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Page 1: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Seminar of Department of Astronomy - University of Belgrade

Stellar mass Primordial Black Holes

Laurindo Sobrinho

Faculdade de Ciências Exatas e da Engenharia da Universidade da MadeiraDepartamento de Matemática

Grupo de Astronomia da Universidade da Madeira (GAUMa)Instituto de Astrofísica e Ciências do Espaço (IA)

March 2, 2021

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 2: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Table of contents

1 PBHs from density uctuations

2 The threshold for PBH formation

3 The fraction of the Universe going into PBHs

4 Conclusions and future work

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 3: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

1 - PBHs from density uctuations

Black Holes may have formed in the Early Universe as a

consequence of the gravitational collapse of density

uctuations (Hawking 1971, Hawking & Carr 1974, Novikov et

al. 1979).

These Primordial Black Holes (PBHs) could have masses

ranging from the Planck mass up to ∼ 1011M. Here we will

consider the (extended) stellar mass range

[0.05M − 500M].

During ination, uctuations of quantum origin are stretched

to scales much larger than the cosmological horizon becoming

causally disconnected from physical processes.

The inationary era is followed, respectively, by

radiation-dominated and matter-dominated epochs during

which these uctuations can re-enter the cosmological horizon.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 4: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

For a given physical scale k, the horizon crossing time tk (i.e. the

instant when that scale re-enters the cosmological horizon) is given

by (Blais et al. 2003):

ck = a(tk)H(tk)

where a(tk) is the scale factor and H(tk) the Hubble parameter.

The collapse that gives rise to the formation of a PBH is now

possible but only if the amplitude of the density uctuation:

δ =∆m

m

is larger than a specic threshold value δc.

δk ≥ δc the expansion of the overdense region will,

eventually, come to a halt, followed by its collapse leading to

the formation of a PBH

δk < δc the uctuation dissipates without forming a PBH

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 5: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The majority of the PBHs formed at a particular epoch have

masses within the order of the horizon mass, MH , at that epoch

(Carr et al. 2003):

MH(t) ∼ 1015(

t

10−23 s

)g.

In the case of perturbations with δ only slightly larger than δc thePBH masses rather obey the scaling law (Niemeyer &

Jedamzik 1999):

MPBH ∝MH (δ − δc)γ

where γ ≈ 0.36 in the case of a radiation-dominated Universe. This

scaling law has been found to hold down to (δ − δc) ∼ 10−10

(Musco & Miller 2013).

Here we assume: MPBH(tk) = MH(tk).

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 6: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The probability that a uctuation crossing the horizon at some

instant tk has of collapsing and forming a PBH can be written as

(e.g. Green 2015):

β(tk) =1√

2πσ(tk)

∫ ∞δc

exp

(− δ2

2σ2(tk)

)dδ

σ2(tk) mass variance at horizon crossing

δc the threshold for PBH formation

The value of β(tk) can also be regarded as the fraction of the

Universe going into PBHs.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 7: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

2 - The threshold for PBH formation

δc = 1/3 simplied model of an overdense collapsing region

for a radiation-dominated Universe (Carr 1975)

δc ' 0.43− 0.47 numerically solving the relativistic

hydrodynamical equations for a radiation-dominated Universe

(Musco & Miller 2013, Harada et al. 2013)

The exact value of δc depends on the perturbation prole. We

consider δc = 0.43 (Mexican-Hat perturbation, a very

representative one).

δc is constant through the radiation-dominated epoch, the

exception occurring during cosmological phase transitions, when

the value of δc decreases (as a consequence of the decrease of the

sound speed). This is relevant, since a lower value of δc favoursPBH formation (Carr 2003).

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 8: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The Standard Model of Particle Physics (SMPP) predicts:

Electroweak (EW) phase transition at temperatures of

∼ 100 GeV (when the age of the Universe was ∼ 10−10 s),responsible for the spontaneous breaking of the EW symmetry.

PBHs formed during the epoch of the EW phase transition

would have ∼ 10−6M.

