17 March 2009
Do some ultra-high-energy cosmic rays originate in
higher-dimensional space-time?
Abstract
I speculate that some ultra-high-energy cosmic rays may originate in another universe in flat (noncompact) higher dimensional space-time. By entering our space-time directly over Earth, the rays
may avoid interacting with the Cosmic Microwave Background Radiation and thus be able to
maintain energies well above the Greisen-Zatsepin-Kuz’min bound.
PACS: 11.27.+d, 95.85.Ry, 96.50.sd, 98.70.Sa
Keywords: ultra-high-energy cosmic rays, higher-dimensional space, φ4 -model, kinks
Ronald Bryan, Physics and Astronomy Dept., Texas A&M University, College Station, Texas 77843, U.S.A
979.777.6609 (cell) , 979.693.5554 (FAX)
1
Introduction
In recent years, a few cosmic rays with energies exceeding 100 EeV (1 EeV = 1018 eV)
have been detected from the the extensive atmospheric showers (EAS) that these particles produce. These ultra high-energy cosmic rays (UHECRs) have been recorded by the
Akeno Giant Air Shower Array (AGASA) [1], the original Fly’s Eye detector [2], the HiRes
Experiment[3], and the Pierre Auger Observatory (PAO)[4]. (For reviews, see [5][6]). The
most energetic UHECR detected so far is still a particle which arrived over Utah, U. S. A.
some 18 years ago with an energy of (320 ± 90) EeV as recorded by the Fly’s Eye detector
[2]. We will call this cosmic ray the “Fly’s Eye event”.
It remains a mystery what the cosmic ray consisted of, how and where it was produced,
and how it could have arrived at Earth with so much energy. Neutrinos have only a small
interaction with Earth’s atmosphere, even at 300 EeV, and photons have not been detected
above 1013 eV, presumably because of pair production in the Cosmic Microwave Background
Radiation (CMBR)[6]. Protons can traverse the cosmos at higher energies, but even they are
limited above ∼ 60 EeV because of the Greisen-Zatsepin-Kuz’min (GZK) effect [7], wherein
they interact with the CMBR to produce neutral pions via the reaction p + γ → ∆(1232) →
p + π 0 . A proton with energy above the GZK bound will lose energy over a few tens of Mpc
until its energy falls below the bound. A light nucleus has an even shorter range due to
photo-disintegration on the infra-red background [8][9].
Thus of the four types of particles considered, a proton seems to be the most likely candidate to have produced the Fly’s Eye event. But even a proton is not likely because of the GZK
1
bound. Indeed, both the HiRes experimenters [10] and the Pierre Auger Collaboration[11]
have recently observed significant attenuation of cosmic-ray events above 50 EeV, presumably because of this bound. So one might wonder how the progenitor of the Fly’s Eye event,
now 18 years old, ever got here. For example, if a proton of the energy of this event originated just 10 Mpc away from Earth, then it would have needed an initial energy of 500 EeV
to arrive at earth with 320 EeV. If it originated farther out, say 50 Mpc away, then it would
have needed an original energy of 1000 EeV! (Fig. 4 in reference[12]).
One might hope to see the source of such an energetic proton by looking back along
its trajectory, since at such a high energy, the bending of the trajectory by galactic and
cosmic magnetic fields is expected to be less than a degree [6]. Active galactic nuclei (AGN),
supernova remnants, and powerful radio galaxies have been suggested as possible sources of
the event [5].Yet no source is seen for the Fly’s Eye event out to 100 Mpc! [13].
Careful studies looking for correlations of other UHECR trajectories with AGN have been
carried out by both the Pierre Auger Collaboration (PAC) and the HiRes (Utah) group. The
PAC claim to see correlations within v 3
◦
of AGN for 20 of their 27 recorded events with
energies above 57 EeV out to a distance of v 70 Mpc [14]. Their observatory is located
in Argentine in the southern hemisphere. However in the United States in the northern
hemisphere, the Utah group does not see a correlation above chance of its recorded UHECR
events’ trajectories with AGN [15].
