Primordial black holes formed in the matter-dominated era . . . - PowerPoint PPT Presentation

. . Primordial black holes formed in the matter-dominated era . . . . . Tomohiro Harada (Rikkyo U) 15/2/2018, GC2018 @ YITP This talk is based on the collaboration: Harada, Yoo (Nagoya), Kohri (KEK), Nakao (OCU) & Jhingan (YGU), 1609.01588 Harada, Yoo, Kohri, & Nakao, 1707.03595 T. Harada (Rikkyo U) PBHs in the MD era GC2018 1 / 13

Introduction Primordial black hole (PBH) PBH = Black hole formed in the early Universe Probe into the early Universe and high-energy physics. γ -rays, X-rays, DM and GWs. (Carr et al. (2010), Carr et al. (2016)) LIGO BBHs may be of primordial origin. (Sasaki et al. (2016), Bird et al. (2016), Clesse & Garcia-Bellido (2017)) BH spins have attracted great attention. (e.g. McClintock (2011), Abbott et al. (2017)) (a) Carr et al. (2016) (b) LIGO Collaboration (2017) T. Harada (Rikkyo U) PBHs in the MD era GC2018 2 / 13

PBH formation in the MD era PBH formation in the matter-dominated (MD) era Pioneered by Khlopov & Polnarev (1980). Early MD phase scenarios such as inflaton oscillations, phase transitions, and superheavy metastable particles. If pressure is negligible, nonspherical effects play crucial roles. The triaxial collapse of dust leads to a “pancake” singularity. (Lin, Mestel & Shu (1965), Zeldovich (1969)) The effect of angular momentum may halt gravitational collapse or spin the formed PBHs. (Peebles (1969), Catelan & Theuns (1996)) T. Harada (Rikkyo U) PBHs in the MD era GC2018 3 / 13

PBH formation in the MD era Anisotropic effect Zeldovich approximation Zeldovich approximation (ZA) (1969) Extrapolate the Lagrangian perturbation theory in the linear order in Newtonian gravity to the nonlinear regime. r i = a ( t ) q i + b ( t ) p i ( q j ) , where b ( t ) ∝ a 2 ( t ) denotes a linearly growing mode. We can take the coordinates in which ∂ p i = diag ( − α, − β, − γ ) . ∂ q j We assume that α , β and γ are constant over the smoothing scale and normalise b so that b / a = 1 at horizon entry t = t H of the scale. T. Harada (Rikkyo U) PBHs in the MD era GC2018 4 / 13

PBH formation in the MD era Anisotropic effect Application of the hoop conjecture Hoop conjecture (Thorne 1972): A BH forms if and only if the circumference C of a mass M satisfies C � 4 π GM / c 2 . Then, we obtain a BH criterion for the pancake collapse.  √ ) 2  α − γ ( α − β 4 π Gm / c 2 = 2 C       h ( α, β, γ ) : = E 1 −  � 1 ,       π  α − γ  α 2    where E ( e ) is the complete elliptic integral of the second kind. T. Harada (Rikkyo U) PBHs in the MD era GC2018 5 / 13

PBH formation in the MD era Spin effect Spin angular momentum Region V : to collapse in the future Angular momentum with respect to the COM in the Eulerian coordinates (∫ ) ∫ ∫ ∫ x δ × u d 3 x − 1 L = ρ 0 a 4 x × u d 3 x + x δ d 3 x × u d 3 x , V V V V V where x : = r / a , u : = aD x / Dt , δ : = ( ρ − ρ 0 ) /ρ 0 , and ψ : = Ψ − Ψ 0 . Linearly growing mode of perturbation ∑ ∑ ∑ δ 1 , k ( t ) e i k · x , ψ 1 = ˆ ψ 1 , k ( t ) e i k · x , u 1 = ˆ u 1 , k ( t ) e i k · x , δ 1 ˆ = k k k a 2 ψ 1 , k = − 2 k 2 0 δ 1 , k = A k t 2 / 3 , ˆ 3 A k t 1 / 3 . ˆ where k 2 A k , u 1 , k = ia 0 ˆ 3 k 2 T. Harada (Rikkyo U) PBHs in the MD era GC2018 6 / 13

