1. Primordial Black Holes from Sound Speed Resonance during Inflation
Yi-Fu Cai, Xi Tong, Dong-Gang Wang, Sheng-Feng Yan [arXiv: / PRL, 121, ]
Good afternoon every one, i’m Sheng-Feng Yan from University of science and technology of China. I’ll talk about our recent work 标题, with Yi-fu my advisor, Xi Tong in Hongkong UST and dong-gang in Leiden university. This presentation is based on this paper.
鄢盛丰 Sheng-Feng Yan
中国科学技术大学天文学系
Department of Astronomy, University of Science and Technology of China

2. Stephen Hawking 8 January 1942 – 14 March 2018 1942-2018
This work is in memory of the giant Prof. Stephen Hawking, who inspired numerous young people to dream about the Universe, and beyond.
8 January 1942 – 14 March 2018

3. Pioneer Works for Primordial Black Holes
S. Hawking
Bernard Carr
Gravitationally collapsed objects of very low mass [Hawking, S.W.(1971)]
Black Holes in the Early Universe [Carr, B.J., Hawking, S.W. (1974)]
The primordial black hole mass spectrum [Carr, B.J.(1975)]
Gamma rays from primordial black holes [Page, D. N., Hawking, S. W.(1976)]
Some cosmological consequences of primordial black-hole evaporations [Carr, B.J.(1976)]
The idea of BHs in the early universe was pointed out by Zeldovich in the 1960s. Then Prof. Hawking and Prof. Carr and their colleagues finished many pioneer works for the primordial black holes. These are their representative papers.

4. Quantum & Astrophysical Effects of BHs
When BHs Form
Quantum & Astrophysical Effects of BHs
So lets have a look at some properties of PBH. The picture in the left hand side shows that the BHs formed in the early universe are PBH. In this period some modes of fluctuations during inflation re-entry the Hubble radius, some of them are potential to form the PBH. The picture in the right hand side shows the mass spectrum of BHs, and the corresponding physical effects that could be produced. We can see PBH span a large fraction of the mass spectrum, from the mass of a dust to a star with lots of observable effects.

5. Primordial black holes can be powerful probe of the early Universe
Due to the era the PBHs are formed, and the observable effects, they can be a powerful probe of the early Universe.

6. The evolution of density perturbations into PBHs
[B. J. Carr, Astrophys. J. 201, 1 (1975)], [ ]
Three-zone model
Over-dense region is part of closed FRW region in flat FRW background
Background
Over-dense region
We here talk about a 3-zone model to describe the PBH formation. The background is a flat FRW spacetime, these are its metric and Friedman equation. And the over-dense region is a part of closed FRW spacetime. This model is just like blowing bubbles. So this is the sketch.

7. The evolution of density perturbations into PBHs
Over-dense region described by closed Friedmann solution for
The areal radius of over-dense region is
Over-density:
Collapse to black hole if max radius exceeds Jeans length
Just kidding. These are the sketches. The areal radius of over-dense region is this capital R, it equals to a sin\kai. That is this dark part, and this circle is the underdense layer, it is matched with the flat background at r equal to r background. The over-density of this region is measured by \delta, it collapse into BH when the maximal radius exceeds Jeans length.

8. The evolution of density perturbations into PBHs
That is when
Threshold for PBH formation
depends on the choice of
PBH mass fraction (assume the density perturbation obeys Gaussian distribution)
A fraction of PBHs against the total dark matter as a function of β
That is Rm larger than R jeans. So there is a threshold for PBH formation \delta c, that means once the over density exceeds \delta c, then it could collapse into a BH. And this quantity depends on the choice of Jeans length. Now we can define a PBH mass fraction as the energy density of PBH against to the total energy density that proportional to an error function. We usually need a fraction of PBHs against the total dark matter like this.

9. Constraints and detections of PBHs
[Sasaki, et al., ]
[Carr, et al., ]
Gravitational lensing:
microlensing, milli-lensing (supermassive PBHs ), femto-lensing ( )
Dynamics:
PBHs affect any astrophysical system by gravitational interactions.
Accretion effects
Now I give a very brief introduction for the constraints and detections of PBH, Sasaki son and Carr both give a review of this part in detail. First of all, PBH can cause gravitational lensing, various in the mass of them, there could be micro lensing, milli-lensing and femto-lensing. And PBH also can affect astrophysical system by gravitational interactions. In the view of BH, the radiation from hot gas around PBH will ionize or heat the gas filling the universe, that may leave imprint on CMB such as spectrum, decoupling time and the ionization history.
Ionizes or heats the gas filling the Universe
Imprint on CMB:
Spectrum,
Decoupling time,
The ionization history
Radiation from heat gas near PBH

10. Constraints and detections of PBHs
[Sasaki, et al., ]
Further constraints
Fast radio bursts (FRB)
FRB signal is lensed by PBHs [ ]
Pulsar timing array (PTA)
The effect of the time delay caused by the intervening PBHs [ ]
21cm lines
[ ], [ ]
There also are some further constrain models to be probed. FRB signal could be lensed by PBH. And the time delay of pulsar timing array also could be caused by the intervening PBH. For 21 cm lines, the radiation from PBH could ionize or heat the neutral hydrogen gas, it will change the spin and kinetic temperature. If there are some PBH minihalos, the gas temperature and density of neutral hydrogen differ from the background value.
Ionizes or heats the neutral hydrogen gas
Changing the spin temperature and kinetic temperature
Radiation from PBH
Gas temperature and density of neutral hydrogen differ from the bgd value
PBH minihalos

11. Constraints and detections of PBHs
Gravitational waves
Electromagnetic waves:
Lensing, CMB, FRB, PTA, 21cm
Instead of the electromagnetic waves, PBH also can be constrained or detected by the gravitational waves. In the last 3 years, there are several BH merger produce GW events. The BHs are probably primordial BHs. We can constrain the mass and the spatial distribution of such PBH by analyzing the GW data.

