Astrophysical implications of hypothetical stable TeVscale black holes
I. INTRODUCTION
One of the most spectacular manifestations of nature realizing certain extradimensional scenarios [1–4] could be the production of microscopic black holes at the LHC [5,6].^{1} These are expected to undergo prompt, quasithermal, Hawking [9] decay to large multiplicities of elementary particles, leading to very characteristic final states. It has been suggested [10,11], however, that black hole decay via Hawking radiation may not be a universal feature, and could, for example, depend on the details of the Planckscale degrees of freedom. While this suggestion is not based on any complete microphysical picture, and furthermore appears contradictory to basic quantummechanical principles^{2}, it does raise a possible question about stability of microscopic black holes that might be produced at the LHC, in TeVscale gravity scenarios. This in turn has led some to express concern about the fate of their evolution: could their accretion pose any threat to the Earth? This is the question addressed in this paper.
The structure of this work, and a succinct summary of our findings, are outlined here. We begin our work by reviewing what are widely regarded as quite convincing arguments for the robustness of the exceedingly rapid decay of microscopic black holes. We then discuss the arguments asserting that macroscopic consequences of new and unknown particles can be ruled out by the lack of evidence for their effects from production by cosmic rays hitting the surface of the Earth or other astronomical bodies. We argue here that charged black holes will lose enough energy to stop when traversing the Earth or the Sun, via standard electromagnetic processes. Since black holes would be typically produced by the collision of quark pairs, whether in cosmicray interactions or at the LHC, they would often be initially charged. To the extent that no mechanism leads to their neutralization, the cosmicray based argument for their being harmless is therefore robust. Their neutralization through the Schwinger mechanism proceeds according to quantum principles like those underlying Hawking radiation. There is therefore no concrete framework where neutralization occurs without Hawking decay taking place as well, leading to a likely contradiction in assuming that stable black holes must be neutral. We nonetheless make the hypothesis that this odd situation could occur, and analyze the possible effects of such neutral and stable black holes, beginning with a review of some essential features of gravity and black holes in $D>4$ dimensions, including both large and warpedextradimensions models.
Next, we develop the formalism to describe the evolution of such black holes trapped inside the Earth or inside dense objects such as white dwarfs and neutron stars. We introduce and discuss accretion scenarios that apply within nuclear, atomic, and macroscopic matter, evaluating the time scales corresponding to various phases in the evolution of a growing black hole. We establish upper and lower limits to the rate at which accretion can take place, building on very basic principles such as conservation laws and classical and quantum dynamics. This is possible since, in order for accretion to become macroscopic, it is necessary that a black hole exerts its influence over distance scales much larger than its event horizon. Such black holes are very small, and their accretion power, if limited to absorbing particles that have impact parameters of the order of the Schwarzschild radius, is typically insufficient to cause macroscopic growth. At large distances the physical processes become independent of the shortdistance properties of the black hole, which only acts through its attractive potential, and as a mass sink. In this longdistance domain, classical or quantum dynamics are well tested, making the study of accretion possible, independently of our detailed knowledge or ignorance about the physics inside the black hole.
We investigate two concrete scenarios, for various configurations of the extra dimensions: growth of hypothetical LHCproduced stable black holes within the Earth, and growth of such black holes produced by cosmic rays impinging on dense stars, such as white dwarfs or neutron stars. In both cases, we make conservative assumptions, namely, fastest possible growth in Earth, and slowest possible growth in a dense star. The first scenario shows that, if the radius of crossover from higherdimensional gravity to fourdimensional gravity is less than about $200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, the natural lifetime of the solar system is too short to allow significant growth of stable black holes that may be captured inside the Earth. In contrast, in a scenario where the crossover radius exceeds the $200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ scale, accretion times could be shorter than the solar time scale. In this case, however, examination of the latter densestar scenario then produces an argument that, given observational data setting the lifetimes of such objects at ${10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ or more, such stable black holes cannot in fact exist for a crossover radius greater than $\simeq 15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $.
These arguments thus conclude with the exclusion of any relevant consequences for Earth due to the evolution of black holes possibly produced by the LHC.
While the main line of the argument is relatively straightforward, being based on wellestablished macroscopic properties of matter, several issues require an indepth study in order to provide a robust basis to our conclusions. When studying accretion, for example, we need to consider the possibility of effects such as the Eddington limit, where radiation emitted during the infall of matter would slow down the accretion via radiative pressure, constraining the accretion rate of a black hole and thus possibly slowing down its growth inside Earth or a dense star. The estimates of the number of black holes produced by cosmic rays need to rely on a solid understanding of the latter’s spectra and composition. Here we therefore consider the worsecase scenarios where, contrary to mounting experimental evidence, the cosmic primaries are dominantly heavy nuclei rather than protons. The loss of acceptance for the highestenergy cosmicray flux hitting dense stars, due to the intense surface magnetic fields, is also an important element that we analyze. Finally, the slowdown and stopping of relativistic black holes inside dense stars requires a detailed study of energy loss in gravitational scattering and absorption processes.
We note that our bounds are conservative. In particular, at each point where we have encountered an uncertainty, we have replaced it by a conservative or “worst case” assumption. For this reason, our bounds can likely be improved through further work removing such uncertainties. Furthermore, we do not exclude that there could be other independent arguments, based, for example, on astrophysical or cosmological production of microscopic black holes, leading to the exclusion of stable black holes, or of their macroscopic effects.
In outline, the next section summarizes existing arguments (and their extension) against risk from TeVscale black holes, namely, the robustness of quantum black hole decay, and constraints from cosmic rays impinging on objects in the solar system. Section III then summarizes aspects of TeVscale gravity scenarios and the corresponding black holes. Section IV discusses black hole accretion in Earth. Section V gives an indepth analysis of the question of stopping of neutral black holes, via gravitational interactions. Section VI describes production of black holes by cosmic rays impinging on white dwarfs. Section VII describes the resulting accretion of a white dwarf, and thus derives constraints on hypothetical TeVscale scenarios that otherwise might have been of concern. Section VIII describes similar constraints arising from black hole production on, and accretion of, neutron stars. A summary and conclusions appear in Sec. IX.
Many of the technical details are treated in a series of Appendices. Appendix A summarizes macroscopic Bondi accretion, which is the canonical framework to deal with the flow of matter into a black hole. For its use in our context, Bondi accretion is generalized here to higherdimensional gravitational fields. Appendix B discusses the effects of radiation emitted during accretion and examines the radiative transport conditions relevant to the question of an Eddington limit for the growth of microscopic black holes. In Appendix C we derive the main equations governing the particle scattering and capture in the field of a $D$dimensional black hole, and Appendix D proves few facts required by the study of stopping by white dwarfs. Appendix E reviews the formalism for the calculation of black hole production at the LHC and in the collisions of cosmic rays, including the case of highenergy neutrinos. Here we discuss the impact of different assumptions about the composition of the highenergy cosmic rays, we evaluate the production rates under the various assumptions, and we establish the most conservative lower limits on such rates. We also discuss the production properties of black holes produced by cosmic rays, to serve as initial conditions for their slowdown in dense stars. In Appendix F we calculate the probability that black holes produced at the LHC are trapped in the Earth’s gravitational field. We argue that, independently of our general conclusion that trapped black holes would not accrete macroscopically, the velocity spectrum of black holes produced during the LHC lifetime in allowed extradimensional scenarios is such that they would typically escape the Earth’s attraction. Appendix G discusses the impact of magnetic fields on the penetration of cosmic rays down to the surface of dense stars. Appendix H addresses the production of black holes by cosmic rays hitting “background” objects, such as neutronstar companions, the interstellar medium, or even dark matter. Here we discuss the conditions for the companion of a neutron star in xray binary systems to efficiently act as a beamdump for cosmic rays, leading to the production of black holes and their capture in the neutron star, unimpeded by the presence of magnetic fields. A final appendix contains useful conversion factors and physical constants that are used throughout the paper.
II. ASSESSMENT OF PREEXISTING ARGUMENTS
We begin by assessing the strengths and weaknesses of previously existing arguments regarding risk from black hole production.
A. Instability of microscopic black holes
One of the most profound steps so far towards the yetincomplete unification of quantummechanical and gravitational principles was Hawking’s discovery [9] that black holes evaporate. This provided an important missing link in the preexisting laws of black hole thermodynamics, by explicitly calculating the temperature that characterizes a radiating black hole.
While Hawking’s result has become nearly universally accepted, it is certainly true that elements of the original derivation of black hole radiance rely on assumptions that are apparently not valid. Notable among these is the use of modes of ultraPlanckian frequencies at intermediate steps in the derivation. This naturally raises the question of the robustness of the result.
Belief in the robustness of Hawking’s prediction of nearly thermal evaporation has been boosted by arguments for the result which have now been produced from several different directions. These derivations have the virtue of either facing headon the issue of the transPlanckian modes, or being independent of them, and the basic effect has survived a number of important consistency checks.
One early approach relying on the trace anomaly and avoiding explicit reference to transPlanckian modes was pioneered by Christensen and Fulling [13]. In this approach, the stress tensor describing the Hawking radiation is found by combining the known trace anomaly in two dimensions, and the constraint that the stress tensor be conserved. This approach has been used to give explicit models of evaporating black holes [14] and has also recently been generalized to higher dimensions in [15] and a number of followup works.
Another approach is to modify the shortdistance physics to remove the offending transPlanckian modes. One summary of such a prescription appears in [16], where it is argued that despite a different origin for the modes, this approach supports the statement that Hawking radiation persists. This approach is also related to analog gravity models.^{3} These are models based on condensed matter systems, where a horizon for sound waves occurs, and exhibits a precise analog of the Hawking effect. For example, the abstract of a recent overview talk [18] on the subject of Hawking radiation states “analog models of gravity have given us a clue that despite the shaky derivation, the effect is almost certainly right.” These models illustrate various detailed features of the corresponding radiation in the analog systems.
One of the puzzles of Hawking radiation stems from the fact that it appears to lead to loss of quantum coherence, when followed to the end state of evaporation [19], and this has led to the “black hole information paradox.”^{4} Many workers feel that the resolution will be that there are subtle corrections to Hawking’s thermal spectrum that lead to unitary evolution. Thus while very few question that black holes Hawking evaporate, it is clear that there are detailed aspects of the evaporation process that we do not understand.
Indeed, the basic result, that black holes evaporate, appears quite robust from a very general perspective. Nature is quantum mechanical, and basic quantummechanical principles dictate that any allowed decay will occur. Thus, stable (or nearly stable) objects must be “protected” by conservation laws, examples being baryon number, lepton number, etc. There is no such conserved charge carried by a black hole that is not carried by ordinary matter, if a black hole can be produced in collisions of partons at the LHC. Thus by basic quantum principles such a heavy black hole should decay into light, ordinary matter, and the only question is the time scale. Since such a black hole can have mass at most around 10 times the higherdimensional Planck mass, ${M}_{D}\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$, the only relevant dimensionful parameter is the corresponding time scale, ${t}_{D}\sim 1/{M}_{D}\sim {10}^{27}s$, and there are no other small dimensionless parameters to suppress decay. Thus, on very general grounds such black holes are expected to be extremely shortlived, as is indeed predicted by the more detailed calculations of Hawking and successors.
Despite these very strong arguments for black hole decay, the possibility of manufacturing microscopic black holes on Earth suggests that one conduct an independent check of their benign nature. For that reason, this paper will test the hypothesis that the statements of this subsection are false by investigating possible consequences of hypothetical black holes that do not undergo Hawking decay.
B. Cosmicray collisions on Earth
Cosmic rays hit Earth with energies known to exceed ${10}^{20}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, corresponding to centerofmass (CM) energies exceeding 100 TeV. Thus, anything that can be made with Earthbased accelerators up to this energy is already being made by nature. This argument can be used to effectively rule out the existence of particles predicted by some speculative scenarios or to exclude possible macroscopic consequences of highenergy collisions, as discussed, for example, in [23,24].
This argument requires, however, more attention if the large momentum imparted to the produced particles by the cosmicray kinematics has an impact on how they evolve after production. Consider the cosmicray collision producing the black hole (or any other particle), at the parton level. ${E}_{1}$ is the energy of the parton inside the cosmicray nucleon and ${E}_{2}<{m}_{p}$ the energy of the parton inside the atmospheric nucleon, where ${m}_{p}$ is the nucleon mass. To produce a particle of mass $M$, we need ${E}_{1}\ge {M}^{2}/2{E}_{2}$, and therefore the minimum energy of the produced black hole in the Earth rest frame is given by ${E}_{1}>{M}^{2}/2{m}_{p}$, or, for the maximum value of $M=14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$ allowed at the LHC,
