Angular Momentum of Dark Matter Black Holes

The putative black holes which may constitute all the dark matter are described by a Kerr metric with only two parameters, mass M and angular momentum J. There has been little discussion of J since it plays no role in the upcoming attempt at detection by microlensing. Nevertheless J does play a central role in understanding the previous lack of detection, especially of CMB distortion. We explain why bounds previously derived from lack of CMB distortion are too strong for primordial black holes with J non-vanishing. Almost none of the dark matter black holes can be from stellar collapse, and nearly all are primordial, to avoid excessive CMB distortion.


Introduction
The Milky Way galaxy in which we reside lies within a large approximately spherical halo of dark matter (DM) which does not experience the strong or electromagnetic interactions, nor as we shall assume here the weak interactions. The popular idea that the dark matter constituent is a WIMP with weak interactions was born out of supersymmetry which lacks any support from extensive LHC data on pp scattering which probed the energy regime where signs of SUSY were most expected. Die-hard SUSY theorists may still have hope, but it is not premature to entertain the assumption that the WIMP does not exist.
With no WIMP one is led to astrophysical MACHOs and then confronted with the constraint from BBN that no more than 20% of the DM can be baryonic. This means that to make 100% of the DM we cannot use compact objects such as white dwarfs, neutron stars, brown dwarfs and unassociated planets. Nor is it possible to use black holes which are the result of gravitational collapse of baryonic stars.
There is, however, a second type of black hole which is formed primordially (PBH) during the radiation era. To form 100% of the DM we must therefore use PBHs. Since the resultant black holes of the two types are indistinguishable, can we use, say, 20% of the gravitational collapse variety and 80% of PBHs? The surprising answer is no. One result of the present paper is that if there is such a mixture of black holes the vast majority, well over 99%, must be PBHs and only a tiny fraction can be the result of gravitational collapse. This strong result comes from a study of X-rays and associated CMB distortion.
Focusing on the Milky Way halo where we can most easily detect the PBHs, we already know from earlier searches, especially [1], that MACHOs with masses M ≤ 20M ⊙ can make up no more that 10% of the halo dark matter. At the high mass end, we know from [2] that MACHOs with M ≥ 10 5 M ⊙ endanger the disk stability. For the Milky Way halo therefore one is led to consider intermediate mass (IM) black holes PIMBHs in the mass range for the DM constituents. This leads to a plum pudding model [3] for the Milky Way halo, named after Thomson's atomic model [4], where for the DM halo the plums are PIMBHs with masses satisfying Eq.(1) and the pudding in this case is rarefied gas, dust and just a few luminous stars.
The formation of PBHs with masses as large as Eq.(1) and much larger is known to be mathematically possible during the radiation era. An existence theorem is provided by hybrid inflationary models [5]. One specific prediction of hybrid inflation is a sharplypeaked PBH mass function. If we need a specific PIMBH mass, we shall use a calligraphic PIMBH defined by M PIMBH ≡ 100M ⊙ exactly. This is merely an example and extension to the whole range of Eq.(1) can also be discussed.
The cosmic time t P BH at which a PBH is formed has been estimated [6] to be so that the PIMBHs in Eq.(1) are formed in the time window 0.0002s ≤ t P IM BH ≤ 1.0s with the special case t PIMBH ≃ 0.001s. In terms of red shift (Z), this corresponds to with the special case Z PIMBH ≃ 2 × 10 11 . The formation of BHs which are not primordial, which we shall denote without an initial P or P, necessarily occurs after star formation which conservatively occurs certainly only for very different redshifts satisfying The sharp difference in the red-shifts of Eq.(3) and Eq.(4) will become important when we discuss the reasons for previous non-detection, the angular momentum of PIMBHs and BHs, and the central issue of possible CMB distortion by X-rays.
As already mentioned, by using the mathematical models in [5], it is possible to form PBHs not only in the PIMBH mass range of Eq. (1) but also Primordial Super Massive Black Hole (PSMBHs) in the mass range where the upper limit derives from the formation time t P SM BH given by Eq. (2) staying within the radiation-dominated era. We shall discuss the higher mass range Eq( 5) later in the paper.
Finally for this Introduction, we recall that in a microlensing experiment, e.g. using the LMC or SMC for convenient sources, microlensing by halo PIMBHs, and assuming a typical transit velocity 200km.s −1 , the time duration of the microlensing light curve can be estimated [7] to be approximately which we note is close to one year and two years, respectively, for lens masses 25M ⊙ and 100M ⊙ . For reference, the highest duration such light curve detected by the MACHO Collaboration which published in the year 2000 [1] corresponded to M P IM BH ≃ 20M ⊙ .
In this paper, we shall need only order-of-magnitude estimates for the rotational period τ and, in the next Section, for the angular momentum J. These will suffice to make our point about concomitant X-ray emission. The solution is axially symmetric and the radius at the pole θ = π 2 is the same as the Schwarzschild radius R = 2M. For other values of θ the black hole radius is smaller than the static one and the rest of the static would-be sphere is filled out by an ergosphere whose equatorial radius is also R = 2M.
To proceed with our estimate we shall take the equatorial velocity of the ergosphere to have magnitude V = 0.1c and use Newtonian mechanics to estimate the rotation period τ as simply For the Sun, we have 2M ⊙ ≃ 3 km so that for a black hole of mass M = ηM ⊙ and therefore radius R ≃ 3η km Eq. (8) is, for V = 0.1c = 3 × 10 4 km.s −1 , Some values of τ , estimated by this method, are shown in the third column of our Table. 3 Angular Momentum J Let us define the dimensionless angular momentun J ≡ J/kg.km 2 .sec −1 . We are interested in order of magnitude estimates of J for the PIMBHs and PSMBHs. The value of J for astrophysical objects is necessarily a large number so to set the scene we shall estimate J for the Earth J ⊕ and for the Sun J ⊙ . The parameters for the Earth are radius R ⊕ ≃ 6300km, period τ ⊕ ≃ 86400s, mass M ⊕ ≃ 6 × 10 24 kg, hence angular velocity ω ⊕ = 2π/τ ⊕ and moment of inertia For the Sun the similar calculation using For the black holes, the value of J is proportional to η 2 where η = (M/M ⊙ ). A similar estimate to that for the Earth and Sun gives J ≃ 7.2×10 34 η 2 , which provides the remaining entries in our Table.