PBH formation in the MD era Spin effect 1st-order effect (∫ ) ∫ x δ × u d 3 x − 1 ∫ ∫ L = ρ 0 a 4 x × u d 3 x + x δ d 3 x × u d 3 x V V V V V If ∂ V is not a sphere, the 1st term contribution grows as ∝ a · u ∝ t . For an ellipsoid with axes ( A 1 , A 2 , A 3 ) , q MR 2 2 (1) ⟩ 1 / 2 ≃ ⟨ L 2 ⟨ δ 2 ⟩ 1 / 2 , √ t 5 15 where r 0 : = ( A 1 A 2 A 3 ) 1 / 3 , R : = a ( t ) r 0 , q Figure: The 1st-order is a nondimensional reduced quadrupole effect can grow if ∂ V moment, and δ is the averaged density is not a sphere. perturbation. (Cf. Catelan & Theuns 1996) T. Harada (Rikkyo U) PBHs in the MD era GC2018 7 / 13

PBH formation in the MD era Spin effect 2nd-order effect (∫ ) ∫ ∫ ∫ x δ × u d 3 x − 1 L = ρ 0 a 4 x × u d 3 x + x δ d 3 x × u d 3 x V V V V V Even if ∂ V is a sphere, the remaining contribution grows as 1st order × 1st order ∝ a · δ · u ∝ t 5 / 3 . 15 I MR 2 (2) ⟩ 1 / 2 = 2 ⟨ L 2 ⟨ δ 2 ⟩ , t where R : = a ( t ) r 0 and we can assume Figure: The 2nd-order I = O (1) . (Cf. Peebles 1969) effect can grow due to the mode coupling. T. Harada (Rikkyo U) PBHs in the MD era GC2018 8 / 13

PBH formation in the MD era Spin effect The application of the Kerr bound Time evolution of V and angular momentum Horizon entry ( t = t H ): ar 0 = cH − 1 , δ H : = δ ( t H ) , σ H : = ⟨ δ 2 H ⟩ 1 / 2 Turn around ( t = t m ): δ ( t m ) = 1 , typically t m = t H σ − 3 / 2 H a ∗ : = L / ( GM 2 / c ) at t = t m √ ∗ (1) ⟩ 1 / 2 = 2 3 ∗ (2) ⟩ 1 / 2 = 2 5 q σ − 1 / 2 5 I σ − 1 / 2 ⟨ a 2 , ⟨ a 2 H H 5 For t > t m , the evolution of V decouples from the cosmological expansion and hence a ∗ is kept almost constant. Consequences ∗ ⟩ 1 / 2 � 1 if σ H � 0 . 1 . This contrasts with small spins Typically ⟨ a 2 ( a ∗ � 0 . 4 ) of PBHs formed in the RD era. (Chiba & Yokoyama (2017)) a ∗ is typically too large for direct collapse to a BH. T. Harada (Rikkyo U) PBHs in the MD era GC2018 9 / 13

PBH formation in the MD era Spin effect Spin distribution Spin distribution of PBHs formed in the MD era 1 1 0.1 0.1 0.05 0.05 0.01 0.01 0 . 8 0 . 8 f BH(1) ( a ∗ ) 0 . 6 f BH(2) ( a ∗ ) 0 . 6 0 . 4 0 . 4 0 . 2 0 . 2 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 a ∗ a ∗ (a) 1st-order effect (b) 2nd-order effect Figure: A Gaussian distribution assumed for the density perturbation. Each curve labelled with σ H . The region with smaller δ H has larger a ∗ . There appears a threshold δ th below which the angular momentum halts the collapse to a BH. T. Harada (Rikkyo U) PBHs in the MD era GC2018 10 / 13

