The observational result of GW150914 can be considered as an indirect signal of dark matter coming from the PBH sub-halo, which is a conversion of a few M⊙ dark matter into grav-itational waves in a massive PHB coalescence. This observed gravgrav-itational wave provides an evidence for the existence of high stellar-mass black holes. In previous section, we have shown that PBHs with the mass about 14M⊙ can be generated during the inflation. While the production of PBHs are sufficiently large, GWs can be produced at horizon crossing as second-order effects in metric perturbation theory [47–50]. In addition to de Sitter vacuum fluctuations, GWs can be directly sourced by the gauge field production, we can use the ap-proximate analytic gauge mode solutions to calculate the relative present GW energy density per logarithmic k interval, given by [36,47,
51, 52]. We will calculate how much amplitude and
frequency gravitational waves the PBH of 14M⊙ can be generated in the early universe.Figure 5.3: Solid line is the total power spectrum of the curvature perturbation with the coupling constant α = 17. The contribution induced by photon production is denoted by the dotted line and the vacuum contribution by the dashed line. The e-folding N denotes the time when the k-mode leaves the horizon. The primordial black hole bound is the short-dashed line.
The GW equation reads where the T T is the transverse and traceless projection of spatial components of the energy-momentum tensor of the gauge field.
We have shown in chapter
4
that the energy densities of inflaton perturbation ρϕ and of gauge quanta ρA during inflation scale as ρϕ/ρA≪ 1. This means that the energy-momentum
tensor that sources the generation of GWs mainly comes from the gauge quanta rather than the inflaton perturbation. More recently, the authors in ref. [47] have confirmed the subdominance of the second-order effect by explicitly computing the GW power spectra induced by the gauge field production as well as by the second-order curvature perturbation. Rather than numerically solving Eq. (5.4) in conjunction with the gauge mode equation (4.9), we use the approximate analytic gauge mode solutions to estimate the present relative GW energy density per logarithmic k interval, given by [36,51].
ΩGW
h
2≃ 3.5 × 10
−7H
2 are functions of time as in Eq. (4.7) and Eq. (4.9), respectively. We plot ΩGWh
2 against the frequency f = 3×10
−18(H/Hi)eN Hz in Fig.5.4, where H
i is the initial value of H. The figure also shows the upper limits on gravitational wave background made by the pulsar timing array experiments EPTA [53], NANOGrav [54], and PPTA [55], and the projected sensitivity of SKA radio telescope [56]. The gravitational waves associated with the production of 14M⊙ PBHs are about an order of magnitude below the current pulsar timing array sensitivity. Slightly lighter PBHs may source gravitational waves whose peak shifts to higher frequencies within the reach of the future SKA observation.Figure 5.4: The blue line is the spectral energy density of the gravitational waves associated with the production of 14M⊙ PBHs. The black lines are the upper limits set by the pulsar timing array experiments EPTA, NANOGrav, and PPTA, and the projected SKA sensitivity.
5.5 Summary
We discuss the formation of primordial black holes in the trapped inflationary scenario, which can provide a successful inflation with a steep potential. In a steep regime of the trapping potential, the primordial black holes can be produced during inflation, and the large non-Gaussian curvature perturbation will be induced. We have used a combination of a flat and a steeper trapping potential, leading to the production of high stellar-mass PBHs with the coupling constant α = 17. The curvature perturbation will lead to the formation of high stellar-mass primordial black holes, which can be dark matter observed by the LIGO detectors through a binary black hole merger. The existence of high stellar-mass PBHs not only solves the dark matter problem, but also provides an important test on the theory of PBH formation in the early universe. We will test more of the relation of the inflaton potential in the axion monodromy inflation, and discussed the gravitational waves sourced by the particle production in next chapter.
Chapter 6
Primordial black holes and associated gravitational waves
The inflation scenario is generally accepted for explaining the observed spatial flatness and homogeneity of the universe. As we have said in the previous chapter, a simple model of the scenario such as the slow-roll inflation driven by a flat inflaton potential predicts quasi de Sitter vacuum fluctuations during the inflation which could give rise to Gaussian and nearly scale-free metric perturbations containing both matter density fluctuations (scalar modes) and gravitational waves (GWs or tensor modes) [11].
6.1 Inflation with a modified monomial potential
To adjust the position of peak of the power spectrum easier, we choose another potential with different formation. The simplest model that provides the flat potential is perhaps the large-field inflation with a monomial potential. We use the method that the inflaton potential has a monomial form with a model-dependent size superimposed sinusoidal modulation in single-field axion monodromy inflation, which is given by [15,
57–59].
