Isotropic charged cosmologies in infrared-modified electrodynamics

TL;DR

This paper investigates whether infrared modifications to electrodynamics could allow for isotropic charged cosmologies, potentially bypassing strict observational limits on the universe's electric charge asymmetry.

Contribution

It introduces a simple cosmological model with infrared-modified scalar electrodynamics, showing how such modifications can produce a cosmological constant-like effect from charge asymmetry.

Findings

Late-time effects mimic a cosmological constant.

Stringent bounds on charge asymmetry if generated during inflation.

Relaxed bounds if charge asymmetry is produced after inflation.

Abstract

It has long been known that the covariant formulation of quantum electrodynamics conflicts with the local description of states in the charged sector. Some of the solutions to this problem amount to modifications of the subsidiary conditions below some arbitrarily low photon frequency. Such infrared modified theories have been shown to be equivalent to standard Maxwell electrodynamics with an additional classical electromagnetic current induced by the quantum charges. The induced current only has support for very small frequencies and cancels the effects of the physical charges on large scales. In this work we explore the possibility that this de-electrification effect could allow for the existence of isotropic charged cosmologies, thus evading the stringent limits on the electric charge asymmetry of the universe. We consider a simple model of infrared-modified scalar electrodynamics in the cosmological context and find that the charged sector generates a new contribution to the energy-momentum tensor whose dominant contribution at late times is a cosmological constant-like term. If the charge asymmetry was generated during inflation, the limits on the asymmetry parameter in order not to produce a too-large cosmological constant are very stringent $\eta_Q <10^{-131}- 10^{-144}$ for a number of e-folds $N=50-60$ in typical models. However if the charge imbalance is produced after inflation, the limits are relaxed in such a way that $\eta_Q<10^{-43}(100 \,\mbox{GeV}/T_Q)$, with $T_Q$ the temperature at which the asymmetry was generated. If the charge asymmetry has ever existed and the associated electromagnetic fields vanish in the asymptotic future, the limit can be further reduced to $\eta_Q<10^{-28}$.

Equations

$$({\nabla }_{\mu }{A}^{\mu }{)}^{(-)}|\Phi ⟩=0$$

$${\partial }_{\mu }{F}^{\mu \nu }-{\partial }^{\nu }({\partial }_{\mu }{A}^{\mu })=J^{\nu }$$

$$({\partial }_{\mu }{A}^{\mu }{)}^{(-)}|\Phi ⟩=0$$

$$\square A_{\mu }=J_{\mu }$$

$$A_{\mu }(x)$$

$${\Delta }^{ret}(x)=\frac{1}{(2\pi {)}^2}\delta (x^2)\delta (x^0)$$

$${\Delta }^{adv}(x)=\frac{1}{(2\pi {)}^2}\delta (x^2)\delta (-x^0)$$

$$A_{\mu }^f(x)=\frac{1}{(2\pi {)}^3}\int \frac{d^{3}k}{2\omega }\left(a_{\mu }(\vec{k})e^{-ikx}+a_{\mu }^{†}(\vec{k})e^{ikx}\right)$$

$$[a_{\mu }(\vec{k}),a_{\nu }^{†}({\vec{k}}^{\prime })]=-{\eta }_{\mu \nu }2\omega {\delta }^3(\vec{k}-{\vec{k}}^{\prime })$$

$${\partial }_{\mu }{A}^{\mu }(x)$$

$$\Delta (x-y)$$

$$(\omega (a_0(\vec{k})-a_{\parallel }(\vec{k}))-\rho (\vec{k}))|\Phi ⟩=0$$

$$\rho (\vec{k})=\frac{1}{(2\pi {)}^{3/2}}\int e^{-i\vec{k}\vec{x}}{J}^0(0,\vec{x})d^{3}x$$

$$\rho (0)=\frac{1}{(2\pi {)}^{3/2}}\int J^0(0,\vec{x})d^{3}x=\frac{Q}{(2\pi {)}^{3/2}}$$

$$|\Phi ⟩=\exp \left(-\frac{1}{2}\int (a_0^{†}(\vec{k})+a_{\parallel }^{†}(\vec{k}))\frac{\rho (\vec{k})}{\omega }\frac{d^{3}k}{2\omega }\right)|\Psi ⟩$$

$$|\Psi ⟩=\mathcal{F}[a_0^{†}(\vec{k})-a_{\parallel }^{†}(\vec{k}),a_{\perp }^{†}(\vec{k})]|0⟩$$

$$⟨\Phi |\Phi ⟩=\exp \left(\int \frac{|\rho (\vec{k}){|}^2}{{\omega }^2}\frac{d^{3}k}{2\omega }\right)$$

