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On the cosmological variation of the fine

structure constant

Antonio F. Rañada
Departamento de F́ısica Aplicada III, Universidad Complutense,

28040 Madrid, Spain∗

November 1, 2001

Abstract

A phenomenological model is proposed to explain the recent ob-
served cosmological variation of the fine structure constant as an effect
of the quantum vacuum, assuming a flat universe with cosmological
constant Λ in the cases (ΩM ,ΩΛ) equal to (0.3, 0.7) and (1, 0). Be-
cause of the fourth Heisenberg relation, the lifetime of the virtual
pairs of the zero-point radiation must depend on the gravitational
potential Φ, so that the quantum vacuum changes its density and ac-
quires a relative permittivity different from one. Since the matter was
more concentrated in the past, the gravitational potential of all the
universe was stronger and the optical density of the vacuum higher,
the electron charge being therefore more renormalized and smaller
than now. The model is based on a first order Newtonian approx-
imation that is valid for the range of the observations, but not for
very high redshift, the prediction being that ∆α/α is proportional to
{ΩM [a(t)−1 − 1] − 2ΩΛ[a(t)2 − 1]}, a(t) being the scale factor. This
agrees with the observations.

PACS numbers: 06.20.Jr, 95.30Sf, 98.80.Es
Short title: Cosmological variation of α

∗E-mail: afr@fis.ucm.es

1

http://arxiv.org/abs/astro-ph/0202224v4


A.F. Rañada, Cosmological variation of α 2

Introduction. The observations of the absorption lines of distant quasars
by Webb et al seem to indicate that the fine structure constant was smaller in
the past [1]. The idea that the fundamental constants are actually changing
goes back to the “large number hypothesis” by Dirac [2], what stimulated
a number of proposals to formulate the variation of the constants, see for
instance ref. [3] in which α is written in terms of a scalar field, or other
theories in which the variation of α is induced by a variation of the light
velocity [4] (see also [5]).

This letter proposes a way to understand this phenomenon as an effect of
the quantum vacuum. More precisely, it is argued here that, because of the
fourth Heisenberg relation, the density of the sea of virtual particles in the
quantum vacuum must change in a gravitational field, with a corresponding
variation of its permittivity and permeability. As a consequence the quan-
tum vacuum was optically denser in the past, when the universe was more
contracted than now and the gravitational potential due to all the universe
was stronger. The observed electron charge e is the result of the renormal-
ization of its bare charge ebare(> e) by the quantum vacuum [6], so that a
denser vacuum implies a smaller observed charge. It would be desirable to
study this question in the frame of a rigorous quantum field theory, but the
quantum vacuum is not understood well enough today in order to do that
(for instance it is not known why its energy density seems to be so small).
The only available alternative (other than waiting for a future theory) is to
try a phenomenological approach, as is done here.

In the model here proposed, the quantum vacuum is treated as a transpar-
ent optical medium characterized by its permittivity and permeability, the
change of α being a consequence of the fourth Heisenberg relation applied to
the gravitational interaction of the virtual pairs in the zero point radiation
with all the universe. The analysis is nonrelativistic (although with the in-
clusion of the rest-mass energy mc2) and only weak Newtonian gravitational
potentials Φ are considered for which |Φ|/c2 ≪ 1.

The fourth Heisenberg relation and the vacuum density. Tradi-
tionally, quantum physics has stated that the sea of virtual pairs that are
created and destroyed constantly in the quantum vacuum, i. e. the zero-
point energy, has infinite density, as follows from the simple application of
its basic principles. However, there is now evidence that this density may be
finite. On the average and phenomenologically, a virtual pair created with
energy E (including rest-mass energy, kinetic energy and electromagnetic en-
ergy) will live during a time τ0 = h̄/E, according to the fourth Heisenberg



