Primordial Black Holes in Cosmology Lectures 1 & 2 : What are PBHs? Do they exist? Massimo Ricotti (University of Maryland, USA) Institute of Cosmos Sciences, University of Barcelona 23/10/2017
What are Primordial Black Holes? May have masses from the Planck mass ~10-5 g to ~10 8 M sun. Masses and formation quite different from standard astrophysical BHs. Collapsed relativistic matter (radiation). Mass comparable to the Horizon mass at the epoch of their formation Form in quasi-linear regime (δ ~ 20%-40%) Tiny collapsed fraction during the radiation era may produce all the dark matter! Institute of Cosmos Sciences, University of Barcelona
Do PBHs exist? PBHs with mass < 10 15 g evaporate in t<t H (Hawking 1975) Abundance of PBHs with mass 1 g<m<10 15 gis β < 10 20 10 22 (e.g., Carr 2003) More massive PBHs are poorly constrained: They may constitute the bulk of the dark matter MACHO collaboration: 20% of Milky-Way halo is in compact objects with M 0.1 1 M (but 2000 result, non confirmed by later data) Institute of Cosmos Sciences, Physics Coll. University Virginia Tech, of 02-08-2008 Barcelona p.9/37
Why do we care? 1. Physics on scale otherwise unaccessible by observations Institute of Cosmos Sciences, Physics Coll. University Virginia Tech, of 02-08-2008 Barcelona p.11/37
Why do we care? 1. Physics on scale otherwise unaccessible by observations 2. The dark matter can be made of PBHs 3. Produce MACHOS, IMBH and ULXs? 4. Seeds for supermassive Black Holes? Institute of Cosmos Sciences, Physics Coll. University Virginia Tech, of 02-08-2008 Barcelona p.11/37
Basic Analytical Calculations Institute of Cosmos Sciences, University of Barcelona
Redshift: z=dλ/λ Scale parameter: a(t)=1/(1+z) Trad=2.7 K /a(t) Early times: a(t)=(t/2.389x10 19 s) 1/2 Matter domination: a(t) ~t 2/3 Institute of Cosmos Sciences, University of Barcelona
Institute of Cosmos Sciences, University of Barcelona
1.1 PBHs formation mechanism: PBH Masses=Horizon mass at formation M c3 t G 1015 What is the Jeans Mass? What is the Particle Horizon Mass? Relationship between BH radius and its mean density. (show derivations on the blackboard) t 10 23 s g: s could span an enormous mass r Institute of Cosmos Sciences, University of Barcelona
1.2 Results of GR simulations and critical collapse M ¼ km H ðδ δ c Þ γ where: constant k, the threshold γ 0.36 [50,52 54,159] δc 0.45 [52 54] k = 3.3 Ref. [42] see Carr, Kühnel, Sandstad 2016 for references Institute of Cosmos Sciences, University of Barcelona
2.1 PBHs Thermodynamics and Evaporation PBHs with masses < 10 15 g evaporate in a time < age of universe It is unknown whether evaporating PBHs disintegrate completely or leave behind stable Planck-mass relics (show derivations on the blackboard) Institute of Cosmos Sciences, University of Barcelona
3.1 Collapsed fraction and relics Ω pbh = β(1 + z f )/(1 + z eq ) ~ βt -1/2 M c3 t t G 1015 10 23 s g: f pbh = Ω pbh Ω dm ( Mpbh 1 M ) 1/2 ( β s could span an enormous mass r 10 9 ) A very small value of β (collapsed fraction) may produce all the dark matter in PBHs (show derivations on the blackboard) Institute of Cosmos Sciences, University of Barcelona
Anisotropies measured by Planck (contrast increased by 106)
Ref: Tegmark et al 2002
Gravitational lensing and microlensing
4.1 Mass function: Press- Schechter formalism Halo mass function: analytical formalism and simulations Ref: Klypin et al 2011 (show derivations of PS on the blackboard) Institute of Cosmos Sciences, University of Barcelona
4.1 Mass function: Press- Schechter formalism PBH with mass M pbh = f Hor M h where f Hor < 1 β(m,z) exp[ (δ cr /2σ(M,z)) 2 ] δ cr w where P = wρ is the cosmic EOS (show derivations on the blackboard) Institute of Cosmos Sciences, University of Barcelona
4.2 Effects of critical collapse on PBHs mass function M ¼ km H ðδ δ c Þ γ constant k, the threshold PðδÞ pffiffiffiffiffiffiffiffiffiffi 1 exp 2πσ 2 δ2 2σ 2 ; β ¼ Z δ c dδ kðδ δ c Þ γ PðδÞ kσ 2γ erfc δc pffiffiffi ; 2 σ Institute of Cosmos Sciences, University of Barcelona