where ${p}_{\mathrm{BH}}$ is the momentum of the resulting black hole. Any argument that such black holes, should they be stable, would undergo macroscopic accretion, must be based on the proof that, while traversing the Earth, they slow down enough to get trapped by the Earth’s gravitational field. Most such TeVscale black holes should initially have color or electric charge, since the incident partons (quarks, gluons) are charged. In the usual picture, where Hawking radiation is present, these rapidly discharge through the Schwinger mechanism [25] of particleantiparticle pair creation in an intense (chromo)electric field, or through breaking/absorbing a QCD string. The time scale for the neutralization is proportional to the black hole Schwarzschild radius $R$. While there is no known example of a consistent microphysics such that Schwinger discharge occurs and Hawking radiation does not, one can point out one difference between them. Specifically, the Schwinger effect can be described in terms of pair production in the gauge field outside the horizon, but Hawking radiation is a transhorizon effect. In the unlikely event that our understanding of the horizon misses some critical element forbidding black hole decay (an assumption that, in our view, appears to contradict the basic quantum principles outlined above), one might imagine that Schwinger discharge nevertheless takes place. (One could parametrize such a scenario by imposing rather artificial boundary conditions at the horizon.)
Thus, we will consider collisions on Earth in two contexts—those which produce hypothetical stable charged black holes, and those which produce hypothetical stable black holes that rapidly neutralize.
Passage of a highenergy charged particle through matter leads to wellunderstood energy loss [26,27]. This is due to longrange electromagnetic effects that have nothing to do with the microphysics associated to the particle itself. Therefore, a muon, or a black hole with the electric charge and the mass of a muon, would be subject to the same energy loss through radiative processes as they move through matter. Since at high energy the radiative properties are mostly determined by the value of the Lorentz $\gamma $ factor of the charged particle, we describe the energyloss properties of a singly charged black hole by rescaling the energy loss of muons to the equivalent $\gamma $ value. For relativistic velocities, below the threshold for ${e}^{+}{e}^{}$ pair production in the field of nuclei, the energy loss is described by the BetheBloch equation. The energy loss in this regime depends on the velocity, with a slow growth proportional to $\mathrm{log}\gamma $. The stopping power for black holes is therefore similar to that of muons, of the order of $2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MeV}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{2}/g$ up to $\gamma \sim {10}^{3}$ [26,27]. For the average composition of Earth, this means an energy loss of about 11 MeV/cm. Above $\gamma \sim {10}^{3}$ (which is the case at the time of production for black holes of mass larger than 1 TeV) pair production, bremstrahlung, and nuclear dissociation appear. These grow approximately linearly with energy (or $\gamma $), with an energy loss of the order of 60 MeV/cm at $\gamma \sim {10}^{4}$, for the average Earth density. With Earthlike densities, the distance scale necessary to slow down from the production energy $E\sim {M}^{2}/{m}_{p}$ to $\gamma \sim {10}^{3}$ is thus of order $M/(6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{keV})$ cm, or, for $M\sim 14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$, more than ${10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, larger than the radius of the Earth. The subsequent slowdown below $\gamma \sim {10}^{3}$ takes place with the constant energy loss of 1 GeV/m, corresponding to $\gtrsim {10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$. The stopping distance grows with $M$, and a more careful estimate shows the Earth provides enough stopping power for black holes with unit electric charge up to a mass of the order of 7 TeV. For larger masses, one can appeal to the stopping power of the Sun. With a core density of about $150\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{3}$, over a radius of the order of 0.2 solar radii, namely $1.4\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, the Sun column density can stop black holes of mass well in excess of 100 TeV. Notice also that our estimates are based on a black hole with unit electric charge. Any minor addition to $Q=1$, due, for example, to accretion of a net charge, would further increase the stopping power. The continued health of the Sun on multibillionyear time scales—during which many such black holes would have been produced and stopped—thus apparently immediately rules out any risk from charged TeVscale black holes.
The above arguments apply to any elementary charged particle. In particular, they apply to possible magnetic monopoles. The stopping power for monopoles moving through matter is about 100 times larger than that of a charged particle [28], since their magnetic charge is at least $1/\alpha $ times greater than the electron electric charge. This means that the Earth itself can stop magnetic monopoles of masses much larger than those that can be produced at the LHC.
Let us next turn to the hypothesis that such black holes are stable but rapidly neutralize via Schwinger production or another mechanism. In this case, the only interactions of a black hole with matter are through its gravitational field. We will give a more careful treatment of such interactions below, but a rough cross section is the geometric size, $\sigma \sim 1/{\mathrm{TeV}}^{2}$. Again using the average density of Earth, we find of order 1 scattering event expected in its transit for this cross section. The maximum momentum loss in a collision with a parton of energy ${E}_{p}$ is of size $\sim {\gamma}^{2}{E}_{p}$. This is achieved when particles are scattered with significant momentum transfer in the black hole frame, for example, in a headon elastic collision, where the target particle is significantly deflected. Since $\gamma \sim M/{E}_{p}$, this would correspond to a loss of a fraction of order 1 of the initial momentum (2.1). However, this is clearly not necessarily typical: due to the shortdistance character of the $D$dimensional gravitational potential, the impact parameter range for hard elastic scattering is restricted, and typically we shall either have a smallangle scattering, or the target particle can be captured by the black hole, leading to a large inelasticity and to a reduced velocity loss. We shall discuss these issues in more detail in Sec. V. The more careful analysis presented there also indicates that the stopping power of the Sun is likewise insufficient.
To summarize, hypothetical stable charged black holes should stop in the Earth for masses up to about 7 TeV, and in the Sun if heavier. The multibillionyear longevity of Earth and Sun apparently provides a good safety guarantee. We have no concrete example of a consistent microphysics such that black holes neutralize via Schwinger discharge but do not Hawking radiate, but our present state of knowledge of quantum black hole processes does not strictly rule out such a possibility. Thus, we will seek alternative bounds on such a scenario, which will also serve the purpose of improving the stringency of the bounds for charged black holes.
III. ESSENTIALS OF HIGHERDIMENSIONAL GRAVITY
In this section we quickly review some features of higherdimensional scenarios realizing TeVscale gravity and of the black holes that exist in these scenarios. A brief overview discussing more aspects of these scenarios and black hole production in them is [7].
A. TeVscale gravity scenarios
The basic idea of TeVscale gravity is that either via large extra dimensions [1,2] or large warping [3], the true Planck scale is lowered to the vicinity of a TeV. To summarize, let the $D$dimensional action be
where ${G}_{D}$ is the $D$dimensional gravitational constant, $\mathcal{R}$ is the Ricci scalar, and $\mathcal{L}$ is the matter Lagrangian. We consider a general compact metric ${g}_{mn}(y)$, possibly together with warp factor $A(y)$,
Here the noncompact coordinates are ${x}^{\mu}$, and the standard model fields are typically taken to lie on a brane spanning these dimensions. Then the relation between the higherdimensional Planck mass,^{5}
and the fourdimensional Planck mass, defined via the fourdimensional gravitational action
is
This equation defines the “warped volume” ${V}_{w}$.
Current lower bounds [26] on ${M}_{D}$ are around 1 TeV. In order for ${M}_{D}$ to be in the TeV vicinity, one must have a large warped volume. This can be achieved by large volume and moderate warping, by large warping, or by some combination of the two. A simplified version of the relationship (3.5), assuming the scale of all extra dimensions is set by a radius ${R}_{D}$, is
where $\Delta A$ is a measure of the relative difference in warping between the region of maximal warp factor and the region in which standard model physics resides.
One can solve for the characteristic $D$dependent size of the extra dimensions,
Of course, larger warping and fixed ${M}_{D}$ means that ${R}_{D}$ is smaller for a given $D$. In particular, for unwarped scenarios ${R}_{5}$ is macroscopic and thus ruled out, but with sufficient warping one finds viable scenarios, such as that of [3].
For ${M}_{D}=1\mathrm{TeV}$ and with no warping, we find the following radii:
For higher ${M}_{D}$ and/or in the presence of warping, these numbers should be multiplied by a factor
B. Higherdimensional black holes
1. Schwarzschild solution
We next turn to properties of black holes in these scenarios. We begin with the $D$dimensional Schwarzschild solution with mass $M$, which takes the form
Here the Schwarzschild radius $R(M)$ is
where the constant ${k}_{D}$ is defined as
and where ${\Omega}_{D2}$ is the volume of the unit $D2$ sphere,
One can likewise write down the higherdimensional version of the Kerr solution [29].
2. Black holes in standard compactifications
Nonrotating black holes with radius much less than the curvature scales and the sizes of the extra dimensions are well approximated by the Schwarzschild solution (3.15). When the Schwarzschild radius reaches a size comparable to that of an extra dimension, one expects an inverse GregoryLaflamme[30] transition to the lowerdimensional black hole, extended over the extra dimension. In this case the solution will be given by the lowerdimensional version of the solution (3.15), with trivial dependence on the compact coordinates.
For the purposes of computing forces due to a black hole, we will need its gravitational potential. In the weakfield regime this is given by
leading to the force on a mass $m$:
where
The attractive gravitational force matches between the lower and higherdimensional expressions in the region $r\sim {R}_{D}$. Specifically, by equating the $D$dimensional force law to that of 4D, we find that the forces match at a crossover radius
whose values are of size $56{R}_{D}$ in cases of interest.
3. Black holes in warped compactifications
As an example of a broad class of warped compactification scenarios, consider a metric of the form
where ${e}^{2y/{R}_{D}}d{x}_{4}^{2}$ describes the $D=4$ part of the metric. Here ${R}_{D}$ is both the characteristic curvature radius associated with the warping and the radius of the remaining extra dimensions, compactified on some compact manifold $X$, whose metric $d{s}_{X}^{2}$ we have taken to have radius $\mathcal{O}(1)$. The coordinate $y$ is taken to have range $(0,L)$. The form (3.23) is representative of many known examples of warped compactifications, such as the truncated ${\mathrm{AdS}}_{5}$ solutions of [3] and the warped flux compactifications of [4,31], although more generally one expects, for example, the metric of $X$ to vary with $y$.
For these compactifications, a black hole whose radius satisfies $R\ll {R}_{D}$ is well described as the $D$dimensional Schwarzschild solution (3.15). On the other hand, a sufficiently large black hole is expected to be described by a fourdimensional Schwarzschild solution, represented by (3.23) with $d{x}_{4}^{2}$ replaced by the fourdimensional Schwarzschild metric. Between these extremes the solutions are not known.
However, approximate forms for the linearized gravitational potential, appropriate to describing the weakfield regime of a concentrated mass such as a black hole, have been derived in [32]. In the region ${R}_{D}\lesssim r\lesssim L$, one finds the linearized perturbation of $d{x}_{4}^{2}$ given by
Here ${j}_{D}={j}_{(D3)/2,1}$ is the first zero of the relevant Bessel function and ${\widehat{k}}_{D}$ is a constant. Thus, the radial gravitational force at $y=0$ is
An important question is at what specific radius ${R}_{C}$ does the gravitational force from (3.24) match onto that for four dimensions. This can be found by equating ${F}_{D,w}$ to ${G}_{4}M/{r}^{2}$. This yields the relation
From this, we see that if ${R}_{C}$ is significantly smaller than the radius given by the unwarped version of (3.6), this correspondingly increases the size of the region between ${R}_{D}$ and ${R}_{C}$ where warping is significant.
4. A general perspective
We close this section by outlining a broader perspective on TeVscale gravity and black holes. Note that a general feature of the above discussion is that below the value ${R}_{C}$ the potential (3.19) crosses over from the fourdimensional form to one that grows more rapidly as $r$ decreases. The flat and warped cases give both power law and exponential growth laws. Moreover, the gravitational potential is proportional to the mass. In order to have a TeVscale model, note that the potential corresponding to a TeVscale mass should reach the value $\varphi \sim 1$ by the time $r$ reaches the value $r\sim 1/\mathrm{TeV}$. We assume that while the gravitational potential is modified at short distances, the dynamics of other forces is fourdimensional, as in braneworld models, in order to agree with experiment.
Moreover, as we will see, many features of the accretion process only depend on the longrange potential, since it is at such long scales that gravity begins to compete with other effects. From this general perspective, a very general definition of a black hole is as an object with such a longrange gravitational potential, and which is allowed to accumulate mass at $\varphi \sim 1$.
One could postulate more general forms for the potential, but clearly the cases we have described are representative of a very wide class of potentials that become strong at the TeV scale.
IV. BLACK HOLE ACCRETION IN EARTH
A. Accretion basics
Our interest is in accretion of black holes trapped inside astronomical bodies such as planets, stars, neutron stars, etc. One parameter governing this accretion is the effective capture radius ${r}_{c}(M)$ of the black hole, for a given mass. This is the radius out to which the gravitational field of the black hole succeeds in attracting matter that will eventually be absorbed. The other parameter is the flux of mass towards the black hole, $F$. Specifically, the black hole mass grows as
This formula neglects reradiation of incident energy, which is discussed in Appendix B, and, if present, can lower the growth rate. The flux can arise either from the motion of the black hole relative to the body, or from the motion of the constituents of the body relative to the black hole. In the case where the dominant effect is the velocity $v$ of the black hole, we have
where $\rho $ is the mass density near the capture radius. This produces an evolution equation
The capture radius ${r}_{c}$ is frequently different from the Schwarzschild radius $R$ and depends on the size and state of motion of the black hole, as well as on properties of the surrounding medium. For example, free particles with velocity $v$ with respect to a black hole have capture radii