A.F. Rañada, Cosmological variation of α 3

relation. A constant number density of pairs is established in this way at a
certain equilibrium between the number of particles created and destroyed
per unit time. This has an important consequence: virtual pairs must live
longer in the gravitational field created by a mass distribution, because they
have an extra negative potential energy EΦ/c2 there (Φ being the Newtonian
potential). Indeed their lifetime must be

τΦ = h̄/(E + EΦ/c2) = τ0/(1 + Φ/c2). (1)

The consequence is clear if unexpected: the number density of pairs N de-
pends on the gravitational potential Φ as

NΦ = N0/(1 + Φ/c2). (2)

In other words, the density of the quantum vacuum depends on the grav-
itational potential because it must have there an extra number density of
pairs depending on Φ/c2. If Φ < 0, the quantum vacuum becomes denser, if
Φ > 0, it becomes thinner. These two cases correspond to the gravity cre-
ated by the mass and the cosmological constant, respectively. However, this
change of the number density can produce no gravitational effects, since the
gravitational mass of the virtual pairs in a volume depends on Φ through the
product of their mass E/c2 by the number density per unit volume and unit
energy N (E). This product does not change since the second factor acquires
a factor (1 + Φ/c2)−1 in a gravitational field, the first a factor (1 + Φ/c2), as
we have seen. The effect that we are considering is optical and dielectric, not
gravitational. There is no contradiction with the usual Lorentz invariance
of the quantum vacuum, since the presence of a mass or any non uniform
gravitational field breaks the Lorentz symmetry. This variation of the vir-
tual pairs lifetime in a gravitational field is not an ad hoc hypothesis, but an
unavoidable consequence of the fourth Heisenberg relation. Note also that
the virtual particles considered here are not created by gravity: they are just
the usual zero point particles that fill the space everywhere, according to
elementary quantum physics: they live a bit longer (or shorter), that is all.

In the following we will deal with the variation of the observed values of
certain quantities between a spacetime point of the universe with potential
Φ and a terrestrial observatory. As they will be expressed at first order in
the potential, (Φ − Φ⊕) will be written instead of Φ, Φ⊕ being the present
gravitational potential of all the universe here at Earth.



A.F. Rañada, Cosmological variation of α 4

Vacuum permittivity and permeability in a gravitational poten-

tial. Following the previous considerations, we will admit as a phenomeno-
logical hypothesis that the quantum vacuum can be considered as a substra-
tum, similar to an ordinary transparent optical medium and characterized
by a permittivity and a permeability that depend on Φ. As Φ becomes more
negative, its density increases and the bare electron charge ebare is renormal-
ized further or, otherwise stated, the observed charge must become smaller.
In such a view the permittivity and the permeability of the quantum vacuum
must change to ǫrǫ0 and µrµ0, the first factor expressing the effect of its thick-
ening (or lightening). As we assume a weak field, we can express the relative
permittivity and permeability at first order in the potential. Since both are
equal to 1 now at Earth, their value at a spacetime point with potential Φ
can be expressed as:

ǫr = 1 − β(Φ − Φ⊕)/c2, µr = 1 − γ(Φ − Φ⊕)/c2, (3)

β and γ being certain coefficients, which must be positive since the quantum
vacuum is dielectric but paramagnetic (its effect on the magnetic field is due
to the magnetic moments of the virtual pairs). It must be stressed that eqs.
(3) are a plausible hypothesis, consequence of the fourth Heisenberg relation.

Because of (3) the observed values of the electron charge and light velocity
at potential Φ must be equal to e/ǫr = e(1 + β(Φ − Φ⊕)/c2) and c/

√
ǫrµr =

c[1+(β +γ)(Φ−Φ⊕)/2c2], at first order (the more negative is Φ, the smaller
the charge and the slower the light). Consequently α = e2/4πh̄cǫ0 must

change to α
√

µr/ǫ3
r . This means that the change in the observed value of

α, as computed at Earth from the light absorbed or emitted at a spacetime
point with potential Φ, must be

