Small-scale clustering of primordial black holes: Cloud-in-cloud and exclusion effects

Author(s)

Auclair, Pierre, Blachier, Baptiste

Abstract

Using an excursion-set approach, we revisit the initial spatial clustering of Primordial Black Holes (PBHs) originating from the Hubble reentry of large Gaussian density fluctuations in the early Universe. We derive the two-point correlation functions of PBHs, properly accounting for the “cloud-in-cloud” mechanism. Our expressions naturally and intrinsically correlate the formation of pairs of PBHs, which is a key difference with the Poisson model of clustering. Our approach effectively includes short-range exclusion effects and clarifies the clustering behaviors at small scale: PBHs are anticorrelated at short distances. Using a scale-independent collapse threshold, we derive explicit expressions for the excess probability to find pairs of PBHs separated by a distance <math display="inline"><mi>r</mi></math>, as well as the excess probability to find pairs with asymmetric mass ratio. Our framework is model independent by construction, and we discuss possible other applications.

Figures

Schematic representation of random walks performed by smoothed overdensities sharing a common past until ``time'' $S_{r}$ and which subsequently evolve in distinct ways, resulting in two different collapses that occur respectively at the first-crossing times $S_{1}$ and $S_{2}$.

Schematic representation of random walks performed by smoothed overdensities sharing a common past until ``time'' $S_{r}$ and which subsequently evolve in distinct ways, resulting in two different collapses that occur respectively at the first-crossing times $S_{1}$ and $S_{2}$.


 : $\nu = 1$

: $\nu = 1$


 : $\nu = 2$ : $\nu = 5$

: $\nu = 2$ : $\nu = 5$


 : $\nu = 10$ : Two-point correlation function of the PBHs spatial distribution with constant density threshold $\delta_c$, using the Press-Schechter solution of Ref.~\cite{Ali-Haimoud:2018dau} (dashed red) and the Excursion-Set formalism of \cref{sec:section3} (solid blue), see in particular \cref{eq:P2_semianalytic}.

: $\nu = 10$ : Two-point correlation function of the PBHs spatial distribution with constant density threshold $\delta_c$, using the Press-Schechter solution of Ref.~\cite{Ali-Haimoud:2018dau} (dashed red) and the Excursion-Set formalism of \cref{sec:section3} (solid blue), see in particular \cref{eq:P2_semianalytic}.


 : Caption not extracted

: Caption not extracted


Schematic representation of random walks, from $S=0$ to the maximum value $\sigma^2$. Most of the trajectories are confined in the gray region which ranges between $\pm \sigma$. The horizontal lines correspond to different values of $\nu = \delta_c / \sigma$, where $\nu$ is a measure of the barrier's height in units of the typical spread of the trajectories. Low values of $\nu$ indicate that barrier crossings are frequent: hence multiple crossings are also frequent. Higher values of $\nu$ indicate that barrier crossings (and therefore PBH collapses) are rare events, and multiple crossings are suppressed.

Schematic representation of random walks, from $S=0$ to the maximum value $\sigma^2$. Most of the trajectories are confined in the gray region which ranges between $\pm \sigma$. The horizontal lines correspond to different values of $\nu = \delta_c / \sigma$, where $\nu$ is a measure of the barrier's height in units of the typical spread of the trajectories. Low values of $\nu$ indicate that barrier crossings are frequent: hence multiple crossings are also frequent. Higher values of $\nu$ indicate that barrier crossings (and therefore PBH collapses) are rare events, and multiple crossings are suppressed.


 : $\lambda = 1$

: $\lambda = 1$


 : $\lambda = 2$ : $\lambda = 5$

: $\lambda = 2$ : $\lambda = 5$


 : $\lambda = 10$ : Excess probability to find pairs of PBHs with masses $S_1, S_2$ at a fixed distance $r$. $w_n = S_r / S_n$ is a measure of the PBH mass, the limit $w_n \to 0$ corresponds to a very small PBH, whereas $w_n \to 1$ corresponds to the maximum allowed PBH mass in the volume contained within the two PBHs.

: $\lambda = 10$ : Excess probability to find pairs of PBHs with masses $S_1, S_2$ at a fixed distance $r$. $w_n = S_r / S_n$ is a measure of the PBH mass, the limit $w_n \to 0$ corresponds to a very small PBH, whereas $w_n \to 1$ corresponds to the maximum allowed PBH mass in the volume contained within the two PBHs.