CMB Distortion
Because of rotational invariance, angular momentum is conserved. The J of a compact astrophysical object will not change dramatically unless there is an extremely unlikely event like a major collision. For example, the Earth and the Sun in the first two rows of our In detecting the dark matter, let us focus on the special case PIMBH with M = 100M ⊙ . The PIMBH was formed, accordng to Eq.(2), at time t = 10 −3 s and rotates with period t ≃ 63ms, thus rotating ∼ 16 times per second and with an absolute angular momentum ∼ 6 × 10 11 times that of the Earth and ∼ 600 times that of the Sun. There is no known reason that J PIMBH would change significantly after its formation.
These remarks about angular momentum are salient to resolving the contradiction between the dark matter proposal in [3] and the limits on halo MACHOs derived earlier by Ricotti, Ostriker and Mack (ROM) in [9] on the basis of X-ray emission and CMB distortion.
In [3] the proposal was made that the Milky Way dark halo is a plum pudding with, as "plums", PIMBHs in the mass range of Eq.(1) making up 100% of the dark matter. On the other hand, in Figure 9 of ROM [9], there is displayed an upper limit of less than 0.01% of the dark matter for this mass range of MACHO. Thus, it would seem that at least one of [3] and [9] must be incorrect? The conclusion of the present paper is that ROM [9] is correct for stellar-collapse black holes but is not applicable to the model of [3] which employs primordial black holes. This issue was discussed also in [10].
This ROM upper limit arises from the lack of any observed departure of the CMB spectrum from the predicted black-body curve or of any CMB anisotropy. ROM calculated the accretion of matter on to the MACHOs, the emission of X-rays by the accreted matter and then the downgrading of these X-rays to microwaves by cosmic expansion and more importantly by Compton scattering from electrons.
A crucial assumption made by ROM [9] is that the accretion on to the MACHO can be modeled as if the MACHO has zero angular momentum J = 0. The justification for this assumption is based on earlier work by Loeb [11] who studied the collapse of gas clouds at redshifts 200 ≤ Z ≤ 1400. Such collapse can form compact objects, eventually black holes, but during the collapse angular momentum is damped out from the electrons by Compton scattering with the CMB.
From Loeb's discussion, the resultant black holes will have J = 0 and this appears to underly why ROM [9] used the Bondi-Hoyle model [12] which presumes spherical symmetry for accretion. This is justified for stellar-collapse black holes by the arguments of Loeb [11] and therefore the upper bounds derived by ROM are applicable.
There is evidence that the Bondi-Hoyle model of accretion is not, by contrast, applicable to spinning PSMBHs, in particular the one at the centre of the large galaxy M87. In [13,14] Bondi-Hoyle [12] is used to calculate the number of X-rays expected from the accreted material near M87. In the case of M87 the X-rays are experimentally measured. The conclusion is striking: that the measured X-rays are less by several orders of magnitude than predicted by Bondi-Hoyle theory.
This supports the idea that the SMBHs such as that in M87 are primordial, so we list PSMBH(M87) in the final row of our Table. The ROM constraints apply to black holes which originate from gravity collapse of baryonic stars. Collecting this fact, together with the ROM limit of ≤ 10 −4 of the dark matter for MACHOs, implies that 99.99% of the dark matter black holes are primordial, formed during the radiation era.

Discussion
The plum pudding model for the dark halo proposed in [3] arose from a confluence of theoretical threads including study of the entropy of the universe and the knowledge of how to form PBHs with many solar masses as in Eqs. (1) and (5). Nevertheless it was the weakening of the argument for WIMPs which was most decisive, The strongest objection to the MACHOs in [3] has been based on the X-rays and the CMB distortion as calculated by ROM [9]. In the present paper we have attempted to lay this criticism to rest by noting that ROM assumed J = 0 and that the putative PIMBHs have not only many times the Solar mass but also many times the Solar angular momentum. This appears to us to render the ROM constraints inapplicable to the PIMBHs. On the other hand, they do apply to stellar-collapse black holes which implies that almost none (≤ 0.01%) of the dark matter black holes are of that type. To decide whether dark matter really is PIMBHs will require their detection by a dedicated microlensing experiment.
Examples of PSMBHs may already have been observed in galactic cores and quasars. Other PSMBHs can play the role of dark matter in clusters and may well be detectable by other future lensing experiments. There is also the upper mass range contained in Eq. (5). Although masses of PSMBHs up to a few times 10 10 M ⊙ may have already been observed in quasars, there are what could be called Primordial Ultra Massive Black Holes (PUMBHs) with masses between 10 11 and 10 17 solar masses which might exist within the visible universe.