PBH formation in the MD era Spin effect PBH production probability Triple integral for production probability β 0 ∫ ∞ ∫ α ∫ β β 0 ≃ d α d β d γθ [ δ H ( α, β, γ ) − δ th ] θ [1 − h ( α, β, γ )] w ( α, β, γ ) , 0 −∞ −∞ where we use w ( α, β, γ ) given by Doroshkevich (1970). Figure: The red lines are due to both angular momentum and anisotropy. The black solid line is solely due to anisotropy. T. Harada (Rikkyo U) PBHs in the MD era GC2018 11 / 13

PBH formation in the MD era Spin effect Discussion of PBH production probability Semianalytic estimate (black dashed line and blue dashed line)     − 0 . 15 I 4 / 3     2 × 10 − 6 f q ( q c ) I 6 σ 2    H exp   ( 2nd-order effect )           σ 2 / 3        H      β 0 ≃ 3 × 10 − 14 q 18  − 0 . 0046 q 4       exp ( 1st-order effect )          σ 4  σ 2       H H    0 . 05556 σ 5 ( anisotropic effect )   H where f q ( q c ) : the fraction of regions with q < q c = O ( σ 1 / 3 H ) . The density fluctuation σ H can be written in terms of the power spectrum P ζ ( k ) of the curvature perturbation ζ as ) 2 ( 2 σ 2 H ≃ P ζ ( k BH ) . 5 T. Harada (Rikkyo U) PBHs in the MD era GC2018 12 / 13

Summary Summary PBHs may form not only in the RD era but also in the (early) MD era from primordial cosmological fluctuations. In the MD era, the effect of anisotropy gives β 0 ≃ 0 . 05556 σ 5 H , while the effect of angular momentum gives further suppression for the smaller values of σ H . PBHs formed in the MD era mostly have large spins ( a ∗ ≃ 1 ) in contrast to the small spins ( a ∗ � 0 . 4 ) of PBHs formed in the RD era. T. Harada (Rikkyo U) PBHs in the MD era GC2018 13 / 13

Anisotropic collapse in the ZA The triaxial ellipsoid of a Lagrangian ball (assumption)  r 1 = ( a − α b ) q      r 2 = ( a − β b ) q    r 3 = ( a − γ b ) q   Evolution of the collapsing region: Horizon entry ( t = t H ): a ( t H ) q = cH − 1 ( t H ) = r g : = 2 Gm / c 2 . Maximum expansion ( t = t f ): ˙ r 1 ( t f ) = 0 giving r f : = r 1 ( t f ) = r g / (4 α ) . Pancake singularity ( t = t c ): r 1 ( t c ) = 0 giving a ( t c ) q = 4 r f = r g /α . T. Harada (Rikkyo U) PBHs in the MD era GC2018 14 / 13

Evolution of the perturbation T. Harada (Rikkyo U) PBHs in the MD era GC2018 15 / 13

Application of the Kerr bound Technical assumption q MR 2 15 I MR 2 2 δ, | L (2) | ≃ 2 ⟨ δ 2 ⟩ 1 / 2 δ. | L (1) | ≃ √ t t 5 15 The above assumption implies √ a ∗ (1) = 2 3 , a ∗ (2) = 2 5 q δ − 1 / 2 5 I σ H δ − 3 / 2 , a ∗ = max ( a ∗ (1) , a ∗ (2) ) . H H 5 The Kerr bound a ∗ ≤ 1 gives a threshold δ th for δ H , where ) 2 / 3 δ th = max ( δ th(1) , δ th(2) ) , δ th(1) : = 3 · 2 2 ( 2 q 2 , δ th(2) : = 5 I σ H . 5 3 T. Harada (Rikkyo U) PBHs in the MD era GC2018 16 / 13