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arXiv:hep-ph/0111311v2 7 Dec 2001
SU-ITP-01/48
hep-ph/0111311
Dimming Supernovae without Cosmic
Acceleration
Csaba Cs´ akia,∗, Nemanja Kaloperband John Terninga
aTheoretical Division T-8, Los Alamos National Laboratory, Los Alamos, NM 87545
bDepartment of Physics, Stanford University, Stanford CA 94305-4060
csaki@lanl.gov, kaloper@stanford.edu, terning@lanl.gov
Abstract
We present a simple model where photons propagating in extra-galactic magnetic
fields can oscillate into very light axions. The oscillations may convert some of the pho-
tons departing a distant supernova into axions, making the supernova appear dimmer
and hence more distant than it really is. Averaging over different configurations of the
magnetic field we find that the dimming saturates at about 1/3 of the light from the su-
pernovae at very large redshifts. This results in a luminosity-distance vs. redshift curve
almost indistinguishable from that produced by the accelerating Universe, if the axion
mass and coupling scale are m ∼ 10−16eV, M ∼ 4 · 1011GeV. This phenomenon may
be an alternative to the accelerating Universe for explaining supernova observations.
∗Address after December 20, 2001: Newman Laboratory of Physics, Cornell University, Ithaca, NY 14853.
E-mail: csaki@mail.lns.cornell.edu
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Current observations of supernovae (SNe) at redshifts 0.3<
fainter than expected from the luminosity-redshift relationship in a decelerating Universe [1].
On the other hand, the large scale structure and CMBR observations suggest that the Uni-
verse is spatially flat, with the matter density about 30% of the critical density [2–6]. It is
therefore usually inferred that the Universe must have become dominated by a dark energy
component, which comprises about 70% of the critical energy density, and has the equation
of state p/ρ<
∼−2/3, implying that our Universe would be accelerating at present. The
dark energy component could either be a small cosmological constant or a time-dependent
quintessence energy [7]. Neither possibility is elegant from the current vantage point of fun-
damental theory because of unnaturally small numbers needed to fit the data: the present
value of the energy density, ρc∼ 10−12eV4, and, in the case of quintessence, the tiny mass
smaller than the current Hubble parameter mQ< H0∼ 10−33eV and sub-gravitational cou-
plings to visible matter to satisfy fifth-force constraints [8]. Further, describing an eternally
accelerating Universe with future horizons is at present viewed as a conceptual challenge for
string theory and more generally any theory of quantum gravity [9–11].
Because the SNe observations probe length scales l ∼ H−1
inaccessible to any particle physics experiments, it is natural to consider alternative expla-
nations to the supernova data without cosmological dark energy. A simple such alternative
is that light emitted by a distant supernova encounters an obstacle en route to us and gets
partially absorbed [12].∗However any mechanism must be very achromatic because the light
from the SNe appears to be dimmed independently of frequency. This would seem to rule
out a medium of matter particles, which can absorb light in the optical spectrum but will
re-emit it in the IR, affecting the CMBR in unacceptable ways.
In this paper we consider a model where the dimming of SNe is based on flavor oscillations.
Flavor oscillations occur whenever there are several degrees of freedom whose interaction
eigenstates do not coincide with the propagation eigenstates. Such particles can turn into
other particles simply by evolution and evade detection. We will consider a model with
an axion with a mass m ∼ 10−16eV, much smaller than the usual [15] QCD axion mass
scale, 10−5eV<
∼max
mQ<
∼10−33eV. This axion couples to electromagnetism through the usual term (a/4M)˜FF,
which leads to energy-dependent mixing of the photon and the axion in the presence of an
external magnetic field B [16]. Hence light traveling in inter-galactic magnetic fields can in
part turn into axions, and evade detection on Earth. A source would then appear fainter even
if the Universe is not accelerating. To satisfy other cosmological constraints we assume that
the Universe is presently dominated by some form of uniform dark energy which does not
clump, but need not lead to cosmological acceleration, e.g. with equation of state p/ρ = −1/3.
