Dark Matter as a consequence of electric charge non-conservation will it remain “dark” forever?
Richard M. Weiner
Laboratoire de Physique Théorique, Univ. Paris-Sud, Orsay, France and
Physics Department, University of Marburg, Germanyx
It is conjectured that dark matter (DM) was produced as a consequence
of electric charge non-conservation during the early stages of
the universe expansion. Charge conservation was restored through a
phase transition of Rochelle-salt type. The corresponding critical
temperature or energy has not yet been reached in the laboratory and
this might explain why the constituents of DM have so far not been
observed. Arguments are presented according to which DM
was created even before inflation, implying that the corresponding
conditions will possibly never be reproduced in the laboratory. The
fact that the observed mass density of DM exceeds that of ordinary
matter gets a most natural explanation.
Dark Matter (DM) constitutes one of the most challenging, unsolved problems
of cosmology[1] and of physics in general: although it represents about 85% of
the total mass density of the universe the mechanism of its creation and its
constituents are still unknown. This is in part due to the fact that it manifests
itself only indirectly, through its gravitational mass, while ordinary matter is
primarily detected directly by its electromagnetic interactions and is thus
accessible to optical devices. This last fact also explains why ordinary matter is
often called “luminous” and also why DM was discovered rather lately.
At present the most popular explanations of DM are based on the assumption
that it is a thermal relic of the Big Bang made of weakly interacting massive
particles (WIMP -s) [2], which are electrically neutral. However the
mechanism through which this happened is unknown, nor is it understood why
DM is the dominant component of the mass density of our universe and why
x
Email address: weiner@staff.uni-marburg.de
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its constituents have not been detected in the laboratory, despite the fact that
other types of neutral weakly interacting particles like neutrinos have been.
The purpose of this note is to suggest that the creation of dark matter might
cease to be a mystery once we give up the assumption, made so far in all
approaches to DM, that one of the laws of physics, that of conservation of
electric charge, which is valid in our universe at present, was also valid
immediately after the Big Bang. In particular we postulate that dark matter was
produced as a consequence of the fact that most of the initially charged
particles lost their electric charge during the expansion of the universe.
Charge conservation was established at a later stage of expansion. This
postulate is used to explain why DM has not yet been seen and why its mass
density exceeds that of ordinary matter
That the assumption of charge non-conservation is not so shocking as it might
appear at a first look can be realized by reminding that charge conservation,
like most other conservation laws, is, according to Noether’s theorem, a
consequence of a symmetry of the Lagrangian, in this case the electromagnetic
gauge symmetry. On the other hand, whether a given solution of the equations
of motion of a Lagrangian exhibits a given symmetry or not depends on
external circumstances like temperature, matter densities or external fields and
on details of the Lagrangian. In particular, the fact that most known
symmetries are conserved at high temperatures led by analogy to the
assumption that this is valid also for evolution of the universe, so that most
conservation laws known at present were valid also immediately after the Big
Bang. However ferromagnets like the Rochelle salt are a well known counter
example of the observation that high temperatures lead to more symmetry [3].
Actually non-conservation of electric charge at high temperatures was
proposed already a long time ago [4] in order to explain the absence of
magnetic monopoles but strangely enough, the most obvious application of the
charge non-conservation hypothesis at high temperatures, that of DM, which
imposes itself almost by definition, was not considered so far.
In the following we will discuss some of the consequences of the assumption
that the gauge symmetry associated with electric charge was spontaneously
broken after the Big Bang and subsequently restored at lower temperatures,
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during the expansion of the universe. This happened through a phase transition
at a critical temperature Tcrit.
As far as we can gather this assumption does not contradict any known
experimental fact and arguments will be presented why it should be considered
as a candidate for the solution of the puzzle of dark matter.
A first implication of charge non-conservation at high temperatures is that in
opposition to what is assumed so far dark matter as observed at present was
created before the symmetry restoring phase transition.
We will consider two possible scenarios of the hypothesis that electric charge
violation (ECV) was at the origin of DM creation. In both cases the
corresponding energies have not yet been reached in the laboratory and this
explains why the constituents of DM have not been observed so far.