Quantum Chromodynamics (QCD) phase transition at

Tc = 170 MeV when quarks and gluons become conned in

hadrons. The QCD phase transition occurred when the

universe was ∼ 10−5 s and that corresponds to MH ∼ 0.5M.

If we want to study the formation of stellar mass PBHs we cannot

neglect the eect of the QCD phase transition. In particular we

need to know how the variation of the sound speed during theQCD phase transition will aect the value of δc.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 9: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

During the radiation-dominated epoch the Universe can be

regarded as a diluted gas with its Equation of State (EoS) written

as (Carr 2003):

p = wρ

where p is pressure, ρ is the cosmological density, and the

dimensionless quantity w (the EoS parameter) is equal to 13 , since

the sound speed is (Schmid et al. 1999):

c2s =

(∂p

∂ρ

)S

= w =1

3

If during the QCD phase transition the Universe becomes

matter-dominated (pressureless gas), then we get w = 0 and

c2s = 0.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 10: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The evolution of the scale factor for the perturbed region (s(τ)) is(Sobrinho 2011; Sobrinho, Augusto & Gonçalves 2016):

(ds

)2

=8πG

3

Ks

1 + δk

(1 + δks(τ)1+3w

− Kk

Ks

δk

a1+3wkk

)

where:

Ks and Kk are constants to be determined

ak is the scale factor value at the horizon crossing time

wk is the EoS parameter at the horizon crossing time

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 11: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The turnaround point (tc) is reached when the perturbed region

stops expanding, i.e., when ds/dτ = 0. Thus, we get:

s1+3wcc =

Ks

Kka1+3wkk

(1 + δkδk

)

which relates the size of the perturbed region at the horizon

crossing time with the respective size at the turnaround point. The

calculation of the relation Ks/Kk is detailed in Sobrinho, Augusto

& Gonçalves (2016).

We assume that whentc is reached a PBH will form. Hence, we are

not taking into account the dynamics between the turnaround point

and the instant when the PHB actually arises with the formation of

an event horizon (which would require to numerically solve the

Hernandez-Misner equations).

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 12: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

If we want to consider the formation of stellar mass PBHs we need

to take into account the QCD phase transition.

More exactly we need to know how the sound speed behaves during

the QCD phase transition.

Unfortunately we don't know exactly which model ts the QCD

phase transition. We have considered three dierent models:

Bag Model (BM)

Lattice Fit Model (LFM)

Crossover Model (CM)

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 13: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

2.1 - Bag Model (BM)

a high temperature region (T > Tc, t < t−) where we have a QuarkGluon Plasma (QGP)

T = Tc = 170 MeV ([t−, t+]): dust-like phase where quarks,gluons, and hadrons coexist in equilibrium at constant pressure andtemperature and c2s = 0

a low temperature region (T < Tc, t > t+) where we have anHadron Gas (HG).

-4.4 -4.2 -4 -3.8 -3.6Log10Ht1sL

0

0.1

0.2

0.3

0.4

cs2

t-

t+

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 14: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Classes of uctuations

Class Horizon crossing Turnaround

A quarkgluon quarkgluon

B quarkgluon mixed

C quarkgluon hadron

D mixed mixed

E mixed hadron

F hadron hadron

Idea: replace δc by δc(1− f)

f (0 ≤ f < 1) fraction of the overdense region spent in the

dust-like phase of the transition (Sobrinho, Augusto & Gonçalves

2016).

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 15: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 1a in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

0 0.2 0.4 0.6 0.8 1∆k

0

0.2

0.4

0.6

0.8

1

∆k

HaL

ABC

∆AB∆BC

d

∆c1

solid line −→ (1− f)δc [red PBH formation allowed: δk ≥ (1− f)δc]

dashed line −→ δk

tk = 4.0× 10−5 s

δc1 = 0.28 new threshold for PBH formation (< 0.43)

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 16: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 1b in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

