Revisiting gravitational wave background from primordial black holes

Heling Deng

邓鹤凌

Arizona State University

2110.02460

Outline

  • Primordial black holes
  • PBH binary merger rate
  • Gravitational wave background from PBH mergers
  • Revisiting PBH binary merger rate
  • Mass accretion

Outline

  • Primordial black holes
  • PBH binary merger rate
  • Gravitational wave background from PBH mergers
  • Revisiting PBH binary merger rate
  • Mass accretion
M_\text{Pl} \newline \sim10^{-5}\ \rm g
M_\text{evp} \newline \sim10^{15}\ \rm g
M_\odot \newline \sim10^{33}\ \rm g
M_\text{PBH}

LIGO BHs

Supermassive black holes

\mathcal O (10) \ M_\odot

SMBHs

...

...

Primordial black holes (PBHs)

\mathcal{O}(10^6 \text{-} 10^{10})M_\odot
M_\text{Pl} \newline \sim10^{-5}\ \rm g
M_\text{evp} \newline \sim10^{15}\ \rm g
M_\odot \newline \sim10^{33}\ \rm g
M_\text{PBH}

LIGO BHs

Supermassive black holes

LIGO black holes

\mathcal O (10) \ M_\odot

SMBHs

...

...

Astrophysical BHs

Primordial black holes (PBHs)

\mathcal{O}(10^6 \text{-} 10^{10})M_\odot

LIGO BHs

M_\text{Pl} \newline \sim10^{-5}\ \rm g
M_\text{evp} \newline \sim10^{15}\ \rm g
M_\odot \newline \sim10^{33}\ \rm g
M_\text{PBH}

LIGO BH

Supermassive black holes

LIGO black holes

Dark matter

\mathcal O (10\text{-} 100) \ M_\odot

SMBH

...

...

Primordial black holes (PBHs)

\mathcal{O}(10^6 \text{-} 10^{10})M_\odot

Astrophysical BHs

Observational constraints of monochromatic PBHs as DM

Outline

  • Primordial black holes
  • PBH binary merger rate
  • Gravitational wave background from PBH mergers
  • Revisiting PBH binary merger rate
  • Mass accretion

PBH binary

PBH binary

free-fall time

Hubble time

When do two neighboring PBHs decouple from Hubble flow?

<

PBH binary

\(y\)

\({x}\)

At dust-radiation equality (\(z_{eq}\sim3000\))

\(\rho\) -- dark matter density

\(M\) -- PBH mass

\(f\) -- fraction of dark matter in PBHs

\(\to\) \(n\sim \bar{x}^{-3} \sim\frac{f\rho}{M}\)

  • \(\bar{x}\) -- average physical separation between two PBHs
  • \({x}\) -- physical distance between two neighboring PBHs without Newton
  • \(y\) -- physical distance from the third nearby PBH to the binary

PBH binary

Two neighboring PBHs decouple from Hubble flow when

 free-fall time < Hubble time

\(\to\)

  • \(z_{dec}>z_{eq}\)
  • physical separation when decouple is \(a=\left(\frac{z_{eq}}{z_{dec}}\right)x=\frac{x^4}{f\bar{x}^3}\)

\(y\)

\({x}\)

PBH binary

\(a\)

Initial semi-major axis: \(a\sim\frac{x^4}{f\bar{x}^3}\)

Coalescence time given by the Peters formula 

P. C. Peters, Phys. Rev. 136, B1224 (1964)

\(t_P \propto \frac{a^4}{M^3}(1-e^2)^{7/2}\)

Initial eccentricity: \(\sqrt{1-e^2}\sim \left(\frac{x}{y}\right)^3\)

\(\propto \frac{x^{37}}{y^{21}}\)

For a random PBH, the probability that a second BH is at \((x, x+\text{d}x)\) and a third BH is at \((y, y+\text{d}y)\) is

\(P(x,y)\text{d}x\text{d}y=\mathcal{O}(10)n^2x^2y^2\text{d}x\text{d}y\)

\(y\)

\({x}\)

PBH binary

\(\to\) For a random PBH, the probability that a merger occurs at \((t, t+\text{d}t)\) is

\(P(t)\text{d}t=\mathcal{O}(10)n^2\int x^2y^2|\frac{\text{d}x}{\text{d}t}|\text{d}y\text{d}t\)