V (ϕ) = V
0+ µ4−pC
V−p[√1 + (CV
ϕ)
2p− 1
]+ Λ4cos [
ϕ
C
f + γ0 ](6.1)
where V0, µ, and γ0are constants, p = 3, 2, 4/3, 1, 2/3, Λ is the energy scale, and Cf is the axion decay constant. The cosmological phenomena like tensor-to-scalar ratio and the resonantly enhanced modulations of the scalar power spectrum and bispectrum due to the sinusoidal modulations of the potential have been studied [16–25]. According to cosmic microwave back-ground experiments [60], we abandon the convex potentials (p > 1) and choose a linear or concave potential (p = 1 or p = 2/3) for the inflation potential (6.1). We selected the ground state is at ϕ = 0 and took the approximation that Λ and f are constants. Then we repeat the calculating in chapter
4
with the modulated potential.6.2 Numerical result
By choosing appropriate parameters, we can find the probability of the observational results of the GWs [61]. To form PBHs in certain mass ranges which associated with GWs of ob-servational interests, we tried three specific cases. Again, we solve the coupled differential equations of motion numerically for inflaton and photon mode functions in Eqs. (4.6), (4.7), and (4.9), and calculate the backreaction due to photon production during the inflation and the power spectrum (4.15).
We try three cases with different initial conditions and potential. In case 1, we set p = 1,
µ = 2.4 ×10
−4, Cf = 0.51, CV = 0.8, Λ4 = 3.7×10
−12, V0 = Λ4, and γ0 = π for the parameters of potential, ϕi =−5.9, (dϕ/dt)
i = 7.1× 10
−8, and the coupling constant α = 11.8. From Eq.(5.2) we can know the seed of PBHs will form at the peak of the scalar power spectrum, in first case that is at N∗ = 38, which corresponds to mass about 19.7M⊙. In case 2, we set p = 2/3,
µ = 2.0 ×10
−4, Cf = 0.64, CV = 1.0, Λ4 = 1.6×10
−12, V0 = Λ4, and γ0 = π for the parameters of potential, ϕi =−7.0, (dϕ/dt)
i = 3.8× 10
−8, and the coupling constant α = 22.3. In case 3, we set p = 1, µ = 2.0× 10
−4, Cf = 1.25, CV = 5.0, Λ4 = 7.7× 10
−12, V0 =−Λ
4, and γ0 = 0 for the parameters of potential, ϕi =−9.6, (dϕ/dt)
i = 1.4× 10
−7, and the coupling constantα = 26.3. Similarly, in second and third cases we can evaluate that the peak of the scalar
power spectrum at N∗≃ 23 and N
∗≃ 8, which corresponds to M
PBH∼ 2.4 × 10
−13M
⊙ andM
PBH∼ 1.2×10
8g, respectively. The results are shown in Figure 6.1, 6.2, and 6.3, respectively.
These figures show the potentials and total power spectrum of the curvature perturbation of
each case. We rescale all dynamical variables in terms of the reduced Planck mass in these figures. In the right panel of Figure
6.1, 6.2, and 6.3, we present the total power spectrum of
the curvature perturbation, which is the sum of the vacuum contribution in Eq. (4.14) and the induced power in Eq. (4.15).Figure 6.1: Case 1. Left panel: All dynamical variables in this figure and in the following figures are rescaled by the reduced Planck mass. Solid curve denotes the inflaton potential V (ϕ) with p = 1, µ = 2.4× 10−4, Cf = 0.51, CV = 0.8,
Λ4= 3.7× 10−12, V0= Λ4, and γ0= π. The dashed and short-dashed curves denote the linear term and the modulation, respectively. Right panel: Solid line is the total power spectrum of the curvature perturbation for V (ϕ) in left panel with the coupling constant α = 11.8. The induced and vacuum contributions are denoted by the dotted and dashed lines, respectively. Short-dashed line is the primordial black hole bound.
Figure 6.2: Case 2. Left panel: Solid curve denotes the inflaton potential V (ϕ) with p = 2/3, µ = 2.0× 10−4, Cf = 0.64, CV = 1, Λ4= 1.6× 10−12, V0= Λ4, and γ0= π.
The dashed and short-dashed curves denote the concave term and the modulation, respectively. Right panel: Solid line is the total power spectrum of the curvature perturbation for V (ϕ) in left panel with the coupling constant α = 22.3. The induced and vacuum contributions are denoted by the dotted and dashed lines, respectively.
Short-dashed line is the primordial black hole bound.
Figure 6.3: Case 3. Left panel: Solid curve denotes the inflaton potential V (ϕ) with p = 1, µ = 2.0× 10−4, Cf = 1.25, CV = 5, Λ4= 7.7× 10−12, V0=−Λ4, and γ0= 0.
The dashed and short-dashed curves denote the linear term and the modulation, respectively. Right panel: Solid line is the total power spectrum of the curvature perturbation for V (ϕ) in left panel with the coupling α = 26.3. The induced and vacuum contributions are denoted by the dotted and dashed lines, respectively.
Short-dashed line is the primordial black hole bound.