It is interesting that the PAC does not see a correlation of their most energetic UHECR at
∼ 150 EeV with an AGN, nor does the HiRes group see a correlation of their most energetic
2
UHECR at ∼ 220 EeV with an AGN. So UHECRs at around 150, 220, and 320 EeV do not
appear to have come from AGN. So what are they, and where did they come from?
One possibility is that these ultra-high energy particles resulted from the decay of
metastable “X” particles with rest-masses exceeding 1020 eV, released by the collapse or annihilation of cosmic topological defects envisioned in Grand Unified Theories. Such decays
could give rise to extremely energetic nucleons, neutrinos, and photons. (This is reviewed
in Ref. [5].) However such “top-down” models have largely been ruled out by experimental
upper limits on the flux of high energy gamma rays [6].
I will propose a more radical idea. I consider the possibility that the Fly’s Eye event
was caused by a particle that originated in a universe in higher dimensional space-time,
perhaps a light-year or more away in a (noncompact) flat dimension orthogonal to the three
of ordinary space. See Fig 1. This model then differs from most other higher dimensional
models in that its extra dimension is not compact.
My model was inspired by a paper by Rubakov & Shaposhnikov (RS)[16] . The RS
model is simply φ4 theory plus a Dirac term gψφψ in five flat dimensions (thus also noncompact) with a domain wall in the extra dimension. In this toy model, if particles have
enough energy then they can escape the wall and propagate freely in all five dimensions.
To account for the Fly’s Eye event, I choose a different solution of the RS lagrangian which
consists of two domain walls separated by a distance d, as in Fig. 1. I assume that a particle
escapes its universe, drawn as Plane B, propagates in the extra dimension and reaches our
universe, Plane A. Here it can hit a knock-on particle which initiates the shower over Earth,
3
as sketched in Fig. 1, or it can glance off another particle and enter our plane to cause the
shower itself. Either way it encounters our plane near Earth and avoids the gauntlet of CMB
photons and the GZK bound.
In this model, the particle causing the shower travels in a different direction than does
the primary particle from plane B, so the primary’s source can not be determined by sighting
along the direction of the shower. In any case, the source is in the extra dimension, and
neither we nor our instruments can look in that direction anyway. This could explain why
no obvious source for the Fly’s Eye event has shown up.
In what follows, I review the Rubakov-Shaposhnikov model and then extend it to two
domain walls.
The Rubakov-Shaposhnikov five-dimensional φ4-model
2
As mentioned in the Introduction, Rubakov and Shaposhnikov[16] constructed a toy model
which generates a scalar boson bound to a domain wall in a five-dimensional flat space
e where M 4 is Minkowski-space and R
e is one-dimensional Cartesian space1 . The
M 4 × R,
domain wall is generated by the φ4 lagrangian
4 X
∂φ
∂φ
LB =
+ 12 m2 φ2 − 14 λφ4 ,
µ
∂x
∂x
µ
µ=0
1
2
(1)
with metric gµν = diag. (+, −, −, −; −) . One of the unusual things about this model is that
bosons with sufficient energy can escape the wall altogether.
1
Symbols relating to M 4 (R̃, M 4 × R̃ ) will be printed normally (with a tilde, in bold-face type).
4
The lagrangian LB of Eq. 1 generates the field equation
4
P
∂ µ ∂ µ φ − m2 φ + λφ3 = 0,
(2)
µ=0
which admits the “kink” solution
√ h
√ i
eA (e
φ = m/ λ tanh m (e
x−x
eA ) / 2 ≡ φ
x) .