 : Caption not extracted

: Caption not extracted


References
  • <ref id="c1"><label>[1]</label><mixed-citation publication-type="journal"><object-id>1</object-id><person-group person-group-type="author"><string-name>S. Hawking</string-name></person-group>, <article-title>Gravitationally collapsed objects of very low mass</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>152</volume>, <page-range>75</page-range> (<year>1971</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/152.1.75</pub-id></mixed-citation></ref>
  • <ref id="c2"><label>[2]</label><mixed-citation publication-type="journal"><object-id>2</object-id><person-group person-group-type="author"><string-name>B. J. Carr</string-name> and <string-name>S. W. Hawking</string-name></person-group>, <article-title>Black holes in the early Universe</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>168</volume>, <page-range>399</page-range> (<year>1974</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/168.2.399</pub-id></mixed-citation></ref>
  • <ref id="c3"><label>[3]</label><mixed-citation publication-type="journal"><object-id>3</object-id><person-group person-group-type="author"><string-name>B. J. Carr</string-name></person-group>, <article-title>The primordial black hole mass spectrum</article-title>, <source>Astrophys. J.</source> <volume>201</volume>, <page-range>1</page-range> (<year>1975</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/153853</pub-id></mixed-citation></ref>
  • <ref id="c4"><label>[4]</label><mixed-citation publication-type="journal"><object-id>4</object-id><person-group person-group-type="author"><string-name>G. F. Chapline</string-name></person-group>, <article-title>Cosmological effects of primordial black holes</article-title>, <source>Nature (London)</source> <volume>253</volume>, <page-range>251</page-range> (<year>1975</year>).<pub-id pub-id-type="coden">NATUAS</pub-id><issn>0028-0836</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1038/253251a0</pub-id></mixed-citation></ref>
  • <ref id="c5"><label>[5]</label><mixed-citation publication-type="journal"><object-id>5</object-id><person-group person-group-type="author"><string-name>B. Carr</string-name>, <string-name>K. Kohri</string-name>, <string-name>Y. Sendouda</string-name>, and <string-name>J. Yokoyama</string-name></person-group>, <article-title>Constraints on primordial black holes</article-title>, <source>Rep. Prog. Phys.</source> <volume>84</volume>, <page-range>116902</page-range> (<year>2021</year>).<pub-id pub-id-type="coden">RPPHAG</pub-id><issn>0034-4885</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1361-6633/ac1e31</pub-id></mixed-citation></ref>
  • <ref id="c6"><label>[6]</label><mixed-citation publication-type="journal"><object-id>6</object-id><person-group person-group-type="author"><string-name>P. Meszaros</string-name></person-group>, <article-title>Primeval black holes and galaxy formation</article-title>, <source>Astron. Astrophys.</source> <volume>38</volume>, <page-range>5</page-range> (<year>1975</year>).<pub-id pub-id-type="coden">AAEJAF</pub-id><issn>0004-6361</issn></mixed-citation></ref>
  • <ref id="c7"><label>[7]</label><mixed-citation publication-type="journal"><object-id>7</object-id><person-group person-group-type="author"><string-name>R. Bean</string-name> and <string-name>J. Magueijo</string-name></person-group>, <article-title>Could supermassive black holes be quintessential primordial black holes?</article-title>, <source>Phys. Rev. D</source> <volume>66</volume>, <page-range>063505</page-range> (<year>2002</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>0556-2821</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.66.063505</pub-id></mixed-citation></ref>
  • <ref id="c8"><label>[8]</label><mixed-citation publication-type="journal"><object-id>8</object-id><person-group person-group-type="author"><string-name>S. Bird</string-name>, <string-name>I. Cholis</string-name>, <string-name>J. B. Muñoz</string-name>, <string-name>Y. Ali-Haïmoud</string-name>, <string-name>M. Kamionkowski</string-name>, <string-name>E. D. Kovetz</string-name>, <string-name>A. Raccanelli</string-name>, and <string-name>A. G. Riess</string-name></person-group>, <article-title>Did LIGO detect dark matter?</article-title>, <source>Phys. Rev. Lett.</source> <volume>116</volume>, <page-range>201301</page-range> (<year>2016</year>).<pub-id pub-id-type="coden">PRLTAO</pub-id><issn>0031-9007</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevLett.116.201301</pub-id></mixed-citation></ref>
  • <ref id="c9"><label>[9]</label><mixed-citation id="c9a" publication-type="journal"><object-id>9a</object-id><person-group person-group-type="author"><string-name>M. Sasaki</string-name>, <string-name>T. Suyama</string-name>, <string-name>T. Tanaka</string-name>, and <string-name>S. Yokoyama</string-name></person-group>, <article-title>Primordial black hole scenario for the gravitational-wave event GW150914</article-title>, <source>Phys. Rev. Lett.</source> <volume>117</volume>, <page-range>061101</page-range> (<year>2016</year>); <pub-id pub-id-type="coden">PRLTAO</pub-id><issn>0031-9007</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevLett.117.061101</pub-id></mixed-citation><mixed-citation id="c9b" publication-type="journal" specific-use="authorjournal"><object-id>9b</object-id><person-group person-group-type="author"><string-name>M. Sasaki</string-name>, <string-name>T. Suyama</string-name>, <string-name>T. Tanaka</string-name>, and <string-name>S. Yokoyama</string-name></person-group><source>Phys. Rev. Lett.</source><volume>121</volume>, <page-range>059901(E)</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PRLTAO</pub-id><issn>0031-9007</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevLett.121.059901</pub-id></mixed-citation></ref>
  • <ref id="c9"><label>[9]</label><mixed-citation id="c9a" publication-type="journal"><object-id>9a</object-id><person-group person-group-type="author"><string-name>M. Sasaki</string-name>, <string-name>T. Suyama</string-name>, <string-name>T. Tanaka</string-name>, and <string-name>S. Yokoyama</string-name></person-group>, <article-title>Primordial black hole scenario for the gravitational-wave event GW150914</article-title>, <source>Phys. Rev. Lett.</source> <volume>117</volume>, <page-range>061101</page-range> (<year>2016</year>); <pub-id pub-id-type="coden">PRLTAO</pub-id><issn>0031-9007</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevLett.117.061101</pub-id></mixed-citation><mixed-citation id="c9b" publication-type="journal" specific-use="authorjournal"><object-id>9b</object-id><person-group person-group-type="author"><string-name>M. Sasaki</string-name>, <string-name>T. Suyama</string-name>, <string-name>T. Tanaka</string-name>, and <string-name>S. Yokoyama</string-name></person-group><source>Phys. Rev. Lett.</source><volume>121</volume>, <page-range>059901(E)</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PRLTAO</pub-id><issn>0031-9007</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevLett.121.059901</pub-id></mixed-citation></ref>
  • <ref id="c10"><label>[10]</label><mixed-citation publication-type="journal"><object-id>10</object-id><person-group person-group-type="author"><string-name>S. Clesse</string-name> and <string-name>J. García-Bellido</string-name></person-group>, <article-title>The clustering of massive primordial black holes as dark matter: Measuring their mass distribution with Advanced LIGO</article-title>, <source>Phys. Dark Universe</source> <volume>15</volume>, <page-range>142</page-range> (<year>2017</year>).<pub-id pub-id-type="coden">PDUHA3</pub-id><issn>2212-6864</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1016/j.dark.2016.10.002</pub-id></mixed-citation></ref>
  • <ref id="c11"><label>[11]</label><mixed-citation publication-type="journal"><object-id>11</object-id><person-group person-group-type="author"><string-name>Y. Ali-Haïmoud</string-name>, <string-name>E. D. Kovetz</string-name>, and <string-name>M. Kamionkowski</string-name></person-group>, <article-title>Merger rate of primordial black-hole binaries</article-title>, <source>Phys. Rev. D</source> <volume>96</volume>, <page-range>123523</page-range> (<year>2017</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.96.123523</pub-id></mixed-citation></ref>
