It has been recently claimed that cosmologies with time dependent speed of
light might solve some of the problems of the standard cosmological scenario,
as well as inflationary scenarios. In this letter we show that most of these
models, when analyzed in a consistent way, lead to large violations of charge
conservation. Thus, they are severly constrained by experiment, including those
where $c$ is a power of the scale factor and those whose source term is the
trace of the energy-momentum tensor. In addition, early Universe scenarios with
a sudden change of $c$ related to baryogenesis are discarded.Comment: 4 page

A. Albrecht

A.I. Shlyakhter

A.M. Wolfe

A.Y. Pothekin

C. M. Will

D. Reusser

E. W. Kolb

E.B. Norman

E.W. Kolb

F.T. Avignone

H. Vucetich

J. Beckenstein

J. Ekkis

J. Magueijo

J. Moffat

J.B. Kim

J.D. Barrow

J.D. Prestage

J.E. Horvath

J.H. Schwartz

J.K. Webb

L.B. Okun

L.L. Cowie

M.B. Green

M.W. McElhinny

O. Klein

P. D. Sisterna

P. Sisterna

R. Alonso

R.H. Hellings

Riazuddin

S. J. Landau

S. Orito

T. Damour

T. Harko

T. Kaluza

English

arXiv

It has been recently claimed that cosmologies with time dependent speed of light might solve some of the problems of the standard cosmological scenario, as well as in?ationary scenarios. In this letter we show that most of these models, when analyzed in a consistent way, lead to large violations of charge conservation. Thus, they are severly constrained by experiment, including those where c is a power of the scale factor and those whose source term is the trace of the energy-momentum tensor. In addition, early Universe scenarios with a sudden change of c related to baryogenesis are discarded.Facultad de Ciencias Astronómicas y Geofísica

Landau, Susana Judith

Sisterna, Pablo D.

Vucetich, Héctor

SEDICI - Repositorio de la UNLP

Charge conservation and time-varying speed of light

It has been recently claimed that cosmologies with time dependent speed of light might solve some of the problems of the standard cosmological scenario, as well as inflationary scenarios. In this letter we show that most of these models, when analyzed in a consistent way, lead to large violations of charge conservation. Thus, they are severly constrained by experiment, including those where c is a power of the scale factor and those whose source term is the trace of the energy-momentum tensor. In addition, early Universe scenarios with a sudden change of c related to baryogenesis are discarded.Facultad de Ciencias Astronómicas y Geofísica

