Primordial Black Holes: the morphology of cosmological - - PowerPoint PPT Presentation

Primordial Black Holes: the morphology of cosmological perturbations

Ilia Musco (CNRS, Observatoire de Paris/Meudon - LUTH)

Gravity and Cosmology workshop YITP - (Kyoto) 13 February 2018

Collaborators: John Miller (Oxford) Alexander Polnarev (London) Chris Byrnes & Sam Young (Sussex)

• In the early universe large amplitude perturbations of the metric can

collapse into Primordial Black Holes (PBHs) [Zeldovich & Novikov (1967); Hawking (1971)] characterized by a wide range of masses (from the Planck mass to 10⁶ M⊙ for PBHs formed at the Nucleosynthesis).

• Hawking evaporation effect (1974) has been inspired by the idea of PBH

formation which could be small as particles and quantum effects need to taken into account. PBHs smaller than 1015 grams would evaporate by now via Hawking evaporation, becoming possible sources of Gamma Ray Burst, Cosmic Rays, evaporation remnants as cold dark matter.

• The threshold amplitude of PBH formation ( δc ∼ cs2 ) measured at horizon

crossing time Carr (1975) tell us if a perturbation collapse into a PBH or bounce and disperse into the surrounding medium. This has been confirmed by full relativistic numerical simulations Nadezin, Novikov & Polnarev (1978), Niemeyer & Jedamzik (1998, 1999); Musco et al. (2005, 2007, 2009, 2013)] suggesting that critical collapse (scaling law) might apply in the early universe, in particular during the radiation dominated era ( cs2 = 1/3 ). Harada, Nakama et al. (2013, 2014, 2015) have calculated an analytical threshold for PBH formation, proposing also a phenomenological parameterisation of PBH threshold in terms of initial density shapes.

PBHs: A bit of history

COSMIC TIME NULL TIME

Dt ≡ 1 a ✓ ∂ ∂t ◆ Dr ≡ 1 b ✓ ∂ ∂r ◆

U ≡ DtR Γ ≡ DrR DtU = −  Γ (e + p)Drp + M R2 + 4πRp

• Dtρ = − ρ

ΓR2 Dr(R2U) Dte = e + p ρ Dtρ DtM = −4πR2pU Dra = − a e + pDrp DrM = 4πR2Γe Γ2 = 1 + U 2 − 2M R Dt ≡ 1 f ✓ ∂ ∂u ◆ Dk ≡ Dr + Dt DtU = − 1 1 − c2

s

 Γ (e + p)Dkp + M R2 + 4πRp + + c2

s

✓ DkU + 2UΓ R ◆ Dtρ = ρ Γ  DtU − DkU − 2UΓ R

• Dte =

✓e + p ρ ◆ Dtρ DtM = −4πR2pU Dk (Γ + U) f

• = −4πR(e + p)f

DkM = 4πR2[eΓ − pU], Γ = DkR − U = 1 + U 2 − 2M R

ds2 = −f 2 du2 − 2fb dr du + R2dΩ

ds2 = −a2 dt2 + b2 dr2 + R2dΩ2

f du = a dt − b dr

• Simulations are performed using a Lagrangian spherically symmetric GR

hydro code with an adaptive grid (AMR).

• We set initial conditions using a cosmic

time coordinate t.

• We transfer those onto a null foliation
• f the space time, then evolved using

an observer time coordinate u.

• The formation of a PBH is seen by a

distant external observer (the singularity is hidden by the asymptotic formation of the apparent horizon).

Numerical Results: the method

Equation of State

• Barotropic fluid (no rest mass density): with
• radiation dominated era:
• matter dominated era:
• Polytropic fluid:
• If the fluid is adiabatic (no entropy change): (constant)

rest mass density specific internal energy (velocity dispersion)

w = 1/3

adiabatic index - particle degree of freedom

w = 0 p = we

w ∈ [0, 1]

p = K(s)ργ

K(s) = K

RADIATION DUST energy density: pressure:

e = ⇢(1 + ✏)

(γ = 4/3) p = ( − 1)⇢✏ (γ = 1)

(γ = 5/3, 4/3, 2)

Numerical Results: PBH formation/bounce

IM, J. Miller - CQG (2005, 2009)

Background model & Curvature profile

• The unperturbed solution, describing an expanding homogeneous universe,

is given by the FRW metric: K = ±1, 0 is the curvature parameter, is the scale factor, and R= r circumferential radial coordinate / areal radius.

• In the linear regime of cosmological perturbations, pure growing modes on

the super horizon scale can be described by a time independent curvature profile (quasi-homogeneous / gradient expansion solution).