  • <ref id="c12"><label>[12]</label><mixed-citation publication-type="journal"><object-id>12</object-id><person-group person-group-type="author"><string-name>G. Hütsi</string-name>, <string-name>M. Raidal</string-name>, <string-name>V. Vaskonen</string-name>, and <string-name>H. Veermäe</string-name></person-group>, <article-title>Two populations of LIGO-Virgo black holes</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>03</issue> (<volume>2021</volume>) <page-range>068</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2021/03/068</pub-id></mixed-citation></ref>
  • <ref id="c13"><label>[13]</label><mixed-citation publication-type="journal"><object-id>13</object-id><person-group person-group-type="author"><string-name>V. De Luca</string-name>, <string-name>G. Franciolini</string-name>, <string-name>P. Pani</string-name>, and <string-name>A. Riotto</string-name></person-group>, <article-title>Bayesian evidence for both astrophysical and primordial black holes: Mapping the GWTC-2 catalog to third-generation detectors</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>05</issue> (<volume>2021</volume>) <page-range>003</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2021/05/003</pub-id></mixed-citation></ref>
  • <ref id="c14"><label>[14]</label><mixed-citation publication-type="journal"><object-id>14</object-id><person-group person-group-type="author"><string-name>G. Franciolini</string-name>, <string-name>V. Baibhav</string-name>, <string-name>V. De Luca</string-name>, <string-name>K. K. Y. Ng</string-name>, <string-name>K. W. K. Wong</string-name>, <string-name>E. Berti</string-name>, <string-name>P. Pani</string-name>, <string-name>A. Riotto</string-name>, and <string-name>S. Vitale</string-name></person-group>, <article-title>Searching for a subpopulation of primordial black holes in LIGO-Virgo gravitational-wave data</article-title>, <source>Phys. Rev. D</source> <volume>105</volume>, <page-range>083526</page-range> (<year>2022</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.105.083526</pub-id></mixed-citation></ref>
  • <ref id="c15"><label>[15]</label><mixed-citation publication-type="journal"><object-id>15</object-id><person-group person-group-type="author"><string-name>G. Franciolini</string-name>, <string-name>I. Musco</string-name>, <string-name>P. Pani</string-name>, and <string-name>A. Urbano</string-name></person-group>, <article-title>From inflation to black hole mergers and back again: Gravitational-wave data-driven constraints on inflationary scenarios with a first-principle model of primordial black holes across the QCD epoch</article-title>, <source>Phys. Rev. D</source> <volume>106</volume>, <page-range>123526</page-range> (<year>2022</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.106.123526</pub-id></mixed-citation></ref>
  • <ref id="c16"><label>[16]</label><mixed-citation publication-type="journal"><object-id>16</object-id><person-group person-group-type="author"><string-name>R. A. Allsman</string-name> <etal/> (<collab>Macho Collaboration</collab>)</person-group>, <article-title>MACHO project limits on black hole dark matter in the 1-30 solar mass range</article-title>, <source>Astrophys. J. Lett.</source> <volume>550</volume>, <page-range>L169</page-range> (<year>2001</year>).<pub-id pub-id-type="coden">AJLEEY</pub-id><issn>2041-8213</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/319636</pub-id></mixed-citation></ref>
  • <ref id="c17"><label>[17]</label><mixed-citation publication-type="journal"><object-id>17</object-id><person-group person-group-type="author"><string-name>P. Tisserand</string-name> <etal/> (<collab>EROS-2 Collaboration</collab>)</person-group>, <article-title>Limits on the macho content of the galactic halo from the EROS-2 survey of the Magellanic clouds</article-title>, <source>Astron. Astrophys.</source> <volume>469</volume>, <page-range>387</page-range> (<year>2007</year>).<pub-id pub-id-type="coden">AAEJAF</pub-id><issn>0004-6361</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1051/0004-6361:20066017</pub-id></mixed-citation></ref>
  • <ref id="c18"><label>[18]</label><mixed-citation publication-type="journal"><object-id>18</object-id><person-group person-group-type="author"><string-name>Y. Ali-Haïmoud</string-name> and <string-name>M. Kamionkowski</string-name></person-group>, <article-title>Cosmic microwave background limits on accreting primordial black holes</article-title>, <source>Phys. Rev. D</source> <volume>95</volume>, <page-range>043534</page-range> (<year>2017</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.95.043534</pub-id></mixed-citation></ref>
  • <ref id="c19"><label>[19]</label><mixed-citation publication-type="journal"><object-id>19</object-id><person-group person-group-type="author"><string-name>M. Ricotti</string-name>, <string-name>J. P. Ostriker</string-name>, and <string-name>K. J. Mack</string-name></person-group>, <article-title>Effect of primordial black holes on the cosmic microwave background and cosmological parameter estimates</article-title>, <source>Astrophys. J.</source> <volume>680</volume>, <page-range>829</page-range> (<year>2008</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/587831</pub-id></mixed-citation></ref>
  • <ref id="c20"><label>[20]</label><mixed-citation publication-type="journal"><object-id>20</object-id><person-group person-group-type="author"><string-name>V. Poulin</string-name>, <string-name>P. D. Serpico</string-name>, <string-name>F. Calore</string-name>, <string-name>S. Clesse</string-name>, and <string-name>K. Kohri</string-name></person-group>, <article-title>CMB bounds on disk-accreting massive primordial black holes</article-title>, <source>Phys. Rev. D</source> <volume>96</volume>, <page-range>083524</page-range> (<year>2017</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.96.083524</pub-id></mixed-citation></ref>
  • <ref id="c21"><label>[21]</label><mixed-citation publication-type="journal"><object-id>21</object-id><person-group person-group-type="author"><string-name>S. Clesse</string-name> and <string-name>J. García-Bellido</string-name></person-group>, <article-title>Seven hints for primordial black hole dark matter</article-title>, <source>Phys. Dark Universe</source> <volume>22</volume>, <page-range>137</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PDUHA3</pub-id><issn>2212-6864</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1016/j.dark.2018.08.004</pub-id></mixed-citation></ref>
  • <ref id="c22"><label>[22]</label><mixed-citation publication-type="journal"><object-id>22</object-id><person-group person-group-type="author"><string-name>G. Ballesteros</string-name>, <string-name>P. D. Serpico</string-name>, and <string-name>M. Taoso</string-name></person-group>, <article-title>On the merger rate of primordial black holes: Effects of nearest neighbours distribution and clustering</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>10</issue> (<volume>2018</volume>) <page-range>043</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2018/10/043</pub-id></mixed-citation></ref>
  • <ref id="c23"><label>[23]</label><mixed-citation publication-type="journal"><object-id>23</object-id><person-group person-group-type="author"><string-name>B. Carr</string-name> and <string-name>J. Silk</string-name></person-group>, <article-title>Primordial black holes as generators of cosmic structures</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>478</volume>, <page-range>3756</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/sty1204</pub-id></mixed-citation></ref>
  • <ref id="c24"><label>[24]</label><mixed-citation publication-type="journal"><object-id>24</object-id><person-group person-group-type="author"><string-name>M. Raidal</string-name>, <string-name>V. Vaskonen</string-name>, and <string-name>H. Veermäe</string-name></person-group>, <article-title>Gravitational waves from primordial black hole mergers</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>09</issue> (<volume>2017</volume>) <page-range>037</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2017/09/037</pub-id></mixed-citation></ref>
  • <ref id="c25"><label>[25]</label><mixed-citation publication-type="journal"><object-id>25</object-id><person-group person-group-type="author"><string-name>J. García-Bellido</string-name> and <string-name>S. Clesse</string-name></person-group>, <article-title>Constraints from microlensing experiments on clustered primordial black holes</article-title>, <source>Phys. Dark Universe</source> <volume>19</volume>, <page-range>144</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PDUHA3</pub-id><issn>2212-6864</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1016/j.dark.2018.01.001</pub-id></mixed-citation></ref>