Servicio de Difusión de la Creación Intelectual

arXiv:astro-ph/0007108v4  18 Jan 2001Charge conservation and time-varying speed oflightS. Landau∗F. C. Astronómicas y Geof́ısicas, U. N. de La PlataP. D. Sisterna Facultad de C. Exactas, U. N. de Mar del Plataand H. Vucetich†F. C. Astronómicas y Geof́ısicas, U. N. de La PlataFebruary 1, 2008AbstractIt has been recently claimed that cosmologies with time dependentspeed of light might solve some of the problems of the standard cosmolog-ical scenario, as well as inflationary scenarios. In this letter we show thatmost of these models, when analyzed in a consistent way, lead to largeviolations of charge conservation. Thus, they are severly constrained byexperiment, including those where c is a power of the scale factor andthose whose source term is the trace of the energy-momentum tensor. Inaddition, early Universe scenarios with a sudden change of c related tobaryogenesis are discarded.Since one of the key hypothesis of special relativity is the frame indepen-dence of the velocity of light c, it is implied in this statement the time andspace independence of this velocity. As well established that it may seem, thisconstancy principle has been recently contested [1, 2] to provide an alternativeaccount of the horizon, flatness and cosmological constant problems present inthe standard big bang scenario. Instead of the superluminal expansion of theUniverse present in inflationary scenarios, a period in which light traveled muchfaster than today would explain the homogeneity we see today in the Universe.Some cosmological models have also been analyzed afterwards [3, 4] to test thedynamical viability of this scenario.These ideas are highly provocative, not only from the observational view-point but also from the conceptual one. Indeed one of the key aspects of Ein-stein equivalence principle is the time-independence of the so called “fundamen-tal constants” of physics [5]. The replacement of these parameters by one ormore dynamical fields can lead to time- as well as space-dependent local fun-damental constants. Unification schemes such as superstring theories [6] andKaluza-Klein theories [7] have cosmological solutions in which the low-energyfundamental constants are functions of time. Usually low-energy phenomena are∗Fellow of FOMEC†Member of CONICET1used to analize the variation rate of the fundamental constants [8]-[26]. If thecosmological dynamics of a field is such that its large-scale value is invariant un-der local Lorentz transformations, or if the local coupling with matter is strongenough so that it depends on the local environment (e.g. electromagnetism withthe absorber condition), then the local field equations will be Lorentz invariant.If on the other hand the cosmological evolution is non-trivial and the field cou-ples softly with the local matter, it will act as an external bath, breaking localLorentz invariance. A variable speed of light theory may belong to the latterset of theories. Any VSL theory poses an additional problem, namely that cis a dimensional constant, and talking about a varying dimensional constant isnot an invariant statement: we can change our units and obtain a different timedependence of such a parameter. Of course, once we fix our units, every claimabout a dimensional parameter is an invariant claim, since we are implicitlyreferring to a dimensionless ratio: that between the parameter and the unit[17, 18].Any scientific theory has to be stated in clear and precise terms. Becken-stein’s theory of a variable fine structure constant was based on Lorentz invari-ance, explicitly protecting charge conservation. In the case of VSL theories,local Lorentz invariance is relaxed, and the inhomogeneous Maxwell equationsare assumed to be [2]1c∂µ (cFµν) = 4πjν , (1)where jµ = (ρ, j/c) is the electric charge current. In reference [2] it was suggestedthat charge is conserved, implying a variation of the fine structure constantα = e2h̄c. The constancy of e can be derived, for instance, from Dirac’s equation,written in Hamiltonian form:ih̄∂tψ = −(ih̄α · ∇ + h.c.)