We find that contrary to the familiar example of neutrino oscillations, in our model both
the flavor mixing and the oscillation length of photons in the optical range are insensitive
to energy, and so our axion will induce strongly achromatic oscillations of optical photons.
On the other hand the small axion mass m ∼ 10−16eV insures that the photon-axion oscil-
∗Other possibilities are that SNe may evolve with time [13], or that there are more than four space-time
dimensions [14].
∼z<
∼1.7 reveal that they are
0
∼ few × 103Mpc which are
QCD<
∼10−1eV, but exponentially larger than the quintessence mass
1
Page 3
lations leave CMBR essentially unaffected. Our task here is to find a small correction to
the luminosity-redshift relation induced by the oscillations. Because the SNe observations at
redshifts z ∼ 0.5 can be explained by a 10%−15% total increase in the distance relative to a
matter dominated universe, the total attenuation of SNe light should be about 30% between
the cosmic string-dominated geometry and flavor oscillations. Hence we need to account for
about a 20% of decrease in the luminosity by the flavor oscillations.
The axion-photon coupling is
Lint=
M
where the scale M characterizes the strength of the axion-photon interactions. This induces
a mixing between the photon and the axion [16,17] in the presence of a background magnetic
field?B (as exists in our Universe [18]). Indeed, working in the Coulomb gauge at distances
short compared to the size of a coherent magnetic domain Ldom, we see that the photon
with electric field orthogonal to?B remains unaffected by mixing. The polarization whose
electric field is parallel to?B mixes with the axion. The field equations are, after rotating
the coordinate axes such that the propagation is along the y-direction,
a
?E ·?B ,
(1)
?d2
dy2+ E2−
?
0iE
B
ML
m2
−iE
B
ML
???|γ?
|a?
?
= 0(2)
where we Fourier-transformed the fields to the energy picture E and introduced the state-
vectors |γ? and |a? for the photon and the axion. Here B = ?? e ·?B? ∼ |?B| is the averaged
projection of the extra-galactic magnetic field on the photon polarization ? e. We will assume
that the averaged value of?B is close to its observed upper limit, and take for the magnetic
field amplitude |?B| ∼ few · 10−9G [18,19]. Therefore its energy density is?B2∼ cH2
where c ∼ few · 10−11and the Hubble parameter is H0 ∼ 10−33eV. The magnetic fields
we will be considering are sufficiently small that we can safely ignore the Euler-Heisenberg
effect [16,20].
We can now define the propagation eigenstates by diagonalizing the mixing matrix in
Eq. (2), which is, using B/M = µ,
0M2
Pl,
M2=
?
0iEµ
m2
−iEµ
?
. (3)
This matrix is the analogue of the usual see-saw matrix for neutrinos, with the only difference
that the off-diagonal terms are imaginary and complex-conjugates of each other. This is
because the mixing arises from the derivative terms in the field equations rather than the
potential terms. Defining the propagation eigenstates |λ−? and |λ+? which diagonalize the
matrix (3), whose eigenvalues are λ∓ =
equation (2). The solutions describing particles emitted by a supernova at a distance y0> 0
and propagating towards us at y = 0 are
µE
?
|a? =
?
2
m2
2∓
?
m4
4+ µ2E2, we can solve the Schr¨ odinger
|γ? =
λ2
−+ µ2E2|λ−? e−i[Et+p1(y−y0)]+
−iλ−
λ2
iµE
?
λ2
++ µ2E2|λ+? e−i[Et+p2(y−y0)],
λ+
λ2
−+ µ2E2|λ−? e−i[Et+p1(y−y0)]+
?
++ µ2E2|λ+? e−i[Et+p2(y−y0)],(4)
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where pk=√E2− λk. It is now clear that as the photon propagates, it mixes with the axion
by an amount depending on the energy of the particle. In the limit E2≫ λi> m2, which
covers all of the applications of interest to us, the mixing angle is
sinθ =
µE
?
λ2
++ µ2E2, (5)
the photon survival probability Pγ→γ= |?γ(y0)|γ(y)?|2is
Pγ→γ= 1 −
4µ2E2
m4+ 4µ2E2sin2
?√m4+ 4µ2E2
4E
∆y
?