0 0.1 0.2 0.3 0.4 0.5∆k

0

0.1

0.2

0.3

0.4

0.5

∆k

HbL

ABC∆AB∆BC

d

∆c1

∆c2

solid line −→ (1− f)δc [red PBH formation allowed: δk ≥ (1− f)δc]

dashed line −→ δk

tk = 1.1× 10−5 s

[δc1, δc2] = [0.15, 0.23] new thresholds (new window) for PBHformation (< 0.43)

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 17: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

δc for a QCD BM

Figure 2 in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

-6 -5.5 -5 -4.5 -4Log10Htk1sL

0.1

0.2

0.3

0.4

0.5

0.6

∆k

t- t+BH formation

No BH formation

∆c1

∆c2

∆c=0.43

Minimum value of δc: 0.097

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 18: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

2.2 - Crossover Model (CM)

The sound speed during a QCD Crossover (Schwarz 1998):

c2s(t) =

3 +∆gT (t)sech

(T (t)−Tc

∆T

)2

∆T(gHG + gQGP + ∆gtanh

(T (t)−Tc

∆T

))−1

T (t) = T0

[exp

(c

√Λ

3(tSN − t0)

)(teqtSN

)2/3(t

teq

)1/2]−1

∆T = 0.1Tc with Tc = 170 MeV; T0 = 2.72548 K

gQGP number of degrees of freedom for the QGP (61.75).

gHG number of degrees of freedom for the HG (17.25).

∆g = gQGP − gHG (see Sobrinho 2011 for details)

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 19: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 3 in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

-4.4 -4.2 -4 -3.8 -3.6 -3.4 -3.2Log Ht1sL

0.1

0.15

0.2

0.25

0.3

0.35

0.4

cs2

t1 t2

f =3

2

(tk

1 + δkδk

)−3/2 ∫ tk1+δkδk

t1

(1− cs(t)

cs0

)√tdt

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 20: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 4 in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

0.2 0.4 0.6 0.8 1∆k

0.32

0.34

0.36

0.38

0.4

0.42

0.44

∆k

∆c1

solid line −→ (1− f)δc [red PBH formation allowed: δk ≥ (1− f)δc]

dashed line −→ δk

tk = 4.2× 10−5 s

δc1 = 0.345 new threshold for PBH formation (< 0.43)

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 21: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

δc for a QCD Crossover

Figure 5 in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

-5 -4.5 -4 -3.5 -3Log10Htk1sL

0.34

0.36

0.38

0.4

0.42

0.44

0.46

∆k

BH formation

No BH formation

t1 t2

∆c1

∆c=0.43

Minimum value of δc: 0.345

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 22: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

2.3 - Lattice Fit Model

Figure 4 in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

-7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5

Log10Ht1sL

0

0.1

0.2

0.3

cs2

t-

t+

t1

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 23: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

δc for a QCD LFM

Figure 4 in New thresholds for primordial black hole formation during the QCD phase transition",Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L., 2016, Monthly Notices of the Royal AstronomicalSociety, Volume 463, Issue 3.

-9 -8 -7 -6 -5 -4 -3

Log10Htk1sL

0.1

0.2

0.3

0.4

0.5

0.6

∆k

∆c1∆c2

∆cA

∆c=0.43

t1 t- t+

BH formation

No BH formation

Minimum value of δc: 0.15

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 24: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

3 - Mass variance

(Sobrinho & Augusto 2020)

σ2(k) =

∫ kek

0x3δ2H(kx)W 2

TH(x)W 2TH(

x√3

)dx

δ2H(k) =

(10

9

)2

δ2H(kc)

(k

kc

)n(k)−1

WTH Top-hat window function

kc = 0.05Mpc−1 ≈ 1.6× 10−24m−1 → pivot scale (Planck mission)

ke ≈ 0.01m−1 → smallest slace generated by ination

δ2H(kc) ≈ 2.198× 10−9 → Planck Collaboration et al. 2016

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 25: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Spectral index:

n(k) = n0 +∑i≥1

ni(i+ 1)!