Peters formula: \(t\propto\frac{x^{37}}{y^{21}} \to x\propto\left(\frac{y^{21}}{t}\right)^{1/37}\)

PBH binary

Peters formula: \(t\propto\frac{x^{37}}{y^{21}} \to x\propto\left(\frac{y^{21}}{t}\right)^{1/37}\)

bounds:

  • \(x<f^{1/3}\bar{x}\) -- ensure the binary is formed during the radiation era
  • \(y<\bar{x}\) -- probability that two PBHs with separation > \(\bar{x}\) is exponentially suppressed

curves with constant \(t\)

Integration largely determined by \(y_{max}\)

\(t_2\)

\(t_3\)

\(t_1>t_2>t_3\)

\(P(t)\propto\int x^2y^2|\frac{\text{d}x}{\text{d}t}|\text{d}y\)

PBH binary

\(P(t)\propto\int x^2y^2|\frac{\text{d}x}{\text{d}t}|\text{d}y\)

results:

\(P(t)\approx\frac{0.005f}{t}\)

{

\(\left(\frac{t}{t_c}\right)^{-1/7},\ t > t_c\)

\(\left(\frac{t}{t_c}\right)^{3/37},\ t < t_c\)

where \(t_c\sim 10^{42}f^7\left(\frac{M}{M_\odot}\right)^{-5/3}\ \text{s}\)

curves with constant \(t\)

\(t_2\)

\(t_3\)

\(t_1>t_2>t_3\)

\(t_2=t_c\)

1603.08338, Sasaki, Suyama, Tanaka and Yokoyama

Example:

\(M=10M_\odot, f=0.1\% \to t_c\sim 10^{19}\ \text{s}\) 

\(M=100M_\odot, f=0.1\% \to t_c\sim 10^{17}\ \text{s}\)

Peters formula: \(t\propto\frac{x^{37}}{y^{21}} \to x\propto\left(\frac{y^{21}}{t}\right)^{1/37}\)

Outline

  • Primordial black holes
  • PBH binary merger rate
  • Gravitational wave background from PBH mergers
  • Revisiting PBH binary merger rate
  • Mass accretion

PBH binary

\(a\)

Coalescence time given by the Peters formula 

P. C. Peters, Phys. Rev. 136, B1224 (1964)

\(t_P \propto \frac{a^4}{M^3}(1-e^2)^{7/2}\)

\(e \to 1, t_p \to 0?\)

free-fall time: \(t_{ff}\sim \sqrt{\frac{a^3}{M}}\)

Coalescence time can be estimated by Peters formula only if \(t_P > t_{ff}\)

PBH binary

\(a\)

Peters formula: \(t_P \propto \frac{a^4}{M^3}(1-e^2)^{7/2}\)

free-fall time: \(t_{ff}\sim \sqrt{\frac{a^3}{M}}\)

\(t\sim t_P+t_{ff}\)

Coalescence time should be

PBH binary

Peters formula: \(t= t_P(x,y) \to x=F(y)\)

curves with constant \(t\)

Integration largely determined by \(y_{max}\)

\(t_2\)

\(t_3\)

\(t_1>t_2>t_3\)

\(P(t)\propto\int x^2y^2|\frac{\text{d}x}{\text{d}t}|\text{d}y\)

PBH binary

\(t= t_P+t_{ff} = \frac{3\rho^4}{170M^7}\frac{x^{37}}{y^{21}} + \frac{\rho^{3/2}}{M^2}x^6\)

\(\to y\propto\left(\frac{x^{37}}{M^2\rho^{-3/2}t-x^6}\right)^{1/21}\)

Integration largely determined by \(y_{max}\)

\(t_1>t_2>t_3\)

**PBH binaries could have a significantly larger merger rate "at some point" than previously expected**

\(P(t)\propto\int x^2y^2|\frac{\text{d}x}{\text{d}t}|\text{d}y\)

Peters formula: \(t\propto\frac{x^{37}}{y^{21}} \to x\propto\left(\frac{y^{21}}{t}\right)^{1/37}\)

Outline

  • Primordial black holes
  • PBH binary merger rate
  • Gravitational wave background from PBH mergers
  • Revisiting PBH binary merger rate
  • Mass accretion