  • <ref id="c26"><label>[26]</label><mixed-citation publication-type="journal"><object-id>26</object-id><person-group person-group-type="author"><string-name>J. Calcino</string-name>, <string-name>J. Garcia-Bellido</string-name>, and <string-name>T. M. Davis</string-name></person-group>, <article-title>Updating the MACHO fraction of the Milky Way dark halowith improved mass models</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>479</volume>, <page-range>2889</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/sty1368</pub-id></mixed-citation></ref>
  • <ref id="c27"><label>[27]</label><mixed-citation publication-type="journal"><object-id>27</object-id><person-group person-group-type="author"><string-name>B. Carr</string-name>, <string-name>S. Clesse</string-name>, <string-name>J. García-Bellido</string-name>, and <string-name>F. Kühnel</string-name></person-group>, <article-title>Cosmic conundra explained by thermal history and primordial black holes</article-title>, <source>Phys. Dark Universe</source> <volume>31</volume>, <page-range>100755</page-range> (<year>2021</year>).<pub-id pub-id-type="coden">PDUHA3</pub-id><issn>2212-6864</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1016/j.dark.2020.100755</pub-id></mixed-citation></ref>
  • <ref id="c28"><label>[28]</label><mixed-citation publication-type="journal"><object-id>28</object-id><person-group person-group-type="author"><string-name>M. Petač</string-name>, <string-name>J. Lavalle</string-name>, and <string-name>K. Jedamzik</string-name></person-group>, <article-title>Microlensing constraints on clustered primordial black holes</article-title>, <source>Phys. Rev. D</source> <volume>105</volume>, <page-range>083520</page-range> (<year>2022</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.105.083520</pub-id></mixed-citation></ref>
  • <ref id="c29"><label>[29]</label><mixed-citation publication-type="journal"><object-id>29</object-id><person-group person-group-type="author"><string-name>M. Gorton</string-name> and <string-name>A. M. Green</string-name></person-group>, <article-title>Effect of clustering on primordial black hole microlensing constraints</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>08</issue> (<volume>2022</volume>) <page-range>035</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2022/08/035</pub-id></mixed-citation></ref>
  • <ref id="c30"><label>[30]</label><mixed-citation publication-type="book"><object-id>30</object-id><person-group person-group-type="author"><string-name>P. J. E. Peebles</string-name></person-group>, <source>The Large-Scale Structure of the Universe</source> (<publisher-name>Princeton University Press</publisher-name>, Princeton, NJ, <year>1980</year>).</mixed-citation></ref>
  • <ref id="c31"><label>[31]</label><mixed-citation publication-type="journal"><object-id>31</object-id><person-group person-group-type="author"><string-name>A. A. Klypin</string-name> and <string-name>A. I. Kopylov</string-name></person-group>, <article-title>The spatial covariance function for rich clusters of galaxies</article-title>, <source>Sov. Astron. Lett.</source> <volume>9</volume>, <page-range>41</page-range> (<year>1983</year>).<pub-id pub-id-type="coden">SALEDW</pub-id><issn>0360-0327</issn></mixed-citation></ref>
  • <ref id="c32"><label>[32]</label><mixed-citation publication-type="journal"><object-id>32</object-id><person-group person-group-type="author"><string-name>N. A. Bahcall</string-name> and <string-name>R. M. Soneira</string-name></person-group>, <article-title>The spatial correlation function of rich clusters of galaxies</article-title>, <source>Astrophys. J.</source> <volume>270</volume>, <page-range>20</page-range> (<year>1983</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/161094</pub-id></mixed-citation></ref>
  • <ref id="c33"><label>[33]</label><mixed-citation publication-type="journal"><object-id>33</object-id><person-group person-group-type="author"><string-name>W. H. Press</string-name> and <string-name>P. Schechter</string-name></person-group>, <article-title>Formation of galaxies and clusters of galaxies by self-similar gravitational condensation</article-title>, <source>Astrophys. J.</source> <volume>187</volume>, <page-range>425</page-range> (<year>1974</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/152650</pub-id></mixed-citation></ref>
  • <ref id="c34"><label>[34]</label><mixed-citation publication-type="journal"><object-id>34</object-id><person-group person-group-type="author"><string-name>N. Kaiser</string-name></person-group>, <article-title>On the spatial correlations of Abell clusters</article-title>, <source>Astrophys. J. Lett.</source> <volume>284</volume>, <page-range>L9</page-range> (<year>1984</year>).<pub-id pub-id-type="coden">AJLEEY</pub-id><issn>2041-8213</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/184341</pub-id></mixed-citation></ref>
  • <ref id="c35"><label>[35]</label><mixed-citation publication-type="journal"><object-id>35</object-id><person-group person-group-type="author"><string-name>H. D. Politzer</string-name> and <string-name>M. B. Wise</string-name></person-group>, <article-title>Relations between spatial correlations of rich clusters of galaxies</article-title>, <source>Astrophys. J. Lett.</source> <volume>285</volume>, <page-range>L1</page-range> (<year>1984</year>).<pub-id pub-id-type="coden">AJLEEY</pub-id><issn>2041-8213</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/184352</pub-id></mixed-citation></ref>
  • <ref id="c36"><label>[36]</label><mixed-citation publication-type="journal"><object-id>36</object-id><person-group person-group-type="author"><string-name>L. G. Jensen</string-name> and <string-name>A. S. Szalay</string-name></person-group>, <article-title>N-point correlations for biased galaxy formation</article-title>, <source>Astrophys. J. Lett.</source> <volume>305</volume>, <page-range>L5</page-range> (<year>1986</year>).<pub-id pub-id-type="coden">AJLEEY</pub-id><issn>2041-8213</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/184673</pub-id></mixed-citation></ref>
  • <ref id="c37"><label>[37]</label><mixed-citation publication-type="journal"><object-id>37</object-id><person-group person-group-type="author"><string-name>C. Porciani</string-name>, <string-name>S. Matarrese</string-name>, <string-name>F. Lucchin</string-name>, and <string-name>P. Catelan</string-name></person-group>, <article-title>Excursion set approach to the clustering of dark matter halos in Lagrangian space</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>298</volume>, <page-range>1097</page-range> (<year>1998</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1365-8711.1998.01704.x</pub-id></mixed-citation></ref>
  • <ref id="c38"><label>[38]</label><mixed-citation publication-type="journal"><object-id>38</object-id><person-group person-group-type="author"><string-name>A. R. Zentner</string-name></person-group>, <article-title>The excursion set theory of halo mass functions, halo clustering, and halo growth</article-title>, <source>Int. J. Mod. Phys. D</source> <volume>16</volume>, <page-range>763</page-range> (<year>2007</year>).<pub-id pub-id-type="coden">IMPDEO</pub-id><issn>0218-2718</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1142/S0218271807010511</pub-id></mixed-citation></ref>
  • <ref id="c39"><label>[39]</label><mixed-citation publication-type="journal"><object-id>39</object-id><person-group person-group-type="author"><string-name>V. Desjacques</string-name>, <string-name>D. Jeong</string-name>, and <string-name>F. Schmidt</string-name></person-group>, <article-title>Large-scale galaxy bias</article-title>, <source>Phys. Rep.</source> <volume>733</volume>, <page-range>1</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PRPLCM</pub-id><issn>0370-1573</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1016/j.physrep.2017.12.002</pub-id></mixed-citation></ref>
  • <ref id="c40"><label>[40]</label><mixed-citation publication-type="journal"><object-id>40</object-id><person-group person-group-type="author"><string-name>H. J. Mo</string-name> and <string-name>S. D. M. White</string-name></person-group>, <article-title>An analytic model for the spatial clustering of dark matter halos</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>282</volume>, <page-range>347</page-range> (<year>1996</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/282.2.347</pub-id></mixed-citation></ref>