DM produced after inflation. The first scenario is based on the model of
Langacker and Pi [4] which addresses the issue of t’Hooft-Polyakov magnetic
monopoles within the Georgi-Glashow SU(5) GUT model, by using the fact
that monopoles appear whenever a U(1) subgroup arises as a consequence of
spontaneous symmetry breaking. Ref. [4] assumes two or more phase
transitions
G →H1→… →Hn → SU(3)c X U(1)em
where U(1)em is not a subgroup of Hn. By choosing the critical temperature
Tcrit at which U(1)em appears below the value mm of the monopole mass, the
production of monopoles is avoided.
Concretely one considers at T=0 the SU(5) model with SU(5) broken to
SU(3) X SU(2) X SU(1) by an adjoint Higgs representation and then to
SU(3)c X U(1)em by one or more five-dimensional Higgs representations.
For 0 ≤ T ≤ MX where MX >>MW ±, Z the problem is reduced to the
SU(2) X U(1) part of the model with n complex Higgs doublets φi. Actually it
turns out that three doublets φi do the job, i.e. n=3. The Higgs potential at
temperature T = 0 reads
3
V = Σi=1 [ - µi2 φi † φi + λi(φi† φi)2] + Σi<j [σij φi † φj φj † φj + ρij φi † φj φj φj † φj
+ η ij (φi † φj)2 + (φj † φi)2 ]
(1)
and then, as shown explicitly in ref. [4] for the case
G ≡ SU(3)c X (SU(2) X U(1), by choosing appropriately the vacuum
expectation values of the scalar fields <φi(T)> at finite temperature T, one can
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construct a model in which G is broken to SU(3)c at T > Tcrit. In other words,
by choosing G = SU(3)c X (SU(2) X U(1), the lower symmetry, i.e. that
corresponding to the non-conservation of electric charge, is realized at high
temperatures. The fact that the violation of local gauge invariance induces the
violation of global invariance is due to the spontaneous nature of the breaking
of the local invariance [5].
An educated guess of Tcrit proposed in ref.[4] in the assumption of three
doublets φi and satisfying the conditions of an existence proof is Tcrit ≥ 1 TeV.
Surprisingly, this value is quite compatible [2] with that which would follow
from the “WIMP miracle” according to which the present mass density of DM
would be due to a weakly interacting particle with a mass of the order of 100
GeV that was produced by falling out of thermodynamical equilibrium in the
early stages of the universe expansion. This would mean that one can hope to
produce DM particles at the CERN LHC. A further experimental consequence
of this scenario is the prediction of at least two more Higgs doublets.
However the above variant is open to criticism:
(a) As pointed out by Bimonte and Lozano [6] (cf. also Senjanovic [3]) next to
leading order corrections might invalidate the approach of ref.[4].
(b) The “derivation” of Tcrit is based on a simple generalization of the GeorgiGlashow model and assumes three Higgs doublets. Up to now there is no
evidence either for this model or for this particular assumption . Actually,
as emphasized also by the authors of [4], any other value for Tcrit is
possible.
(c) The scenario based on ref. [4] does not explain the experimental
observation
ρDM ≈ 5 ρord ,
(2)
where ρDM and ρord are the mass densities of DM and ordinary matter,
respectively. The fact that these densities are of the same order of
magnitude has lead some authors [7] to conjecture that the two types of
matter have a common origin. Neither does this first scenario explain at
least qualitatively why the dark matter mass density exceeds that
of ordinary matter.
(d) The Langacker-Pi model, proposed before the invention of inflation,
assumes the existence of a phase transition at Tcrit ≈ 100 GeV above which
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electric charge is not conserved. In the evolution of the universe there is no
experimental evidence for any phase transition except possibly inflation
itself. (Actually, in the version of inflation traded under the name “chaotic”
[8] there is no phase transition.)
Since the inflation temperature Tinfl >> 100 GeV this puts under further
question marks the estimate of the critical temperature of ref.[4] but, as
will be shown below, at the same time suggests a different scenario for
the creation of DM as a consequence of ECV, namely that DM was created
before inflation.
Dark Matter produced before inflation. This possibility not only takes care of
problem (d) but shifts problem (a) to the very early stages of the universe
evolution, when gravitational effects are important, too. And although the
quantum field theory of gravitation is also non-renormalizable, almost
everything we know today about the beginning of the universe is based,
besides the theory of general relativity, on classical gravitation. We assume
that this applies also for the following considerations.
In particular we postulate that after the Big Bang the universe consisted of
charged and neutral particles and during the following expansion most or all
charged particles lost their charge. This happened during inflation, which acts
as a filter for charge. Conservation of electric charge was established only after
inflation.
