The threshold for primordial black holes and the impact of thermal history 147

distinguishing between cosmological perturbations collapsing into PBHs (d[d_c) and those ones bouncing back into the surrounding medium (d\d_c). This is a fundamental parameter because the resulting PBH abundance is exponentially sensitive to its value. The analysis of the gravitational collapse of curvature perturbations to form PBHs and the appropriate threshold condition has been an active line of research in the past years (Kopp et al.2011; Harada et al.2013; Young et al.2014). It has been estimated using analytical methods (Harada et al.2013) but

the best approach is to use fully relativistic simulations of PBH formation in spherical symmetry (Musco et al.2005). Important results having emerged from recent studies are that its exact value is impacted by non-linear and non-Gaussian effects, and that it depends on the radial profile of the overdensity (Musco2019; Young et al.2019; De Luca et al.2019; Kehagias et al. 2019) as well as on the shape of the primordial power spectrum (Yoo et al.2018; Germani and Musco2019).

Very recently, a new semi-analytical method tested against simulations in numerical relativity has been proposed in Musco et al. (2021), computingd_cfrom the shape of the power spectrum, applied to a few particular cases (power-law spectra, log-normal or Gaussian peak, ...). In the radiation dominated universe the typical range of the threshold lies within 0:4\d_c\0:6, with the larger values corresponding to a more peaked shape of the peak of the power spectrum.

On super horizon scales the non-linear amplitude of the curvature profile f is important for the formation of PBHs, and the energy density contrastdq=q_b, when the universe is radiation dominated, is expressed in terms offas

dq

q_b qðr;tÞ q_bðtÞ

q_bðtÞ ¼ 1 a²H²

8

9e^5fðrÞ=2r²e^fðrÞ=2: ð147Þ It can be shown that the amplitudedis a quadratic function of the curvature profile (see for example (Musco2019) for more details)

d¼ 2

3f⁰ðr_mÞ½2þr_mf⁰ðr_mÞ ¼dG 13 8dG

; ð148Þ

wheredG 4

3r_mf⁰ðr_mÞis the linear component of the amplituded. The value ofr_m is defined by the location of the peak of the compaction functionC2DM=R(where DM ¼MMbis the excess of mass with respect the background) and is given by fðrmÞ þrmf⁰ðrmÞ ¼0. Given the value ofd_c the threshold value of the linear com- ponent dG is included within 0:5.d_c;G.0:9, and from Eq. (148) we see that dmax¼8=3, corresponding to the maximum value ofdG, above which d becomes negative, and does not describe a cosmological perturbation of our universe.

The thresholddcis also sensitive to the EoS at the time of formation. For example, the QCD phase transition makes the EoS to drop, increasing the production of PBHs of mass OðMÞ(Jedamzik1997; Byrnes et al. 2018). The reheating at the end of inflation should havefilled the universe with radiation. In the absence of extensions beyond the SM, the universe remains dominated by relativistic particles with an energy density decreasing as the fourth power of the temperature as the universe expands. The number of relativistic degrees of freedom remains constant (g¼106:75) until around 200 GeV, when the temperature of the universe falls to the mass thresholds of SM particles.

Thefirst particle to become non-relativistic is the top quark atT ’m_t¼172 GeV, followed by the Higgs boson at 125 GeV, and theZandWbosons at 92 and 81 GeV, respectively. These particles become non-relativistic at nearly the same time and this induces a significant drop in the number of relativistic degrees of freedom down to g¼86:75. There are further changes at theband cquark ands-lepton thresholds but these are too small to appear in Fig.30. Thereafter g remains approximately

constant until the QCD transition at around 200 MeV, when protons and neutrons condense out of the free light quarks and gluons. The number of relativistic degrees of freedom then falls abruptly tog¼17:25. A little later the pions become non- relativistic and then the muons, givingg¼10:75. Thereafter,g remains constant untile^þe annihilation and neutrino decoupling at around 1 MeV, when it drops to g¼3:36.

Whenever the number of relativistic degrees of freedom suddenly drops, it changes the effective EoS parameterw. There are thus four periods in the thermal history of the universe when w decreases. After each of these, w resumes its relativistic value of 1/3 but each sudden drop modifies the probability of gravitational collapse of any large curvaturefluctuations present at that time, as shown in Fig.30.

As illustrative examples, we have computed the PBH mass functions for two models with an (almost) scale-invariant power spectrum and two different values of the spectral index,ns¼0:97 (Model 1) andns¼1 (Model 2). We assumedc¼0:8 in both cases. The imprints of the thermal history on the PBH mass function are clearly visible. It is worth noticing that these features rely on known physics and are therefore unavoidable for any PBH model with a wide mass function. The former case corresponds to the scenario proposed in Carr et al. (2021b), Clesse and García- Bellido (2022) and the latter in Byrnes et al. (2018), De Luca et al. (2021b). They can both account for the totality of the DM and somehow explain some LIGO/Virgo GW events, but produce different abundances in the stellar mass range:fPBHðMÞ 0:8 in thefirst case,fPBHðMÞ 10⁴in the second case where the peak lies in the sub- lunar range. We stress that the second example avoids the bounds in the LIGO/Virgo range.