4.2 Effects of critical collapse 1 0.500 Ref: Carr, Kühnel, Sandstad 2016 f 0.100 0.050 f 0.010 0.005 0.001 10 20 50 100 MM 11 Institute of Cosmos Sciences, University of Barcelona
Ref: Carr, Kühnel, Sandstad 2016 0.050 f 0.010 0.005 0.001 0 10 20 50 100 MM 11 Institute of Cosmos Sciences, University of Barcelona
4.3 Caveats Non-sphericity: for galaxy formation (triaxiality) this leads to small modifications of the Press- Schechter formalism (Sheth-Tormen) Non-Gaussianity: peak formalism instead of Press- Schechter is probably more precise Institute of Cosmos Sciences, University of Barcelona
5.1 Evolution of PBHs in an expanding universe 1. Primordial binaries (Nakamura et al., 1997) 2. Poisson halos (Afshordi, McDonald, Spergel 2003) 3. Clothing halos (Mack, Ostriker, Ricotti 2007), from theory of secondary infall (Bertshinger 1985) Institute of Cosmos Sciences, University of Barcelona
Clothing Halos (secondary infall) Mack, Ostriker, Ricotti 2007 / r o radius z þ 1 1 M h (z) ¼ i M PBH ; 1000 φ i =3 α = -2.25 M 1=3 h 1 þ z 1 r h ¼ 0:019 pc ; 1 M 1000
6.1 Sub-critical perturbations and ultra-compact minihalos (UCMHs) Lensing from PBHs differs from point mass due to the clothing halo Ricotti & Gould 2009 UCMHs can exist without central PBHs and should be much more common than PBHs. Figure 1. Left: time sequence of the growth of the mass profile of a spherical halo due overdensity δ(z eq ) = 10 and excess mass Institute δm of = Cosmos 100 MSciences,. The solid University lines show of Barcelona the time delineates the lens equation. The two dotted lines show power-law mass profiles with l
Microlensing signal from clothed PBHs and UCMHs Ricotti & Gould 2009 Clothed PBH UCMH, without PBH
Ultra-compact minihalos (UCMHs) UCMHs are a new probe of the high-z universe (Ricotti & Gould 2009): they differ from normal small mass halos produced by DM candidates and inflationary perturbations (low angular momentum of halos). Observable trough: 1. Microlensing 2. DM annihilation For gamma-ray limits see: Scott & Silvertsson 2009 Laki & Beacom 2010 (conclude: either all DM is in PBHs or a small fraction otherwise overproduce UCMHs) Institute of Cosmos Sciences, University of Barcelona
Some models for PBH formation 1. Softening of the EOS during phase transitions: QCD (1 M )ore + e annihilation (10 5 M )(Kholopov & Polnarev 80; Jedamzik 97) 2. Collapse of rare density peaks: depend on the shape of inflaton potential (e.g., potential as in Kawasaki et al 06 produces 100 M PBHs) 3. Collapse of cosmic string loops (e.g., Polnarev & Zemboricz 88; Hawking 89; Brandenberger & Wichoski 98) 4. Bubble collisions (e.g., Crawford & Schramm 82; La & Steinhardt 89) 5. Collapse of domain walls (Berezin et al 83; Ipser & Sikivie 84; Rubin et al. 00) Institute of Cosmos Sciences, Physics Coll. University Virginia Tech, of 02-08-2008 Barcelona p.8/37
Summary (I): Newtonian limit easy to understand: During radiation era c s c/ 3 hence R J R Sch R h Any linear perturbation δ > δ cr 0.1 0.7 collapses into PBH with mass M pbh = f Hor M h where f Hor < 1 Collapsed fraction depends on the power spectrum of initial density fluctuations and the cosmic equation of state: δ cr w where P = wρ is the cosmic EOS β(m,z) exp[ (δ cr /2σ(M,z)) 2 ] (assuming Gaussian fluctuations) Institute of Cosmos Sciences, Physics Coll. University Virginia Tech, of 02-08-2008 Barcelona p.6/37
Radiation is redshifted away, PBHs are not: Example: Ω pbh = β(1 + z f )/(1 + z eq ) f pbh = Ω pbh Ω dm ( Mpbh 1 M ) 1/2 ( β 10 9 During QCD phase transition at t = 10 5 sec M pbh M h =1M if β = 10 9 f pbh 1: all the dark matter is made of PBHs Summary (II): ) Institute of Cosmos Sciences, Physics Coll. University Virginia Tech, of 02-08-2008 Barcelona p.7/37
Next: astrophysical constraints on PBHs (part I). Institute of Cosmos Sciences, University of Barcelona