(3)
Here I have taken the “kink” or domain-wall solution to be realized in the fifth dimension
eA , etc.)
e, x4A ≡ x
x4 at point x4A . (I will denote x4 ≡ x
e minimizes the action locally, and if the field φ is expanded around it, then φ − φ
e ≡
φ
A
A
η satisfies
4
P
2
2
e
∂ ∂µ + 3λφA − m η = 0,
µ
(4)
µ=0
with cubic and quartic terms in η to be treated by perturbation theory. If we set η =η (xµ ) e
η (e
x) ,
then Eq. 4 separates into the equations
e2 − m2 e
η = M 2e
η
−∂ 2 /∂e
x2 + 3λφ
A
(5)
+ M 2 η = 0;
(6)
and
here M is the separation constant, taken to be the particle’s mass, and is the d’Alembertian
in M 4 . Eq. 5 is seen to be a one-dimensional Schrödinger-like equation with the potential
e2 − m2 . This potential is plotted in Fig. 2. It admits three solutions: e
Ve = 3λφ
η0 =
A
eA /de
dφ
x with eigenmass M 2 = 0 (which merely corresponds to a translation of the soliton),
e −X
eA / cosh2 X
e −X
eA with M 2 = 3 m2 , which represents a genuine boson
e
η 3/2 = sinh X
2
5
trapped in the well, and continuum solutions e
η c for M 2 ≥ 2m2 which represent scalar bosons
q
q
e = 1 me
eA = 1 me
free to propagate in all five dimensions. Here X
x
and
X
xA . I have plotted
2
2
e
η 3/2 in Fig. 2.
In their classic paper, Rubakov and Shaposhnikov also added to bosonic Lagrangian LB ,
the Dirac terms
LD =
4
P
ψ γ µ ∂µ ψ + gψφψ.
(7)
µ=0
In the presence of the wall, the Dirac particle satisfies, to a first approximation, the field
equation
−i
4
P
eA ψ = 0,
γ µ ∂µ ψ − g φ
(8)
µ=0
where γ µ γ ν + γ ν γ µ = 2gµν , µ, ν = 0, 1, . . . , 4. If we choose the “chiral” set of 4 × 4 gamma
matrices






 i 0 
 0 σj 
 0 −1 
j
,
 , j = 1, 2, 3, & γ 4 = 



,
γ
=
γ =





−1 0
−σ j 0
0 −i
0
(9)
e , where ψ is a wavefunction in M 4 for a massless
then Eq. 8 admits a solution ψ =ψ 0,L ψ
A
0,L
left-helical particle (or massless right-helical antiparticle) and
h
i−g√2/λ
e = cosh X
e −X
eA
ψ
.
A
(10)
e is plotted
(There is no right-helical massless particle solution for positive g in Eq.8.) ψ
A
in Fig. 2 for g =
√
λ and XA = 0. The Dirac particle is seen to be confined to a slab of
6
√
thickness v 4/m. There are also unconfined Dirac states of mass ≥ gm/ λ. Thus Dirac
particles as well as Bose particles can escape the domain wall if they have sufficient energy.
3
The Rubakov-Shaposhnikov model generalized to two
domain walls
A static solution with two domain walls does not exist for field Eq. 2 of the bosonic φ4 lagrangian (Eq. 1). However there exists a static two-domain-wall wavefunction which very
nearly satisfies it[17]. This wavefunction is
m
eA − φ
e +√
eD ,
φ
≡φ
B
λ
(11)
eB is given by Eq. 3 with x
where φ
eB replacing x
eA .
e satisfies the RS field Eq. 2 extremely well for wall-separations
It turns out that φ
D
e in Eq. 2 yields
eA − X
eB ' 20. Inserting φ
X
D
m3 e
e − m2 φ
e + λφ
e3 = √
∂ µ ∂µ φ
∆,
D
D
D
λ
µ=0
(12)
e
e
e
e
e
∆ = 3 ΦA − ΦB ΦA + 1 ΦB − 1 ,
(13)
4
P
where
√
eA =
with Φ
λe
φ
m A
√
eB =
and Φ
λe
φ .
m B
e should be zero to satisfy Eq. 12, it can be very
While ∆
e contains strong cancellations and is more accurately calculated
nearly zero, as I will show. ∆
using the alternate form
n
h io
eA − X
eB
1
−
exp
−2
X
e = 3h
∆
i2 h
i2 .