Black holes whose production is accessible at the LHC have an initial radius of the order ${\mathrm{TeV}}^{1}$. As they absorb matter, their physical and capture radii grow.^{6} In atomic matter, there are three possible domains where the properties of black hole interactions with matter vary. The first phase is that where ${r}_{c}$ is smaller than the nucleon size, ${r}_{N}\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}$. A second phase is that where ${r}_{N}\lesssim {r}_{c}\lesssim a$ (where $a\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ is the atomic radius). The third phase is ${r}_{c}>a$. Similar phases are present for growth inside a white dwarf, but in the case of growth inside nuclear matter of a neutron star, the two latter phases are replaced by a single phase with ${r}_{c}\gtrsim {r}_{N}$. The details of the evolution during these phases will vary, depending on where ${r}_{c}$ is relative to ${R}_{D}$ and ${R}_{C}$, the distances characterizing crossover from the $D$dimensional force law to that of four dimensions.
This section will discuss evolution, first at the atomic level, and then, for largerscale black holes, from macroscopic matter. The latter is described by Bondi evolution; we also briefly discuss, and argue against, the presence of an Eddington limit, which would be relevant if emitted radiation were sufficient to slow accretion. Our goal will be to estimate, under the most pessimistic assumptions, namely, of fastest possible growth, the time scale required for accretion of black holes to macroscopic size. In subsequent sections, we will also perform similar calculations for accretion of white dwarfs and of neutron stars.
B. Subatomic accretion in Earth
1. Competition with electromagnetic binding
As is described in Appendix F, most LHCproduced black holes would be produced with large velocity as compared to the Earth’s escape velocity, ${v}_{E}=11\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}/\mathrm{s}$, due to imbalanced kinematics of the initialstate partons, initial and final state radiation, etc. However, those that are downward directed will accrete matter and slow down while passing through the Earth. Appendix F estimates this effect based on closely related calculations for black holes created by cosmic rays in Sec. V. For present purposes, we will simply make the most conservative assumption that some of these black holes do become gravitationally bound to Earth. Given the escape velocity and that the minimum mass of such a black hole would be $\mathcal{O}(\mathrm{TeV})$, typical kinetic and gravitational potential energies would thus be $\gtrsim \mathcal{O}(\mathrm{keV})$. This means that on the occasions where a black hole and nucleus bind, the black hole’s energy overcomes the (atomic) binding energy of the nucleus to the surrounding material, and the combined system continues to fall. Thus the black hole’s motion should initially be dictated by the net gravitational field of the Earth.
In a collision of the black hole with a nucleus, binding depends on the size of the impact parameter $b$; we will largely neglect separate capture of electrons since their capture rates are much smaller due to their smaller masses and higher velocities. The black hole’s effects are significant at impact parameters where its gravity competes with the electromagnetic binding forces of the surrounding medium. The latter are estimated by noting that if a nucleus is displaced from its equilibrium position by a small displacement $d$, one will find a restoring force of the form
for some constant $K$. This is justified, for example, by considering the force acting on an ion^{7} of charge ${Z}^{\prime}$ as it is displaced by a distance $d$ from the center of charge of its electron cloud:
where we assumed the electron charge to be uniformly distributed in the atomic volume. The black hole will exert a competing gravitational force which is maximum at the point of closest approach. For a $D$dimensional force law (3.20), and with the nucleus displaced by $d$ towards the black hole, it is
where the nuclear mass is given in terms of the mass number and proton mass as $m\simeq A{m}_{p}$. The nucleus can become bound to the black hole if this force dominates ${F}_{E}$ for all $d$ over the range $(0,b)$. This amounts to the condition that, for all $d$,
Maximizing the righthand side (RHS) with respect to $d$, we find the binding condition:
which defines the electromagnetic capture radius ${R}_{\mathrm{EM}}$. To simplify subsequent expressions, we rewrite this as
where we define
The ratio of electromagnetic to Schwarzschild radii is given by
Since $K$ is governed by atomic scales, ${\beta}_{D}\gg 1$, and ${R}_{\mathrm{EM}}$ exceeds $R$ for subatomic $R$.
For low relative velocities, nuclei entering this radius can become bound to the black hole. Note that this will not be the case for sufficiently high relative velocity $v$, as the freeparticle capture radius $R/v$ is smaller than ${R}_{\mathrm{EM}}$ for large $v$. However, we will consider sufficiently small velocities that ${R}_{\mathrm{EM}}<R/v$, where competition with electromagnetic binding described by ${F}_{E}$ dominates.
While subsequent total absorption of a captured nucleus is not guaranteed,^{8} the most conservative assumptions for the purposes of discussing accretion on Earth are those that lead to the fastest accretion. We will thus assume that all of the mass of the nucleus is absorbed. In the absence of other effects, this black hole would also have the charge of the nucleus. This charge may discharge through the Schwinger mechanism, or be retained, depending on assumptions. If it is retained, the next time that the black hole encounters a nucleus within ${R}_{\mathrm{EM}}$, this charge is insufficient to prevent absorption, but with sufficient charge buildup repulsion could become an important effect. A positively charged black hole will also have an enhanced absorption rate for electrons, which works toward neutralization. So, while charge effects could possibly somewhat slow the absorption rate, we will make the conservative assumption that they do not and that sufficient neutralization is automatic.
To determine actual capture sizes, one needs the parameter $K$. There are different ways of estimating a typical $K$. One approach is to estimate the dipole force when one separates an ion in a crystal from the electron cloud of the bonding orbitals, as suggested earlier by Eq. (4.6). Assuming there that all but the outermost electrons move coherently with the nucleus yields the value
where $a$ is the atomic radius, $\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $. Another method is to use the relation to the Debye frequency ${\omega}_{D}$,
where $\chi $ is an $\mathcal{O}(1)$ constant that depends on the material. Corresponding Debye temperatures, ${T}_{D}={\omega}_{D}$, fall in the range 300–600 K for typical materials forming the Earth’s interior ( ${T}_{D}^{\mathrm{Fe}}=460\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{K}$, ${T}_{D}^{\mathrm{Si}}=625\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{K}$, ${T}_{D}^{\mathrm{Mg}}=320\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{K}$). In this case, one finds a typical K of size:
(4.15)
consistent with (4.13). Here we have introduced the TeV mass scale,
which frequently provides a useful normalization scale. In Earth one also has semisolid or semifluid layers, but these have characteristic values of $K/m$, given by pairing potentials of similar size to those of solids.
Since $K/m$ is directly related to ${T}_{D}$, we will in fact parametrize results in terms of this temperature. For example,
With this prelude, we are now prepared to discuss the subatomic phase of accretion.
2. Subatomic growth laws
As we have described, in principle there can be three regimes depending on the size of the capture radius relative to nuclear and atomic scales. The capture radius in atomic matter is given by ${R}_{\mathrm{EM}}$; let us estimate this for the minimum size black hole, with $M\sim {M}_{D}$. For $D=11$, we find from (4.10), (4.11), and (4.14) that
with larger values for smaller $D$. Thus the subnuclear growth phase is nearly negligible, and we will (conservatively) set the corresponding time to zero.
We therefore turn directly to evolution from ${R}_{\mathrm{EM}}\sim {r}_{N}$ up to the atomic radius, $a\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $. Combining the general evolution equation (4.3) with the expression (4.10) for the electromagnetic radius, the growth law for a black hole moving with velocity $v$ takes the form
This expression integrates to give a distance
for growth to a mass $M$, where we introduce the characteristic distance, given via TeV units (4.16)
Using the average density for Earth, ${\rho}_{E}=5.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{3}$, one finds ${d}_{0}=3\times {10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}=9s$, which is much bigger than the Earth’s radius. The distance (4.20) is governed by the upper limit of the mass, and the lower limit has thus been dropped. The expression (4.20) can also be written in terms of the final ${R}_{\mathrm{EM}}$, using (4.10), as
From Eqs. (3.8),(3.9),(3.10),(3.11),(3.12),(3.13) we see that this evolution applies to all values of ${R}_{\mathrm{EM}}<a$ if $D\le 7$. For $D\ge 8$, instead, the radius ${R}_{D}$ of the extra dimensions is smaller than $a$, and therefore as ${R}_{\mathrm{EM}}$ grows larger than ${R}_{D}$ the dimension governing the force law changes, and one should then set $D=4$ in Eq. (4.22). We therefore must treat these cases separately.
In performing the following estimates, using the formula (4.22), we will assume that the black hole uniformly travels at the escape velocity ${v}_{E}$. Were its velocity higher, it would not be gravitationally bound to Earth. This is clearly a conservative assumption, as the black hole will slow down as it accretes; if needed one could model this slowdown by integrating the evolution of the mass and potential energy over the lifetime of the black hole. (We expect that the bounds of this paper might be tightened by a more complete treatment of this slowdown.) We will also assume a uniform density ${\rho}_{E}$ for Earth. Of course, if the black hole spent most of its time near the center of the Earth, one should use a higher central density. But, correspondingly, the black hole velocity would be lower, scaling linearly with the distance from the Earth’s center. Since the Earth’s density in the deep core is at most a factor of 2–3 higher than its average value, the estimates using the average density and the escape velocity should tend to (conservatively) overestimate the accretion rate.
3. Time scales for ${r}_{N}\lesssim {R}_{\mathrm{EM}}\lesssim a$ in $D=6,7$
The relevant time scales for growth to atomic sizes for $D=6$, 7 can be found by setting ${R}_{\mathrm{EM}}\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ in in Eq. (4.22), assuming constant velocity ${v}_{E}$, and substituting the values of the other parameters. This results in the following time scales:
These times are quite short, compared to geologic time scales, and this phase for $D=6$, 7 will therefore be regarded as negligible.
4. Time scales for ${r}_{N}\lesssim {R}_{\mathrm{EM}}\lesssim {R}_{D}$ in $D\ge 8$
For $D\ge 8$, the black hole would first evolve from ${R}_{\mathrm{EM}}\sim {r}_{N}$ up to ${R}_{\mathrm{EM}}\sim {R}_{D}$ via the evolution law (4.19). The relative time scales can be obtained by equating ${R}_{\mathrm{EM}}$ in Eq. (4.22) with the expression of ${R}_{D}$ given by Eq. (3.7), and assuming the uniform velocity ${v}_{E}$, resulting in:
The corresponding times are short compared to e.g. the solar lifetime and become shorter in higher dimensions, since the values of ${R}_{D}$ in these cases become smaller and smaller, approaching ${r}_{N}$ and reducing the available evolution range. As we show next, this, however, means that there will be more range for the 4dimensional evolution, which is typically slower because of the weaker gravitational coupling in 4 dimensions.
5. Time scales for ${R}_{D}\lesssim {R}_{\mathrm{EM}}\lesssim a$ in $D\ge 8$
Once ${R}_{\mathrm{EM}}$ reaches $\mathcal{O}({R}_{D})$, the distance at which the black hole’s gravity competes with electromagnetic binding forces is in the region where the black hole’s field transitions to the lowerdimensional form. For concreteness, let us first neglect warping and assume that all radii are the same so that this is a transition to the fourdimensional regime. One reaches this regime at the crossover radius given by (3.22).
The distance required to reach a given ${R}_{\mathrm{EM}}>{R}_{C}$ is then, from the distance (4.22) and using the formula (4.17) for ${\beta}_{4}$,
We are interested in the corresponding time scale to reach ${R}_{\mathrm{EM}}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, where evolution begins to cross over to “macroscopic.” Introducing the numerical values for our parameters, along with the escape velocity ${v}_{E}$ and taking ${R}_{\mathrm{EM}}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, we then find a time of the order of ${10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$
for evolution to the “macroscopic” crossover.
In the case of $D=8$, ${R}_{C}$ is close to $1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $. One may therefore fear that, should the effective ${R}_{C}$ be an underestimate by a factor of 2–3, there will be no room for this phase of evolution. However, notice that, aside from the factor $(D1)/(D3)$, the distance (4.20) is the characteristic distance for one efold growth of the mass. Since the higherdimensional evolution law must match onto the fourdimensional one in this region (as can be seen explicitly), and since the growth of the radius over a single efold in the mass is small ( ${e}^{1/(D1)}$, or ${e}^{1/3}$ in four dimensions), the efold time for ${R}_{\mathrm{EM}}\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ sets a lower bound on the evolution time.
C. Macroscopic accretion, ${R}_{\mathrm{EM}}\gtrsim a$
1. Bondi accretion basics
Once the electromagnetic capture radius of the black hole, ${R}_{\mathrm{EM}}$, grows beyond the atomic radius $a$ , accretion becomes a macroscopic process, with multiple atoms falling in, and with the gravitational range of the black hole exceeding the mean free path. In this regime, two effects counter the free fall of matter: the cohesion forces that keep atoms together and matter’s finite compressibility. We shall not attempt to provide a simple model to describe the effect of cohesion forces on a macroscopic scale, due to the varied composition of matter inside the Earth, with crystalline, semisolid, and liquid phases at various depths. Moreover, it is possible that once the accretion rate reaches a certain threshold, radiation emitted from the accreting matter can melt the surrounding material. (This reradiation effect is discussed in Appendix B, where it is found not to be important until ${R}_{\mathrm{EM}}\gg 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $.) Since our aim is to be conservative and consider the fastest conceivable evolution, we shall neglect the slowdown due to cohesion forces, and treat the inside of the Earth as a nonviscous fluid, free to fall into the black hole, subject only to the general laws of hydrodynamics, such as the continuity equation and energy conservation. The compressibility of the medium, which limits the amount of matter that can be funneled towards the black hole, is accounted for by macroscopic hydrodynamic properties of the medium, such as its sound speed.
The description of accretion under these conditions was developed by Bondi, Hoyle, and Lyttleton [33]. We review the derivation of the resulting evolution equation and extend it to incorporate the $D$dimensional force law in Appendix A; it is
where we define the Bondi radius in terms of the black hole radius $R$,
${\lambda}_{D}$ is a numerical constant depending on $D$ and on the polytropic index $\Gamma $, given in (A.20), which can range between $3<{\lambda}_{D}<18$, and $\rho $ and ${c}_{s}$ are the density and sound speed within the matter asymptotically far from the black hole.
2. Matching microscopic and macroscopic regimes
Before estimating corresponding time scales, let us first compare the sub and superatomic regimes of growth at the transition point, ${R}_{\mathrm{EM}}\sim a$.
Using Eq. (4.14) and the relation between Debye frequency and sound velocity ${c}_{s}\simeq {\omega}_{D}a={T}_{D}a$,^{9} we can rewrite the expression for ${R}_{\mathrm{EM}}$ given in Eq. (4.10) as
where
From Eqs. (4.32) and (3.16), we also find:
The evolution equations for the two regimes can therefore be equivalently rewritten as
As before, we use ${v}_{\mathrm{EM}}\sim {v}_{E}$; notice that ${c}_{s}$ inside the Earth has a comparable value. So, up to an overall factor of order 1, ${M}_{a,D}$ turns out to be the mass value at which the two evolution rates are the same, and the capture radii for the subatomic and the Bondi accretion regimes coincide. The subatomic growth is faster when $M<{M}_{a,D}$, while Bondi’s growth is faster when $M>{M}_{a,D}$. This means that for the purpose of being conservative, it is justified to use the former accretion model below ${M}_{a,D}$, and the latter above ${M}_{a,D}$.
3. Time evolution with Bondi accretion
We can split the numerical analysis for the Bondi accretion into the case of ${R}_{D}<a$ ( $D\ge 8$), where all the evolution for ${R}_{\mathrm{EM}}>a$ is four dimensional, and the case with ${R}_{D}>a$ ( $D\le 7$), where we need to consider both phases.
The $D$dimensional Bondi evolution equation (4.31) is straightforward to integrate. Since we are interested in times for evolution to given radii, it is most useful to convert it to an equation for the Bondi radius ${R}_{B}$, using (4.32). This gives the following times, for evolution from an initial Bondi radius ${R}_{B,i}$ to a final Bondi radius ${R}_{B}$:
Recall that ${d}_{0}$ was defined in (4.21). A transition from the $D$ to 4dimensional Bondi behavior will occur when ${R}_{B}$ is in the range of ${R}_{D,C}$. It is easy to check that this transition is continuous, namely, the values of the $D$ and 4dimensional Bondi radii coincide, when ${R}_{B}={R}_{C}$. At this radius, the mass is given by
The evolution time to the slightly smaller radius ${R}_{B}={R}_{D}$ is found from (4.39),
4. Macroscopic time scales: $D\ge 8$
We start with the case ${R}_{D}<a$, so that evolution for ${R}_{B}>a$ is purely fourdimensional Bondi accretion. When the black hole enters the macroscopic regime, ${R}_{B}\approx a$, Eq. (4.32) implies that its mass is of order ${10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$, therefore it is still small in geologic terms. Thus by this time it should have settled deep within the Earth. The quantity ${c}_{s}{d}_{0}$, relevant to the Bondi evolution formulae, can be estimated from the approximately linear relation, known as Birch’s law [34], between sound speed and density. This has been tested experimentally for Fe [35,36] up to the densities of $12\mathrm{gr}/{\mathrm{cm}}^{3}$ found in the Earth’s core, giving
which is then density independent. (One also finds values comparable to this using, e.g., sound speeds and densities for materials such as iron at low pressure.) Equation (4.41) then gives the following time to double the radius from ${R}_{B}\approx a$ to ${R}_{B}\approx 2a$ (and the mass from $M\approx {10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$ to $M\approx 2\times {10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$):
As shown in Appendix A, the value of ${\lambda}_{4}$ is in the range 4–18; for $\Gamma =5/3$ (namely the adiabatic index of a nonrelativistic electron gas), ${\lambda}_{4}=4$. Notice that the parameter dependence of (4.45) is identical to that of Eq. (4.29), once we use ${c}_{s}\simeq {T}_{D}a$. This is also reflected in the similarity of the time scales, Eqs. (4.30) and (4.45).
As a side note, we remind the reader of discussions of the possibility that primordial black holes remain from the early universe; in the standard quantum scenario only those with masses $\gtrsim {10}^{15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$ would have not yet evaporated. With the present formalism, we can provide a bound on the lifetime of Earth, should a minimummass primordial black hole be captured within its gravitational field. Using the parameters in this section, and the evolution law (4.41), we find a bound on the accretion time $t\gtrsim 47\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Myr}$, with shorter times for highermass black holes. These, and corresponding accretion times we will find for white dwarfs and neutron stars, may allow one to set limits on galactic densities of primordial black holes.
5. Macroscopic time scales, $D=6,7$
In the cases $D=6$, 7, ${R}_{D}>a$ and so we have $D$dimensional Bondi evolution up to ${R}_{B}\sim {R}_{D}$, and then fourdimensional Bondi evolution from ${R}_{B}={R}_{C}$ up to infinity. To be conservative, we model the phase with ${R}_{D}<{R}_{B}<{R}_{C}$ by assuming $D$dimensional evolution with a constant Bondi radius, with ${R}_{B}={R}_{C}$, until the black hole mass grows to the point that the respective ${R}_{B}$ exceeds ${R}_{C}$. For $D=7$ the mass at ${R}_{B}\approx a$ is of order ${10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$; for $D=6$ it is much smaller. For our approximate estimates, we again use the value of ${c}_{s}{d}_{0}$ given in (4.44).
The times for the evolution up to ${R}_{D}$ are thus given by (4.43), leading to
The following phase, between ${R}_{D}$ and ${R}_{C}$, then has a time scale given by
The subsequent evolution to large sizes has a time scale determined by the initial radius, ${R}_{B}={R}_{C}$ in (4.41), and results in an expression identical to (4.48), with ${\lambda}_{D}$ replaced by ${\lambda}_{4}$. The time scales for these two phases are given by
The $D=6$ time is short as compared to geologic time scales. Using ${\lambda}_{7}=4$ (independent of $\Gamma $) and ${\lambda}_{4}=4$ (for $\Gamma =5/3$) we obtain in $D=7$ a combined time scale of approximately (6.4, 20, 40, 65, 94) billion years for ${M}_{D}=1,\dots ,5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$. Since, as we discuss in Appendix E, a conservative threshold for black hole formation is ${M}_{\mathrm{min}}=3{M}_{D}$, these values correspond to minimum black hole masses ${M}_{\mathrm{min}}=(3,6,9,12,15)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$.
D. Warped evolution
In order to parametrize more general evolutions, we consider the case of a warped scenario, as described in Sec. III B 3. In this case, we have $D$dimensional evolution up to capture radius ${R}_{D}$, then a warped evolution up to ${R}_{C}$, then fourdimensional evolution from then on. These scales are related by an expression of the form (3.26). Combining this with the maximum value of the warping, found from (3.6) by taking ${R}_{D}$ to be the minimum possible value, ${M}_{D}^{1}$, we find that
so there is still not wide disparity between these scales.
As the preceding discussion has illustrated, a key question is the location of ${R}_{C}$ with respect to the atomic scale $a$. Macroscopic evolution in the fourdimensional regime is dominated by the time scale from the lower end point at ${R}_{B}={R}_{C}$, and if ${R}_{C}\lesssim 200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, this, via (4.41), yields a safe time scale in excess of $3\times {10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. The warped growth below ${R}_{C}$ should also yield a similar time scale, as in the preceding discussion.
On the other hand, accelerated growth is possible for ${R}_{C}$ significantly larger than $200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $. We particularly saw this in the case $D=6$. Warped evolution presents another extreme (but finely tuned) scenario. Specifically, consider the case $D=5$ with radius just below the experimental bound, say ${R}_{C}\approx 0.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. In this case, one finds fivedimensional evolution through subnuclear, subatomic, and Bondi phases. If, in line with our discussion of scales, we take ${R}_{D}\approx 0.02\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, the evolution time up to this radius follows from (4.40). This yields an estimate ${t}_{B,5}\approx 5\times {10}^{3}s$. At this point, the black hole has a mass ${M}_{B,5}$ around 0.1 gr. Next, one evolves through the warped regime. The precise form of the evolution in this regime is not completely understood. The form of the linearized potential [32] suggests that the radius grows as the logarithm of the energy, but Ref. [32] also points out that the corresponding solutions are possibly unstable and instead spread out more widely on the visible brane. Reference [37] worked out details of a possible picture of the resulting solution, which would be the gravitational dual of a plasma ball of QCD. Whatever the precise evolution law is, it is quite slow, since at the end of this phase the mass is of size ${M}_{B,4}\approx 2\times {10}^{17}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$ while the radius has only changed from ${R}_{D}$ to ${R}_{C}$. Conservatively, one can take the fastest time scale for evolution, namely, that with a constant capture radius ${R}_{B}\approx {R}_{C}$. Finally, the time for the fourdimensional evolution from ${R}_{B}={R}_{C}$ to the mass of the Earth has a time scale given by (4.41). Both of these phases yield time scales
While this time scale as well as that of the $D=6$ case only represent lower bounds on the accretion time, since we have made various conservative assumptions such as that of accretion from a fluid, these times are too short to provide comfortable constraints.
E. An Eddington limit?
If evolution is sufficiently rapid, as is particularly exhibited by the $D=6$ case and the extreme $D=5$ warped case, and in four dimensions, one is naturally led to question whether there is an Eddington limit. This would occur if radiation from the rapidly accreting matter produced sufficient pressure to inhibit accretion.
In particular, one can parametrize the total luminosity of the outgoing radiation in terms of the rate of mass accretion by an efficiency parameter $\eta $,
If this radiation is in photons, it will exert a force on the infalling atomic matter of Earth which can be approximated as given by the Thomson cross section $\sigma $:
The question is whether this can balance the force of gravity pulling the matter inwards.
We examine this question in more detail in Appendix B. In short, we do not find evidence for such an Eddington limit; this is connected to the known result (see e.g. [38]) that spherical accretion onto a black hole is inefficient at producing a large luminosity.^{10} Nonetheless, we note that if a mechanism to produce such an Eddington limit were found, this would lengthen the shorter accretion time scales considerably. In particularly, as we review in Appendix B, fourdimensional Eddington evolution produces exponential growth of the mass with time, with time constant
where $m$ is the average atomic mass per electron, $m\approx 2{m}_{p}$. This produces an efold time scale of size ${t}_{\mathrm{Edd}}\approx 2.3\eta \times {10}^{8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$.
F. Summary of growth on Earth
This section has modeled growth laws on Earth, using conservative assumptions.
The resulting growth times for $D\ge 8$ are bounded below by the manybillionyear time scales given in Eqs. (4.30) and (4.45); these arise from the subatomic and superatomic regimes, respectively, and are times that are long as compared to the expected natural lifetime of the Sun. At these times, the black holes masses are still in the range of ${10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}$, and have a growth rate of the order of 300 kW, which gives a bound on possible power output that is totally negligible in the geologic context.
The case of $D=7$ also gives times longer than the Sun’s lifetime, with an overall time scale for the evolution, Eq. (4.50), ranging from 6 to over $80\times {10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. At such time scales, the growth rate reaches approximately 10 GW, about ${10}^{7}$ the solar flux on Earth, and much smaller than the $\sim 40\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TW}$ heat flow from the interior of the planet; this bounds any possible thermal impact to be negligible. The case $D=6$, with time scale (4.49), even though very long by human standards, is much shorter than the natural lifetime of the solar system. As we saw, certain warped scenarios are also similarly potentially problematic. Therefore, in order to constrain these scenarios we turn to their consequences for other astronomical bodies, particularly white dwarfs and neutron stars.
V. STOPPING OF COSMIC RAYPRODUCED BLACK HOLES
Collisions with centerofmass energies comparable in energy to LHC occur frequently in the universe. The best known and directly measured process is the collision of highenergy cosmic rays (CRs) with the nucleons in the Earth’s atmosphere. For the collision of a CR of mass $A{m}_{p}$ to exceed the nucleonnucleon centerofmass (CM) energy of ${E}_{\mathrm{LHC}}=14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$, the CR energy should be at least ${E}_{\mathrm{min}}(A)=A{E}_{\mathrm{LHC}}^{2}/2{m}_{p}\sim ({10}^{17}A)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, well below the maximum value of measured CR energies. A simple estimate of the number of nucleonnucleon interactions above LHC energies can be obtained from an approximate flux relation, derived from the current data [40–42]:
which provides a lower bound to the measured CR spectra in the interesting region ${E}_{\mathrm{CR}}>{10}^{17}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, up to the GZK cutoff [43,44] of ${E}_{\mathrm{GZK}}\sim 5\times {10}^{19}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$. Confining ourselves to the part of the spectrum below the GZK cutoff, we obtain the following integrated flux:
where $\sqrt{s}$ is the CM energy of a nucleonnucleon collision. This corresponds to about $1/A\times {10}^{22}$ collisions above the LHC energy at the surface of the Earth during the course of its existence. This number greatly exceeds the total number of collisions in the course of the LHC operations at its highest intensity (about ${10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{s}}^{1}$ over a ${10}^{8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{s}$ period), even assuming a cosmic ray flux dominated by Fe nuclei.
If black holes can be produced at the LHC, they will therefore be copiously produced in such CRinduced collisions with astronomical bodies. Thus, stability of such bodies on astronomically long time scales offers the prospect of ruling out rapid accretion scenarios. In order to provide such bounds, one needs to check that a CRproduced black hole will slow sufficiently to be trapped in such an object, so that it begins accreting. After that, one needs to check the relevant accretion time scale. This section will focus on the former question. We will find that while, as briefly described in Sec. II B, we cannot guarantee that Earth is an efficient target for trapping hypothetical CRproduced black holes in all scenarios, white dwarfs and neutron stars do provide very useful targets.
A. Production kinematics
Let $E$ be the energy of a cosmic ray nucleon hitting an astronomical target. Black hole production would arise from collisions of two partons, with centerofmass momentum fractions ${x}_{1}$ and ${x}_{2}$ for incident parton and target parton, respectively. The mass $M$ of the resulting black hole is given by
where $y\le 1$ is an efficiency factor, parametrizing inelasticity (energy loss) due to radiation in the collision process, and ${m}_{p}$ is the proton mass. The bulk of the production is at ${x}_{1}\sim {x}_{2}\sim x=M/(y\sqrt{2E{m}_{p}})$. After being produced, the black hole in the fixedtarget frame will carry energy ${x}_{1}yE\sim M\sqrt{E/2{m}_{p}}$, and thus has a Lorentz $\gamma $ factor
Since ${x}_{1},{x}_{2},y\le 1$, Eq. (5.3) gives
The black hole will therefore be highly relativistic. These boosts range up to typical values of size $\gamma \sim 3M/{m}_{p}$; for example, in the extreme case of interest for the LHC, $M=14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$, Fig. 7 shows significant production at $\gamma \lesssim 4.5\times {10}^{4}$.
If the black hole is very weakly interacting, it could travel across the object unimpeded, thus preventing limits from being set. We therefore concentrate first on this issue.
B. Stopping: neutral black holes
