primordial black holes as dark matter Caption: "The formation of the observable universe without and with primordial black holes (PBHs)." (Slightly edited.)

    Features:

    1. PBHs are a possible form of dark matter. However, if they are a significant part of all dark matter, they must have formed before the Big Bang nucleosynthesis era (cosmic time ∼ 10--1200 s ≅ 0.17--20 m) during the Big Bang since Big Bang nucleosynthesis (BBN) rules out that they could have formed during over after the Big Bang nucleosynthesis era (cosmic time ∼ 10--1200 s ≅ 0.17--20 m).

    2. We will NOT explicate Image 1 here, but we do list some keywords:

        Primordial black hole keywords (i.e., primordial black holes (PBHs) keywords): asteroid window for primordial black hole masses (∼ 10**17--10**23 g), Big Bang, Big Bang nucleosynthesis (BBN), Big Bang nucleosynthesis era (cosmic time ∼ 10--1200 s ≅ 0.17--20 m), dark ages (∼ 377 kyr to ∼< 200 Myr), dark matter, dark matter halos, galaxy, galaxy formation and evolution, James Webb Space Telescope (JWST, 2021--2041?), large-scale structure of the universe, Laser Interferometer Space Antenna (LISA, c.2037--?), observable universe, Population III stars (i.e., first stars), primordial black holes (PBHs), recombination era t = 377,770(3200) y, structure formation (AKA large-scale structure formation), Wikipedia: Chronology of the universe, etc.

    3. We will elaborate on the keyword asteroid window for primordial black hole masses (∼ 10**17--10**23 g) which has NOTHING to do with asteroids, except they have masses comparable to PBHs that may make up the dark matter (see the next item).

    4. Various observations, but primarily of gravitational microlensing of stars by various possible astronomical objects (i.e., black holes, brown dwarfs, dim stars, neutron stars, white dwarfs), indicate that if PBHs make up all or nearly all dark matter (which BBN rules CANNOT be baryonic matter), their masses must lie in the asteroid window for primordial black hole masses (∼ 10**17--10**23 g) (see, e.g., Auffinger 2022; Wikipedia: Primordial black holes: Observational limits and detection strategies).

      Vesta and other asteroid collage

    5. Image 2 Caption: A collage image showing the comparative sizes of 9 asteroids. For more explication of Image 2, see Asteroid file: asteroid_collage.html.

      In order to make the asteroid window for primordial black hole masses (∼ 10**17--10**23 g) somewhat intelligible, we consider the masses of asteroids below with the masses of Ceres (NOT shown in Image 2), Vesta, and Itokawa as examples.

    6. It may seem odd that we CANNOT tell whether dark matter is exotic hypothetical particles (i.e., microscopic scale particles that are NOT baryonic matter and interact with each and baryonic matter very weakly, except via gravity) or macroscopic scale primordial black holes (PBHs) in asteroid window for primordial black hole masses (∼ 10**17--10**23 g). But we CANNOT. All we see is the gravity effect of some sort of zero (or nearly zero) pressure matter entities (that are NOT baryonic matter) that clumps to form dark matter halos. There is NO limits on the masses of the exotic hypothetical particles and only the asteroid window for primordial black hole masses (∼ 10**17--10**23 g) limits the masses of PBHs.

    7. To make the asteroid window for primordial black hole masses (∼ 10**17--10**23 g) somewhat intelligible, we can write a fiducial value formula for asteroid mass:

      M_asteroid = [(4π/3)*R**3]*ρ = [(1.57 ...)*10**21 g]*[(D/100 km)**3]*(ρ/3 g/cm**3) ,

      where R is mean radius, D is mean diameter, ρ is mean density, 100 km is a fiducial value for an asteroid mean diameter, and 3 g/cm**3 is a fiducial value for a rocky asteroid mean density.

      For rocky asteroids, the mean density does NOT vary much from 3 g/cm**3. For rocky-icy asteroids with ∼ 50% rock and ∼ 50% water ice by mass, the mean density does NOT vary much from 2 g/cm**3.