,(6)
and the oscillation length is
LO=
4πE
√m4+ 4µ2E2. (7)
In the limit E ≫ m2/µ, the mixing is maximal, while the oscillation length is completely in-
dependent of the photon energy: sinθ ∼
optical frequencies E ∼ 10 eV as we will see) oscillate achromatically.
On the other hand, in the low energy limit E ≪ m2/µ, the mixing is small, and the
oscillation length is sensitive to energy: sinθ ∼
dispersive, due to the energy-dependence of both the mixing angle and the oscillation length.
But the probability to find axions Pγ→a = 1 − Pγ→γ is small, bounded from above by
Pγ→a< sin2(2θ) ≤ 4µ2E2/m4.
In our Universe the magnetic field is not uniform. Assuming that a typical domain size for
the extra-galactic magnetic field is Ldom∼ Mpc [18,19], it is straightforward to numerically
solve for the quantum mechanical evolution of unpolarized light in such magnetic domains
with uncorrelated field directions. An analytic calculation shows that in the case of maximal
mixing, with cos(µLdom) > −1/3, the survival probability is monotonically decreasing:
Pγ→γ=2
1
√2, LO∼2π
µ. Thus high-energy photons (including
µE
m2, LO ∼
4πE
m2. The oscillations are very
3+13e−∆y/Ldecay
(8)
where the inverse decay length is given by
Ldecay=
Ldom
ln
?
4
1+3cos(µLdom)
? .(9)
For µLdom≪ 1 this reduces to
Ldecay=
8
3µ2Ldom
.(10)
Thus we see that with a random magnetic field the problem becomes essentially classical
and after the traversal of many magnetic domains the system is equilibrated between the
two photon polarizations and the axion. This leads to the generic prediction that on average
one-third of all photons converts to axions after large traversed distances.
3
Page 5
We can now estimate the axion mass and coupling needed to reproduce SN observations.
To take the oscillations into account, in the luminosity-distance v.s. redshift formula we
should replace the absolute luminosity L by an effective one:
Leff= L Pγ→γ.
The optical photons must oscillate independently of their frequency. For them, the oscilla-
tions should reduce the flux of incoming photons by about 20% for SNe at z ∼ 0.5. This
requires Ldec<
∼H−1
that this is above the experimental exclusion limit for M. The experimental bound on M
quoted by the PDG [21] is M ≥ 1.7 · 1010GeV [22]. However, for ultralight axions there
is [22,23] a more stringent (though also more model dependent) limit from SN1987A given
by M ≥ 1011GeV, which is still lower than the value required here.
If the microwave photons had fluctuated a lot in the extra-galactic magnetic field, their
anisotropy would be very large due to the variations in the magnetic field. To avoid affecting
the small primordial CMBR anisotropy, ∆T/T ∼ 10−5, the axion mass should be large
enough for the mixing between microwave photons and the axion to be small. In this limit,
we can ignore the averaging over many random magnetic domains and simply treat each
domain as a source of CMBR fluctuation. The photon-axion mixing and the oscillation
length in that case are given by sinθ ∼
controlled by the transition probability into axions Pγ→a≤ 4B2E2
expression for B is
Pγ→a≤ 4 · 10−11M2
For microwave photons E ∼ 10−4eV, and so Pγ→a ≤ 2.5 · 10−70(eV)4/m4. Therefore for
m ∼ few × 10−16eV we find Pγ→a≤ 10−7, which is smaller than the observed temperature
anisotropy. For this mass scale, the oscillation length of microwave photons is LO∼ 10−4H−1
which is of order of the coherence length of magnetic domains Ldomand so a lot shorter than
the horizon size. This is harmless since the oscillation amplitude is so small. Thus we see
that if the axion scales are
(11)
0 /2. Thus the mass scale M for this should be M ∼ 4 · 1011GeV. Note,
µE
m2, LO ∼
4πE
m2. The disturbances of CMBR are
m4M2, which using the explicit
PlH2
M2m4
0E2
. (12)
0,
m ∼ 10−16eV,M ∼ 4 · 1011GeV,(13)
the mixing could produce the desired effect of reducing the flux of light from SNe while
leaving the primordial CMBR anisotropy unaffected. We stress here that while at early
times the CMBR photons were much more energetic there were no sizeable extra-galactic
magnetic fields yet, since their origin is likely tied to structure formation [24]. Hence we can
get a rough estimate of the influence of our effect on CMBR using their current energy scale.