12. Sound speed resonance of curvature perturbations can produce PBHs
From now on, I ‘ll introduce our works.

13. Question: Are there other ways to form PBHs during inflation?
For multi-field inflation, the effective sound speed is modified.
The power spectrum is inversely proportional to cs. To greatly enhance the power spectrum, cs must become extremely small.
Or not. There are more efficient ways to amplify the fluctuations.
A hint from the swings: Parametric Resonance
When f_swinging=2 f_swing, the swing can be swung higher and higher
There are many models to produce PBHs, most of them are trying to suppress slow-roll parameters to make an enhancement in the power spectrum. Such as the single field chaotic new inflation proposed by Junchi Yokoyama son, and the inflection point inflation. But in multi-field inflation, the sound speed could be modified, and the spectrum is inversely proportional to cs. So we can use the same trick that suppress cs to enhance the power spectrum. Or, we guess parametric resonance of cs can do the same thing. This process just like the swings. When we shake our body periodically, and if the frequency of body equals to the double of the swing frequency, the swing can be swung higher and higher. So we propose a model independent mechanism to produce PBH.

14. Our model Suppose we have a time-dependent sound speed
The sound speed squared is oscillating in the conformal time
Change to the Mukhanov-Sasaki variable,
First, we have a time dependent cs square, it is oscillating in the conformal time like this. Where \xi is the amplitude of the oscillatory feature, k-star is a specific mode which the resonance start around this mode. Then the equation of motion becomes, that is the Mukhanov-Sasaki equation with various speed of sound. Here z and \epsilon are defined as these.
The EOM becomes:
where

15. Our model Thus the EOM can be written in the standard form of
the Mathieu equation
with
and
Near the scale k*, the mode grows exponentially.
Away from the scale k*, the modes are unaffected.
Substitute the oscillatory cs into the Mukhanov-Sasaki equation, that can be written in the standard form of the Mathieu equation. This is the basic equation for the parametric resonance which is also the equation of motion in preheating phase. Near the scale k-star, the mode grows exponentially in this way, while the modes away from k-star are unaffected. Here the blue curved line shows the evolution of the k-star mode in the de-Sitter approximation, it is enhanced by this mechanism. The green line is this analytic result. The orange line is a mode which is not relative to the mode of k-star, so there is no resonance. The dashed gray line is in the Starobinski inflation model, that fits the dS approximation result well.

16. Resonace peaks in the power spetrum
Narrow resonance has multiple peaks.
But only the first peak is dominant.
k* must be small enough to avoid affecting CMB.
Perturbation theory requires Pζ<1
This picture shows the power spectrum, in fact there are multiple very narrow resonance peaks. But only the first peak is dominant. So we use the Dirac \delta function to approximate this spectrum in later simulations. In the meanwhile, k-star must be small enough to avoid affecting the CMB.

17. PBHs as dark matter PBH mass: PBH fraction against total DM
Perturbation theory bound Pζ<1
The Schwarzschild radius of PBH is related to the physical wave length, so the mass of PBH proportional to XXXXX. And we can write down the energy density fraction against to Dark matter. Due to the perturbation bound is less than 1, so the standard deviation of PBH is less than this value.

18. PBHs as dark matter
So this is our result, the upper shadow area are the constraints by the observations. And the dashed red narrow peaks correspond to the fPBH with different choose of resonance frequency k-star. this is a distinctive feature of PBHs formed by sound speed resonance from PBHs formed by other processes, for which the mass distribution is usually more spread out. By varying the value of k , the peaks form a 1-parameter family enveloped by a yellow solid curve that mainly depends on the amplitude parameter ξ.

19. Constraints on the parameters
Constraints come from ratio of the density of PBHs against total dark matter density, namely, fPBH.
Here ΔN is the e-folding numbers from τ0 to the horizon-exit of the characteristic mode.
The regions obove the contours are excluded.
In our model, the fPBH mostly depends on the parameter amplitude \xi, and the e-folding numbers from τ0 to the horizon-exit of k-star mode. So we showed the contours of these parameters. We can see the sound speed resonance has a large parameter space, left to be probed by future observations.

20. Summary
PBHs are expected to be a new probe of the very beginning of the universe
GW opens a new window to probe PBHs
Sound speed resonance provides a new phenomenological mechanism to produce PBHs
Our model motivates theoretical investigations on the possible inflation models from theoretical viewpoints, which could yield oscillating behaviors in the sound speed squared.
So now let me make a summary.

21. Thank You!