  • <ref id="c41"><label>[41]</label><mixed-citation publication-type="journal"><object-id>41</object-id><person-group person-group-type="author"><string-name>T. Baldauf</string-name>, <string-name>U. Seljak</string-name>, <string-name>R. E. Smith</string-name>, <string-name>N. Hamaus</string-name>, and <string-name>V. Desjacques</string-name></person-group>, <article-title>Halo stochasticity from exclusion and nonlinear clustering</article-title>, <source>Phys. Rev. D</source> <volume>88</volume>, <page-range>083507</page-range> (<year>2013</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>1550-7998</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.88.083507</pub-id></mixed-citation></ref>
  • <ref id="c42"><label>[42]</label><mixed-citation publication-type="journal"><object-id>42</object-id><person-group person-group-type="author"><string-name>T. Baldauf</string-name>, <string-name>S. Codis</string-name>, <string-name>V. Desjacques</string-name>, and <string-name>C. Pichon</string-name></person-group>, <article-title>Peak exclusion, stochasticity and convergence of perturbative bias expansions in <inline-formula><math display="inline"><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math></inline-formula> gravity</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>456</volume>, <page-range>3985</page-range> (<year>2016</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/stv2973</pub-id></mixed-citation></ref>
  • <ref id="c43"><label>[43]</label><mixed-citation publication-type="journal"><object-id>43</object-id><person-group person-group-type="author"><string-name>J. R. Chisholm</string-name></person-group>, <article-title>Clustering of primordial black holes: Basic results</article-title>, <source>Phys. Rev. D</source> <volume>73</volume>, <page-range>083504</page-range> (<year>2006</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>1550-7998</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.73.083504</pub-id></mixed-citation></ref>
  • <ref id="c44"><label>[44]</label><mixed-citation publication-type="journal"><object-id>44</object-id><person-group person-group-type="author"><string-name>Y. Ali-Haïmoud</string-name></person-group>, <article-title>Correlation function of high-threshold regions and application to the initial small-scale clustering of primordial black holes</article-title>, <source>Phys. Rev. Lett.</source> <volume>121</volume>, <page-range>081304</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PRLTAO</pub-id><issn>0031-9007</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevLett.121.081304</pub-id></mixed-citation></ref>
  • <ref id="c45"><label>[45]</label><mixed-citation publication-type="journal"><object-id>45</object-id><person-group person-group-type="author"><string-name>V. Atal</string-name>, <string-name>A. Sanglas</string-name>, and <string-name>N. Triantafyllou</string-name></person-group>, <article-title>LIGO/Virgo black holes and dark matter: The effect of spatial clustering</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>11</issue> (<volume>2020</volume>) <page-range>036</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2020/11/036</pub-id></mixed-citation></ref>
  • <ref id="c46"><label>[46]</label><mixed-citation publication-type="journal"><object-id>46</object-id><person-group person-group-type="author"><string-name>J. M. Bardeen</string-name>, <string-name>J. R. Bond</string-name>, <string-name>N. Kaiser</string-name>, and <string-name>A. S. Szalay</string-name></person-group>, <article-title>The statistics of peaks of Gaussian random fields</article-title>, <source>Astrophys. J.</source> <volume>304</volume>, <page-range>15</page-range> (<year>1986</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/164143</pub-id></mixed-citation></ref>
  • <ref id="c47"><label>[47]</label><mixed-citation publication-type="journal"><object-id>47</object-id><person-group person-group-type="author"><string-name>V. Desjacques</string-name> and <string-name>A. Riotto</string-name></person-group>, <article-title>Spatial clustering of primordial black holes</article-title>, <source>Phys. Rev. D</source> <volume>98</volume>, <page-range>123533</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.98.123533</pub-id></mixed-citation></ref>
  • <ref id="c48"><label>[48]</label><mixed-citation publication-type="journal"><object-id>48</object-id><person-group person-group-type="author"><string-name>A. Moradinezhad Dizgah</string-name>, <string-name>G. Franciolini</string-name>, and <string-name>A. Riotto</string-name></person-group>, <article-title>Primordial black holes from broad spectra: Abundance and clustering</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>11</issue> (<volume>2019</volume>) <page-range>001</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2019/11/001</pub-id></mixed-citation></ref>
  • <ref id="c49"><label>[49]</label><mixed-citation publication-type="journal"><object-id>49</object-id><person-group person-group-type="author"><string-name>V. De Luca</string-name>, <string-name>V. Desjacques</string-name>, <string-name>G. Franciolini</string-name>, and <string-name>A. Riotto</string-name></person-group>, <article-title>The clustering evolution of primordial black holes</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>11</issue> (<volume>2020</volume>) <page-range>028</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2020/11/028</pub-id></mixed-citation></ref>
  • <ref id="c50"><label>[50]</label><mixed-citation publication-type="journal"><object-id>50</object-id><person-group person-group-type="author"><string-name>M. Sasaki</string-name>, <string-name>T. Suyama</string-name>, <string-name>T. Tanaka</string-name>, and <string-name>S. Yokoyama</string-name></person-group>, <article-title>Primordial black holes—perspectives in gravitational wave astronomy</article-title>, <source>Classical Quantum Gravity</source> <volume>35</volume>, <page-range>063001</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">CQGRDG</pub-id><issn>0264-9381</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1361-6382/aaa7b4</pub-id></mixed-citation></ref>
  • <ref id="c51"><label>[51]</label><mixed-citation publication-type="journal"><object-id>51</object-id><person-group person-group-type="author"><string-name>M. Shibata</string-name> and <string-name>M. Sasaki</string-name></person-group>, <article-title>Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity</article-title>, <source>Phys. Rev. D</source> <volume>60</volume>, <page-range>084002</page-range> (<year>1999</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>0556-2821</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.60.084002</pub-id></mixed-citation></ref>
  • <ref id="c52"><label>[52]</label><mixed-citation publication-type="journal"><object-id>52</object-id><person-group person-group-type="author"><string-name>I. Musco</string-name></person-group>, <article-title>Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations</article-title>, <source>Phys. Rev. D</source> <volume>100</volume>, <page-range>123524</page-range> (<year>2019</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.100.123524</pub-id></mixed-citation></ref>
  • <ref id="c53"><label>[53]</label><mixed-citation publication-type="journal"><object-id>53</object-id><person-group person-group-type="author"><string-name>K. Jedamzik</string-name> and <string-name>J. C. Niemeyer</string-name></person-group>, <article-title>Primordial black hole formation during first order phase transitions</article-title>, <source>Phys. Rev. D</source> <volume>59</volume>, <page-range>124014</page-range> (<year>1999</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>0556-2821</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.59.124014</pub-id></mixed-citation></ref>
  • <ref id="c54"><label>[54]</label><mixed-citation publication-type="journal"><object-id>54</object-id><person-group person-group-type="author"><string-name>I. Hawke</string-name> and <string-name>J. M. Stewart</string-name></person-group>, <article-title>The dynamics of primordial black-hole formation</article-title>, <source>Classical Quantum Gravity</source> <volume>19</volume>, <page-range>3687</page-range> (<year>2002</year>).<pub-id pub-id-type="coden">CQGRDG</pub-id><issn>0264-9381</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/0264-9381/19/14/310</pub-id></mixed-citation></ref>
  • <ref id="c55"><label>[55]</label><mixed-citation publication-type="journal"><object-id>55</object-id><person-group person-group-type="author"><string-name>T. Nakama</string-name>, <string-name>T. Harada</string-name>, <string-name>A. G. Polnarev</string-name>, and <string-name>J. Yokoyama</string-name></person-group>, <article-title>Identifying the most crucial parameters of the initial curvature profile for primordial black hole formation</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>01</issue> (<volume>2014</volume>) <page-range>037</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2014/01/037</pub-id></mixed-citation></ref>