Having determined the scales, we can check that the approximations we have been using are
appropriate for optical and microwave photons, respectively. In the former case, the mixing
angle and the oscillation length receive energy-dependent corrections ∝
while in the latter case the energy-dependent corrections are ∝
the validity of our approximations.
m4
optical∼ 10−5,
∼ 10−6, confirming
µ2E2
µ2E2
CMB
m4
4
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- 1.5 - 1 - 0.50
log z
- 1
- 0.75
- 0.5
- 0.25
0
0.25
0.5
∆ (m - M)
Figure 1: The luminosity-distance vs. redshift curve for several models, relative to the curve with
Ωtot= 0 (dotted horizontal line). The dashed curve is a best fit to the supernova data assuming the
Universe is accelerating (Ωm= 0.3, ΩΛ= 0.7); the solid line is the oscillation model with Ωm= 0.3,
ΩS= 0.7, M = 4 · 1011GeV, m = 10−16eV; the dot-dashed line is Ωm= 0.3, ΩS= 0.7 with no
oscillations, and the dot-dot-dashed line is for Ωm= 1 again with no oscillations.
To compare our model with observations, we assume that the constraint on the total
energy density of the Universe Ωtot≃ 1 is satisfied because the Universe contains some form
of dark energy which does not clump, but it need not lead to cosmological acceleration.
A simple example is dark energy with the equation of state w = p/ρ = −1/3 and energy
density ΩS = 0.7, which could originate from a network of frustrated strings with small
mass per unit length. Note that because the scale factor a of the universe obeys ¨ a/a =
−
w = p/ρ is greater than −1/2.1 ≃ −0.48 the Universe would presently not be accelerating.
These forms of dark energy do not appear to be excluded either by the position of the
first acoustic peak in the CMBR measurements [26] or by combined CMBR+large scale
structure fits [27]. In Fig. 1 we have plotted the typical prediction of the oscillation model
in a spatially flat Universe with Ωm = 0.3 and ΩS = 0.7 against the best fit model for
the accelerating Universe with a cosmological constant (Ωm = 0.3 and ΩΛ = 0.7). The
two curves are practically indistinguishable. We note that the oscillation model predicts
limited attenuation of the SN luminosities, unlike some other alternatives to the accelerating
Universe. The total attenuation is limited to about 1/3 of the initial luminosity, as we
have explained above. Since for larger values of z the Universe becomes matter dominated,
and the disappearance of photons is saturated in the oscillation model, the two curves will
continue lying on top of each other for higher values of z. Thus simply finding higher z
supernovae [25] will not distinguish between the two models. The main difference between
the two is that the curve for the oscillation model is an averaged curve, with relatively large
standard deviations. Therefore it may be much easier to explain outlying events than in the
case of the accelerating Universe.
4π
3M2
PL(ρtotal+ 3ptotal), and assuming Ωm= 0.3 and Ωdark= 0.7, then as long as the ratio
5
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2000 4000 6000 8000distance (Mpc)
Figure 2: The effect of the oscillation on the polarization of the photons. We assume that the
photon emitted is totally polarized, and show how much polarization remains as a function of
physical distance traveled.
