficient deposition dep / a 7 on-the-spot deposition dep inj ρ no feedback strong feedback

25 µ 1.4 y 1 4 CMB spectral distortions Z Z heat dep d ln(1 + z), H cmb heat dep d ln(1 + z) H cmb hli e.g. Chluba 2016 µ apple f pbh max z y 0.02 f pbh hli L Edd z 200 L Edd. Undetectable by FIRAS (µ, y ~10-5 ), or even by PIXIE (µ, y ~10-8 )

26 Effect on recombination history We take the simple prescription of Chen & Kamionkowski 2004: T gas = 2 1+2x e 3n tot 3 dep heating ẋ direct e = 1 x e 3 dep E I n H direct ionizations ẋ 2 = 1 3 x e dep E 2 n H excitations

27 M,f pbh = M,f pbh = M,f pbh =1 Δ - - Computed with modified HYREC (Ali-Haïmoud & Hirata 2011)

28 Effect on anisotropies M,f pbh =1 Δ / ( ) M,f pbh = M,f pbh = Computed with modified CLASS (Blas, Lesgourgues & Tram 2011)

29 Effect on anisotropies 10 4 M,f pbh = 10 4 Δ / ( ) 10 3 M,f pbh = M,f pbh =1 Computed with modified CLASS (Blas, Lesgourgues & Tram 2011)

30 Planck data analysis We explicitly use the Planck high-l (l > 30) Plik_lite best-fit and covariance matrix for the TT, TE and EE spectra. Add a prior on τreion = ± to account for l < 30 (Planck intermediate results, 2016) We perform a χ 2 analysis to fit simultaneously for (H 0, b h 2, c h 2,A s,n s, reio,f pbh ) This also gives (co)variances of all 7 parameters

31 Note We only consider M < 10 4 Msun: for larger masses the steady-state approximation breaks down (Ricotti 2007). Larger masses are excluded by Lyman-α clustering anyway (Afshordi & Spergel 2003): Poisson noise from PBH clustering would lead to fluctuations larger than observed.

32 micro-lensing Planck (strong feedback) wide binaries Planck (no feedback) ultra-faint dwarfs ( ) ROM - - Ali-Haïmoud & Kamionkowski arxiv: /

33 Did LIGO detect dark matter? with Bird, Cholis, Kamionkowski, Kovetz, Muñoz, Raccanelli & Riess, PRL 2016

34 Basic idea ~ 30 Msun PBHs are not excluded by CMB limits. Wide-binary bound is weak. Ultra-faint dwarf bound is assumption-dependent (see Brandt 2016). If 2 PBHs pass close enough to one another, they may lose enough energy through GW emission to become bound. GM c 2 2 (v/c) 18/7 (Quinlan & Shapiro 1989) Merger rate per halo: = 1 Z dv h vi 2 M 2 GM h Total merger rate per unit volume: R = c 2 G h c (v h/c) 11/7 Z dm h dn h dm h (M h )

35 Assume a ~ Press-Schechter mass function and a massconcentration relation ρh(mh) extrapolated from simulations Cutoff the diverging integral at Mmin ~ 400 Msun s.t. evaporation time ~ Hubble time. We get R 1Gpc 3 yr 1 Roughly consistent with LIGO s inferred merger rate R LIGO =2 53 Gpc 3 yr 1 (90%credible range) This is an order-of-magnitude estimate but still, an interesting rate coincidence!

36 Some follow-ups (non-exhaustive list) Raccanelli et al. (2016): estimate the cross-correlation between galaxy catalogs and GW events Stellar, z=0.35 PBH, z=0.35 Stellar, z=1.0 PBH, z=1.0 ( +1) C /2π Assuming merger scenario of Bird et al., spatial distribution of GW events from PBHs should be less biased than stellar-origin mergers wrt large-scale matter distribution

37 Some follow-ups (non-exhaustive list) Cholis et al. (2016): compute the distribution of eccentricities of PBH binaries, assuming the same scenario as Bird et al initial PDF of initial eccenticity e 0 for PBH binaries m 1 =m 2 =30 M ~entry in LIGO band PDF of eccenticity at r p =22 R Sch Orbit, e 22, for PBH binaries M vir =10 12 (M /h) m 1 =m 2 =30 M M vir =10 9 (M /h) M vir =10 6 (M /h) PDF(e0) PBH M vir =10 12 (M /h) M vir =10 9 (M /h) PDF(e22) PBH M vir =10 6 (M /h) e e 22 A small fraction (~ 1%) of PBH binary mergers should have high eccentricities. Prospects for LIGO are weak, better with Einstein Telescope

38 Some follow-ups (non-exhaustive list) Muñoz et al. (PRL 2016): lensing of Fast Radio Bursts (FRBs) by PBHs can be used to set stringent bounds on PBH abundance, once large number of FRBs are detected (e.g. by CHIME) MACHO WB fdm EROS Δt = 0.3 ms Δt = 1 ms Δt = 3 ms M L [M ]

39 Alternative PBH merger scenario Bird et al. 16: assume PBHs merge through energy loss by GW radiation (analogous to atomic free-bound radiation) Sasaki et al. 16: assume PBHs binaries are primordially formed, arising from random distribution of PBHs. Vastly different merger rate!

40 Conclusions CMB spectral distortions (present and future) do not set any bounds to PBHs with M M CMB temperature and polarization anisotropy data from Planck exclude PBHs to be the dominant dark matter for M & 10 2 M These statements rely on conservative, physicallymotivated estimates of the accretion luminosity, but are subject to large theoretical uncertainty. PBH binaries merge with a rate roughly comparable with LIGO s estimate, if they make up ~100% of the DM (Bird et al. 2016) or ~0.1% of it (Sasaki et al. 2016)