2
e
e
e
e
cosh X − XA
cosh X − XB
7
(14)
e ≡
Denoting φ
D
m e
√
Φ ,
λ D
e D and ∆
e in Fig. 3 for the case X
eA − X
eB = 20. One may
I plot Φ
e reaches a maximal value of about 10−16 for X
e between X
eA and X
eB , and falls off
see that ∆
exponentially on either side. (The precise maximum is 1.02×10−16 .) If we approximate either
well’s width by
4
m
eA − X
eB = 20
v 10−18 meter (the maximal size of a Dirac particle), then X
∼
∼
eD is
corresponds to xA − xB = 7.5 · 10−18 meter. Thus even when the walls are this close, φ
a good approximation to Eq. 2. If we now assume that the walls are a full light-year apart,
then
m
eA − X
eB = √
X
(e
xA − x
eB ) v 1018 × 1016 = 1034 .
2
e with the same shape as that in Fig. 3, but with a central maximum v
This yields a ∆
4
1/ (cosh 1034 ) v 1/ exp (10136 ) , truly a small number.
e satisfies the RS field
Clearly φ
D
equation to an accuracy satisfactory for our purposes.
e very nearly minimizes the action locally, and if φ is expanded around it, then
φ
D
e ≡ η satisfies
φ−φ
D
4
P
µ
∂ ∂µ +
e2
3λφ
D
−m
2
η = 0,
(15)
µ=0
with cubic and quartic terms in η again to be treated by perturbation theory. If we set
η =η (xµ ) e
η (e
x) , then Eq. 15 separates into the equations
e2 − m2 e
−∂ 2 /∂e
x2 + 3λφ
η = M 2e
η
D
(16)
and
+ M 2 η = 0.
e2 − m2 with two wells which I plot in Fig. 4. Because
Eq. 16 includes the potential Ve = 3λφ
D
walls A and B are so well separated, each acts essentially independently, giving Eq. 16
8
2
e /de
e
e
e
e
confined solutions e
η 0,C = dφ
x
and
e
η
=
sinh
X
−
X
/
cosh
X
−
X
C
C , with
C
3/2,C
C = A or B. More interesting from the standpoint of UHECR are solutions for M 2 ≥ 2m2
which we shall denote e
η cAB ; the associated bosons can propagate freely in all five dimensions.
The full RS lagrangian LB + LD also predicts Dirac particles propagating freely in all
√
five dimensions if they have masses ≥ gm/ λ. These particles appear for solutions of Eq. 8
eD replacing the single domain-wall φ
eA , i.e.,
with the double domain-wall φ
−i
4
P
eD ψ = 0.
γ µ ∂µ ψ − g φ
(17)
µ=0
Massless Dirac particles are also trapped on the two domain walls. For the choice of gammamatrices given in Eq. 9 and positive g, a zero-mass negative-helicity Dirac particle (or
positive-helicity antiparticle) is trapped on wall A, and a zero-mass positive-helicity Dirac
particle (or negative-helicity antiparticle) is trapped on wall B. If the walls are separated by
eA − X
eB = 20, then their respective wavefunctions are almost exactly
a distance exceeding X
h
i−g√2/λ
e
e = cosh X
e −X
eA
≡ψ
ψ
A
0−
(18)
i−g√2/λ
e = cosh X
e .
e −X
eB
ψ
≡ψ
B
0+
(19)
and
h
e 0− and ψ
e 0+ are plotted in Fig. 4 for X
eA = 10 and X
eB = −10.
ψ
If a Dirac particle in domain-wall B of this toy model were accelerated to enough energy
to escape, then it might propagate to domain-wall A (our universe) to bring about a shower.
A sufficiently energetic Bose particle could behave similarly.
