Primordial Black Holes and the QCD Phase Transition
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1 Primordial Blak Holes and te QCD Pase ransition odd Springer and Joe Kapusta University of Minnesota arxiv: DSU 07 June 8 t, 007
2 Outline Formation of PBH s Desription of te QCD pase transition Impat on PBH formation Cosmologial impliations Mass spetrum Present day abundane
3 Formation Qualitative Desription Adapted from an approa by Carr. We also losely follow Cardall and Fuller [astro-p/ ]. Over-dense region rosses te orizon Expansion Region stops expanding: urnaround Depending on onditions, region will ten eiter ollapse, or disperse
4 Classifying Over-dense Regions Regions are speified by Wen tey enter te orizon eir over-density δρ δ ρ At orizon rossing: ρ ρ ρ ρ ( 1+ δ
5 Evolution of Region Model te region as sperial and omogeneous Apply metri for a losed FRW universe Mat inner/outer regions at orizon rossing ds( τ 8πG τ 3 R d [ ρ τ τ ρ δ ] ( S ( ρ Energy density S Sale fator for region τ ime oordinate for region κ Constant (from mating Expansion stops wen: * ρ S * ρr δ
6 Collapse Condition Density flutuation satisfies equations of General Relativity Size of region must exeed relativisti Jeans Lengt at turnaround d * > R J π vs 4G(1 + 3v ( ρ + s P v ρ s P Speed of sound Energy density Pressure For a onstant sound speed: δ 8π v ( s vs 0.8 for v 3 3 (1 + v (1 + 3 s s vs 1 3
7 Determining Critial Over-density is is rougly onsistent wit iemeyer and Jedamzik (1999 wo found δ 0.70 ± 0.01 in a numerial approa.
8 Determining Critial Over-density is is rougly onsistent wit iemeyer and Jedamzik (1999 wo found δ 0.70 ± 0.01 in a numerial approa. Green et al. (004 and Muso et al. (005 found (onsidering growing mode only δ 0.45 ± 0.0 δ + ~ t (growing mode δ (t A dδ ( t0 0 dt t t 0 + B A δ B ~ t t t 0 1 (deaying mode (50% mixture Perturbations wi are pure growing modes would require only ½ te initial over-density
9 Models of Degrees of Freedom Bag Model Softened First Order/ Seond Order Rapid Crossover < > ( ( ( q < Δ + > Δ ( exp ( exp ( q α β ( Δ + + q q tan 1 ( ( 11.15(1 13( 11.15( pions 17.5 (leptons, gluons (u, d,s quarks, q nd st nd st Δ β α
10 Softening of te Equation of State 4π s( 90 P( Redution in te speed of sound ( s( d 0 ρ( P + s 3 v s dp dρ dp d dρ d
11 Critial δ First Order ransitions mixed pase at orizon rossing enters mixed pase at turn-around exits mixed pase at turn-around ρ is te density at orizon rossing Yellow region denotes te mixed pase
12 Critial δ Higer Order ransitions ρ is te density at orizon rossing
13 Spetrum of Perturbations Assume Gaussian distribution 1 δ P( δ, M exp[ πσ ( M σ ( M ] σ M 10 M 1 n 4 σ is COBE normalized variane wi depends on te spetral index n. [Green and Liddle: astro-p/970451, 1997] Power spetrum of flutuations P ( k ~ k n tis WMAP suggests n 1but pertains to larger lengt sales.
14 Cumulative umber Density umber density formation rate n today dn dt d dt Horizon rossing rate 1 ( t V ( t ε t 0 ( dn R( t m m dt R( t t ( m 0 > 3 dt Probability tat over-density at orizon rossing leads to ollapse Dilution from formation to present time t is te orizon rossing time wi first results in PBHs of mass m dn today dm d ( m today dm ( n ( m > m
15 PBH Mass Spetrum Results
16 PBH Mass Spetrum Results
17 PBH Mass Spetrum Results
18 PBH Mass Spetrum Results
19 PBH Mass Spetrum Results
20 PBH Abundane (QCD ransition Ω PBH n1.4 n1.3 n1.5 n1. n1.1 n1.0 n0.95 Bag Model 7.8x x x x x x x10 7 First Order 1.3x x10 1.3x10 7.3x10 7.3x10 7 nd Order 1.8x x x10-6 ~ 0 ~ 0 Crossover 1.8x x x10-6 ~ 0 ~ 0 Bag Model 6.4x x10 1.5x x10 8.6x x10 10 First Order 1.0x x x x x10 8 nd Order 1.5x x x10-14 ~ 0 ~ 0 Crossover 1.5x x x10-14 ~ 0 ~ 0.3x10 7.x10 7 ~ 0 ~ 0 ~ 0 ~ 0 Fixed Speed 1.8x x x10-6 ~ 0 ~ 0 ~ 0 ~ 0 umber/p 3 n1.4 n1.3 n1.5 n1. n1.1 n1.0 n x x x10 9 ~ 0 ~ 0 ~ 0 ~ 0 Fixed Speed 1.5x x x10-14 ~ 0 ~ 0 ~ 0 ~ 0
21 PBH Abundane (EW ransition Ω PBH n1.4 n1.3 n1.5 n1. n1.1 n1.0 n0.95 L x x x x x x x10 8 L x x x x x x x10 9 L x x x x x x x10 9 Analysis also arried out for a Bag Model type Eletroweak pase transition wit: 100 GeV L 0.5 4, 1.0 4, (latent eat density
22 Summary Enaned blak ole formation for all models of a QCD pase transition. QCD equation of state, spetrum of perturbations, PBH abundane, and PBH mass distribution are diretly related. Experimental data on one or more of tese quantities an elp determine te oters. In te ontext of our model, first order pase transitions in te early universe may be inompatible wit te observed lak of primordial blak oles.
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