Some Astrophysical Implications of Compact Extra Dimensions

Michael J. Longo

University of Michigan, Department of Physics, Ann Arbor MI 48109-1120


There have been many suggestions that there are extra spatial dimensions "outside" of our

normal (3+1)-dimensional space. Generally it is assumed that electromagnetic and

hadronic fields are restricted to the normal dimensions, while gravity can extend into the

extra dimensions. Thus a hadron or lepton is confined in a potential well, and excited states

should exist. I explore the possibility that ordinary hadrons and leptons have excited states

("compactons"), some of which may have electromagnetism and possibly gravity confined

to the extra spatial dimensions. Some of these may be long-lived. If sufficiently long-

lived, relic compactons may exist as dark matter. Black holes may be the source of new

particles with bizarre properties that appear as UHE cosmic rays.




1. Introduction


There has been considerable discussion on the possibility of extra spatial dimensions,

which are compactified on some distance scale R [1]. In gravity-mediated theories the

associated energy scale is M ~ R1 ~ 1013 GeV. In weak-scale compactification theories [2]

the energy scale is ~1 TeV. It has been suggested that the compactification of the extra

dimensions may occur on scales  1 mm with energy scales also in the TeV range [3]. In
all theories, because of the success of the Standard Model, electroweak and hadronic fields

must be confined to the 3 ordinary spatial dimensions, at least down to scales approaching

the Planck scale. Gravity, however, can penetrate into the extra dimensions, and the

strength of the gravitational interaction increases dramatically on scales  R.
In this paper I assume there are extra compactified dimensions with R somewhere

between the Planck scale and 1 mm. I speculate on some of the possible ramifications,

particularly in astrophysics.


2. Some astrophysical implications of large-scale compactification


Given a theory with a built-in length scale R and the greatly increased strength of the

gravitational force for distances  R, it is plausible (though not necessarily compelling) that
black holes with dimensions ~R and masses ~R1 would be formed in the early universe. If

R ~0.1 mm, this would correspond to black holes with masses comparable to that of the

moon. Thus much of the dark matter in the universe might now consist of black holes with

mass smaller than that of the Moon. These would be very difficult to observe. If their

average mass density near the solar system were comparable to that for ordinary matter,

the chance of a close encounter with Earth would be comparable to that for the Earth to

encounter a Moon-sized normal object. If R ~ 1 fm, the corresponding black hole mass is

~1012 kg. Such less massive versions would be more plentiful, but correspondingly more

difficult to observe. They might, for example, pass through the Earth with very little

effect. In (3+1)-dimensional general relativity, black holes with mass  1012 kg would
evaporate due to Hawking radiation with lifetimes much less than the age of the universe.

However, the much greater strength of gravity in (4+n) dimensions for distances  R would
slow their evaporation, so primordial black holes with masses ~104 kg might still exist.[4]

If gravity were significantly stronger at short distance scales, the evolution of the

universe would be profoundly affected while its size was  R. Depending on R, this could
cause difficulties with inflation. In effect we are translating our previous ignorance of

what happens below the Planck scale to the compactification scale.




3. Do ordinary particles have counterparts with different compactified dimensions?


Given the existence of compactified extra dimensions, we can picture an ordinary

hadron or lepton as having the electroweak and hadronic parts of its wave function

confined almost completely to ordinary 3-dimensional space. In other words there is very

little penetration into the compactified dimensions. (The success of the standard model up

to energy scales ~ 1 TeV proves that it is valid to distance scales ~1019 m.) The particle is

effectively confined to a potential well 1019 m wide in the compactified dimensions.