GWB from PBH mergers

\(\Omega_{GW}(\nu_d)=\frac{\nu_d}{\rho_c}\int N(z)\frac{\text{d}E_{GW}(\nu_s)}{\text{d}\nu_s}|_{\nu_s=\nu_d(1+z)}\text{d}z\)

  • \(N(z)\text{d}z\) -- comoving number density of events at \((z,z+\text{d}z)\)
  • \(v_d\) -- GW frequency in detector
  • \(\nu_s\) -- GW frequency in source
  • \(\frac{\text{d}E_{GW}(\nu_s)}{\text{d}\nu_s}\) -- GW energy spectrum from a single event

for PBH merger:

\(N(z)\text{d}z=nz_{eq}^{-3}P(t)\text{d}t\)

\(\frac{\text{d}E_{GW}(\nu_s)}{\text{d}\nu_s}\) from numerical relativity

\(\Omega_{GW}\) has a peak near \(\nu \sim 10^4(M/M_\odot)^{-1}\ \text{Hz}\)

1903.05924, Wang, Terada and Kohri

Examples of \(\Omega_{GW}\) from PBH mergers

\(\Omega_{GW}\) has a peak near \(\nu \sim 10^4(M/M_\odot)^{-1}\ \text{Hz}\)

\(t_1>t_2>t_3\)

**PBH binaries could have a significantly larger merger rate "at some point" than previously expected**

Examples of \(\Omega_{GW}\) from PBH mergers taking into account the "free-fall time" effect

More stringent constraints on PBHs if we don’t see GWB in the future 

Observational constraints of monochromatic PBHs as DM

Observational constraints of large PBHs as DM

  • Primordial black holes
  • PBH binary merger rate
  • Gravitational wave background from PBH mergers
  • Revisiting PBH binary merger rate
  • Mass accretion

Outline

PBHs absorb ambient gas and dark matter after dust-radiation equality and could grow by orders of magnitude

However, details are still unclear

We consider accretion from \(z=30\) to \(z=10\) (\(t\sim 10^{16}\ \text{s}\))

\(P(t)\approx\frac{0.005f}{t}\)

{

\(\left(\frac{t}{t_c}\right)^{-1/7}-f\left(\frac{t}{t_{last}}\right)^{3/8},\ t > t_c\)

\(\left(\frac{t}{t_c}\right)^{3/37},\ t < t_c\)

PBH merger rate

where \(t_{last}\sim 10^{42}\left(\frac{M}{M_\odot}\right)^{-5/3}\ \text{s}\)

Example: for \(M=100M_\odot\), we have \(t_{last}\sim 10^{39}\ \text{s}\)

\(P(t_{last})=0\)

In the presence of accretion at \(t\sim 10^{16}\ \text{s}\), the change in black hole mass alters the binary orbit, and thus the PBH merger rate 

\(a\)

\(\frac{\dot{a}}{a}+3\frac{\dot{M}}{M}=0\)

\(t_P\propto \frac{a^4}{M^3}\)

Coalescence time decreases from \(t\) to

\(t_{acc}=\left(\frac{M_f}{M_i}\right)^{-15}t\)

 \(\to a \propto M^{-3}\)

eccentricity remains const

2005.05641, De Luca, Franciolini, Pani and Riotto

could be tiny!

Example: for \(M=100M_\odot\), we have \(t_{last}\sim 10^{39}\ \text{s}\)

for \(M_i=100M_\odot\) and \(M_f=10000M_\odot\), we have \(t_{acc}=\left(\frac{M_f}{M_i}\right)^{-15}t_{last}=10^9\ \text{s}\)!

**Aggregated mergers during accretion**

Examples of \(\Omega_{GW}\) taking into account mass accretion

Conclusions and discussion

  • Peters formula underestimates merger rate of large PBHs
  • GWB spectrum for mergers of PBH with \(M>10^5M_\odot\) develops an extra peak
  • If future missions do not see such a background, the fraction of dark matter in PBHs is constrained to \(f < 10^{−6} \text{-}10^{−4}\) within the mass range \(10\text{-}10^9M_\odot\)
  • Mass accretion at \(z\sim 10\) could significantly affect GWB from PBH mergers
  • Some assumptions in this work: monochromatic PBHs, no PBH clustering, simple accretion models...
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