Phys. Rev. D 106, 123519 (2022) - Primordial black hole dark matter from inflation: The reverse engineering approach

Primordial black hole dark matter from inflation: The reverse engineering approach

Gabriele Franciolini and Alfredo Urbano

Phys. Rev. D 106, 123519 – Published 23 December 2022

Abstract

Constraining the inflationary epoch is one of the aims of modern cosmology. In order to fully exploit current and future small-scale observations, it is necessary to devise tools to directly relate them to the early Universe’s dynamics. We present here a novel reverse engineering approach able to connect fundamental late-time observables to consistent inflationary dynamics and, eventually, to the inflaton potential. Employing this procedure, we are able to describe which conditions can give rise to a raised plateau in the power spectrum of curvature perturbations at small scales, which are not constrained by CMB observations. Within this new phenomenologically driven approach, we find that inflation can generate a raised plateau in the spectrum of curvature perturbations that potentially connects three fundamental observables; a dominant component of the dark matter in the form of asteroid-mass/atomic-size primordial black holes, detectable signals in stochastic gravitational waves, and a subdominant fraction of stellar-mass primordial black holes mergers.

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Received 13 October 2022
Accepted 23 November 2022

DOI:https://doi.org/10.1103/PhysRevD.106.123519

Physics Subject Headings (PhySH)

Classical black holes Dark matter Gravitational waves Inflation

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Gabriele Franciolini ^* and Alfredo Urbano ^†

Dipartimento di Fisica, “Sapienza” Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy and INFN sezione di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy

^*gabriele.franciolini@uniroma1.it.
^†alfredo.urbano@uniroma1.it.

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Vol. 106, Iss. 12 — 15 December 2022