(lnk

kc

)i

n0 = 0.9476n1 = 0.001n2 = 0.022

Planck (e.g. Erfani 2014)

n3n4

work on the plane (n3, n4)

We are assuming ni = 0 when i ≥ 5

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 26: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

PBHs in signicant numbers =⇒ n(k) > 1(blue spectrum)

-10 -5 0 5 10 15

log10Htk

1 sL

0.25

0.5

0.75

1

1.25

1.5

1.75

2

nHtkLnmax=1.803, log10Htkmax1sL=-3

n3 = 0.0099, n4 = −0.0033.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 27: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

We are interested in situations for which n(k) shows a local maximumat some point k = k+ with n+ = n(k+) > 1.

n+ = n0 +n1

2ln

k+kc

+n2

6

(ln

k+kc

)2

+n3

24

(ln

k+kc

)3

+n4

120

(ln

k+kc

)4

dn(k)

dk

∣∣k=k+ = 0⇔ n1

2+

n2

3ln

k+kc

+n3

8

(ln

k+kc

)2

+n4

30

(ln

k+kc

)3

= 0

Given a pair of values (n+, k+), or equivalently (n+, t+), we candetermine the corresponding pair of values (n3, n4):

n3 =

−4(24n0 − 24n+ + 9n1 ln

k+kc

+ 2n2

(ln

k+kc

)2)

(ln

k+kc

)3

n4 =

20

(18n0 − 18n+ + 6n1 ln

k+kc

+ n2

(ln

k+kc

)2)

(ln

k+kc

)4

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 28: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The path for the calculus of β

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 29: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Constraints on PBHs for a variety of evaporation, dynamical, lensing,large-scale structure and accretion eects (Carr et al. 2016)

f(M) ≈ 4.11× 108β(M)

(M

M

)−1/2

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 30: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 2a in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

1.5 1.6 1.7 1.8 1.9 2

nmax

-9

-7

-5

-3

-1

logHtkmax1sL

HaL Crossover Model

No PBH formation

Forbidden region

Below the solid curve, PBH formation is not allowed since it wouldviolate the observational constraints. Above the dashed line, PBHformation is allowed although in negligible numbers (less than one PBHwithin the observable Universe).

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 31: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 2b in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

1.5 1.6 1.7 1.8 1.9 2

nmax

-9

-7

-5

-3

-1

logHtkmax1sL

HbL Bag Model

No PBH formation

Forbidden region

For a given value of tkmaxthe fraction of the Universe going into PBHs,

β(tk), will be maximum if the corresponding value of nmax is the onelocated over the solid curve

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 32: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 2c in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

1.5 1.6 1.7 1.8 1.9 2

nmax

-9

-7

-5

-3

-1

logHtkmax1sL

HcL Lattice Fit Model

No PBH formation

Forbidden region

Selected cases: (a) tkmax= 10−9 s; (b) tkmax

= 10−7 s;(c) tkmax

= 10−5 s; (d) tkmax= 10−3 s, and (e) tkmax

= 10−1 s.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 33: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 3e in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

-2.5 -2 -1.5 -1 -0.5 0 0.5 1

log10Htk1sL

-40

-30

-20

-10

0

log10ΒHtkL

BM

LFM

CM

oc

HeL log10Htkmax1sL=-1.0

The uctuations crossed the horizon suciently after the QCD epoch.All the three QCD models share the same curve (with nmax = 1.920),characterized by a radiation peak.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 34: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

The present day value of the PBH density parameter:

ΩPBH(t0) = a(teq)

∫ t′

t∗

β(tk)

a(tk)dtk

where

ΩPBH(t0, tk) = β(tk)a(teq)

a(tk)

The present day value of the PBH number density for PBHs

formed between two given instants t1 and t2:

nPBH(t0) = ρc(t0)

∫ t2

t1

ΩPBH(t0, tk)

MH(tk)dtk

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 35: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

1 2 3 4 5

log10HMPBH

5 ML

0

2

4

6

8

10

12

log10HNGpc3L

log10Htkmax1sL=-1.0

Intermediate mass PBHs (5× 102 − 5× 104M) with a radiation peaklocated at 5× 103M, giving ∼ 1011 PBHs/Gpc3.