In line with our discussion of Sec. II A, we will explore the assumption that black holes interact only via their gravitational field. Collisions can slow such a black hole via two mechanisms. First, in a typical collision the black hole will gravitationally scatter a particle in its asymptotic field, thus losing some of its momentum. We henceforth refer to this gravitational mechanism as Coulomb slowdown. Second, for smaller impact parameters, a black hole can absorb a particle, in the process possibly emitting some radiation, and typically increasing its mass. This also slows the black hole, and will be referred to as accretion slowdown. In what follows we denote by ${E}_{i}$ and ${p}_{i}$ the energy and momentum of the black holes at the time of production by the cosmic ray.
1. Accretion slowdown
For the smallest black holes, such as those in the early stage of the accretion, the target particles should be thought of as partons, moving relativistically inside the nucleon. In the black hole rest frame, denoted by a prime, incident partons have energy and momentum
where ${E}_{p}$ and ${p}_{p}$ are the parton energy and parallel momentum component in the star rest frame. Note that for the first few collisions ${E}_{p}^{\prime}\sim M$, since ${E}_{i}/M\sim M/{m}_{p}\sim M/{E}_{p}$. But as the black hole slows, ${E}_{p}^{\prime}$ becomes much smaller than $M$.
If black holes can capture matter via their gravitational fields and accrete within Earth, then their gravitational fields will similarly capture and scatter matter while moving through any astronomical object, and we will consistently work within such a framework. Some general features of gravitational scattering and/or capture of relativistic particles in the field of a $D$dimensional black hole are reviewed in Appendix C. There we find the minimum impact parameter, ${\widehat{b}}_{\mathrm{min}}R$, below which relativistic particles enter the capture regime. This can be calculated classically, or defined quantummechanically in terms of the capture cross section,
with closely corresponding results in the cases of interest. A capture collision should result in the parton transferring its momentum and is expected to result in much of the parton energy also being absorbed, although it is also likely that some of the parton energy will be radiated in the process. The change in the black hole mass and momentum are thus parametrized as
We expect ${c}_{ac,p}\simeq 1$, and have $0<{c}_{ac,M}<\mathrm{min}(1,{c}_{ac,p})$ parametrizing the fraction of energy absorbed.^{11} Back in the star frame, the momentum change will then be
where the terms proportional to ${p}_{p}$ contribute zero when averaged over different parton momenta. One can combine these equations with the capture cross section to determine the accretion contribution to the momentum and mass variation. For a black hole of velocity $v\sim 1$ in a matter distribution with parton density $n$, one finds:
where
This leads to the following evolution as a function of path length $\ell $:
where we introduced the energy density $\rho \simeq n\u27e8{E}_{p}\u27e9$. We see that in the limit ${c}_{ac,M}={c}_{ac,p}$ (“perfect accretion”), there is no average momentum transfer.
2. Coulomb slowdown
Consider now the case of gravitational elastic collisions. For the earliest collisions, $R\gg 1/{p}_{p}^{\prime}$, which is the regime of classical particle scattering. Once $\gamma $ has decreased by a factor of approximately ten, the wavelength becomes longer than $R$ and we enter the quantum regime.
Thus, we consider incident partons outside the capture regime, classically described by impact parameter $b>{\widehat{b}}_{\mathrm{min}}R$. The momentum loss of the black hole is
where $\theta $ is the CM scattering angle. (For all but the first collisions, the CM frame is well approximated by the black hole frame.) One can then sum over collisions, as in the accretion case, to obtain the differential momentum loss. In terms of the differential cross section, this takes the form
where ${\theta}_{c}\sim 1$ represents the maximum angle avoiding capture. Here the parton momenta have averaged to zero. In parallel with (5.13), we write this as
where
the corresponding classical expression is
The parameter ${c}_{sc}$ is estimated in Appendix C, yielding for the quantum case the values (0.5, 0.25, 0.17) for $D=5\u20137$. Note that as a result of ${\widehat{b}}_{\mathrm{min}}>1$, gravitational Coulomb scattering plays a subdominant role.
3. Slowdown to $\gamma \sim 1$
We now combine the Coulomb and accretion stopping rates to determine the length required to slow down the black hole to the nonrelativistic regime. As noted, initially $E{E}_{p}\sim {M}^{2}$, but as the energy falls, the mass term in (5.12) dominates. Then the Coulomb and accretion stopping formulas, (5.17) and (5.13), have the same form. Moreover, dividing (5.14) by their sum, and defining
we find in this regime
with solution
This, together with the dependence $R(M)={R}_{i}(M/{M}_{i}{)}^{1/(D3)}$, then allows us to integrate the sum of (5.17) and (5.13) and deduce the distance $d$ to evolve to a given final momentum $p$ and mass $M$ related by (5.22):
The lefthand side defines the column density, $\delta (d)$, as a function of $d$.
The momentum where the black hole becomes nearrelativistic, $p\approx M={M}_{\mathrm{NR}}$, is obtained from Eq. (5.22) as
Since, as we have seen, the initial boost ${\gamma}_{i}={p}_{i}/{M}_{i}$ is large, growth of the mass in this phase can be significant. The corresponding column density is
For rough benchmarks we can replace the righthand side by ${M}_{0}^{3}/\pi $. If we work with constant density, we see that the stopping distance is then approximated by $d\sim {d}_{0}={M}_{0}^{3}/\pi \rho $, as defined in (4.21).
In view of the value for Earth ${d}_{0}(E)\approx 3\times {10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$, these mechanisms cannot efficiently slow down neutral CRproduced black holes in Earth, or in other bodies such as planets and ordinary stars.^{12} For the same reason, typical black holes produced at the LHC are expected not to be captured by the Earth (see Appendix F), posing no risk; however, there is small but finite probability for them to be produced with velocities small enough to become gravitationally bound to the Earth and, in the hypothetical case of stability, to begin accreting.
On the other hand, for a neutron star with densities surpassing ${10}^{14}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{3}$, one has ${d}_{0}(NS)\lesssim 0.01\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$. Thus neutron stars can promptly slow down such black holes, and then quickly bring them to below the escape velocity, which for a neutron star is close to $v\sim 1$. Finally, for white dwarfs, whose central density can exceed $\rho ={10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{3}$, one finds ${d}_{0}(WD)\sim 1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, compared to radii in the ${10}^{3}\u2013{10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$ range. Thus, in order to establish stopping in white dwarfs, we need to make a complete numerical analysis, considering also the nonrelativistic phase of the slowdown.
C. Stopping in white dwarfs
Stopping scales in white dwarfs are not enormously far from their radii, motivating a more complete treatment. We begin by noting that once a black hole reaches the nearrelativistic regime, $p\sim M$, it must be further slowed to below the escape velocity in order to be trapped. Thus we must understand nonrelativistic slowing.
We consider white dwarfs with masses $M\approx {M}_{\odot}$, which have radii ${R}_{\mathrm{WD}}\approx 5500\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$. Such a white dwarf has an escape velocity ${v}_{\mathrm{WD}}\sim 2\times {10}^{2}$. The material in such white dwarfs is described as a fluid of degenerate electrons in which are embedded atomic nuclei, for example, carbon and oxygen. (Later, at time scales $\sim \mathrm{Gyr}$, this material can crystallize.)
1. Nonrelativistic stopping
At the end of the relativistic regime, the black hole has a mass ${M}_{\mathrm{NR}}\approx p$ given by (5.24). In addition to $R$, capture dynamics can in principle be governed by the capture radius for free particles, $R/v$ or ${R}_{\mathrm{EM}}$, defined using white dwarf parameters, ${a}_{\mathrm{WD}}\sim {10}^{10}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$. However, in the regime $v\gtrsim .02$, these radii are all less than or of order ${r}_{N}$ (see Appendix D), so the capture is dominantly subnuclear.
As the black hole moves through the stellar material, it collides with nucleons at a rate
When a collision occurs, it moves nonrelativistically through the nucleon. As it does so, it is bombarded by the relativistic partons within the nucleon. The average mass and momentum collected during such a transit is given using (5.10) and (5.11), times the average time $\Delta t=4{r}_{n}/3v$ of the transit; combining with (5.26) gives
Note two possible caveats to these formulas. First, in a collision with a single nucleon, $\Delta M$ cannot be bigger than ${m}_{p}$. We show that this is true in Appendix D. Second, there could be enhancements of $\Delta M$ due to the fact that if the black hole captures a parton, the QCD string can pull in more energy, whether or not it breaks. (Moreover, since most of the nucleons are in nuclei, it may even be that it pulls in more of the nucleus.) Indeed, one might expect a minimum energy captured of $\sim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MeV}$ if the BH captures one parton. Let us, however, stick with this simple and conservative estimate.
Coulomb stopping may be suppressed in this regime, and we conservatively neglect it. As before, one can find an equation relating $M$ and $p$:
Thus, for “perfect accretion,” $p$ remains a constant. We also find
One can then integrate (5.28) to find the column density to a given final mass ${M}_{f}$:
This can be evaluated using the scaling (3.16) of $R$ with $M$, to find the distance travelled to reach mass ${M}_{f}$:
(5.32)
an evolution law governed by larger velocities. Note that this gives the same scale as the evolution to ${M}_{\mathrm{NR}}$, as in (5.25). This is of course a conservative scale, since we have completely neglected any momentum loss due to scattering, and also have neglected possible enhancements due to binding effects with nuclear fragments.
2. Stopping bounds
From these equations we can compute the column densities necessary for stopping. We first note that, by virtue of the fact that ${\gamma}_{i}={p}_{i}/{M}_{i}$ is large, the stopping distance grows with increasing ${c}_{ac,M}/{c}_{ac,p}$, due to the exponential dependence in (5.24). Therefore, we set it to its maximum value, ${c}_{ac,M}={c}_{ac,p}$.^{13}
Notice also that stopping distance increases with decreasing ${c}_{sc}$. Thus, one is tempted to set this to zero. However, even small ${c}_{sc}$ plays an important role. Specifically, consider the bound on the nonrelativistic stopping,
To test its sensitivity to our physical expectation ${c}_{ac,p}\simeq 1$, let us see how much the column density for given ${c}_{sc}$ changes if we take ${c}_{ac,p}=1/4$, as compared to its value for ${c}_{ac,p}=1$ and ${c}_{sc}=0$. Note from ${c}_{ac,M}\le {c}_{ac,p}$ that this would correspond to a reduction in the mass accretion rate of $1/4$. From (5.33) we easily find less than 25% variation in the resulting bound on ${\delta}_{\mathrm{NR}}$ as long as
The tightest constraints on stopping parameters arise for largest ${M}_{i}$ and thus, as we see from (5.5), large boost. Using that equation, or alternately from Fig. 7, we see that there is a large production efficiency for ${\gamma}_{i}\lesssim 3{M}_{i}/{m}_{p}$, or with ${M}_{i}=14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$, around ${\gamma}_{i}\sim 4.5\times {10}^{4}$. Thus, we find the variation of the stopping distance bounded in this fashion so long as ${c}_{sc}>(.12,.07)$ for $D=6$, 7. Moreover, this bound neglects the fact that any reduction of ${c}_{ac,p}$, corresponding to a reduction of accretion, should lead to an increase of scattering, parametrized by ${c}_{sc}$, thus improving the bound. These features arise from the exponential dependence in (5.24); a similar statement is slightly stronger for ${\delta}_{R}$, as a consequence of its additional dependence on ${c}_{sc}$ through its denominator. Rough values of ${c}_{sc}$ for $D=6$, 7 are given (see Appendix C) by (.25, .17). Thus, in addition to the physical expectation ${c}_{ac,p}\simeq 1$, we find the statement that even for small ${c}_{sc}$, one does not increase the stopping distance by varying ${c}_{ac,p}$ over a wide range. We thus take ${c}_{sc}=0$ and ${c}_{ac,p}=1$, and will rely on the resulting column densities to not be more than 25% higher, although we expect that they could be significantly lower.
Taking these values and combining our bounds from (5.25) and (5.33), we find
(5.35)
The column density ${M}_{0}^{3}$ converts to $4.6\times {10}^{12}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{2}$. For our numerical estimates here we shall confine ourselves to black holes produced with ${\gamma}_{i}\lesssim 3{M}_{i}/{m}_{p}$, as above. Since the evolution is dominated by the phase where the wavelength of the incident particle is large compared to the black hole radius, we use the values of the parameters ${\widehat{b}}_{\mathrm{min}}$ corresponding to the quantum absorption, as given in Appendix C 2. To maximize the needed column density, we also use the maximum value of ${M}_{D}$ corresponding to a given mass ${M}_{i}$, namely ${M}_{D}={M}_{i}/3$. Equation (5.35) then leads to the maximum column densities shown in Table I for various black hole masses.
TABLE I.
Column densities ${\delta}_{T}$, in units of ${10}^{15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{2}$, required to stop a black hole of given masses.
Integration of the column density of a ${M}_{\mathrm{WD}}={M}_{\odot}$ white dwarf along a diameter, using the density profiles shown in Fig. 1 [46], yields a column density ${\delta}_{\mathrm{WD}}=2{\int}_{0}^{{R}_{\mathrm{WD}}}\rho d\ell =13\times {10}^{15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{2}$. This number increases to ${\delta}_{\mathrm{WD}}=21\times {10}^{15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{2}$ for ${M}_{\mathrm{WD}}=1.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$, and to ${\delta}_{\mathrm{WD}}=38\times \phantom{\rule{0ex}{0ex}}{10}^{15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{2}$ for ${M}_{\mathrm{WD}}=1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$. The systematic uncertainty on these values, determined by varying the parameters of the white dwarf such as temperature and composition, is of the order of 10% [47].
A comparison between the required stopping column densities and the available stopping power of white dwarfs is shown in Fig. 2.
We thus conclude that a solarmass white dwarf can efficiently stop black holes. In the case of $D=5$ and 6, black holes corresponding to the most conservative LHC scenario, with masses up to 14 TeV and ${M}_{D}$ accordingly large, will stop within a fraction of the maximum column depth for white dwarfs at or above ${M}_{\odot}$. In the case of $D=7$, one needs stars heavier than approximately 1.1 solar masses in order to achieve stopping up to 14 TeV. However, as indicated in Sec.IV C 5, $D=7$ black holes above 6 TeV give rise to accretion lifetimes on Earth in excess of $20\times {10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ years. Our calculated stopping column densities for masses below this are well below the column density of a solarmass white dwarf.
As shown in Table I, the column densities required to stop the heaviest black holes for $D\ge 8$ exceed the stopping power of even the most massive white dwarfs, and therefore we shall only state empirical constraints on such scenarios when discussing the neutron stars case.
VI. BLACK HOLE PRODUCTION ON WHITE DWARFS
In this section we briefly describe the expected rates of black hole production on white dwarfs; for more details see Appendix E. Before discussing such production rates, however, we must discuss the effects of magnetic fields.
A. Magnetic screening
White dwarfs and neutron stars are known to have significant magnetic fields, which can have important effects on the charged cosmicray primaries. (For more details see Appendix G.) For example, in the case of a dipole field of polar strength ${B}_{p}$, an incident ray of charge $Ze$ and momentum $p$ perpendicular to the magnetic axis will have a Larmor radius that depends on the distance $r$ from the center of the object as follows:
where ${R}_{0}$ is the radius at the surface. While even for neutron stars ${r}_{L}$ can be greater than the radius of the star, as shown in Appendix G, with the very highenergy cosmic rays required one must consider the effects of synchrotron radiation. This is studied in more detail in Appendix G, with the result that a ray at incident angle $\theta $ and with mass number $A$ will have an energy at ${R}_{0}$ determined in terms of its incident energy ${E}_{\infty}$ by
where
Thus, ${E}_{\mathrm{max}}$ sets an effective maximum energy for cosmic rays that penetrate to the surface of the star; for protons, and normalizing to typical white dwarf parameters, we find that
Since, as we will find, optimal bounds come from considering cosmicray energies up to $\sim {10}^{20}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, we see that in order to avoid significant magnetic screening, we must consider white dwarfs with magnetic fields ${B}_{p}\lesssim \mathrm{few}\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$. Bounds from stars with larger fields are still achievable, since cosmic rays incident at angles closer to the magnetic poles will experience reduced energy loss, but this leads to a reduction of rates for acceptable cosmic rays (see Appendix G). We will return to the question of magnetic fields for neutron stars, where we will find them to be somewhat more problematic.
B. Production rates
Production rates can be computed using the known fluxes of cosmic rays, together with cross sections obtained by convolving partonlevel cross sections with parton distribution functions. We will briefly summarize these calculations here; more details are supplied in Appendix E.
The basic partonlevel production cross section is of the form $\sigma \approx \pi {R}^{2}(\sqrt{\widehat{s}})$, where $\sqrt{\widehat{s}}$ is the CM energy of the pair of partons forming the black hole. This is only an estimate; one must take into account that not all the collision energy is captured by the black hole, and even less is captured as impact parameters grow comparable to $R(\sqrt{\widehat{s}})$. Thus, to be more precise, we account for this via an inelasticity factor, $y$; we conservatively summarize trappedsurface calculations of the inelasticity [48–51]^{14} by a simple dependence like that in [8]
To be conservative for cosmicray production rates, we use a lower than expected value, $y=0.5$ [8], for the inelasticity. This corresponds to a limit of 7 TeV for the maximum value of black hole mass that can be produced at the LHC. The actual value of $y$ may be higher, and its reach could be slightly extended due to quantum fluctuations. For this reason we also consider values in the range $0.5<y<1$ (the upper limit being an unrealistic extreme [8,51,53]) to allow for black hole production at the LHC all the way up to the largest available energy, namely 14 TeV. In making the estimates of cosmicray production rates we shall conservatively choose the value of $y$ corresponding to the smallest possible inelasticity compatible with production of a given mass value at the LHC, namely $y={M}_{\mathrm{min}}/14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$. Furthermore, again to be conservative in our rate estimates, we take the minimum black hole mass to be ${M}_{\mathrm{min}}=3{M}_{D}$; for example, [5] used as a benchmark the lessconservative value $M=5{M}_{D}$.
The resulting cross sections are then convoluted with the CTEQ6M [54] parton distribution functions. The resulting nucleon cross sections are, in turn, convolved with the measured ultrahighenergy cosmicray flux, extracted from the latest Auger spectra [40]. In the case of cosmicray primaries that are nuclei of mass number $A$, one must also use a nucleon energy reduced by this factor. The rate calculations of Appendix E are carried out for two test cases: that of incident protons, and that of incident iron nuclei ( $A=56$). Partial results are summarized in Table II, for production on a white dwarf of radius 5400 km, and more results are presented in Appendix E. (In that appendix we also present figures resulting from a 20% hypothesized overestimate of cosmicray energies, to model the quoted Auger energy resolution of $\pm 20\%$ [40,55].)
TABLE II.
Black hole production rates, per million years, induced by cosmic rays impinging on a $R=5400\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$ white dwarf. ${N}_{p}$ refers to the case of 100% proton composition, ${N}_{\mathrm{Fe}}$ refers to 100% Fe. ${M}_{D}={M}_{\mathrm{min}}/3$ and $y={M}_{\mathrm{min}}/14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$.
According to the stopping calculations of Sec. V, not all cosmicrayproduced black holes are stopped by a white dwarf. In order to allow for sufficient column density, we must require the cosmic rays to reach the white dwarf at an angle sufficiently close to the azimuth as to force the black hole to traverse a sufficient fraction of the full column density of the star. The angular dependence of the column density is shown in the right plot of Fig. 1. In the case of $D=5$, our calculated column densities required for stopping are a mere few percent of those available and we can therefore easily accept a reduction to 10% of the available column density. In the cases of $D=6$ and 7, to be conservative we can allow for at least 30% and 80% of the full available column density, leading to a reduction of the useful cosmicray flux to a level of 10% and 2%, respectively. Multiplying these efficiencies by the number of events given in Table II, and integrating over a period of ${10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$, leads to total numbers of accumulated black holes larger than 5000, even in the totally extreme case of 100% Fe composition of the cosmic rays. Notice that even in the most conservative case of rates obtained for $M=14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$ with $y=0.5$ (see Table VII in Appendix E 2), the number of events in $5\times {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ still exceeds 100 for each value of $D$.
VII. BLACK HOLE CATALYSIS OF WHITE DWARF DECAY
The conclusion of the preceding sections is that white dwarfs with solarsize masses, as well as neutron stars, have sufficient ability to stop a cosmicrayproduced black hole, and that significant production rates for such black holes can, in particular, be achieved on white dwarfs. We will seek bounds from the statement that any stable black holes that could be produced on Earth will be produced and trapped in cosmicray collisions with these astronomical objects. The remaining problem is to examine the subsequent evolution.
We have, in particular, argued in Sec. V C that a black hole will be brought to zero velocity, largely through accretion and scattering, deep within a white dwarf. By this time the mass of the black hole may grow significantly from its initial value. The surrounding medium consists of nuclei, for example, carbon and oxygen, embedded in a sea of relativistic degenerate electrons. A lower bound on the relative velocities between the black hole and these nuclei is given by the thermal velocity; for temperatures $\gtrsim {10}^{7}K$, these are at least of size ${v}_{T}\gtrsim 3\times {10}^{4}$. At times $\mathcal{O}(0.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr})$ white dwarf cores begin to crystallize, but we will find accretion to be relevant well before this time.
A. Subatomic accretion
A small black hole will only exert an influence when within the nucleus, as described by our “bagmodel” discussion in Sec. V C 1. However, once larger it will have a longerscale influence. In general its gravitational force must compete with electrostatic forces between the nuclei and their surroundings. We can estimate these forces as in the discussion of atomic matter on Earth—a displacement of a nucleus from its equilibrium position is expected to produce a dipole force resulting from interactions with the ambient electron cloud, and this is estimated to be of a size given by (4.6), where one uses the typical internuclear separation ${a}_{\mathrm{WD}}\approx {10}^{10}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$ corresponding to $\rho \approx {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{3}$. This leads to a characteristic electromagnetic capture radius ${R}_{\mathrm{EM}}$ as given in (4.10), with ${\beta}_{D}$ given by (4.11), together with the approximate value
For a black hole evolving from $M\sim {M}_{D}$, the radius ${R}_{\mathrm{EM}}$ will initially be smaller than both the nuclear scale and the velocitydominated capture radius $R/v$. One can readily check when ${R}_{\mathrm{EM}}$ exceeds the latter; this happens at
Thus, for $D\ge 6$, by the time ${R}_{\mathrm{EM}}$ reaches $\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}$ it governs capture, though for $D=5$ there is a brief phase where $R/{v}_{T}>1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}$ governs the capture rate. Both the subnuclear phases and this phase are governed by very short time scales.
We will focus on cases where ${R}_{D}>{a}_{\mathrm{WD}}$, as they include scenarios with short evolution times on Earth. These, in particular, include the nonwarped cases of $D=6$, 7. The evolution time up to the scale ${R}_{\mathrm{EM}}={a}_{\mathrm{WD}}$ is then given by $d/{v}_{T}$, with $d$ the electromagnetic evolution distance given in (4.22). This yields time scales for this phase
which are quite short.
B. Bondi accretion
Once ${R}_{\mathrm{EM}}>{a}_{\mathrm{WD}}$, one enters the Bondi accretion phase. Since, by assumption, ${R}_{D}>{a}_{\mathrm{WD}}$, this is initially $D$dimensional up to ${R}_{D}$, and then fourdimensional beyond ${R}_{C}$. In cases with warping, there can also be a warped phase intermediate between ${R}_{D}$ and ${R}_{C}$.
Consider first the unwarped cases, with $D=6$, 7. The relevant time scales are given by Eqs. (4.43) and (4.48) for evolution up to ${R}_{D}$, and from ${R}_{C}$ onward. To be conservative, we now model the transition phase between ${R}_{D}$ and ${R}_{C}$ by assuming the slowest evolution, namely, a 4dimensional growth, with a constant Bondi radius equals to ${R}_{D}$, until the mass reaches the value corresponding to ${R}_{B}={R}_{C}$. This gives an evolution time scale:
The evolution time scales are determined by the white dwarf parameters (for a $M={M}_{\odot}$ white dwarf [56]) ${d}_{0}\approx 1.4\times {10}^{6}s$ and ${c}_{s}\approx 1.4\times {10}^{2}$. The combination ${d}_{0}{c}_{s}$ has a value that is approximately $1.5\times {10}^{4}$ times the Earth’s value (4.44). The resulting time scales will therefore be comparatively shorter:
Taking the largest value of relevance for the LHC, ${M}_{D}\sim 4.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$, and using the value of ${\lambda}_{D}$ appropriate for a relativistic electron gas, the longest possible phase of the evolution in $D=7$ does not exceed $8\times {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. We recall that, in this largest $M$ condition, the Earth’s lifetime for $D=7$ exceeds ${10}^{10}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ years by a large margin. We also notice that if we consider the case of more massive and thus denser white dwarfs, the time scales are reduced. For example, the central densities of white dwarfs of mass $M=1.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$ and $M=1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$ are 2 and 4 times, respectively, larger than for $M={M}_{\odot}$ [56], leading to accordingly shorter evolution times.
From this analysis we conclude that Bondi accretion time scales for the unwarped $D=6$, 7 scenarios are quite short, especially as compared to known white dwarf lifetimes that exceed 1 Gyr. We have argued in Appendix B that these are not modified by an Eddington limit, at least until accretion macroscopically disrupts the star. If there were an Eddington limit, Appendix B argues that one is even more likely to find one for Earth. Moreover, as discussed in Appendix B, radiation of an ensemble of black holes at Eddington fluxes would interfere with white dwarf cooling, providing an independent argument against this possibility. We note, parenthetically, that in the true macroscopic regime, when the black hole starts to exert largescale effects on its host body, the evolution may well not be Bondi, but in any case would disrupt the object in question.
C. Generalized scenarios
We next consider a more general warped scenario, with ${R}_{C}\gtrsim 15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $. In this regime, we find from (3.26) that ${R}_{C}/{R}_{D}\lesssim 20$, so ${R}_{D}>{a}_{\mathrm{WD}}$. The growth will then be governed by Bondi evolution that is $D$dimensional up to ${R}_{D}$, warped between ${R}_{D}$ and ${R}_{C}$, and fourdimensional between ${R}_{C}$ and large mass.
For ${R}_{D}\ll {R}_{C}$, the bound on the first phase is significantly smaller than the other two. As discussed in the case of accretion in Earth, the evolution in the warped regime in general involves a large change in mass in a relatively small change of radius. For such a slowgrowth law, the growth time is governed by the upper limit of the radius range, ${R}_{B}={R}_{C}$. This implies that the time scale for growth through this phase is given by
The time for growth from ${R}_{C}$ to large mass is the same, from (4.41). For ${R}_{C}\gtrsim 15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, this yields time scales ${t}_{\mathrm{WD}}\lesssim 5\times {10}^{6}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. In fact, even in the unrealistic case of evolution via a constant radius relationship ${R}_{B}={R}_{D}$ over the entire warped range up to just below ${R}_{C}$, this would only enhance this time scale by $({R}_{C}/{R}_{D}{)}^{2}\sim 400$, and thus the slightly larger value ${R}_{C}\gtrsim 30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ would still be in the range constrained by experimental bounds ( $\lesssim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr}$). Indeed, one can directly compare the white dwarf evolution in the range near ${R}_{B}={R}_{C}$ to that on Earth; from (4.41) we find
Taking for example $\Gamma =5/3$ for Earth and $\Gamma =4/3$ for a white dwarf, this yields a ratio of accretion times
D. D. Summary of white dwarf constraints
This and the preceding sections have argued that in the hypothetical TeVscale gravity scenarios possibly relevant to LHC, (1) cosmic rays will produce significant numbers of black holes on white dwarfs of low ( $\lesssim \mathrm{few}\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$) magnetic fields, on time scales short as compared to known white dwarf lifetimes; (2) such black holes, even if neutral and with the highest masses accessible at the LHC, will be stopped on white dwarfs with masses $M\gtrsim {M}_{\odot}$, by accreting and scattering the dense matter of the star during their transit; and (3) the white dwarf will then be accreted. Accretion of a white dwarf has been argued to be more rapid than that of Earth. Different considerations reinforce this statement. First, before the $\sim \mathrm{Gyr}$ time scale, whitedwarf matter is in a liquid form, in contrast to significant solidity of matter in Earth. Second, white dwarfs pack the mass of the Sun into a region the size of the Earth, so are much more dense, and have much higher internal pressures, assisting accretion.
Several surveys of lowmagneticfield white dwarfs exist in the literature. The use of Zeeman spectropolarimetry, in particular, has allowed detection of fields down to the level of a few kG. White dwarf masses are determined by spectral measurements of surface gravities, with parallax and gravitational redshifts serving as crosschecks; see e.g. [57,58]. Ages are determined through white dwarf cooling; a textbook account appears in [59], with further discussion in [58,60,61]. Several known white dwarfs satisfying our criteria of mass $M\gtrsim {M}_{\odot}$, ${B}_{p}\lesssim \mathrm{few}\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and age $T\gtrsim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Myr}$ can be found, for example, in [62–65]. When not given explicitly, the ages can be inferred from the masstemperature relations, as discussed e.g. in [60,61]. Some examples of relevant stars are^{15}