0.2
0.4
0.6
0.8
1
Polarization
Let us now consider photons which may pass through the magnetic field of a galaxy, or
just skim it. The galactic magnetic fields are much stronger than the extra-galactic ones,
BG ∼ µG ∼ 103B. However, the density of baryons (and therefore also of electrons) is
large enough in such regions that refraction has to be taken into account, which introduces a
diagonal element M11for the photon in (3).†A simple estimate [16] for this term gives M11∼
10−23(eV)2for 10 eV photons traveling within a galaxy, while the off-diagonal terms are of the
order 10−27(eV)2. Therefore this term will dominate the mixing matrix, and the oscillations
will be highly suppressed while passing through the magnetic field of a galaxy. However,
since there is no evidence for the presence of gas uniformly distributed between clusters, this
effect is likely negligible for most of extra-galactic space. This is because a simple estimate
shows, that even assuming the worst-case scenario where all matter is uniformly distributed
and totally ionized, M11would be ∼ 10−29(eV)2for Ωbaryon= 0.05, which would somewhat
suppress the mixing. However since matter is not uniformly distributed and definitely not
ionized in the inter-cluster voids (which make up most of space) this effect should be negligible
for our results.
While it is natural to wonder if there are laboratory constraints on our mechanism, a
simple order of magnitude estimate shows that it would be difficult to observe in a lab. Since
we have assumed that the extra-galactic magnetic field is ∼ 10−13T, for a uniform magnetic
field about 1014times larger the corresponding oscillation length would be LO/1014∼ 6·1012
cm, which is about a thousand times the circumference of the Earth. The current direct
experimental bounds [28] quoted by the PDG [21] on the coupling of an axion-like particle
(with a mass less than 0.03 eV) to?E ·?B is M > 1.6 × 109GeV.
Another question is whether the oscillations may cause any observable polarization effects
on the light arriving from the SNe. If the orientation of the extra-galactic magnetic field
†We thank Georg Raffelt for pointing out this effect.
6
Page 8
were constant, and the field perfectly homogeneous, light from the SNe would be partially
plane-polarized. However, since the coherence length of the extra-galactic magnetic field is
of order ∼ Mpc, the direction of the magnetic field is effectively random, and thus no strong
polarization effects are expected for faraway SNe. Rather, the converse effect of depolarizing
incoming light is more important, since the oscillations in a random magnetic field may
deplete existing photon beam polarizations. Because there are distant sources which are
partially polarized, with the polarization direction correlated with the shape of the source,
it is important to show that the photon-axion mixing does not completely depolarize light
from a polarized source. The observed polarization as a function of distance is shown in
Fig. 2, where we see that the polarization decrease is rather slow. This should be expected
because the degradation of polarization occurs after an axion produced by a polarized photon
conversion regenerates a photon of a different polarization, after the orientation of the?B
has changed. This is a second-order effect, and so polarization is depleted more slowly than
intensity. As a result the existing measurements of polarized optical photons from distant
sources can be accommodated in this model.
An axion with the scales which are needed for our model can for example arise from
the spontaneous breaking of an axial lepton number symmetry. Suppose that it couples
to the electroweak gauge theory in the standard way. Specifically it would couple to the
electromagnetic field like the QCD axion [15]. Then the mass scale M would be related to
the scale of the spontaneous breaking of axial symmetry faby
M =8π
α
fa
ξ,
(14)
where α = g2/4π is related to the gauge coupling constant, and ξ is a dimensionless number
depending on the precise couplings to fermions. We will take α ∼ 1/30 and ξ = O(1) in
what follows for simplicity. Hence fa∼ 10−3M. In perturbation theory the shift symmetry
a → a + c protects the axion from acquiring a mass term (more generally any potential).
This symmetry is broken by nonperturbative effects induced by instantons, which give rise
to the axion potential [15]. Because by assumption our axion couples to electroweak gauge
fields, a possibility to generate the potential is via the electroweak instantons. In particular
the axion potential will be of the form
1 + cos(a
V (a) = Λ4?
fa)
?