  • <ref id="c56"><label>[56]</label><mixed-citation publication-type="journal"><object-id>56</object-id><person-group person-group-type="author"><string-name>A. Escrivà</string-name></person-group>, <article-title>Simulation of primordial black hole formation using pseudo-spectral methods</article-title>, <source>Phys. Dark Universe</source> <volume>27</volume>, <page-range>100466</page-range> (<year>2020</year>).<pub-id pub-id-type="coden">PDUHA3</pub-id><issn>2212-6864</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1016/j.dark.2020.100466</pub-id></mixed-citation></ref>
  • <ref id="c57"><label>[57]</label><mixed-citation id="c57a" publication-type="journal"><object-id>57a</object-id><person-group person-group-type="author"><string-name>T. Harada</string-name>, <string-name>C.-M. Yoo</string-name>, and <string-name>K. Kohri</string-name></person-group>, <article-title>Threshold of primordial black hole formation</article-title>, <source>Phys. Rev. D</source> <volume>88</volume>, <page-range>084051</page-range> (<year>2013</year>); <pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>1550-7998</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.88.084051</pub-id></mixed-citation><mixed-citation id="c57b" publication-type="journal" specific-use="authorjournal"><object-id>57b</object-id><person-group person-group-type="author"><string-name>T. Harada</string-name>, <string-name>C.-M. Yoo</string-name>, and <string-name>K. Kohri</string-name></person-group><source>Phys. Rev. D</source><volume>89</volume>, <page-range>029903(E)</page-range> (<year>2014</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>1550-7998</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.89.029903</pub-id></mixed-citation></ref>
  • <ref id="c57"><label>[57]</label><mixed-citation id="c57a" publication-type="journal"><object-id>57a</object-id><person-group person-group-type="author"><string-name>T. Harada</string-name>, <string-name>C.-M. Yoo</string-name>, and <string-name>K. Kohri</string-name></person-group>, <article-title>Threshold of primordial black hole formation</article-title>, <source>Phys. Rev. D</source> <volume>88</volume>, <page-range>084051</page-range> (<year>2013</year>); <pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>1550-7998</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.88.084051</pub-id></mixed-citation><mixed-citation id="c57b" publication-type="journal" specific-use="authorjournal"><object-id>57b</object-id><person-group person-group-type="author"><string-name>T. Harada</string-name>, <string-name>C.-M. Yoo</string-name>, and <string-name>K. Kohri</string-name></person-group><source>Phys. Rev. D</source><volume>89</volume>, <page-range>029903(E)</page-range> (<year>2014</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>1550-7998</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.89.029903</pub-id></mixed-citation></ref>
  • <ref id="c58"><label>[58]</label><mixed-citation publication-type="journal"><object-id>58</object-id><person-group person-group-type="author"><string-name>A. Escrivà</string-name>, <string-name>C. Germani</string-name>, and <string-name>R. K. Sheth</string-name></person-group>, <article-title>Universal threshold for primordial black hole formation</article-title>, <source>Phys. Rev. D</source> <volume>101</volume>, <page-range>044022</page-range> (<year>2020</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.101.044022</pub-id></mixed-citation></ref>
  • <ref id="c59"><label>[59]</label><mixed-citation publication-type="journal"><object-id>59</object-id><person-group person-group-type="author"><string-name>A. Escrivà</string-name>, <string-name>C. Germani</string-name>, and <string-name>R. K. Sheth</string-name></person-group>, <article-title>Analytical thresholds for black hole formation in general cosmological backgrounds</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>01</issue> (<volume>2021</volume>) <page-range>030</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2021/01/030</pub-id></mixed-citation></ref>
  • <ref id="c60"><label>[60]</label><mixed-citation publication-type="journal"><object-id>60</object-id><person-group person-group-type="author"><string-name>I. Musco</string-name>, <string-name>V. De Luca</string-name>, <string-name>G. Franciolini</string-name>, and <string-name>A. Riotto</string-name></person-group>, <article-title>Threshold for primordial black holes. II. A simple analytic prescription</article-title>, <source>Phys. Rev. D</source> <volume>103</volume>, <page-range>063538</page-range> (<year>2021</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.103.063538</pub-id></mixed-citation></ref>
  • <ref id="c61"><label>[61]</label><mixed-citation publication-type="journal"><object-id>61</object-id><person-group person-group-type="author"><string-name>A. Escrivà</string-name></person-group>, <article-title>PBH formation from spherically symmetric hydrodynamical perturbations: A review</article-title>, <source>Universe</source> <volume>8</volume>, <page-range>66</page-range> (<year>2022</year>).<issn>2218-1997</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.3390/universe8020066</pub-id></mixed-citation></ref>
  • <ref id="c62"><label>[62]</label><mixed-citation publication-type="journal"><object-id>62</object-id><person-group person-group-type="author"><string-name>J. A. Peacock</string-name> and <string-name>A. F. Heavens</string-name></person-group>, <article-title>Alternatives to the Press-Schechter cosmological mass function</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>243</volume>, <page-range>133</page-range> (<year>1990</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/243.1.133</pub-id></mixed-citation></ref>
  • <ref id="c63"><label>[63]</label><mixed-citation publication-type="journal"><object-id>63</object-id><person-group person-group-type="author"><string-name>R. G. Bower</string-name></person-group>, <article-title>The evolution of groups of galaxies in the Press-Schechter formalism</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>248</volume>, <page-range>332</page-range> (<year>1991</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/248.2.332</pub-id></mixed-citation></ref>
  • <ref id="c64"><label>[64]</label><mixed-citation publication-type="journal"><object-id>64</object-id><person-group person-group-type="author"><string-name>J. R. Bond</string-name>, <string-name>S. Cole</string-name>, <string-name>G. Efstathiou</string-name>, and <string-name>N. Kaiser</string-name></person-group>, <article-title>Excursion set mass functions for hierarchical Gaussian fluctuations</article-title>, <source>Astrophys. J.</source> <volume>379</volume>, <page-range>440</page-range> (<year>1991</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/170520</pub-id></mixed-citation></ref>
  • <ref id="c65"><label>[65]</label><mixed-citation publication-type="book"><object-id>65</object-id><person-group person-group-type="author"><string-name>S. Redner</string-name></person-group>, <source>A Guide to First-Passage Processes</source> (<publisher-name>Cambridge University Press</publisher-name>, Cambridge, England, <year>2001</year>).</mixed-citation></ref>
  • <ref id="c66"><label>[66]</label><mixed-citation publication-type="journal"><object-id>66</object-id><person-group person-group-type="author"><string-name>M. Maggiore</string-name> and <string-name>A. Riotto</string-name></person-group>, <article-title>The halo mass function from excursion set theory. I. Gaussian fluctuations with non-Markovian dependence on the smoothing scale</article-title>, <source>Astrophys. J.</source> <volume>711</volume>, <page-range>907</page-range> (<year>2010</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/0004-637X/711/2/907</pub-id></mixed-citation></ref>
  • <ref id="c67"><label>[67]</label><mixed-citation publication-type="journal"><object-id>67</object-id><person-group person-group-type="author"><string-name>P. Auclair</string-name> and <string-name>V. Vennin</string-name></person-group>, <article-title>Primordial black holes from metric preheating: Mass fraction in the excursion-set approach</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>02</issue> (<volume>2021</volume>) <page-range>038</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2021/02/038</pub-id></mixed-citation></ref>