      The mean diameters and hence the masses of asteroid do vary widely. Table: Asteroid Mass below gives some example asteroid masses from established measurements (the M values) and those calculated from the fiducial value formula (the M_fid values).

      We see that the fiducial value formula is better than factor of 2 accurate when compared to the masses of real asteroids. The main reason for the disagreement between the M values and M_fid values is density. Recall, all M_fid values were calculated with the fiducial value ρ = 3 g/cm**3. To explicate:

      1. Ceres has actual mean density 2.162(8) g/cm**3. This low value is probably because Ceres is 1/4 water ice by mass (see Wikipedia: Ceres: Geology).
      2. Vesta has actual mean density 3.456(35) g/cm**3. This value a bit higher than our fiducial value just because Vesta has somewhat denser rock on average than indicated by our fiducial value.
      3. Itokawa has actual mean density 1.95(14) g/cm**3. This low value is primarily because Itokawa is a rubble pile asteroid which means there are cavities in its interior (see Wikipedia: 25143 Itokawa: Physical characteristics).

      We also see that our fiducial values for mean diameter and mean density give a fiducial value mass 1.57*10**21 g that is larger than the logarithmic mean mass of the asteroid window for primordial black hole masses (∼ 10**17--10**23 g) which is (17+23)/2 = 20 (i.e., in antilogarithm form mass 10**20 g) by more than 1 dex (i.e., in antilogarithm form by more than a factor of 10).

        Table: Asteroid Mass 
           Aste-  Name        Year of   Mean     Mass M            Mass M_fid from the fiducial    Ratio
   roid               Discov-   Diameter                   value formula with fiducial     Mass_fid 
   No.                ery       (km)     (g)               density ρ = 3 g/cm**3: (g)  /Mass

        
     1    Ceres       1801      939.4    9.3835*10**23     1.30*10**24                     1.39 
     4    Vesta       1807      525.4    2.59076*10**23    2.28*10**23                     0.879
   ...    M_upper     ...       399.3    ...               1.0*10**23
   ...    D = 100 km  ...       100.0    ...               1.57*10**21
   ...    M_lower     ...         3.993  ...               1.0*10**17    
 25143    Itokawa     1998        0.330  3.35*10**13       5.64*0**13                      1.59
   ...    D = 1 m     ...         0.001  ...               1.57*10**7
        Notes:
        1. Reference: Wikipedia: List of exceptional asteroids: Largest asteroids by diameter; Wikipedia: List of exceptional asteroids: Spacecraft targets).
        2. 1 Ceres ⚳ is actually oblate because of the centrifugal force caused by its relatively fast rotation with rotation period 9.074170(2) hours. From accurate/precise measurements, Ceres has mean equatorial diameter 963.2 km and polar diameter 891.2 km.
        3. 4 Vesta ⚶ is approximately triaxial with dimensions: 572.6 x 557.2 x 446.4 km. Thus, even bodies larger than 300 km in size scale do NOT have to be exactly spherical or oblate spherical. The centrifugal force and rigid body forces can partially withstand gravity for Vesta-size astronomical objects.
        4. M_upper has the upper limit mass of the asteroid window for primordial black hole masses (∼ 10**17--10**23 g).
        5. D = 100 km has the fiducial value diameter.
        6. M_lower has the lower limit mass of the asteroid window for primordial black hole masses (∼ 10**17--10**23 g).
        7. 25143_Itokawa: An example of relatively small asteroid. It is a rubble pile asteroid.
        8. D = 1 m is has the diameter of the smallest size asteroid by definition (see Wikipedia: Asteroid).

    Images:
    1. Credit/Permission: © European Space Agency (ESA), 2021 (uploaded to Wikimedia Commons by User:Dabed, 2021) / Creative Commons CC BY-SA 3.0.
      Image link: Wikimedia Commons: File:PBHs-formation.png.
    2. Credit/Permission: NASA / JPL-Caltech / JAXA / ESA, 2011 (uploaded to Wikipedia by User:Eumeldingens, 2011) / Public domain.
      Image link: Wikipedia: File:Asteroidsscale.jpg.
    Local file: local link: black_hole_primordial.html.
    File: Black hole file: black_hole_primordial.html.