.(15)
For example, in a particular supersymmetric (SUSY) model [29], the scale Λ is
Λ4= e−
2π
α2(MPl)ǫ10M3
SUSYMPl, (16)
where MSUSY is the soft SUSY-breaking mass scale, ǫ a flavor symmetry breaking parameter
and α2(MPl) the electroweak gauge coupling strength at the Planck scale. It is straightfor-
ward to verify that for MSUSY ∼ few TeV, ǫ = O(1) and α2(MPl) = 1/23, we get Λ ∼ 10
eV. Since the axion mass is
m ≃Λ2
fa
, (17)
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we see that for fa∼ 4 · 108GeV we find m ∼ 10−16eV. As we have seen above, these are
roughly the scales most interesting for photon-axion oscillations in extra-galactic magnetic
fields.
It is important to stress that while our axion must be light it is not light enough to be
quintessence. Cosmologically, the axion particles with mass m ∼ 10−16eV are relativistic
throughout the history of the Universe, and so would behave like warm dark matter. Because
they are weakly coupled, with M−1∼ 10−12GeV−1, they are out of equilibrium from a very
early time. If they are not significantly produced during reheating after inflation, their
abundance can be harmlessly small. On the other hand, the homogeneous axion background
field a(t) will oscillate around its minimum, with its energy density scaling as cold dark
matter at late times. Thus one may worry about the cosmological moduli problem which
such fields usually lead to. However in our case this does not happen because fa ∼ 108
GeV and m ∼ 10−16eV. In the early Universe, the background field will satisfy the slow
roll conditions, and remain frozen until the Hubble scale comes down to H ∼ 10−16eV,
when the Universe cools to the temperature Ti∼ 100 keV. At that moment, the field may
start rolling. Its kinetic and potential energy will rapidly virialize, after which the energy
density stored in it will scale as ρ ∼ ρi(T/Ti)3. The initial energy density is determined by
the initial displacement of the axion from its minimum, which is of order fa. Therefore the
energy density will be of order given in Eq. (16), Λ4∼ (few×1eV)4. This would not compete
with radiation until the temperature comes down to T ∼ ρi/T3
that even if the axion was displaced from the minimum it would remain tiny for a long time
into the future. Furthermore, while an axionic sector can give rise to both domain walls and
cosmic strings in the early Universe, because the axion scales in the model we discuss are so
low, these defects may remain negligible well into the future of our Universe [30].
In summary, we have presented an alternative explanation of the observed dimming of
SNe at large distances. The effect is based on a quantum mechanical oscillation between
the photon field and a hypothetical axion field in the presence of extra-galactic magnetic
fields. This would result on average in about a third of the photons emitted by distant SNe
oscillating into axions. This is, roughly, the right amount needed to explain the supernova
observations. If the average magnetic field is of the order 10−9G, and the average domain
size is of order ∼ Mpc, one would need an axion whose coupling to the photon is given
by M ∼ 4 · 1011GeV, and mass m ∼ 10−16eV. With these parameters the luminosity-
distance vs. redshift curve is almost indistinguishable from the curve of an accelerating
Universe with Ωm= 0.3,ΩΛ= 0.7. Since the precise value of the luminosity-distance for a
particular supernova depends on the details of the inter-galactic magnetic field, we expect
more variations in the observed luminosity, and thus this model may easily incorporate
outlying data points. However, distinguishing this model from the accelerating Universe
paradigm will likely be easier through improving the bounds on the couplings of ultra-light
axions, by understanding the details of the intergalactic magnetic field, or by a precise
independent determination of the equation of state for the dark energy component, for
example through the DEEP survey [31].
i∼ 10−15eV, which means
8
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Acknowledgements
We thank T. Bhattacharya for explaining to us the proper procedure to average over the
magnetic field, to A. Albrecht, S. Dimopoulos, J. Erlich, C. Grojean, S. Habib, M. Kapling-
hat, L. Knox, A. Linde and R. Wagoner for useful discussions, and to G. Raffelt for comments
on the first version of this paper. N.K. thanks the members of the T-8 group at Los Alamos
for their hospitality where this work was initiated. C.C. is an Oppenheimer fellow at the Los
Alamos National Laboratory, and is supported in part by a DOE OJI grant. C.C. and J.T.
are supported by the U.S. Department of Energy under contract W-7405-ENG-36. N.K. is
supported in part by an NSF grant PHY-9870115.
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