∆α/α = ξ(Φ − Φ⊕)/c2, (4)

where ξ = (3β − γ)/2 and Φ⊕ is the potential at Earth.
On the gravitational potential of all the universe. As said before,

this work considers only weak gravitational potentials that verify |Φ|/c2 ≪ 1
and can be studied therefore in the Newtonian limit [7]. When that condi-
tion is no longer verified, the gravitational energy of a body is comparable
to its rest energy and a relativistic theory must be used instead. This point,
however, needs a clarification, since there are two independent different as-
pects in a Newtonian approximation. One is that the force on a particle
must be weak and equal to −∇Φ, the potential Φ being determined up to



A.F. Rañada, Cosmological variation of α 5

an additive constant, other is the value of the potential energy of a particle.
When studying a bounded system, as the Earth-Moon pair, a galaxy or a
cluster, the effect of the faraway bodies can be neglected in the study of
its motion, taking only the potential Φ caused by the nearby masses. The
reason is clearly that the distant bodies contribute to Φ with a term that
is almost independent of the space coordinates, at the scale of the system
studied, and has therefore a negligible effect on the forces on the system.
This is correct for all practical purposes concerning the motion. However,
the gravitation being a long range force, one must be careful to include all
the faraway matter that could have an effect on the problem if something
depends on its gravitational potential energy. For this reason, the potentials
Φ and Φ⊕ that appear in eqs. (1)-(4) will be taken in this work as being due
to all the universe. Note that this potential must include the contribution
of matter, either ordinary or dark, as well as the effect of the cosmological
constant.

Indeed the gravitational potential energy of a body is defined as the
energy needed to bring it from the infinity to its actual position, without
changing its kinetic energy or any other nongravitational energy. It could be
argued that a body can not be brought actually from the infinity, since this
is farther away than the horizon of the visible universe. However, it is clear
that less energy is needed to create a virtual pair if Φ is negative than if it
is zero, the difference playing the same role as the gravitational energy when
bringing a body to a spacetime point at potential Φ. Therefore, the variation
of the lifetime of the virtual pairs is due in this model to the gravitational
interaction of the virtual pairs of the quantum vacuum with all the matter
and radiation in the universe, and also with all the quantum vacuum itself.

It is assumed here that the universe is flat (i.e. k = 0) and consists in
ordinary plus dark matter (with zero pressure) and dark energy due to the
cosmological constant Λ. The densities over the critical density at present
time are noted as usual ΩM , ΩΛ (with ΩM + ΩΛ = 1). Taking pΛ = −ρΛc2

as the equation of state of the quantum vacuum, the source of its gravity
in the Newtonian approximation is ρΛ + 3pΛ/c2 = −2ρΛ. Consequently,
the space average potentials at present time due to the matter and to the
quantum vacuum are Φav,M = ΩMΦ0 and Φav,Λ = −2ΩΛΦ0, respectively,
where Φ0/c

2 = − ∫RU

0
Gρcr4πrdr/c2 ≃ −0.3 is the potential that would be

created by a mass distribution with the critical density, RU = 3, 000 Mpc
being taken as the visible universe radius. Most of these potentials is due
to the larger distances. This means that the average gravitational potential



A.F. Rañada, Cosmological variation of α 6

produced by the visible universe is now equal to Φav = Φ0(ΩM − 2ΩΛ). The
effect of the inhomogeneities can be neglected. For instance, the contributions
to the potential of the Sun, of Earth and of the Galaxy at a terrestrial
laboratory are, respectively, ≃ −10−8c2, ≃ −7× 10−10c2 and ≃ −6× 10−7c2,
which are much weaker than Φav.

