Page 1

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

Page 2

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)

Page 4

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

Page 6

- 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

Page 7

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)

7

Page 9

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

Page 10

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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