9
4
Discussion
There still seems to be no satisfactory conventional explanation for the most energetic cosmic
ray ever measured by physicists. (I will refer to it as the “Fly’s Eye” event). This cosmic
ray, recorded by the Utah Fly’s Eye Collaboration in 1991, had an energy of 320 EeV which
was approximately six times the energy of the GZK bound. (1 EeV = 1018 eV.) This limit,
due to the interaction of cosmic ray flux with the CMBR, has recently been verified in
measurements of ultra-high-energy cosmic rays (UHECRs) by the two leading international
cosmic-ray collaborations: the Utah High-Resolution Fly’s Eye Collaboration (HiRes)[10]
and the Pierre Auger Collaboration (PAC)[11]. Both groups see a sharp falloff of cosmic
rays above v 60 EeV. So how did the Fly’s Eye event ever get here with 320 EeV? Looking
back along its trajectory, researchers see no source. Yet if it had originated as close as, say
50 Mpc, it still would have needed an initial energy of some 1000 EeV! In such a case its
source might be expected to stand out. Moreover, HiRes and PAC see no obvious sources
for their most energetic cosmic rays either, at 220 EeV[15] and 150 EeV[14], respectively.
In this paper I suggest that the 320 EeV Fly’s Eye event, and possibly the 220 and
150 EeV events, may have originated in another universe, separated from our universe by a
distance in a higher (fifth) dimension; a linear (not curved) distance greater than a few fermis
and possibly extending to light years. Such ultra-high-energy cosmic rays might have entered
our lower four dimensions directly over Earth and never had to run the gauntlet of CMB
photons. I illustrate this idea with a toy φ4 model in five dimensions, generalizing a model
by Rubakov & Shaposhnikov[16]. In the RS model, particles in our universe are confined to
10
a “kink” or plane in the fifth dimension, except that sufficiently energetic particles might
escape, such as in e+ + e− → two Dirac particles propagating freely in five-dimensional spacetime. In this paper, I generalize the RS model to a model with two kinks or confining planes,
one representing our universe and the other representing another universe, the source of the
UHECRs.
Dirac particles leaving “upper” plane B might no longer be quarks or leptons; rather,
they might be undifferentiated Dirac particles. (In a model I have constructed, quark and
lepton properties depend on how the particle is trapped in the plane[18].) Furthermore,
when particles leave the upper plane, they probably are not restricted by Einstein’s speed
limit. Time might be very different between the planes, if it is defined at all, so particles
reaching our plane might enter with speeds greatly exceeding c. In that case, any particle
should set up a shock wave, whether or not it interacts with ordinary matter at subluminal
speeds.
Furthermore, there might even be a significant “potential drop” from the other universe
to ours.
I would be remiss if I didn’t point out another possibility for particles entering our space
with superluminal speeds. A particle might be produced by a Supernova in our own universe,
say, but with enough energy to escape and skate just above our lower four dimensions with
speeds exceeding c. (This might be somewhat akin to a light wave in an optical fiber being
reflected internally from the walls as it travels.) Then as the particle comes above our Solar
system, it reenters M 4 and scatters into Earth, creating the shock wave and initiating the
11
shower. Meanwhile the rest of the Supernova is left behind, rumbling along at the speed of
light to arrive here later. And we would see no Supernova because its light and neutrinos
hadn’t reached us yet.
I would like to take this opportunity to encourage experimentalists, when they check
incoming particles’ speeds to rule out micrometeorites and the like, to also check for incoming
speeds up to several times the speed of light.
Acknowledgments
I would like to thank Pierre Sokolsky and Charlie Jui for interesting conversations and
helpful information. I would also like to thank my colleagues Robert Webb and Ronald
Schorn for useful comments. Finally I must thank Joseph McMoneagle for an important
observation.
References
1. M. Takeda et al., “Extension of the cosmic-ray energy spectrum beyond the predicted
Greisen-Zatsepin-Kuz’min cutoff”, Physical Review Letters 81 (1998) 1163.
2. D. J. Bird et al., “Detection of a cosmic ray with measured energy well beyond the
expected spectral cutoff due to cosmic microwave radiation”, Astrophysical Journal
441 (1995) 144.
3. P. Sokolsky and G. B. Thomson, “Highest-energy cosmic rays and results from the
HiRes Experiment”, Journal of Physics G 34 (2007) R401-R429.