Cosmological variation of α. Webb et al measured lines absorbed
by distant gas clouds at high redshift, so that Φ in (4) must be the po-
tential at the absorption time t, which can be taken to be approximately
uniform, except for small scale inhomogeneities. In the past, when the uni-
verse was more compact and dense, the distances varied as the scale factor
a(t), the mass density as a(t)−1 while the density of the quantum vacuum
was constant (except for its variation due to eq. (2) that would give a sec-
ond order effect). Consequently, the space average potential at time t was
Φav(t) = Φ0(ΩM/a(t)−2ΩΛa2(t)). Introducing this expression in eq. (4) and
neglecting the local inhomogeneities, it turns out that (αz − α)/α, can be
expressed as

∆α

α
= ξ

Φ0

c2

[

ΩM

(

1

a(t)
− 1

)

− 2ΩΛ

(

a2(t) − 1
)

]

. (5)

Assuming a flat dust universe, the time evolution of the scale factor is

a(t) =
(

ΩM

ΩΛ

)1/3

sinh2/3

[

(3Λ)1/2t

2

]

, (6)

with Λ = 8πGΩΛρcr (eq. (6) reduces to a(t) = (t/t0)
2/3 if Λ = 0, t0 being the

age of the universe). Equations (5)-(6) give the main result of this work: the
time dependence of ∆α/α. Note that, since ξΦ0 < 0, ∆α < 0 for t < 0, so
that the quantum vacuum must have been optically denser in the past (the
electron charge was more renormalized than now since Φav(t)−Φav(t0) < 0).

The thick line in Figure 1 shows the relative change of the fine structure
constant given by eqs. (5)-(6) in the case (ΩM = 0.3, ΩΛ = 0.7) as compared
with the observations by Webb et al [1] (in units of 10−5) versus the look
back-time (in units of the age of the universe), with ξ = 1.3 × 10−5. This is
the value that gives the best fit (remind that Φ0/c

2 ≃ −0.3); it was obtained
by minimizing χ2, the minimum value being 0.63 per point. The thin line
shows the same result for (ΩM = 1, ΩΛ = 0). The best fit was obtained here
for ξ = 1.9×10−5, the value of χ2 being of 0.56 per point. Although the latter
gives a slightly better fit, both curves fit well the observational points taking



A.F. Rañada, Cosmological variation of α 7

into account the large error bars, their difference being small. As explained
before, the light velocity must be affected also by the quantum vacuum and
be time dependent. A similar argument shows that the refractive index at
time t is n(t) = c/c(t) = 1− (β +γ)Φ0[Ωm(1/a−1)−2ΩΛ(a2 −1)]/2c2 (> 1),
so that the light velocity is an increasing function of time.

Validity of the model. Being based on a first order approximation in
Φ, this model is valid only if ΩM |Φ0|/a(t)c2 ≪ 1, what means that the
gravitational potential energy of a particle is much smaller than its rest
energy. This condition is verified at present time, approximately at least,
but looses progressively its validity when a(t) decreases so that the model
can not be extended arbitrarily towards the past (in its present nonrelativistic
formulation).

It must be stressed that the singularity of (5) at t = 0 has no physical
meaning since it is outside of the applicability of the model. Neither it can
be said that α is predicted to change too rapidly or to vanish near time zero.
One can not apply these ideas, therefore, to the recombination era, which
happened at a very high redshift. But the interval of validity does cover the
range of the observations by Webb et al, at least as a good approximation.
To extend this model farther away in the past, it must be reformulated in
relativistic terms.

This is not to say that the Newtonian approximation can not be used
farther away in the past, for higher redshift. It can certainly be used whenever
it is not necessary to consider the gravitational interaction of a system with
all the universe.

On the Oklo and other data. Damour and Dyson [8] analyzed the
data from the natural reactor which operated 1.8 billion years ago at Oklo
(Gabon) and concluded that the relative change of α from then to now is in
the interval (−0.9×10−7, 1.2×10−7) (assuming that other constants like the
Fermi constant do not vary). Equation (5) gives for that time the relative
change ∆α/α ≃ −1.3 × 10−6 (resp. ≃ −5.4 × 10−7) if ΩM = 0.3, ΩΛ = 0.7
(resp. ΩM = 1, ΩΛ = 0), values which are outside but not far from that
interval.