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If ordinary quantum mechanics is applicable in the compactified dimensions, there

must be excited states of ordinary hadrons and leptons that correspond to the excited states

of the classic problem of a particle in a potential well. (We assume, of course, that the

"wall" does not have zero thickness [5].) Some of these might have quite different

symmetries than the ground states. As discussed below, they are likely to have bizarre

properties compared to ordinary hadrons. They might, for example, be the supersymmetric

partners of the ordinary fermions. For definiteness, I shall refer to these states as

compactons. Some compacton states should be at least quasi-stable, similar to the situation

in alpha decay where there is only a large potential well separating the states. This could

give a huge range of lifetimes for different compacton states, but the lowest energy states

are likely to be quite long-lived due to the large potential barrier separating the normal

state and compacton counterpart. Compacton masses could be modest (and possibly

negative!), but they would be very difficult or impossible to produce at accelerators. If

stable or very long-lived, they might exist as relics of the big bang, or they may be

produced in the present universe near or even inside the Schwarzchild radius of black

holes. If they can occasionally escape, they might be the source of UHE cosmic rays or

other cosmic ray phenomena, which are otherwise very difficult to explain.

It is difficult to even speculate on compacton properties. An assumption of theories

with large extra compactified dimensions is that the standard-model fields are confined to

the 3 ordinary spatial dimensions for distances >>R, while gravity can penetrate into the

extra dimensions. Correspondingly, there could be compacton states with strong and

electroweak fields not confined to ordinary dimensions as depicted in Figure 1. Fig. 1(a)

shows the electric field around a normal particle with the x-axis a normal dimension and

the y-axis a compactified dimension. Fig. 1(b) illustrates a possible compacton state with

the electric field extending into the "extra" dimensions and restricted from normal

dimensions. Such a particle, though charged, would have almost no electromagnetic

interactions with normal particles. Formally, a normal state can be transformed into a new

compacton state by rotating the gauge fields in (3+n) spatial dimensions from an ordinary

spatial dimensions into (normally) compactified ones.





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Compactified dimension



Normal
dimension





(a) (b)


Figure 1 (a) A normal charged particle with its electric field confined to ordinary spatial
dimensions; (b) A compacton state with the electric field confined to extra dimensions. Such a
state would have little electromagnetic interaction with normal particles, but might have normal
gravitational interactions.




States might also exist with gravity confined to the extra dimensions [though these

could not be formed by a simple rotation in (3+n) space]. It is thus plausible that the

gravitational interactions of some compacton states are quite different than for ordinary

particles, while their other interactions may be similar. They could, for example, have no

or very weak gravitational interaction with ordinary matter or possibly even repulsive

gravity. Thus once they are formed in a black hole they could easily escape. Anti-gravity

would help to explain the acceleration mechanism for UHE cosmic rays.

The possibility of particles with intrinsic angular momentum components in the extra

dimensions gives even more degrees of freedom.


4. The ultimate symmetry


In a sense this is the ultimate symmetry  symmetry with respect to compactification.

 Some compacton states might have electromagnetic and/or hadronic fields that are

compactified in "ordinary" dimensions and long-range in what we take to be compactified

dimensions. I propose in particular that compacton states exist for which gravity is

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confined to the extra dimensions for distances >>R, just as the standard model fields are

confined to normal dimensions for ordinary particles. This automatically gives these

compactons no or very weak large-scale gravitational interactions with ordinary matter, so

that they would be able to escape a black hole's gravity. Given this scenario, production of

compactons in a black hole is possible. For example, protons falling into a rotating black

hole along the axis of rotation will gain large amounts of kinetic energy and occasionally

collide with matter falling through the disc from the other side. If the black hole is

rotating, the singularity is thought to be disc-shaped, so that the protons can approach

arbitrarily close without actually entering the singularity [6], and thus they could collide

with almost arbitrarily large energies.

As described above, rotating the electromagnetic field (in 3+n dimensions) out of the

normal dimensions into the extra ones could produce a compacton with no long-range

electromagnetic field in normal dimensions [7]. Relic long-lived compacton states with no

electromagnetic or hadronic fields in normal dimensions could be a major component of

dark matter. Another possibility is that compacton pairs could be produced from black

holes by a mechanism similar to Hawking radiation.