4. K.-H. Kampert, “Ultra high-energy cosmic ray observations”, Journal of Physics
Conference Series 120 (2008) 062002
5. P. Bhattacharjee and G. Sigl, “Origin and propagation of extremely high energy
cosmic rays”, Physics Reports 327 (2000) 109. [astro-ph/9811011].
6. A. V. Olinto, “Highest energy cosmic rays”, AIP Conference Proceedings 745 (2005)
48. [astro-ph/0410685]
12
7. K. Greisen, “End to the cosmic ray spectrum?”, Physical Review Letters 16 (1966)
748; G. T. Zatsepin and V. A. Kuz’min, “Upper limit on the spectrum of cosmic rays”,
Pis’ma Zh. Eksp. Teor. Fiz. 4 (1966) 114 [JETP Lett. 4 (1966) 78].
8. J. L. Puget, F. W. Stecker and J. H. Bredekamp, “Photonuclear interactions of ultrahigh energy cosmic rays and their astrophysical consequences”, Astrophysical Journal
205 (1976) 638.
9. L. N. Epele and E. Roulet, “Comment on ‘On the origin of the highest energy cosmic
rays’ ”, Physical Review Letters 81 (1998) 3295.
10. R. U. Abbasi et al. (High Resolution Fly’s Eye Collaboration), “First observation of the Greisen-Zatsepin-Kuzmin suppression”, Physical Review Letters 100 (2008)
101101. [astro-ph/0703099]
11. J. Abraham et al. (Pierre Auger Collaboration), “Observation of the suppression of
the flux of cosmic rays above 4 · 1019 eV ”, Physical Review Letters 101(2008) 061101.
[arXiv: 0806.4302]
12. J. W. Cronin, “Cosmic rays: the most energetic particles in the universe”, Reviews
of Modern Physics 71 (1999) S165.
13. Pierre Sokolsky, private communication.
14. J. Abraham et al. (Pierre Auger Collaboration), “Correlation of the highest-energy
cosmic rays with the positions of nearby active galactic nuclei”, Astroparticle Physics
29 (2008) 188. [arXiv: 0712.2843]
15. R. U. Abbasi et al. (The High Resolution Fly’s Eye Collaboration), “Search for
correlations between HiRes stereo events and active galactic nuclei”, Astroparticle
Physics 30 (2008) 175. [arXiv: 0804.0382]
16. V. A. Rubakov and M. E. Shaposhnikov, “Do we live inside a domain wall?”,
Physics Letters B125 (1983) 136.
17. D. K. Campbell, J. Schonfeld and C. Wingate, “Resonant structure in kink-antikink
interactions in φ4 theory”, Physica 9D (1983) 1.
18. R. A. Bryan, “Are quarks and leptons dynamically confined in four flat extra
dimensions?”, Nuclear Physics B 523 (1998) 232.
Figure captions
13
1. Fermion escaping domain-wall B (another universe) and propagating to domain-wall
A (our universe) where it strikes a particle and continues on past A. The knock-on particle
streaks to Earth where it initiates a shower in the upper atmosphere. Not drawn to scale.
2
∼
2. Ṽ = 3λφ̃A − m2 plotted vs. mx̃ for X A = 0. Superimposed is the bosonic excitation
q
2 e
1
e
e
η̃ 3/2 = sinh X/ cosh X, where X ≡
mx̃; this excitation has eigenmass M 2 = 32 m2 .
2
−g√2/λ
√
∼
e
The Dirac wavefunction ψ̃ A = cosh X
is also plotted, for g = λ and X A = 0. A
continuum of unbound states lies above Ṽ = 2m2 .
eD = Φ
eA − Φ
e B + 1 plotted vs. X,
e for X
eA = 10
3. The double domain-wall wavefunction Φ
eB = −10. The discrepancy function ∆
e is also plotted.
and X
e2 − m2 plotted vs. X
e for X
eA = 10 and X
eB = −10. Also plotted are the Xe
4. Ve = 3λφ
D
e 0− and ψ
e 0+ centered at X
eA and X
eB , respectively.
dimensional Dirac-particle wavefunctions ψ
14