The Oklo study is thought to give the most powerful method to determine
the variation of α with geochemical data but, according to Uzan [9] “one has
to understand and to model carefully the correlations of the variations of αw

and gs as well as the effect of µ (= me/mp). This difficult but necessary task
remains to be done”. There are several others studies which set bounds on
the variation of |∆α|, using a number of different data [9]. The results of the



A.F. Rañada, Cosmological variation of α 8

present model are compatible with these other bounds, except for the one set
by Olive et al using data of β-decay in meteorites [10], |∆α/α| < 3 × 10−7

during the past 4.6 Gyr (redshift about 0.45), while this model gives ≃
3× 10−6 (resp. ≃ 1.7× 10−6) if ΩM = 0.3, ΩΛ = 0.7 (resp. ΩM = 1, ΩΛ = 0).
However, this bound by Olive et al “can also be altered if the neutrinos are
massive”, according to Uzan [9].

In any case, this model gives a fair account of the observations by Webb
et al. Taking everything into account, it seems worth of consideration to
explore its consequences.

Comparison with the gravitational redshift. The effect described
here produces a frequency shift which may seem similar to the gravitational
redshift ∆ω/ω = ∆Φ/c2. The two effects are different, however. The one
observed by Webb et al is due to the variation of the fine structure constant.
The principal part of the frequencies emitted by an atom is proportional to
(e2/4πh̄ǫ0)

2, not exactly to α2 while the relative separation between lines in
the relativistic fine structure splitting does depend on α2 = (e2/4πh̄cǫ0)

2.
On the other hand, the gravitational shift affects equally to all the lines in
a multiplet. In experiments that disregard the width of the multiplets, the
contribution of the change of α to the frequency shift is obtained from (4),
but with 2β instead of ξ, i.e. ∆ω/ω = 4β∆Φ/c2. In that case, the observed
shift would be ∆ω/ω = (1 + 4β)∆Φ/c2, i.e. the addition of the two effects.
The best confirmations of the gravitational redshift, those by Pound, Rebka
and Snider, agree with the prediction of General Relativity up to about 1 %
[11], but they refer to nuclear levels in which the electromagnetism plays only
a part. This means that the results of this work are certainly compatible with
the experiments on the gravitational redshift if β ≤ 2.5×10−3, an inequality
that this work assumes to be satisfied. However, there is a problem: as
ξ < 3β/2, a bound on β is also a bound on ξ but the converse is not true.
In any case, a necessary condition for the compatibility of the results of this
work and the gravitational redshift experiments is that ξ < 4 × 10−3, while
this model predicts that ξ is equal to about 1.3× 10−5 and 1.9× 10−5 in the
two cases considered.

Two final comments. First, note that the dressed electron charge and
the fine structure constant are not treated here as universal constants but as
the result of the interaction of point charges with the quantum vacuum, their
variations being an indirect consequence of the universal expansion, through
the modification of their renormalization.

Second, because of the conservation of the charge conjugation, the pairs



A.F. Rañada, Cosmological variation of α 9

are created with L = S = 0, so that their energy in a magnetic field is
E − 2µBB with probability 1/2, E being the non-magnetic energy. For
magnetic field close to B0 = mc2/µB ≃ 8.8 × 1013 gauss, some pairs would
have zero total energy, their lifetime being infinite according to the fourth
Heisenberg relation). If B approaches B0 from below, the creation of the pairs
would begin to be energetically free. There would be a threshold to some
peculiar phenomenon, although it is not clear what this can be (note that
the inequality B ≪ B0 is a necessary condition for the validity of the Euler-
Heisenberg Lagrangian to study in QED the interactions of the quantum
vacuum with a magnetic field [12]). Intriguingly, it turns out that B0 is
close to the highest magnetic field measured in magnetars, see Kouveliotou
et al [13]. Commenting ref. [13], Kulkarni and Thompson [14] say, “our
failure to detect radio pulsars with magnetic fields greater than 2 × 1013

gauss is because, at such high fields, quantum electrodynamic effects help to
damp the radio emission” this being “direct evidence of magnetic fields strong
enough to perturb the very structure of the vacuum”. This gives support to
the present work: the upper limit for the magnetic field of magnetars could
be just a manifestation of the fourth Heisenberg relation similar to the one
considered here.