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Figure 1
(left) Time evolution (in terms of the number of $e$ -folds $N$ defined by $d N = H d t$ ) of the inverse comoving Hubble horizon $R_{H}^{- 1} \equiv a H$ throughout the history of our observable Universe. We start from $N = 0$ , defined as the time at which the CMB pivot scale $k_{⋆} = 0.05 {Mpc}^{- 1}$ crossed the Hubble horizon, $k_{⋆} = a (0) H (0)$ . We assume instantaneous reheating and, after inflation, standard $Λ CDM$ cosmology. The three horizontal bands mark the milestones of our phenomenological analysis. The region shaded in green ( $0.005 ≲ k [{Mpc}^{- 1}] ≲ 0.2$ ) represents the range of comoving wave numbers (horizontal lines with constant $k$ in the figure) constrained by CMB anisotropy measurements, the region shaded in red ( $1.7 \times 10^{6} ≲ k [{Mpc}^{- 1}] ≲ 1.7 \times 10^{7}$ ) corresponds to the range of comoving wave numbers for which curvature perturbations, after reentering the cosmological horizon, have the chance of generating solar-mass PBHs ( $1 ≲ M_{PBH} [M_{⊙}] ≲ 100$ ), and the region shaded in blue ( $1.5 \times 10^{13} ≲ k [{Mpc}^{- 1}] ≲ 1.5 \times 10^{14}$ ) is the same as the red band but corresponds to asteroid-mass PBHs ( $10^{- 16} ≲ M_{PBH} [M_{⊙}] ≲ 10^{- 12}$ ). Right part. We plot the power spectrum of scalar perturbations $P_{R} (k)$ as a function of the comoving wave number $k$ . The plot is rotated in such a way as to share the same $y$ -axis with the left part of the figure. We plot the region excluded by CMB anisotropy measurements, Ref. [1], the FIRAS bound on CMB spectral distortions, Ref. [29] and the bound obtained from Lyman- $α$ forest data [30]. If $P_{R} (k) ≳ 10^{- 2}$ the abundance of PBHs overcloses the Universe (that is, their abundance would be larger than the cold dark matter density of the Universe). The black dashed line is the typical prediction of slow-roll inflationary models. The solid black line, on the contrary, is characterized by the presence of an USR phase [concretely, it corresponds to model (1) in Sec. 2 and Ref. 31].
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Figure 2
Schematic evolution of $η (N)$ and $ε (N)$ . The former is given by our analyticalansatz in Eq. (2); the latter follows from Eq. (1). We label the USR phase characterized by a negative friction and the plateau in $ε$ in the phase of vanishing $η_{III}$ .
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Figure 3
Power spectrum (solid green) corresponding to the model described in the second column of Table 1. The region shaded in gray corresponds to the region constrained by CMB spectral distortions [29]. To guide the eye, we indicate in red the frequency range $2.5 \times 10^{- 9} < f [Hz] < 1.2 \times 10^{- 8}$ characterising the NANOGrav signal and in blue the mass range $10^{- 16} < M_{PBH} [M_{⊙}] < 10^{- 12}$ in which PBHs may comprise the totality of DM.
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Figure 4
Time evolution (in terms of the $e$ -fold number) of $k^{3 / 2} | u_{k} (N) / z (N) |$ for three different modes with $k_{1, 2, 3} = 10^{10, 9, 8} {Mpc}^{- 1}$ obtained by numerically solving the MS equation (solid lines). The dashed lines correspond to the absence of the USR phase. The vertical lines labeled with $N_{k}$ mark the $e$ -fold time of horizon crossing for the mode with comoving wave number $k$ . As shown in the right panel (colored dots, one for each $k_{1, 2, 3}$ ), these modes contribute to the plateau of the power spectrum. In this plot we also show the power spectrum corresponding to the absence of the USR phase (dotted) and the one computed by means of the slow-roll approximation (dashed).
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Figure 5
Power spectrum for different choices of $δ N_{II}$ (left panel) and $δ N_{III}$ (right panel). All other parameters are fixed according to the first column in Table 1. We zoom in the transition regions at the two edges of the plateau.
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Figure 6
Left panel. We plot the time evolution of the quantity $k^{3 / 2} | u_{k} / z |$ for $k = 8 \times 10^{6} {Mpc}^{- 1}$ for two different values of $δ N_{II}$ (left-side $y$ -axis, lines in red). All other free parameters are fixed to the values collected in the second column of Table 1. We superimpose the time evolution of the Hubble parameter $η$ for the same two values of $δ N_{II}$ (right-side $y$ -axis, lines in blue). The vertical dashed line marks the horizon crossing time while (from Table 1) we have $N_{II} = N_{I} + Δ N_{USR} = 17.9$ . In the region shaded in blue we have that $η (N)$ for $δ N_{II} = 0.2$ is larger than $η (N)$ for $δ N_{II} = 0.8$ . This region highlights the difference between the two choices of $δ N_{II}$ in terms of the evolution of $η (N)$ ; the USR phase lasts for a slightly longer time if we consider smaller $δ N_{II}$ (that is a sharper transition at $N_{II}$ ). Right panel. Same as in the left panel but for $k = 1.75 \times 10^{14} {Mpc}^{- 1}$ and for two different values of $δ N_{III}$ (with all other free parameters kept fixed according to the second column of Table 1). In the region shaded in blue we have that $η (N)$ for $δ N_{III} = 0.1$ is larger than $η (N)$ for $δ N_{III} = 1.2$ . This region highlights the difference between the two choices of $δ N_{III}$ in terms of the evolution of $η (N)$ ; the mode experiences, before horizon crossing, a stronger exponential suppression for increasing $δ N_{III}$ , as discussed in Eq. (19).
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Figure 7
Left panel. Same as in the left panel of Fig. 6 but for the time evolution of the plateau mode with $k = 10^{8} {Mpc}^{- 1}$ . Right panel. Same as in the right panel of Fig. 6 but for the time evolution of the mode with $k = 2 \times 10^{14} {Mpc}^{- 1}$ .
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Figure 8
Same as in Fig. 6 but now with $δ N_{II} = δ N_{III} = 0.5$ fixed while we vary $δ N_{I} = 0.2$ (dot-dashed lines) and $δ N_{I} = 1$ (solid lines). Left panel. Time evolution of the mode with $k = 8 \times 10^{6} {Mpc}^{- 1}$ (one of the modes that contribute to the bumplike feature at the left-side edge of the plateau). Right panel. Time evolution of one of the plateau modes with $k = 10^{8} {Mpc}^{- 1}$ .
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Figure 9
Fraction of DM in the form of PBHs with mass $M_{PBH}$ . We plot the most stringent constraints (meshed regions, cf. the SM) and the mass function resulting from three benchmark realizations of our model [labeled as (1), (2), (3), see Table 1]. The yellow band corresponds to the allowed region for a PBH mass function $\propto M_{PBH}^{- 0.5}$ consistent with the Subaru Hyper Suprime-Cam (HSC) microlensing candidate event [107] (see also [118]). The gray line indicates the minimum PBH abundance required to have at least one PBH merger event per year at the Einstein Telescope (ET) experiment, see Ref. [13].