(i) $\mathrm{WD}0346011$ [62,64], with parameters $M=1.25\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$, ${B}_{p}<1.2\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\sim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Myr}$;

(ii) $\mathrm{WD}1022301$ [64], with $M=1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$, ${B}_{p}<1.2\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\gtrsim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Myr}$;

(iii) $\mathrm{WD}1724359$ [64], with $M=1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$, ${B}_{p}<1.2\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\sim 150\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Myr}$;

(iv) $\mathrm{WD}2159754$ [64], with $M=1.17\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$,^{16} ${B}_{p}<3\times {10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\sim 2.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr}$;

(v) $\mathrm{WD}0652563$ [64], with $M=1.16\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$, ${B}_{p}<2.7\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\sim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Myr}$;

(vi) $\mathrm{WD}1236495$ [64], with $M=1.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$, ${B}_{p}<3\times {10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\gtrsim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr}$;

(vii) $\mathrm{WD}2246+223$ [62], with $M=0.97\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$ [65], ${B}_{\ell}=1.5\pm 13.8\times {10}^{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\sim 1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr}$;

(viii) $\mathrm{WD}2359434$ [63], with $M=0.98\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}_{\odot}$ [64], ${B}_{\ell}=3\times {10}^{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$, and $T\sim 1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr}$.
The above arguments thus state that comparison of respective accretion rates, together with survival of white dwarfs to observed time scales $\gtrsim {10}^{8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$, implies survival of Earth for a significantly longer time, and, in particular, longer than the natural solar lifetime.
Finally, as in Sec. IV C 4, we can estimate the lifetime of a white dwarf, should it capture a minimummass primordial black hole. With Bondi evolution from the corresponding initial radius, one finds a lifetime $\approx 1.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gyr}$.
VIII. BOUNDS FROM NEUTRON STARS
A. Production on neutron stars
Neutron stars are very common in the Universe, and in fact provide robust examples of longlived objects in other galaxies. They also represent the highest known densities of matter that have not undergone gravitational collapse to a black hole. Since they are particularly close to densities beyond which black holes are expected to form, one might expect that introduction of a microscopic stable black hole into a neutronstar would rapidly catalyze its decay into a macroscopic black hole. The known stability of NSs, with lifetimes significantly exceeding ${10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$, therefore offers the prospect of limits on microscopic black hole stability and accretion power.
However, known neutron stars have strong magnetic fields, which are observed to range upwards from $\sim {10}^{8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$. In the case of a field of ${10}^{8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{G}$ and a radius ${R}_{0}=10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, (6.4) yields a maximum energy $1.8\times {10}^{17}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$ for protons impinging perpendicular to the field axis, giving collisions just above the LHC CM energy; the maximum energy is only about 16 times higher for heavy elements. One can avoid this limit for cosmic rays incident near the magnetic poles, but the acceptance for protons of energies in the optimal range of $\sim 5\times {10}^{18}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$ is estimated in Appendix G to lead to a reduction of acceptable flux by a factor of approximately ${10}^{3}$, considerably weakening the resulting bounds.
1. Production in binary systems
This suppression suggests we consider a more reliable way to inject CRproduced black holes into a neutron star. Many NS binaries are known, and, in particular, parameters and evolution of lowmass binaries are well understood. Moreover, in such binaries, the companion to the NS can subtend a significant solid angle in the sky of the NS, as described in Appendix H 1. Cosmic rays which would hit the NS but whose direction intersects the companion will therefore scatter on the companion. In our TeVscale gravity scenarios, part of the time they will therefore convert to black holes, which then impact the neutron star. Since we only need bounds if stable black holes are neutral, the magnetic field of the NS is irrelevant for these. Such a production mechanism produces a “fullcoverage equivalent” (FCE) given by
where $\Delta \Omega $ is the solid angle subtended by the companion, and where we allow for time dependence due to evolution of the binary system. In order to compute the actual production rate on the neutron star, we use the uncorrected rates of Appendix E, times the number of years of FCE. A survey of known classes of binary systems (see Appendix H 1) reliably yields FCE’s in the 2 Myr range, resulting from systems with a 1 Gyr lifespan. The neutronstar production rates are exhibited in Table IX and in Fig. 8 of Appendix E 2. A summary of that table, focusing on the most interesting cases of $D\ge 8$, is shown here in Table III. We find that in the example of a flux of even only 10% protons, we have a rate for the extreme case of 14 TeV black holes that is $\approx 5/\mathrm{Myr}$, and so 2 Myr of FCE would indicate that typical such systems have experienced sufficient black hole production to initiate the accretion. Lessextreme (and still quite robust) binary scenarios provide significantly higher rates. However, a greater dominance of heavy elements reduces the range of such bounds.
TABLE III.
Summary of black hole production rates, per million years, induced by proton cosmic rays impinging on a $R=10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$ neutron star. ${M}_{D}={M}_{\mathrm{min}}/3$ and $y={M}_{\mathrm{min}}/14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$.
At energies below the GZK cutoff, there are indications of a significant component of heavy elements. There are both theoretical and experimental indications that one transitions to a significant proton component at the GZK cutoff. On the theoretical side one can cite both the match of the observed spectrum to that from models of proton acceleration, and expectations that gammaray bursters (GRBs) and active galactic nuclei (AGN) primarily accelerate protons (see [67–69] for more discussion.) On the experimental side, indications of a predominantly light composition include both measured penetration depths of showers [70,71] and, most recently, correlations of arrival directions with known AGN [72,73] (see however [74]). Thus, while not all scenarios are definitively eliminated by such a bound, it appears likely that these bounds will be strengthened with future data on composition.
2. Production via cosmic neutrinos
Primary cosmic rays propagating through the 3K cosmic microwave background photons will experience significant interactions above the GZK energy $\approx 5\times {10}^{19}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$. These interactions produce a “guaranteed” flux of neutrinos (see e.g. [68,75,76]), which avoid the synchrotron losses of charged cosmic rays. Using these fluxes, Appendix E calculates production rates on neutron stars. For example, in the very conservative scenario of requiring black holes to have 14 TeV mass, and using both $D=5$ and our most conservative inelasticity assumptions, $y=0.5$, one finds production rates $\gtrsim 5000/\mathrm{Myr}$, as shown in Table X.
These thus suggest a very robust bound for production on neutron stars. While we believe it is quite good, we will not take this bound with absolute certainty, for two reasons. First, while the physics of the GZK effect is quite robust, and moreover appears to be in the process of being experimentally confirmed via correlations of the highestenergy cosmic rays with AGN [72,73], experiments have not reached sensitivity sufficient to measure the cosmic neutrino flux.^{17} Second, there exist proposals that baryon number conservation is enforced in higherdimensional braneworld models through reduced interactions between neutrinos and quarks by virtue of these living on different branes [78]. While these models are not compelling, they would seem to raise a small possibility that neutrino cosmic rays would not produce black holes the same way that nucleons do.
B. Catalysis of neutronstar decay
Because of the immense pressures inside a neutron star, one expects introduction of even a microscopic black hole to rapidly catalyze its decay. To understand this process, we note that neutron stars have different layers, a crust extending to a depth of $\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, and under this, matter at nuclear densities. Since treatment of accretion is simplest in this inner region, we would like to understand whether a BH can penetrate to this distance.
1. Penetration to core
The slowing distance (5.25), together with a subsequent phase of slowdown to subescape velocities, $v\lesssim 0.1c$, may or may not permit penetration to depths $\gtrsim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{km}$, depending on details. Note that the characteristic distance ${d}_{0}$ can be rewritten in appropriate nuclear units as
and that crustal densities range from ${10}^{6}\u2013{10}^{1}{m}_{p}/{\mathrm{fm}}^{3}$.
Even if penetration does not occur during slowdown, there is a different argument that it takes place on rapid time scales. To see this, instead assume that a sufficiently slow black hole could become temporarily bound in the crust, by absorbing a parton and thus binding to the medium by strong forces. This binding should, however, be temporary, as the scenario we wish to constrain is that where black holes do not remain charged, but instead discharge through Schwinger production. Even ignoring this, if a black hole is bound to a nucleon via the color force, it will absorb the remaining partons of the nucleon, and thus become color neutral, on a relatively short time scale. One can readily estimate this time scale. A nucleon has a parton density of order $1/{\mathrm{fm}}^{3}$, and partons within the nucleon travel at speeds $\approx c$. With the smallest possible absorption cross section, of order $\sigma \sim \pi /{\mathrm{TeV}}^{2}$, we find a characteristic absorption time ${t}_{\mathrm{abs}}\sim {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}$.
The neutralized black hole will then continue to fall in the net gravitational field of the neutron star until another such binding/neutralization event. The characteristic distance between such events is
where $x$ parametrizes the typical nucleon energy fraction per parton, and we have expressed the capture radius ${r}_{c}$ in ${\mathrm{TeV}}^{1}$ units. The corresponding time between such collisions is given by
where ${g}_{\mathrm{NS}}$ is the gravitational acceleration near the surface of the neutron star,
This characteristic time scale is much longer than ${t}_{\mathrm{abs}}$, and thus sets the speed with which the black hole can penetrate the crust. The corresponding average velocity is
The mean free path ${d}_{\mathrm{free}}$ reaches a minimum near the bottom of the crust, and consequently the average velocity is slowest there. We can therefore bound the crust penetration time by using $\rho ={m}_{p}/{\mathrm{fm}}^{3}$ to derive a minimum gravitational drift velocity and time:
Thus the black hole should rapidly penetrate the crust and enter the neutronfluid region of the core.
2. Accretion from within a neutron star
Like in atomic matter, one might expect different phases for black hole accretion within a NS, depending on the relative range of the black hole’s influence as compared to the radii ${R}_{D}$, ${R}_{C}$ representing crossover to lowerdimensional behavior, and as compared to 1 fm, the characteristic separation between nucleons, which delineates the crossover from microscopic to macroscopic absorption in the NS context.
To better understand these points, let us begin by comparing the force due to a black hole to typical forces between the neutrons in the NS, which are of size GeV/fm. Specifically, the force on a nucleon of mass ${m}_{p}$ at distance $r$ is of order
Equating this to the typical nuclear force, we find that gravity beats such a force at scales
which are only moderately larger than the Schwarzschild radius in the regime $R<{r}_{N}$. Thus a conservative (i.e., for the purposes of NS evolution, slow) estimate of the evolution is given by simply taking the capture radius in (4.1) to be ${r}_{c}=R$ in the subnuclear regime $R<{r}_{N}$. To estimate the accretion rate, one needs the flux $F$. This receives contributions both from the velocity of the black hole, and from the Fermi motion of the partons in the nuclei. The latter produces a flux $F\sim \rho $, in units where $c=1$, and thus a geometric rate law