  • <ref id="c68"><label>[68]</label><mixed-citation publication-type="journal"><object-id>68</object-id><person-group person-group-type="author"><string-name>M. Musso</string-name> and <string-name>R. K. Sheth</string-name></person-group>, <article-title>The importance of stepping up in the excursion set approach</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>438</volume>, <page-range>2683</page-range> (<year>2014</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/stt2387</pub-id></mixed-citation></ref>
  • <ref id="c69"><label>[69]</label><mixed-citation publication-type="journal"><object-id>69</object-id><person-group person-group-type="author"><string-name>F. Nikakhtar</string-name>, <string-name>M. Ayromlou</string-name>, <string-name>S. Baghram</string-name>, <string-name>S. Rahvar</string-name>, <string-name>M. R. Rahimi Tabar</string-name>, and <string-name>R. K. Sheth</string-name></person-group>, <article-title>The excursion set approach: Stratonovich approximation and Cholesky decomposition</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>478</volume>, <page-range>5296</page-range> (<year>2018</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/sty1415</pub-id></mixed-citation></ref>
  • <ref id="c70"><label>[70]</label><mixed-citation publication-type="journal"><object-id>70</object-id><person-group person-group-type="author"><string-name>P. F. Hopkins</string-name></person-group>, <article-title>An excursion-set model for the structure of GMCs and the ISM</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>423</volume>, <page-range>2016</page-range> (<year>2012</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1365-2966.2012.20730.x</pub-id></mixed-citation></ref>
  • <ref id="c71"><label>[71]</label><mixed-citation publication-type="journal"><object-id>71</object-id><person-group person-group-type="author"><string-name>P. F. Hopkins</string-name></person-group>, <article-title>The stellar IMF, core mass function, &amp; the last-crossing distribution</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>423</volume>, <page-range>2037</page-range> (<year>2012</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1365-2966.2012.20731.x</pub-id></mixed-citation></ref>
  • <ref id="c72"><label>[72]</label><mixed-citation publication-type="journal"><object-id>72</object-id><person-group person-group-type="author"><string-name>P. F. Hopkins</string-name></person-group>, <article-title>Why do stars form in clusters? An analytic model for stellar correlation functions</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>428</volume>, <page-range>1950</page-range> (<year>2013</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/sts147</pub-id></mixed-citation></ref>
  • <ref id="c73"><label>[73]</label><mixed-citation publication-type="book"><object-id>73</object-id><person-group person-group-type="author"><string-name>D. Higdon</string-name></person-group>, <article-title>Space and space-time modeling using process convolutions</article-title>, in <source>Quantitative Methods for Current Environmental Issues</source> (<publisher-name>Springer</publisher-name>, New York, <year>2002</year>), pp. <page-range>37–56</page-range>.</mixed-citation></ref>
  • <ref id="c74"><label>[74]</label><mixed-citation publication-type="journal"><object-id>74</object-id><person-group person-group-type="author"><string-name>M. G. Genton</string-name></person-group>, <article-title>Classes of kernels for machine learning: A statistics perspective</article-title>, <source>J. Mach. Learn. Res.</source> <volume>2</volume>, <page-range>299</page-range> (<year>2001</year>).<issn>1532-4435</issn></mixed-citation></ref>
  • <ref id="c75"><label>[75]</label><mixed-citation publication-type="journal"><object-id>75</object-id><person-group person-group-type="author"><string-name>P. Auclair</string-name>, <string-name>C. Caprini</string-name>, <string-name>D. Cutting</string-name>, <string-name>M. Hindmarsh</string-name>, <string-name>K. Rummukainen</string-name>, <string-name>D. A. Steer</string-name>, and <string-name>D. J. Weir</string-name></person-group>, <article-title>Generation of gravitational waves from freely decaying turbulence</article-title>, <source>J. Cosmol. Astropart. Phys.</source> <issue>09</issue> (<volume>2022</volume>) <page-range>029</page-range>.<pub-id pub-id-type="coden">JCAPBP</pub-id><issn>1475-7516</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/1475-7516/2022/09/029</pub-id></mixed-citation></ref>
  • <ref id="c76"><label>[76]</label><mixed-citation publication-type="journal"><object-id>76</object-id><person-group person-group-type="author"><string-name>D. Inman</string-name> and <string-name>Y. Ali-Haïmoud</string-name></person-group>, <article-title>Early structure formation in primordial black hole cosmologies</article-title>, <source>Phys. Rev. D</source> <volume>100</volume>, <page-range>083528</page-range> (<year>2019</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>2470-0010</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.100.083528</pub-id></mixed-citation></ref>
  • <ref id="c77"><label>[77]</label><mixed-citation publication-type="journal"><object-id>77</object-id><person-group person-group-type="author"><string-name>P. Meszaros</string-name></person-group>, <article-title>The behaviour of point masses in an expanding cosmological substratum</article-title>, <source>Astron. Astrophys.</source> <volume>37</volume>, <page-range>225</page-range> (<year>1974</year>).<pub-id pub-id-type="coden">AAEJAF</pub-id><issn>0004-6361</issn></mixed-citation></ref>
  • <ref id="c78"><label>[78]</label><mixed-citation publication-type="journal"><object-id>78</object-id><person-group person-group-type="author"><string-name>B. J. Carr</string-name></person-group>, <article-title>The statistical clustering of primordial black holes</article-title>, <source>Astron. Astrophys.</source> <volume>56</volume>, <page-range>377</page-range> (<year>1977</year>).<pub-id pub-id-type="coden">AAEJAF</pub-id><issn>0004-6361</issn></mixed-citation></ref>
  • <ref id="c79"><label>[79]</label><mixed-citation publication-type="journal"><object-id>79</object-id><person-group person-group-type="author"><string-name>K. Freese</string-name>, <string-name>R. Price</string-name>, and <string-name>D. N. Schramm</string-name></person-group>, <article-title>Formation of population III stars and galaxies with primordial planetary-mass black holes</article-title>, <source>Astrophys. J.</source> <volume>275</volume>, <page-range>405</page-range> (<year>1983</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/161542</pub-id></mixed-citation></ref>
  • <ref id="c80"><label>[80]</label><mixed-citation publication-type="journal"><object-id>80</object-id><person-group person-group-type="author"><string-name>B. J. Carr</string-name> and <string-name>J. Silk</string-name></person-group>, <article-title>Can graininess in the early universe make galaxies?</article-title>, <source>Astrophys. J.</source> <volume>268</volume>, <page-range>1</page-range> (<year>1983</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/160924</pub-id></mixed-citation></ref>
  • <ref id="c81"><label>[81]</label><mixed-citation publication-type="journal"><object-id>81</object-id><person-group person-group-type="author"><string-name>N. Afshordi</string-name>, <string-name>P. McDonald</string-name>, and <string-name>D. N. Spergel</string-name></person-group>, <article-title>Primordial black holes as dark matter: The power spectrum and evaporation of early structures</article-title>, <source>Astrophys. J. Lett.</source> <volume>594</volume>, <page-range>L71</page-range> (<year>2003</year>).<pub-id pub-id-type="coden">AJLEEY</pub-id><issn>2041-8213</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/378763</pub-id></mixed-citation></ref>
  • <ref id="c82"><label>[82]</label><mixed-citation publication-type="journal"><object-id>82</object-id><person-group person-group-type="author"><string-name>D. N. Limber</string-name></person-group>, <article-title>Analysis of counts of the extragalactic nebulae in terms of a fluctuating density field. III</article-title>, <source>Astrophys. J.</source> <volume>125</volume>, <page-range>9</page-range> (<year>1957</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1086/146280</pub-id></mixed-citation></ref>