Contribution to CDM: ≈ 0.001%.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 36: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 3a in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

-12 -10 -8 -6 -4

log10Htk1sL

-40

-30

-20

-10

0

log10ΒHtkL

BMLFM

CM

BM

ocHaL log10Htkmax1sL=-9.0

For both the CM and LFM we have the same β(tk) curve (left), whichcorresponds to a radiation peak. These uctuations crossed the horizonsuciently before the QCD epoch. If we consider a BM instead, then wecannot neglect the contribution from the QCD and we have, in addition,a QCD peak.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 37: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

log10HMPBH

5 ML

024681012141618202224

log10HNGpc3L

log10Htkmax1sL=-9.0

BM

PBHs with sub-stellar masses (5× 10−10 − 5× 10−4M)Peak: ∼ 5× 10−7M (∼ 1026 PBHs/Gpc3).Total: ∼ 1026 PBHs/Gpc3

QCD peak (BM only): ∼ 1012Gpc3 PBHs with 0.5M.

Contribution to CDM: ≈ 29%.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 38: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 3b in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

-10 -8 -6 -4

log10Htk1sL

-40

-30

-20

-10

0

log10ΒHtkL

BM

LFM

CM

ocHbL log10Htkmax1sL=-7.0

CM - radiation peak

LFM - radiation peak + QCD peak

BM - dominated by a QCD peak

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 39: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

-7 -6 -5 -4 -3 -2 -1 0

log10HMPBH

5 ML

0

2

4

6

8

10

12

14

16

18

20

22

log10HNGpc3L

log10Htkmax1sL=-7.0

HaL LFM

-7 -6 -5 -4 -3 -2 -1 0

log10HMPBH

5 ML

0

2

4

6

8

10

12

14

16

18

20

22

log10HNGpc3L

log10Htkmax1sL=-7.0

HbL CM

LFM Radiation peak: 5× 10−4M, ∼ 1022 PBHs/Gpc3

QCD peak: 0.5M, ∼ 1018 PBHs/Gpc3

Contribution to CDM: ≈ 2.3%

CM Radiation peak: 5× 10−4M, ∼ 1023 PBHs/Gpc3

Contribution to CDM: ≈ 31%

BM QCD peak: 0.5M, ∼ 1020 PBHs/Gpc3

Contribution to CDM: ≈ 0.4%

Laurindo Sobrinho Stellar mass Primordial Black Holes

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Fluctuations The threshold δc β(tk) Conclusions

Figure 3c in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

-7 -6 -5 -4 -3 -2

log10Htk1sL

-40

-30

-20

-10

0log10ΒHtkL

BM

LFM CM

ocHcL log10Htkmax1sL=-5.0

-4 -3 -2 -1 0 1

log10HMPBH

5 ML

0

2

4

6

8

10

12

14

16

18

20

log10HNGpc3L

log10Htkmax1sL=-5.0

CM Radiation peak: 0.05M, ∼ 1020 PBHs/Gpc3

QCD peak somehow hidden on the 0.5M bar

Contribution to CDM: ≈ 12%

BM & LFM QCD peak: 0.5M, ∼ 1018 PBHs/Gpc3

Contribution to CDM: ≈ 0.6%

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 41: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

-5 -4 -3 -2 -1

log10Htk1sL

-40

-30

-20

-10

0

log10ΒHtkL

BM LFM

CM

oc

HdL log10Htkmax1sL=-3.0

-2 -1 0 1 2 3

log10HMPBH

5 ML

0

2

4

6

8

10

12

14

16

18

log10HNGpc3L

log10Htkmax1sL=-3.0

HaL BM

-2 -1 0 1 2 3

log10HMPBH

5 ML

0

2

4

6

8

10

12

14

16

18

log10HNGpc3L

log10Htkmax1sL=-3.0

HbL LFM

-2 -1 0 1 2 3

log10HMPBH

5 ML

0

2

4

6

8

10

12

14

16

18

log10HNGpc3L

log10Htkmax1sL=-3.0

HcL CM

BM QCD peak: 0.5M, ∼ 1019 PBHs/Gpc3, CDM: ≈ 2%

LFM QCD peak: 0.5M, ∼ 1018 PBHs/Gpc3, CDM: ≈ 2%

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 42: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Figure 4 in Stellar mass Primordial Black Holes as Cold Dark Matter", Sobrinho, J. L. G.; Augusto, P.,2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 1.