ψ +mc2ψwhich implies that Q = e∫Vψ∗ψd3x is conserved. The above form is, however,not unique since powers of c can be introduced in the equation in several ways,followed by appropriate symmetrization.However, from equation (1) it can be easily seen that jµ is no longer aconserved current, but satisfies the equation:4π∂µ(cjµ) = ∂µ∂ν(cFµν) (2)The right hand side is not null, because the partial derivatives do not com-mute:[∂0, ∂i] =c,ic2∂tIt is usually assumed that c is a function of a scalar field and that it dependsonly on time in the comoving cosmological frame. In the local frame of thesolar system, moving with a velocity v with respect to the cosmological frame,a small space dependence will arise, with gradients O(v/c) with respect to thetime derivative, which can be neglected for the present purposes. So, the right2hand side of equation 2 is effectively zero. The left hand side, however, is not afour divergence, because ∂x0 = 1/c(t)∂t. The fully expanded expression is:∂tρ+ċcρ+ ∇ · j = 0 (3)If equation (3) is integrated over a volume V containing the charges, weobtainQ̇Q= −ċc(4)where Q =∫Vρd3x is the total electric charge. This is our main result.Equation (4) provides very stringent tests of the variation of c, since therehave been many experiments to test the conservation of charge [28]. Dependingon the details of the theory, several cases arise.If we assume, as it is usually done in this context [2, 3] that the electroncharge e is constant, charge conservation can only be broken by processes thatchange charge discontinuously, such as the dissapearance of electrons or thetransformation of neutrons into protons. A generic model for these processeshas been proposed in references [19, 20], under the assumption of total energyconservation. The first three entries of Table 1 show some sample upper limitsobtained from these processes with different techniques and hypothesis. Theseupper limits on ċ/c are much smaller than those obtained by a direct measure-ment, for instance in reference [21], namely | ċ/c |< 10−13 yr−1.On the other hand, if e varies continuously in such a way that ce is conserved,then α ∝ c−3 and strict limits can be obtained from geophysical or astronomicaldata, such as the Oklo phenomenon [15, 16] or the line spectra of distant quasars[13, 14]. Furthermore, evidence for the time variation of the fine structureconstant has been claimed by Webb et al. [25] and confirmed while we werecorrecting this paper [26]. Thus, if the systematic errors are well estimated,the requirement of conservation of charge suggests that, the mechanism for αvariation should not be driven by the change in the speed of light.These limits discard a great number of cosmological models with varyingvelocity of light. For instance, the family introduced in [3] parameterizes lightvelocity in the form: c = c0(aa0)nwith a the cosmological scale factor. Eventhough, [3] was not proposing that this behaviour exists at all times, [22] sug-gests that this solutions could be extended to radiation and matter-dominateduniverses. It was shown in reference [3] that n < −1/2 is necessary to solve theflatness and horizon problems, and n < −3/2 solves the cosmological constantproblem.On the other hand, any complete and consistent VSL theory will predictdynamically the value of c via a wave-like equation for ψ = ln cc0, whose sourceterm is proportional to the trace of the energy-momentum tensor T . Thus, inthe neighborhood of a quasi-static system, such as a star or a virialized galaxycluster, a generic expresion for a varying c will be [23]:c = cc(t)(1 −λGMc20r)(5)3Here λ is a constant that depends on the specific VSL theory. Furthermore,in the limit r → ∞ , the expression for c reduces to the cosmological one(c→ cc(t)).For the purposes of this paper, our interest is focused on violation of chargeconservation. Expanding equation(2) we obtain:4π∂µjµ = −4π∂µccjµ − ∂tc~∇c. ~Ec3+~∇.cc∂0 ~E (6)In a static situation ~j = 0 and ∂0E = 0, thus the effects of ~∇c are of secondorder and can be neglected. Hence, there are two contributions to the violationof charge conservation, one accounts for time-variations over cosmological time-scales and the other for the motion of the Earth with respect to massive bodiessuch as clusters or galaxies:ċc=(ċc)cosmological+(ċc)local= nȧa+λGMc20r3~r.~v (7)where ~v is the velocity of the Earth with respect to the lump. (~v ∼ 1000 kmsegforthe Virgo supercluster).From the first term of equation (7) we find the limits of table 1 on n, whichcontradict the above requirements. (We use H0 = 65 km/s/Mpc = 6.65 ×10−11 yr−1). Similar bounds can be obtained for other similar models, such asthose studied in reference [22] (See also [32]).For the Virgo supercluster, GMc20r∼ 2 × 10−7. The last three results in table1 for λ are obtained from the second term of equation (7) . These results ruleout essentially any VSL whose source of variation is the trace of the energy-momentum tensor T .Finally, models similar to the original Albrecht-Magueijo one [2], involvinga sudden change of c between two different constant values in the very earlyUniverse, are not affected by the above limits. Orito and Yoshimura [33] ob-served that if charge conservation is broken in the very early Universe, a largecharge excess should have been formed through a mechanism similar to that ofbaryogenesis: violation of Q, C and CP conservation while the system is out