Summary and conclusions. In the phenomenological model presented
here, the cosmological variation of the fine structure constant is due to the
combined effect of the fourth Heisenberg relation and the gravitational in-
teraction of the virtual pairs in the zero-point radiation with all the uni-
verse. The problem is studied in the Newtonian approximation. The quan-
tum vacuum is treated as an optical medium characterized by its relative
permittivity and permeability that depend on the average gravitational po-
tential of the universe. The model predicts that ∆α/α is proportional to
{ΩM [a(t)−1 − 1] − 2ΩΛ[a(t)2 − 1]} (a(t) being the scale factor). The results
for the cases ΩM = 0.3, ΩΛ = 0.7 and ΩM = 1, ΩΛ = 0 agree with the obser-
vations by Webb et al [1] on the cosmological variation of the fine structure
constant, as is seen in figure 1. The argument goes as follows.

(i) Because of the fourth Heisenberg relation, the lifetime of the virtual
pairs of the zero-point radiation depends on the gravitational potential Φ.
This causes the permittivity and the permeability of the quantum vacuum,
the observable electron charge and the light velocity to depend also on Φ.
The consequent change of the observed fine structure constant is expressed
at first order in terms of the average gravitational potential of the universe,
and of a parameter ξ related to the renormalization effects of the quantum



A.F. Rañada, Cosmological variation of α 10

vacuum.
(ii) As the universe was more compact in the past, its average gravita-

tional potential was more negative (or less positive). Consequently, the lines
of the spectra of distant quasars were absorbed with a more renormalized
value of the electron charge than now. As a result, ∆α/α is given by eq.
(5), which is plotted in fig. 1. The agreement seems good. Note that, in
this model, the optical density of the quantum vacuum increases towards the
past and decreases along the universe history.

(iii) The model can not be extended arbitrarily towards the past, since
it is based on a first order Newtonian approximation that is valid for the
recent past, including the range of the observations by Webb et al, but is no
longer applicable to higher redshift. In particular, it can never be applied to
the recombination era. A relativistic approach should be followed in order
to extend the model to the remote past. Note finally that, according to this
model, the light is also affected by the gravitational potential so that it was
slower in the past.

The conclusion of this paper is that the combined effect of the gravitation
of all the universe and of the fourth Heisenberg relation on the density of the
zero-point radiation and the corresponding cosmological variations of α, e
and c proposed here should be further investigated.

I am grateful to Profs. Claudio Aroca, Ana Inés G. de Castro, Elóısa
López, Juan M. Usón and José L. Trueba for discussions.

References

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[5] C.W. CHURCHILL, quoted in M. CHOWN, New Scientist (28 March
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A.F. Rañada, Cosmological variation of α 11

[6] P. W. MILONNI, The Quantum Vacuum (Academic Pres, Boston) 1994.

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[12] E. ZAVATTINI, Comments At. Mol. Phys. 33 (1996) 83.

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http://arxiv.org/abs/hep-ph/0205340
http://arxiv.org/abs/hep-ph/0205269


A.F. Rañada, Cosmological variation of α 12

Figure 1: ∆α/α (times 10−5) vs fractional look-back time, predicted by
this work in the cases ΩM = 0.3, ΩΛ = 0.7 (eqs. (5)-(6), thick line) and
ΩM = 1, ΩΛ = 0 (thin line), as compared with the data by Webb et al [1]
(explanation in the text).



0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fractional look-back time

-1.5

-1

-0.5

0

0.5

1

1.5

2