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Figure 10
Left panel: Zoom in on the left-side edge of the plateau of the power spectrum in Fig. 3. We highlight in magenta the values of $k$ such that the horizon crossing condition $k = a (N_{k}) H (N_{k})$ is solved for $N_{I} \leq N_{k} \leq N_{II}$ . Right panel: Dynamical evolution of the perturbation modes with $k$ in the red, blue and green part of power spectrum [cf. the use of different colors in the left panel: the red (green) part of the power spectrum corresponds to the $\sim k^{4}$ ( $\sim k^{0.7}$ ) growth while the blue part lies in between]. The vertical lines mark the horizon crossing time, and make clear that for the blue and green modes we have $N_{I} \leq N_{k} \leq N_{II}$ .
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Figure 11
Left panel: Zoom in on the left-side edge of the plateau of the power spectrum in Fig. 3; we consider two variations of model (1) that have different values of $N_{I}$ and $δ N_{II}$ (cf. the plot legend for details). Right panel: Fraction of DM in the form of PBHs with mass $M_{PBH}$ . We zoom in on the solar-mass range and show the abundance corresponding to the three models discussed in the left panel.
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Figure 12
Left panel: The dashed black line is the power spectrum shown in the left panel of Fig. 10; the solid black line is the power spectrum that is obtained taking $Δ N_{USR} = 2.62$ (instead of the benchmark value $Δ N_{USR} = 2.9$ ) and $η_{II} ≃ 3$ (instead of the benchmark value $η_{II} ≃ 2.7$ ); the dot-dashed black line is the power spectrum obtained for the same parameters given above but with an anticipated USR phase ( $N_{I} = 14.38$ instead of $N_{I} = 15.5$ ). The red part of the power spectrum corresponds to the $\sim k^{4}$ growth while the green part is approximately flat; the blue part lies in between. Central panel: Dynamical evolution of the perturbation modes with $k$ in the red, blue and green part of power spectrum discussed in the left panel. The vertical lines mark the horizon crossing time, and make clear that the for blue and green modes we have $N_{I} \leq N_{k} \leq N_{II}$ . Right panel. The dashed black line corresponds to the PBH mass distribution in model (1). The solid black line is obtained taking a shorter URS phase with $Δ N_{USR} = 2.6$ , $η_{II} = 3.014$ and $δ N_{II} = 0.565$ [but with the same $N_{I} = 15.5$ as in model (1)]. The dot-dashed black line corresponds to the same model that gives the solid black line but with an anticipated USR phase, $N_{I} = 14.38$ .
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Figure 13
Fraction of the energy density in GWs relative to the critical energy density of the Universe as function of the frequency. We show the power-law integrated sensitivity curves [140] of future ground- and space-based GW experiments (as derived in Appendix C of Ref. [141]) as well as previous Parkes Pulsar Timing Array (PPTA) constraint [142], NANOGrav putative band [25] and SKA projected sensitivity [143]. We plot the signals predicted by our model in the three realizations proposed in Table 1.
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Figure 14
Left panel: Reconstructed potential for the model in the second column of Table 1. In the inset plot, we zoom in the transition region, and we indicate with different colors the various stages of the classical inflaton dynamics. It should be noted that during the time interval $N_{II} < N < N_{III}$ (shown as dashed vertical line) the inflaton field has almost zero velocity and it remains approximatively stuck in field space (with this part of the dynamics that lasts for about $Δ N_{plateau} ≃ 18$ $e$ -folds) before entering the last stage that ends inflation. Right panel: Phase-space classical dynamics of the inflaton field. Different gradations of blue correspond, according to the inset legend, to increasing $e$ -fold time $N$ starting from $N_{ref} = 0$ (lighter) to the end of inflation $N_{IV} = 55$ (darker). The red dot corresponds to $N = 45$ . On the $y$ -axis, the field has units of reduced Planck mass and we use the notation $ϕ^{'} = d ϕ / d N$ . In the inset plot, we plot the modulus of the velocity in units of $H / 2 π$ as function of $N$ (with the corresponding field values on the top $x$ -axis), and we focus on the USR phase and the subsequent phase during which $η_{III} = 0$ . We note that, in units of $H / 2 π$ , we have $| ϕ^{'} (N) | ≫ 1$ during the whole dynamics with $| ϕ^{'} (N) | = O (10)$ during the phase with $η_{II} = 0$ .
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Figure 15
Left panel: We compare the reconstructed potential (solid black line) with a benchmark potential featuring an approximate stationary inflection point (anchored to the field value at which $V^{'} (ϕ_{0}) = V^{''} (ϕ_{0}) = 0$ with $ϕ_{0}$ chosen to be the field value at $N = N_{II}$ ). The latter is given by Eq. (37) with $n = 4$ and $c_{3, 4} = 0$ (solid red line). The region shaded in red is obtained for nonzero $c_{3, 4}$ at the percent level. Right panel: We model the reconstructed potential (solid black line) with a combination of the potential in Eq. (38) (dashed red, label $V_{SR}$ ) and Eq. (39) (dashed green, label $V_{BSR}$ ). For the potential $V_{SR} (ϕ)$ we use the values of the parameters given in Table 1 (since this is the actual analytical solution of the system in Eq. (34) during the initial slow-roll phase). In the case of $V_{BSR} (ϕ)$ , we take $V_{0} ≃ 0.988$ , $λ ≃ 0.714$ , $c_{4} ≃ - 0.748$ and $ϕ_{0} ≃ 3.054$ ; we extract these values from a fit of the reconstructed potential considering the region $ϕ < ϕ_{I} = ϕ (N_{I})$ . In the reconstructed potential we take $ϕ_{ref} = 3.5$ .
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Figure 16
Schematic representation of the inflationary trajectory in field space.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Primordial black hole dark matter from inflation: The reverse engineering approach

Gabriele Franciolini and Alfredo Urbano

Phys. Rev. D 106, 123519 – Published 23 December 2022

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Physical Review D

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Primordial black hole dark matter from inflation: The reverse engineering approach

Gabriele Franciolini and Alfredo Urbano

Phys. Rev. D 106, 123519 – Published 23 December 2022

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