  • <ref id="c83"><label>[83]</label><mixed-citation publication-type="journal"><object-id>83</object-id><person-group person-group-type="author"><string-name>J. Mercer</string-name></person-group>, <article-title>XVI. functions of positive and negative type, and their connection the theory of integral equations</article-title>, <source>Phil. Trans. R. Soc. A</source> <volume>209</volume>, <page-range>415</page-range> (<year>1909</year>).<pub-id pub-id-type="coden">PTRMAD</pub-id><issn>1364-503X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1098/rsta.1909.0016</pub-id></mixed-citation></ref>
  • <ref id="c84"><label>[84]</label><mixed-citation publication-type="journal"><object-id>84</object-id><person-group person-group-type="author"><string-name>C. J. Paciorek</string-name> and <string-name>M. J. Schervish</string-name></person-group>, <article-title>Spatial modelling using a new class of nonstationary covariance functions</article-title>, <source>Environmetrics</source> <volume>17</volume>, <page-range>483</page-range> (<year>2006</year>).<pub-id pub-id-type="coden">ENVCEE</pub-id><issn>1099-095X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1002/env.785</pub-id></mixed-citation></ref>
  • <ref id="c85"><label>[85]</label><mixed-citation publication-type="book"><object-id>85</object-id><person-group person-group-type="author"><string-name>I. S. Gradshteyn</string-name> and <string-name>I. M. Ryzhik</string-name></person-group>, <source>Table of Integrals, Series, and Products</source> (<publisher-name>Elsevier/Academic Press</publisher-name>, Amsterdam, <year>2007</year>).</mixed-citation></ref>
  • <ref id="c86"><label>[86]</label><mixed-citation publication-type="journal"><object-id>86</object-id><person-group person-group-type="author"><string-name>M. Maggiore</string-name> and <string-name>A. Riotto</string-name></person-group>, <article-title>The halo mass function from excursion set theory. II. The diffusing barrier</article-title>, <source>Astrophys. J.</source> <volume>717</volume>, <page-range>515</page-range> (<year>2010</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/0004-637X/717/1/515</pub-id></mixed-citation></ref>
  • <ref id="c87"><label>[87]</label><mixed-citation publication-type="journal"><object-id>87</object-id><person-group person-group-type="author"><string-name>M. Maggiore</string-name> and <string-name>A. Riotto</string-name></person-group>, <article-title>The halo mass function from excursion set theory. III. Non-Gaussian fluctuations</article-title>, <source>Astrophys. J.</source> <volume>717</volume>, <page-range>526</page-range> (<year>2010</year>).<pub-id pub-id-type="coden">ASJOAB</pub-id><issn>0004-637X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1088/0004-637X/717/1/526</pub-id></mixed-citation></ref>
  • <ref id="c88"><label>[88]</label><mixed-citation publication-type="journal"><object-id>88</object-id><person-group person-group-type="author"><string-name>A. De Simone</string-name>, <string-name>M. Maggiore</string-name>, and <string-name>A. Riotto</string-name></person-group>, <article-title>Conditional probabilities in the excursion set theory. Generic barriers and non-Gaussian initial conditions</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>418</volume>, <page-range>2403</page-range> (<year>2011</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1365-2966.2011.19638.x</pub-id></mixed-citation></ref>
  • <ref id="c89"><label>[89]</label><mixed-citation publication-type="journal"><object-id>89</object-id><person-group person-group-type="author"><string-name>M. Musso</string-name> and <string-name>A. Paranjape</string-name></person-group>, <article-title>Non-Gaussian halo abundances in the excursion set approach with correlated steps</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>420</volume>, <page-range>369</page-range> (<year>2012</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1365-2966.2011.20040.x</pub-id></mixed-citation></ref>
  • <ref id="c90"><label>[90]</label><mixed-citation publication-type="journal"><object-id>90</object-id><person-group person-group-type="author"><string-name>M. Musso</string-name>, <string-name>A. Paranjape</string-name>, and <string-name>R. K. Sheth</string-name></person-group>, <article-title>Scale dependent halo bias in the excursion set approach</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>427</volume>, <page-range>3145</page-range> (<year>2012</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1365-2966.2012.21903.x</pub-id></mixed-citation></ref>
  • <ref id="c91"><label>[91]</label><mixed-citation publication-type="journal"><object-id>91</object-id><person-group person-group-type="author"><string-name>M. Musso</string-name> and <string-name>R. K. Sheth</string-name></person-group>, <article-title>One step beyond: The excursion set approach with correlated steps</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>423</volume>, <page-range>L102</page-range> (<year>2012</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1111/j.1745-3933.2012.01266.x</pub-id></mixed-citation></ref>
  • <ref id="c92"><label>[92]</label><mixed-citation publication-type="journal"><object-id>92</object-id><person-group person-group-type="author"><string-name>M. Musso</string-name> and <string-name>R. K. Sheth</string-name></person-group>, <article-title>The excursion set approach in non-Gaussian random fields</article-title>, <source>Mon. Not. R. Astron. Soc.</source> <volume>439</volume>, <page-range>3051</page-range> (<year>2014</year>).<pub-id pub-id-type="coden">MNRAA4</pub-id><issn>0035-8711</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1093/mnras/stu165</pub-id></mixed-citation></ref>
  • <ref id="c93"><label>[93]</label><mixed-citation publication-type="journal"><object-id>93</object-id><person-group person-group-type="author"><string-name>A. A. Starobinsky</string-name></person-group>, <article-title>Stochastic de Sitter (inflationary) stage in the early universe</article-title>, <source>Lect. Notes Phys.</source> <volume>246</volume>, <page-range>107</page-range> (<year>1986</year>).<pub-id pub-id-type="coden">LNPHA4</pub-id><issn>0075-8450</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1007/3-540-16452-9_6</pub-id></mixed-citation></ref>
  • <ref id="c94"><label>[94]</label><mixed-citation publication-type="journal"><object-id>94</object-id><person-group person-group-type="author"><string-name>A. S. Goncharov</string-name>, <string-name>A. D. Linde</string-name>, and <string-name>V. F. Mukhanov</string-name></person-group>, <article-title>The global structure of the inflationary universe</article-title>, <source>Int. J. Mod. Phys. A</source> <volume>02</volume>, <page-range>561</page-range> (<year>1987</year>).<pub-id pub-id-type="coden">IMPAEF</pub-id><issn>0217-751X</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1142/S0217751X87000211</pub-id></mixed-citation></ref>
  • <ref id="c95"><label>[95]</label><mixed-citation publication-type="journal"><object-id>95</object-id><person-group person-group-type="author"><string-name>A. A. Starobinsky</string-name> and <string-name>J. Yokoyama</string-name></person-group>, <article-title>Equilibrium state of a self-interacting scalar field in the de Sitter background</article-title>, <source>Phys. Rev. D</source> <volume>50</volume>, <page-range>6357</page-range> (<year>1994</year>).<pub-id pub-id-type="coden">PRVDAQ</pub-id><issn>0556-2821</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1103/PhysRevD.50.6357</pub-id></mixed-citation></ref>
  • <ref id="c96"><label>[96]</label><mixed-citation publication-type="journal"><object-id>96</object-id><person-group person-group-type="author"><string-name>V. Vennin</string-name> and <string-name>A. A. Starobinsky</string-name></person-group>, <article-title>Correlation functions in stochastic inflation</article-title>, <source>Eur. Phys. J. C</source> <volume>75</volume>, <page-range>413</page-range> (<year>2015</year>).<pub-id pub-id-type="coden">EPCFFB</pub-id><issn>1434-6044</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1140/epjc/s10052-015-3643-y</pub-id></mixed-citation></ref>
  • <ref id="c97"><label>[97]</label><mixed-citation publication-type="journal"><object-id>97</object-id><person-group person-group-type="author"><string-name>J. Grain</string-name> and <string-name>V. Vennin</string-name></person-group>, <article-title>Unavoidable shear from quantum fluctuations in contracting cosmologies</article-title>, <source>Eur. Phys. J. C</source> <volume>81</volume>, <page-range>132</page-range> (<year>2021</year>).<pub-id pub-id-type="coden">EPCFFB</pub-id><issn>1434-6044</issn><pub-id pub-id-type="doi" specific-use="suppress-display">10.1140/epjc/s10052-021-08932-0</pub-id></mixed-citation></ref>