It seems then natural to assume that the critical temperature Tcrit at which the
symmetry restoring phase transition took place coincides with the temperature
Tinfl at which inflation occured, i.e.
Tcrit = Tinfl .
(3)
The question then arises how dark matter created before inflation survived the
exponential expansion characteristic for inflation and which is expected to
wash out any conserved quantity. Here gravitation comes into play through
the “mimetic” mechanism of Chamseddine and Mukhanov [9] (for a
generalization of ref. [9] cf. e.g. [10]) who described dark matter by a scalar
field Φ coupled to the inflaton field φ through a term of the form ΦF(φ) where
F is a slow function of φ.
These authors prove that at the end of the exponential expansion
a ≡ H –1 exp (Ht),
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where H is the Hubble constant and a defines the flat metric
ds2 = dt2 - a2 (t) δikdxidxk,
the difference between the trace G of the Einstein tensor Gµν = Rµν - ½Rgµν and
the trace T of the energy momentum tensor of matter Tµν can be approximated
at the end of inflation by
G – T ≈ - F(φ)/3H ≠ 0,
(4)
which means that dark matter as defined above survives inflation.
According to our present understanding of inflation, for ordinary matter such a
conservation mechanism does not exist. It follows that ordinary matter has been
created after inflation. This means that inflation is the very mechanism
through which the initially charged particles created immediately after the Big Bang
lost their charge and confirms the assumption that dark matter and ordinary matter
have a common origin. Moreover the fact that in this second scenario dark matter
started to be produced before ordinary matter, explains why the dark matter density
exceeds that of ordinary matter, an observation which obtains in this way fundamental
importance. It is amusing to note that in almost all explanations of DM based
on the popular WIMP models the similarity between ρDM and ρord is considered
accidental [7].
Furthermore this scenario predicts a lower limit for the ratio Tinfl / Treheat between the
inflation temperature Tinfl and the reheating temperature Treheat.
Indeed in the assumption that dark matter starts to be produced before inflation i.e. at
temperatures T > Tinfl and ordinary matter after reheating i.e. at T> Treheat we may
describe the temperature dependence of the corresponding mass densities by
ρDM > ginfl Tinfl4
(5)
and
ρord < greheat Treheat4 ,
(6)
where gi are functions of the corresponding number of degrees of freedom. Taking
into account that inflation does not change significantly the number of degrees of
freedom we have
ginfl ≈ greheat.
(7)
From the observed relation between the mass densities
ρDM ≈ 5 ρord
it then follows
Tinfl / Treheat > 1.5.
(8)
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Given the fact that on the average at high temperatures T the masses of
particles created by a system in thermodynamical equilibrium at that
temperature are of the order of T, it follows that the masses of DM particles,
most of which are created before inflation, are on the average much larger than
those of ordinary matter, which are created after the symmetry restoring phase
transition, i.e. after inflation. It then follows from Eq. (3) and the value of
Tinfl ≈ 10-3-10-4 MPlanck , that in this second scenario the constituents of DM will
most probably never be accessible to direct laboratory experiments, at least
with the techniques known so far. That, among other things, distinguishes this
model of DM from models based on anomalous gauge symmetry [11], despite
the fact that in these models gravitation and inflation also play a major role.
Furthermore the existence of weakly interacting particles with masses up to the
inflation temperature Tinfl could contribute to solving the hierarchy problem,
i.e. filling the gap between the Higgs mass characteristic for the electroweak
interaction and the Planck mass characteristic for gravitation. On the other
hand there are certain specific, possible consequences of the above model
which appear experimentally accessible:
(A) In the era preceding inflation due to ECV massive photons might have been
created. While in conventional QED photons are massless, in QED with
broken gauge invariance photons may acquire mass. The experimental upper
limit of such a mass is [12] at present 10-18 eV.
(B) The creation of dark matter might have been accompanied by the bremsstrahlung
of longitudinal photons associated with the disappearance of the respective charges [13].
For a comprehensive review of these issues cf. ref. [14].
In conclusion, if we accept the hypothesis that in the very early universe charge
was not conserved, the mystery of the origin of dark matter might become an
open secret and although dark matter may remain “dark” forever in the sense
that it will never be reproduced in the laboratory, it represents one of the
strongest signals of the early universe.
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