10.3 Stochastic gravitational-wave background sourced at second order by curvature fluctuations

If PBHs are generated by the collapse of large density perturbations, they are unavoidably associated to the emission of induced GWs at second order by the same scalar perturbations due to the intrinsic nonlinear nature of gravity (Acquaviva et al.

Fig. 30 Left panel: Evolution of the relativistic degrees of freedomgas a function of the temperature. The grey vertical lines correspond to the masses of the electron, pion, proton/neutron,W,Zbosons and top quark, respectively. Right panel: Effect of the evolution ofgon the critical overdensityd_cleading to PBH formation, as a function of the Hubble horizon mass (related to the PBH mass byM¼cm_H)

2003; Mollerach et al. 2004). The phenomenological implications have been investigated in various contexts also associated to PBHs Ananda et al. (2007), Baumann et al. (2007), Bugaev and Klimai (2010), Saito and Yokoyama (2010), García-Bellido et al. (2017), Ando et al. (2018), Bartolo et al. (2018), Bartolo et al.

(2019c), Bartolo et al. (2019b), Clesse et al. (2018), Ünal (2019), Chatterjee and Mazumdar (2018), Wang et al. (2019), Domènech (2020), Domènech et al. (2020), Pi and Sasaki (2020), Ragavendra et al. (2021), Fumagalli et al. (2021a). If the enhancement of the scalar power spectrum responsible for the generation of PBHs occurs around characteristic scales associated with frequencies between 10⁷ and 10²Hz, this SGWB becomes detectable by GW experiments like LISA. It is worth emphasizing that contrary to the PBH abundance that is exponentially sensitive to the power spectrum, this SGWB depends on the power spectrum amplitude to the second power. This way, LISA will even be able to exclude the existence of an extremely tiny fraction of DM made of PBHs (even a single PBH in our universe) (Clesse et al.

2018), within a wide mass range.

Figure31presents the PBH density fraction at formationb_form(left panel) and the corresponding PBH mass function f_PBH today (right panel) for two models with a power-law power spectrum (see the caption of the Figure for details). The SGWB associated with one of these two models is shown in Fig.32, where it is confronted with several current or forecasted experimental limits. The SGWB covers a wide frequency range. In the ultra-low frequency range, around nHz, PTA experiments like PPTA (Shannon et al.2015), NANOGrav (Arzoumanian et al.2018) and EPTA (Lentati et al. 2015) give the most stringent constraints on the GWs abundance.

Future experiments like SKA (Dewdney et al.2009) (see also Moore et al.2015) will greatly improve the sensitivity. In the LIGO/Virgo frequency range, an additional constraint has been set by the non-observation of a SGWB after O1-O2 (Abbott et al.

2019c) and O3 runs (Abbott et al.2021g). All these searches can be translated into a constraint on the amplitude of the comoving curvature perturbation at the

Fig. 31 PBH density fraction at formationb_form(left panel) and the corresponding PBH mass functionf_PBH today (right panel), neglecting the effect of PBH growth by accretion and hierarchical mergers, for two models with a power-law power spectrum and including the effects of thermal history: Model 1 from Carr et al. (2021b), Clesse and García-Bellido (2022) with spectral indexns¼0:97; Model 2 from De Luca et al. (2021b), Byrnes et al. (2018) withns¼1:and a cut-off mass of 10¹⁴M. The transition between the large-scale and small-scale power spectrum isfixed atk¼10³Mpc¹. The power spectrum amplitude is normalized such that both models produce an integrated PBH fractionfPBH¼1, i.e. PBH constitute the totality of DM. A value ofc¼0:8 was assumed

corresponding scales (Bugaev and Klimai2010; Byrnes et al. 2019; Inomata and Nakama 2019; Ünal et al. 2021). Those bounds are also affecting the maximum allowed PBHs fraction of DM with the hypothesis that they originate from the collapse of density perturbations. Detailed studies with the LIGO/Virgo data affecting the mass rangeM 2½10²⁰;10¹⁸Mare reported in Kapadia et al. (2020), while very tight bounds in the mass rangeM 2½10³;1M are obtained in Chen et al. (2020) using the latest NANOGrav data; see also Cai et al. (2019b) where the dependence of the result to non-Gaussianities is also investigated,finding that local non-Gaussianity can for example alleviate the bounds (see Sect.10.2.3for details).

Finally, the next generation multimessenger experiments, CMB distortion (PIXIE) and PTA-SKA, can test the PBH scenario over solar mass robustly, namely they can conclusively detect or rule out the PBHs over solar-mass and the intriguing proposal that the seeds of the MBHs are formed by PBHs (Ünal et al.2021) independent of i) statistical properties of perturbations, ii) accretion and merger history and iii) clustering effects.