5. Compactons as UHE cosmic rays


If symmetry with respect to compactification is a (badly broken) symmetry of

nature, it is plausible that compactons can have a "gravitational charge", just as ordinary

particles have electromagnetic or color charge. A neutral or negative gravitational charge

(mass) would facilitate their escape from black holes.1 The gravitational field of the black

from distances arbitrarily close to the singularity they can occasionally be ejected with

extremely high energies. If their electromagnetic fields are compactified in ordinary

dimensions, their interactions with the microwave background radiation would be quite

weak, while their hadronic interactions could be similar to ordinary hadrons. Thus they



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1 The repulsive gravity of long-lived negative mass compactons surviving from the big bang
or produced later in black holes thus could conceivably account for the cosmological constant.



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hole would accelerate a negative mass compacton. Because the compactons could come

would be able to traverse intergalactic distances without significant energy loss and

interact in our atmosphere much like ordinary hadrons. They might account for cosmic

rays observed with energies >1020 eV [8] that appear to interact in the atmosphere like

ordinary hadrons, but apparently do not interact as expected with the intervening cosmic

microwave background radiation. Compacton states with different hadronic interactions

might also account for other exotic phenomena in cosmic rays such as Centauro events [9].




6. Conclusions



I have explored the provocative possibility that ordinary hadrons and leptons have

counterparts with gravity and/or electromagnetic fields confined to the extra dimensions

rather than the ordinary 3 spatial dimensions. These should exist as excited states of

normal particles whose standard model fields are confined in (3+n) dimensions. Some of

these states may be long-lived.

Compacton states with standard model fields confined to the extra dimensions would

have little interaction with ordinary matter. If their long-range gravitational fields are

confined to extra dimensions, their gravitational interaction with ordinary matter could be

very weak or repulsive. These states, which I call compactons, may be produced in black

holes. They may have quite bizarre properties such as negative or zero gravitational mass

that would allow them to escape from a black hole. Black holes might then be the source

of UHE cosmic rays.

In the early universe the increased strength of the gravitational force for distances

below R would naturally lead to the formation of black holes with dimensions ~R and

masses ~R1. Due to the increased strength of the gravitational interaction on scales <<R,

these black holes are unlikely to have evaporated.





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References



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Witten, Nucl. Phys. B 471 (1996) 135; J.D. Lykken, Phys. Rev. D 54 (1996) 3693.

[2] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429 (1998) 263.

[3] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436 (1998)

257.

[4] Argyres, Dimopoulos, and March-Russell [Phys. Lett. B441 (1998) 96] estimate

lifetimes for black hole evaporation with n compactified dimensions. Their Eq. 16

gives lifetimes comparable to the age of the universe for black holes of mass ~104 kg

for n=2 and M* =10 TeV.
[5] A potential well treatment of compactification is discussed for example by V. A.

Rubakov and M.E. Shaposhnikov, Phys. Lett. 125B (1983) 136. Arkani-Hamed and

Schmaltz [Phys. Rev. D61, (2000) 033005] speculate that the standard-model brane

has finite thickness and the wave function for different fermion species might have

different locations within the extra dimension.

[6] See, for example, S.L. Shapiro and S. A. Teukolsky, Phys. Rev. Letters 66 (1991)

994.

[7] A. Davidson and D.A. Owen [Phys. Lett. 155B (1985) 247] conclude that the extra

dimension becomes fully visible to observers in normal spatial dimensions near the

Schwarzschild horizon of a black hole, so that black holes are windows to the extra

dimensions.

[8] See T.K. Gaisser, T Stanev, Cosmic Rays (Rev. Part. Physics 1998), and references

cited there. Published in Eur. Phys. J. C3 (1998) 132.

[9] Brazil-Japan Collaboration of Chacaltaya Emulsion Chamber Experiment. ICRR-390-

97-13B, Jul 1997. Published in Proc. 25th International Cosmic Ray Conference

(ICRC 97), Durban, South Africa.





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