-2 -1 0 1 2 3

log10HMPBH

5 ML

0

2

4

6

8

10

log10HNMpc3L

log10Htkmax1sL=-3.0

10-6%

44% 32%

10-3%

CM extended mass spectrum (plateau formed by the radiation andQCD peaks):

∼ 1012 PBHs/Gpc3 (0.5M)

∼ 1019 PBHs/Gpc3 (5M)

∼ 1018 PBHs/Gpc3 (50M)

∼ 1013 PBHs/Gpc3 (500M)Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 43: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Conclusions

Stellar mass PBHs might have formed in the Early Universe

Stellar mass PBHs could be an important fraction of CDM

At least some of the BHs mergers observed (gravitational

waves) could be of primordial origin

A monochromatic peak at ∼ 0.5 M will favour a BM or LFM

model, while a broader mass spectrum (550M) will suggesta CM for the QCD.

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 44: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Some ideas for future work

Formation of PBH binaries (at the formation epoch)

Formation of PBH binaries (during the evolution of the

Universe)

Concentration of PBHs around galactic haloes

Formation of clusters of PBHs

Find out scenarios that t with the observed mergers

Consider other mass ranges (sub-stellar and supermassive)

.....

Laurindo Sobrinho Stellar mass Primordial Black Holes

Page 45: Stellar mass Primordial Black Holes

Fluctuations The threshold δc β(tk) Conclusions

Some (other) ideas for future work

Some ideas that can help to rene the results obtained in the

previous topics:

Consider the scaling law for the PBH masses

Consider non-gaussian distributions for the uctuations

Consider other kinds of spcetra for the uctuations

Consider other PBH formation mechanisms

.....

Laurindo Sobrinho Stellar mass Primordial Black Holes

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Fluctuations The threshold δc β(tk) Conclusions

References

Sobrinho, J. L. G. (2011), The Possibility of Primordial Black Hole DirectDetection, Sobrinho, L.; Tese de Doutoramento em Matemática (especialidadede Física-Matemática), Universidade da Madeira, pp. 319.

Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L. (2016), New thresholds forprimordial black hole formation during the QCD phase transition, MonthlyNotices of the Royal Astronomical Society (MNRAS), 463, 2348.

Sobrinho, J. L. G.; Augusto, P.; Gonçalves, A. L. (2020), Stellar massPrimordial Black Holes as Cold Dark Matter, Monthly Notices of the RoyalAstronomical Society (MNRAS), 496, 60.

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Carr B. J., Hawking S. W., 1974, MNRAS, 168, 399

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Carr B. J., 2003, in D. Giulini, C. Kiefer and C. Lämmerzahl, eds, Lecture Notes in Physics, Vol.631, Quantum Gravity: From Theory to Experimental Search. Springer, Berlin, p. 301

Niemeyer J. C., Jedamzik K., 1999, Phys. Rev. D, 59, 124013

green2015 Green A. M., 2015, in Calmet X., ed., Quantum Aspects of Black Holes. Springer,London, p. 129

Musco I., Miller J. C., 2013, Class. Quantum Grav., 30, 145009

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Carr B., Kühnel F., Sandstad M., 2016, PhRvD, 94, 083504. doi:10.1103/PhysRevD.94.083504

Laurindo Sobrinho Stellar mass Primordial Black Holes

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Fluctuations The threshold δc β(tk) Conclusions

Hvala na panºji!

(c) Grupo de Astronomia da Universidade da Madeira 2021

Laurindo Sobrinho Stellar mass Primordial Black Holes