ofthermodynamic equilibrium [34, 35]. In the above mentioned models, the netcharge excess will be produced by way-out-of-equilibrium production and decayof heavy mesons [35].Let X be an unstable meson that produces a mean baryon number ǫB and amean charge excess ǫQ per decay, and assume that matter is created during thecharge transition period. Then, the equations for the evolution of the numberdensities of X,B,Q will be [2, 3, 35]:(a3nX)•+ λX(a3nX) =3Kcċ4πGmXa˙(a3nB)•= ǫBλXnXa3˙(a3nQ)•= ǫQλXnXa34In these equations, K is the curvature parameter, mX is the mass of the Xmeson, and all densities are effective densities (particle - antiparticle). Let usassume that the change in c occurs in a short interval of time tc << τ <<1λx.Charge conservation will be broken only during this interval, but the X mesondecay will always produce a baryon excess. With these hypothesis, the aboveequations have the following solutions:a3nX ≃3K(c20−c2P)8πGmXa(0)e−λt = a3(0)n0Xe−λta3nB ≃ ǫBa3(0)n0X(1 − e−λt)→ ǫBa3(0)n0Xa3nQ ≃ ǫQλXτa3(0)n0X ≃ǫQǫBλXτa3nBAfter the transition, nQ will be fixed but nB will be diluted from the aboveestimate by thermal processes [35]. So we finally get a lower bound on thecharge excess:∣∣∣∣nQnB∣∣∣∣∼ǫQǫBλXτ (8)As we have mentioned before, we expect on general grounds that τ > tPl,while ǫQ ∼ ǫB, since these fractions depend both on the C and CP breakingterms in the lagrangean. Thus, equation (8) predicts a firm lower limit for thecharge excess. Orito and Yoshimura [33] have given limits on any charge excessin the Universe, shown in table 1. These limits are many orders of magnitudebelow the prediction of equation (8). The last column of the table shows roughestimates of τ taken from the observational limits, assuming 1/λX ∼ tGUT .Although these results do not rule out all varying velocity of light theories,they put very stringent bounds on them through the conservation of chargerequirement. Moreover, similar bounds will hold for any theory with varyingspeed of light velocity in the early universe. These bounds, which may belowered through improvements in the experimental techniques [38], will leadinto deeper understanding of these interesting theories.The authors want to thank Diego F. Torres for many interesting discussionsand advice. The authors acknowledge partial economic support through theproject 011/G035, Universidad Nacional de La Plata.References[1] J. Moffat, Int. J. of Phys. D 2 351 (1993); Foundations of Physics 23 411(1993)[2] A. Albrecht and J. Magueijo, Phys. Rev. D 59 043516 (1999)[3] J. D. Barrow, Phys. Rev. D 59 043515 (1999)[4] R. H. Brandenberger and J. Magueijo, hep-ph/9912247 (1999)[5] C. M. Will Theory and experiment in gravitational physics 2nd edition.(Cambridge University Press, Cambridge, 1993)5[6] J. H. Schwartz Phys. Rep. 89 223 (1982); M. B. Green and J. SchwartzPhys. Lett. 199B 117 (1984)[7] T. Kaluza Sitz. Preuss. Akad. Wiss. Phys. Math. K11 966 (1921); O. KleinZ. Phys. 371 895 (1926)[8] J. Ekkis, E. Gabathuler and M. Karliner Phys. Lett. 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Magueijo Class. and Quantum Grav.16 1435 (1999)[23] J.Magueijo astro-ph/0010591 (2000)[24] J. B. Kim, J. H. Kim and H. K. Lee Phys. Rev D 58 027301 (1998)[25] J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater and J.D. Barrow Phys. Rev. Lett. 82 884 (1999)[26] J.K. Webb, M.T. Murphy, V.V. Flambaum, V.A. Dzuba, J.D. Barrow,C.W. Churchill, J.X. Prochaska and A.M. Wolfe astro-ph/0012539[27] J. Beckenstein Phys. Rev. D 25 1527 (1982)[28] L. B. Okun in Review of particle properties Phys. Rev D 45 Part II, VI.10(1992)6Process Ref. Datum | Q̇/Q | (y−1) Param.τ (y) | n |71Ga →71 Ge [29] ≥ 3.5 × 1026 ≤ 2.9 × 10−27 < 5 × 10−17e→ νeγ [30] ≥ 2.4 × 1025 ≤ 4.2 × 10−26 < 7 × 10−16e→ any [31] ≥ 2.7 × 1023 ≤ 3.7 × 10−24 < 6 × 10−14| ∆α/α |Oklo phenomenon [16] ≤ 1.2 × 10−7 ≤ 2.0 × 10−16 < 3 × 10−6Quasar absorption systems [13] ≤ 3.5 × 10−4 ≤ 1.18 × 10−13 < 2 × 10−3Quasar absorption systems [14] ≤ 6 × 10−5 ≤ 5 × 10−15 < 7 × 10−5nQ/nB τ/tPlCoulomb force smaller thanNewton force in stars [33] < 10−18 — < 10−7CMB anisotropy [33] < 2 × 10−20 — < 2 × 10−9Cosmic ray isotropy [33] < 10−29 — < 10−18τ (y) | λ |71Ga →71 Ge [29] ≥ 3.5 × 1026 ≤ 2.9 × 10−27 < 2 × 10−10e→ νeγ [30] ≥ 2.4 × 1025 ≤ 4.2 × 10−26 < 2 × 10−9e→ any [31] ≥ 2.7 × 1023 ≤ 3.7 × 10−24 < 2 × 10−7Table 1: Upper limits on charge non-conservation. The columns show the pro-cess considered, the corresponding references, the observational data, the chargenon-conservation upper bound and the limits for the model parameters.[29] E. B. Norman, J. N. Bahcall and M. 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Charge conservation and time-varying speed of light

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