LISA will be able to provide insights in the intermediate frequencies, and corresponding masses. Since the emission mostly comes when the corresponding scales cross the horizon, one can relate the GWs frequency to the PBHs massMas (see for example (Saito and Yokoyama2010; García-Bellido et al.2017))

f ’3 mHz c 0:2

1=2 M

10¹²M

₁₌₂

; ð149Þ

where the factorcis capturing the relation between the horizon mass at formation and

Fig. 32 SGWB sourced at second order by the density perturbations at the origin of PBH formation, for Model 2 of Fig.31. On top of the plot, we show the PBH mass associated to a given GW frequency as in Eq. (149). The LISA sensitivity (Amaro-Seoane et al.2017) and the hint for a detection by NANOGrav 12.5 yr (Arzoumanian et al.2020) are represented, as well as the constraints coming EPTA (Lentati et al.

2015), PPTA (Shannon et al.2015), NANOGrav 11 yrs (Arzoumanian et al.2018; Aggarwal et al.2019) and future sensitivity curves for planned experiments like SKA (Zhao et al.2013), DECIGO/BBO (Yagi and Seto2011), CE (Abbott et al.2017b), ET (Hild et al.2011), Advanced LIGO?Virgo collaboration (Abbott et al.2017f), Magis-space (MS) and Magis-100 (M100) (Coleman2019), AEDGE (El-Neaj et al.

2020) and AION (Badurina et al.2020). Image reproduced with permission from De Luca et al. (2021b)

the PBH mass after the collapse. Notice that the peak frequencies fall within the LISA sensitivity band for PBH masses around MOð10¹⁵10⁸ÞM and for this mass range, the PBHs can constitute the totality of the DM. Hence, García- Bellido et al. (2017) proposed PBHs in this mass range as DM and further found that density perturbations forming PBHs lead to GWs detectable by LISA. This proposal has been studied in more detail in Cai et al. (2019a), Bartolo et al. (2019b), Bartolo et al. (2019c), Ünal (2019).

The computation of the resulting SGWB spectrum was originally performed in Ananda et al. (2007). We provide here the main result, assuming a generic form for the power spectrum of curvaturefluctuation. The current GW abundance can then be obtained as

XGWðg₀;kÞ ¼a⁴_fqGWðg_f;kÞ

q_rðg₀Þ X_r;0¼gðg_fÞ gðg₀Þ

g_Sðg₀Þ gSðg_fÞ

!4=3

X_r;0XGWðg_f;kÞ;

ð150Þ in terms of the present radiation energy density fraction Xr;0 if the neutrinos were massless. The crucial quantity is XGWðg_f;kÞ, that is the fractional GW energy density for log interval at the emission epochg_f, related to the critical energy density of a spatiallyflat universeq_c¼3H²M_p². Assuming that the scalar perturbationsfare Gaussian, it can be calculated as

qGWðg;~xÞ

q_cðgÞ Z

dlnkXGWðg;kÞ

¼2p⁴M_p² 81g²a²

Z d³k1d³p1

ð Þ2p ⁶ 1 k⁴₁

p²₁ ðk~₁~p1Þ²=k₁²

h i₂

p³₁k~1~p1³ Pfðp1ÞPfðjk~1~p₁jÞI²_cðk~₁;~p₁Þ þI²_sðk~₁;~p₁Þ

;

ð151Þ

where the functionsIc;sare found in Espinosa et al. (2018), Kohri and Terada (2018).

The integrals need to be done numerically for general power spectra (see Saito and Yokoyama2010; Bugaev and Klimai2010for analytical calculations in the specific case of a monochromatic or Gaussian curvature spectrum).

For the frequencies of interest, usingf ’8 mHzðg=10Þ¹⁼⁴ðT=10⁶GeVÞ, one can show that the emission of GWs takes place atg_f well before the time at which top quarks start annihilating, above which we can assume a RD universe with constant effective degrees of freedom.

The non-linear coupling with the curvature perturbation naturally leads to an intrinsically non-Gaussian GWs signal imprinted in phase correlations. However, the coherence is washed out by the propagation of the waves in the perturbed universe mainly due to time delay effects originated from the presence of large scale variations of gravitational potential (Bartolo et al.2018,2019b,c; Margalit et al.2020). This is simply a consequence of the central limit theorem applied to a number

Nðk_Hg₀Þ²o1 of independent lines of sight (Bartolo et al. 2019b),k_H being the characteristic perturbation wave-number roughly proportional to the the inverse horizon size at GW emission. Possible small deformations smearing the GWs spectrum can also arise from similar effects (Domcke et al.2020c).

An interesting primordial signal that is potentially observable is related to scenarios where the scalar power spectrum presents oscillations of sufficiently large amplitude, characteristic of large particle production mechanisms, leading to oscillatory Oð10%Þ modulations in the frequency profile of the scalar-induced SGWB (Fumagalli et al.2021a; Braglia et al.2021), see Sect.8.3for details.

10.4 Resolved sources and stochastic gravitational-wave backgrounds