The evolution equations are of the same form as Eq. (4.31), with the replacements ${c}_{s}=1$, ${\lambda}_{D}=1$, and ${R}_{B}\to R$. The resulting time is analogous to Eq. (4.39), and evaluated at $R={r}_{N}\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}$ gives:
leading to time scales ranging from a fraction of a second to at most a few weeks for $6\le D\le 11$.
When the black hole enters the regime $R\gtrsim {r}_{N}$, the absorption becomes macroscopic—the black hole is capable of absorbing multiple nuclei, and its gravitational range exceeds mean free paths. As Appendix B describes, an Eddington limit would be even more problematic for a neutron star, given its high density and opacity, and so evolution is described as Bondi until the black hole reaches a scale where it disrupts the star.
The corresponding growth laws are those given in Sec. IV. In unwarped scenarios, we find for evolution from ${R}_{B}={r}_{N}$ to ${R}_{D}$ the growth time (4.43), and for evolution from ${R}_{C}$ up to large scales the comparable time scale (4.48), with ${\lambda}_{D}\to {\lambda}_{4}$. For the stage in between, we use, as in the case of the white dwarfs, the conservative time in Eq. (7.6). Taking a typical value $\rho =2\times {10}^{14}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr}/{\mathrm{cm}}^{2}$ gives ${d}_{0}=7\times {10}^{3}s$, and the speed of sound is of magnitude ${c}_{s}\sim 0.1$. The time scales are therefore about ${10}^{6}$ times smaller than those of the white dwarf. The longest evolution corresponds to the 4dimensional phase from ${R}_{D}$ to ${R}_{C}$ in $D=11$ dimensions, with a time scale of ${10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$; for $D\le 7$, times are $\lesssim 50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$.
In the more general warped case, the growth times are, as in the white dwarf case, dominated by the upper end of the warped phase and its crossover to fourdimensional accretion, with time scale (7.13). For neutronstar parameters and a value ${R}_{C}\ge 5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, this yields times of size ${t}_{\mathrm{NS},w}\sim 20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$.
Combining the results of this and the preceding subsections, we find that the ratelimiting step to destroy a neutron star is the time required for a black hole to be produced and reach the surface of the neutron star. Once it reaches the core, the accretion times are very rapid compared to the neutron star’s lifetime, $\mathcal{O}$ (Gyr). These bounds appear quite challenging to avoid. In order to do so, one would need a significant deficit of light cosmicray primaries, together with a heavy ( $\gtrsim 7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{TeV}$) minimum black hole mass and only systems with low FCE, and one would have to have a neutrino flux that is either suppressed by unknown mechanisms or is unusually nonreactive.
As a final note, in our framework we can estimate the lifetime of a neutron star that captures a primordial black hole of mass $\mathcal{O}({10}^{15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{gr})$. Our parameters yield a time of order $3\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. We note that such processes have been proposed as the origin of some gammaray bursts [79], with roughly comparable accretion times. The present analysis, in addition to giving the analogous accretion time scales for Earth and white dwarfs, lends further detail to such a possibility through our description via Bondi accretion, and through our arguments against an Eddington limit.
IX. IX. SUMMARY AND CONCLUSIONS
In this paper we have studied accretion of hypothetical stable TeVscale black holes in two primary contexts: the Earth, and compact stars—white dwarfs and neutron stars.
For Earth, we identified two main evolution domains: that where the black hole’s gravitational range of influence is less than the atomic scale, and that where it is greater. An important distinction occurs depending on where the crossover radius ${R}_{C}$ to fourdimensional behavior lies relative to the atomic scale. In particular, if ${R}_{C}$ is in the subatomic regime, evolution is fourdimensional at both subatomic and macroscopic scales; this case includes unwarped scenarios with $D\ge 8$. In this case we have argued that this evolution occurs on times longer than the expected natural solar lifetime, in two different ways: via a microscopic argument, and via a macroscopic, hydrodynamic argument. In both approaches, we used conservative assumptions, leading to the largest accretion rates and to fastest evolution. At the end of the first phase, at times $\mathcal{O}({10}^{11})$ years, the mass of the black hole is still small, with a mass of less than a megaton. Such statements extend to more general warped scenarios, and crossover scales up to $\simeq 200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ lead to accretion times longer than the Earth’s natural lifetime.
On the other hand, those cases where ${R}_{C}\gtrsim 15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ have been treated by deriving bounds from white dwarfs, and also from neutron stars. In particular, we have argued that in such scenarios cosmic rays will produce black holes on such astronomical objects, and that these objects will stop even these very highmomentum black holes. We then studied accretion, showing that accreting black holes will disrupt such objects on time scales short as compared to their observed lifetimes. In particular, we found a general relationship (7.15) between accretion times for Earth and for white dwarfs, which, when combined with white dwarf ages exceeding ${10}^{9}$ years, provides a very strong constraint. Thus, the implication of these arguments is that such scenarios, where Earth would be disrupted on time scales short as compared to its natural lifetime, are ruled out.
We summarize here the origin of our constraints, as a function of dimensionality $D$ and of black hole mass $M$.

$D=5$: The evolution scales on Earth for $D=5$ in the case of maximum allowed crossover radius ${R}_{C}$ are quite short. This is a result of the higherdimensional force law extending well into the macroscopic regime. On the other hand, the greater interactivity of $D=5$ black holes makes it possible for those produced by cosmic rays to get promptly trapped in both white dwarfs and neutron stars. In white dwarfs they are produced abundantly, with buildup time scales of the order of few thousand years even assuming a cosmicray composition of 100% Fe, and at the largest Planck mass of interest. After being produced and trapped, in these extreme scenarios they quickly accrete to masses comparable to that of the star on time scales that can be short when measured in years. This would make it impossible for any white dwarf with a mass of the order of 1 solar masses to have survived longer than a few thousand years, contrary to observations. Scenarios with increased warping have correspondingly lower ${R}_{C}$ and longer accretion times. In particular, once ${R}_{C}\lesssim 200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, accretion times on Earth exceed its future lifespan.

$D=6$: The evolution times on Earth for $D=6$ are of the order tens of thousand years, thus short on geological time scales. Once again the main reason is the large extradimensional radius, and the high capture rates. As in the $D=5$ cases, such black holes produced by cosmic rays can be stopped inside dense stars. The production rates are even larger than in $D=5$, and the star accretion time scales for unwarped $D=6$ are comparable to the maximum${R}_{C}$ version of that scenario. With increased warping, again ${R}_{C}$ decreases and accretion times increase.

$D=7$: The time scale for their macroscopic evolution on Earth is in the range of 6–$80\times {10}^{9}$ years, depending on the black hole mass. Furthermore, $D=7$ black holes would be produced plentifully by cosmic rays on white dwarfs, and be stopped inside their surface, if the white dwarf mass is larger than 1.1 solar masses. The evolution times would be longer than in $D=5$ and 6, reflecting the lower growth rate that keeps them microscopic for ${10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ inside the Earth. But within the very conservative estimate of $8\times {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ (for a $M={M}_{\odot}$ star, and shorter in the more massive cases) their accretion process of the white dwarf would be completed. Massive white dwarfs older than a few hundred million years would therefore be ruled out in these scenarios, once again contrary to observation. Again, warping decreases ${R}_{C}$, thus increasing accretion times.

$D\ge 8$: For these black holes the evolution time on Earth is extremely long, with times of size ${10}^{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. This is due to the radius of the extra dimensions being smaller than $1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $, thus forcing most of the evolution to take place in 4 dimensions, where gravity is a totally negligible force. Warping only magnifies this effect. In spite of this slow growth, these black holes would still grow fast enough inside a neutron star to consume it within about ${10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$. The significant production rates on neutron stars when $D\ge 8$, and the existence of billionyear old xray binary systems, provide therefore additional evidence that such black holes either do not exist, or decay promptly.
We also note that these bounds likely extend in case other objects are imagined that could result from highenergy collisions in the relevant energy ranges, that have weakscale cross sections, and that could threaten the longterm stability of matter.
We conclude by first summarizing the conditions needed for our bounds to be necessary to rule out a possible risk. In order for our bounds to have relevance, a sequence of unlikely things would have to be true. First, TeVscale gravity, with a Planck scale no higher than a few TeV, would have to be correct, so that black holes can be produced at LHC. Most workers consider this to be a fascinating possibility, but also a somewhat unlikely possibility. Second, black hole radiance, which has been deeply studied from a number of theoretical perspectives, would have to be wrong, and more general quantummechanical arguments for black hole instability would have to be wrong. Most workers consider this to be an exceedingly improbable, if not impossible, scenario. Finally, one would need a mechanism to shut off the quantum effects responsible for Hawking radiation, but still leave intact either the quantum effects responsible for Schwinger discharge, or some other neutralization mechanism that acts to discharge the resulting stable black holes. It is very difficult to conceive of a consistent physical framework that provides such a mechanism.
In the event that all these conditions are satisfied, one can turn to the considerations of this paper to assess the possible impact on Earth. This paper has argued that in order for such a scenario to have an impact on Earth at time scales short as compared to the natural lifetime of the solar system, in the $5.0\times {10}^{9}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{yr}$ range, the configuration of extra dimensions would have to be such that gravity does not transition to fourdimensional behavior until around the $200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ scale. This apparently requires additional finetuning, reducing the likelihood even further. But beyond that, this paper has argued that such scenarios are ruled out by the longevity of known white dwarfs, on billionyear time scales. In such a scenario, cosmicrayproduced black holes should have catalyzed white dwarf destruction on significantly shorter time scales.
Moreover, decay of observed neutron stars would also have been catalyzed, unless both of two unlikely possibilities are realized, namely, that the composition of ultrahighenergy cosmicray primaries is dominantly heavy elements, and ultrahighenergy cosmicray neutrinos either are not produced, or have suppressed gravitational interactions with partons. To summarize, the present study argues for the following additional layers of safety, beyond those that would have to fail to make this study relevant:

(1) Only in scenarios such that the crossover scale to fourdimensional gravity is larger than about $200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\AA $ does one have significant accretion at times short as compared to the natural lifetime of Earth. This is a priori unlikely, due to the additional finetuning required to realize such a TeVscale gravity scenario.

(2) In these scenarios where black hole accretion time on Earth is short as compared to natural time scales, white dwarfs would likewise be accreted, on much shorter time scales, in contradiction to observation.

(3) Unless cosmic rays have dominantly a very heavy composition, and moreover either the expected neutrino flux does not exist or has unusual gravitational couplings to hadronic matter, neutronstar decay would likewise be catalyzed on time scales contradicting observation.
In short, this study finds no basis for concerns that TeVscale black holes from the LHC could pose a risk to Earth on time scales shorter than the Earth’s natural lifetime. Indeed, conservative arguments based on detailed calculations and the bestavailable scientific knowledge, including solid astronomical data, conclude, from multiple perspectives, that there is no risk of any significance whatsoever from such black holes.
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