K(r) or ζ(˜ r)

r = ˜ reζ(˜

r)

s(t)

s(t)

ds2 = −dt2 + s2(t)  dr2 1 − Kr2 + r2dΩ2

• K(r)r2 = −˜

rζ0(˜ r) [2 + ˜ rζ0(˜ r)]

• Defining the scale of the cosmological perturbations as R0 and the cosmological

horizon scale as RH := 1/Hb . In the linear regime of supra horizon growing modes we can construct a small parameter (t) << 1 as:

• Pure growing modes are given by
• In the linear regime, when << 1, the curvature profile is time independent

because pressure gradients are negligible, and can be used as the only independent source of perturbations.

✏ ✏

PBH formation: setting the problem

✏(t) := RH R0 ∝ ✓ t t0 ◆ (1+3w)

3(1+w)

f(r, t) = f0 h 1 + ✏2(t) ˜ f(r) i ✏ ⌧ 1

a = 1 + ✏2˜ a b = @rR p 1 − K(r)r2 ⇣ 1 + ✏2˜ b ⌘ R = s(t)r ⇣ 1 + ✏2 ˜ R ⌘

e = eb

• 1 + ✏2˜

e

• U = HR

⇣ 1 + ✏2 ˜ U ⌘

M = 4 3⇡ebR3 ⇣ 1 + ✏2 ˜ M ⌘

H2 = 8π 3 eb ⇒ RH = 1 H

• The density profile is expressed in terms of the curvature profile :

Density profile & perturbation amplitude

δe eb = ✓ 1 sH ◆2 3(1 + w) 5 + 3w h K(r) + r 3K0(r) i

• The perturbation amplitude can measured as the mass excess inside a

certain radius ( normalized with respect ε = 1 ) :

δ(r) := 1 V Z r 4π δe eb r02 dr0 V = 4 3πr3

δe eb = − ✓ 1 sH ◆2 2(1 + w) 5 + 3w e2ζ(˜

r)

 ζ00(˜ r) + ζ0(˜ r) ✓2 ˜ r + 1 2ζ0(˜ r) ◆

• The radius of the perturbation is identified as the location where the

perturbation reaches its maximum compactness:

rm : C0(rm) = 0 ⇒ δ(rm) + rm 2 δ0(rm) = 0

f(w) = 3(1 + w) 5 + 3w

C := 2[M(r, t) − Mb(r, t)] R(r, t) = f(w)K(r)r2

K(rm) + rm 2 K0(rm) = 0 ζ0(˜ rm) + ˜ rmζ00(˜ rm) = 0

δm = f(w)K(rm)r2

m = −f(w)˜

rmζ0(˜ rm)[2 + ˜ rmζ0(˜ rm)]

δe eb (rm) = δm 3

K(r) = A ⇣ r ∆ ⌘n exp  −1 2 ⇣ r ∆ ⌘2α

δe eb = f(w)  1 + n 3 − α 3 ⇣ r ∆ ⌘2α K(r)

n ≥ 0 α > 0

δm = f(w) ✓n + 2 α ◆(n+2)/2α exp ✓ −n + 2 2α ◆ A∆2

Specifying the curvature profile

r0 rm = ✓n + 3 n + 2 ◆1/2α

measuring the steepness of the profile

PBH threshold against profile steepness

¯ K = K(rm)r2

m

¯ ζ = − ˜ rmζ0(˜ rm) δc = f(w) ¯ Kc = f(w)¯ ζc[2 − ¯ ζc]

To c h a r a c t e r i z e t h e pressure gradients of a density profile we need more than one parameter!!!

δc  r0 rm , δe eb ✓ rp rm ◆

IM - in preparation

Conclusions & Future perpectives

• With the Misner-Sharp equations (cosmic time slicing) we have studied initial

condition for PBH formation corresponding to pure growing modes in the early Universe (radiation dominated era).

• The choice of the equation of state determines the final virialized structure of the
• collapse. Pressure and curvature profiles plays a key role determining the

particular value of the threshold for PBH formation.

• PBH formation is characterised by non liner curvature profile, the linear

approximation used by Green, Liddle, Malik, Sasaki (2004) does not gives accurate

• results. In terms of ζ(r) the threshold is given by its first derivative at the

length scale of the perturbation.

• The shape effects need to be described by more than one parameter.
• Cosmological consequences of numerical results: S.Young, IM, C.Byrnes - in

preparation.

δc  r0 rm , δe eb ✓ rp rm ◆

¯ ζ = −˜ rmζ0(˜ rm)