$$\left(\omega (a_0(\vec{k})-a_{\parallel }(\vec{k}))-\rho (\vec{k})+Q{f}_c(\omega )\right)|\Phi ⟩=0$$

$$\begin{gathered} \mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{\lambda }{2}({\nabla }_{\mu }{A}^{\mu }{)}^2 \\ +(D_{\mu }\phi {)}^{\ast }(D^{\mu }\phi )-V(|\phi |), \end{gathered}$$

$$S[A_{\mu },\phi ,g_{\mu \nu }]=\int \sqrt{\mathfrak{g}}\mathcal{L}{d}^{4}x,$$

$${\nabla }_{\alpha }{F}^{\alpha \mu }+\lambda {\nabla }^{\mu }({\nabla }_{\alpha }{A}^{\alpha })=J^{\mu }$$

$$D_{\mu }(\sqrt{\mathfrak{g}}{D}^{\mu }\phi )+\frac{1}{2}\sqrt{\mathfrak{g}}\frac{V^{\prime }(|\phi |)}{|\phi |}=0,$$

$$J^{\mu }=iq[{\phi }^{\ast }(D^{\mu }\phi )-\phi (D^{\mu }\phi {)}^{\ast }]$$

$$\square ({\nabla }_{\alpha }{A}^{\alpha })=0$$

$$({\nabla }_{\alpha }{A}^{\alpha }{)}^{(-)}(x)|\Phi ⟩=b^{(-)}(x)|\Phi ⟩$$

$$\square b(x)=0$$

$$\int \sqrt{g_{\Sigma }}{\partial }_{\mu }b(x)d{\Sigma }^{\mu }=\frac{q}{\lambda }$$

$$⟨\Phi |({\nabla }_{\alpha }{A}^{\alpha })(x)|\Phi ⟩=b(x)$$

$${\nabla }_{\alpha }{F}^{\alpha \mu }=J^{\mu }-\lambda {\nabla }^{\mu }b$$

$$d{s}^2=d{t}^2-a^{2}d{\vec{x}}^2,$$

$$a^3(t)J_0(t)=\kappa ,$$

Full text

I Introduction

One of the long-standing questions in cosmology, dating back to the works of Bondi and Lyttleton in the late fifties Lyttleton , is the possibility that the universe could have a net electric charge density. It soon became apparent that this kind of charged cosmologies, even respecting large-scale homogeneity, are necessarily anisotropic. Indeed, it is well-established in the context of Maxwell electrodynamics that the presence of a non-vanishing charge density $\rho$ generates electric fields such that ${\bf\nabla}\cdot{\bf E}=\rho$. Even if we assume that both, the electric field and the charge density, are spatially uniform within our Hubble horizon, the corresponding electromagnetic energy-momentum tensor is nonetheless anisotropic. This implies a departure from the Robertson-Walker geometry which can conflict with the observed isotropy of different backgrounds. Thus, introducing the charge asymmetry parameter as $\eta_{Q}=n_{Q}/n_{\gamma}$, where the charge density is $|e|n_{Q}$, it has been shown that the isotropy of the cosmic microwave background imposes a limit \eta_{Q}\,\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}\,10^{-30}, whereas the isotropy of the observed cosmic ray distribution sets \eta_{Q}\,\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}\,10^{-39} Orito . On the other hand, the electromagnetic interaction with the electrostatic potential generated by the net charge density induces an effective mass shift for any charged particle present in the cosmic plasma. These shifts introduce changes in the nucleosynthesis mechanism which can be translated into very stringent limits on the charge asymmetry \eta_{Q}\,\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}\,10^{-43} Masso . The mentioned constrains were obtained ignoring the high conductivity of the cosmic plasma. When conductivity effects are taken into account the limits can be improved, setting \eta_{Q}\,\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}\,10^{-35} for the CMB case Ferreira .

On the theory side, several models have been proposed to generate a cosmological charge imbalance. In gauge1 ; gauge2 ; gauge3 spontaneous breaking of gauge invariance was considered, the symmetry being restored at late times in order to comply with present experimental results. Other mechanisms involve the generation of charge by the presence of a photon mass Dolgov or prior to the GUT phase transition in brane-world scenarios. Quantum fluctuations of charged fields during inflation have also been considered in scalar1 ; scalar2 ; scalar3 to generate charge fluctuations on super-Hubble scales.

Even though the possibility of having a charge asymmetry compatible with current observations seems to be very limited in the context of Maxwell electrodynamics, this is not the case in modified electromagnetic theories. Thus, Barnes Barnes realized for the first time that the Proca generalization of electrodynamics, which propagates and additional longitudinal polarization for the photon, admits the possibility that the universe could possess a net electric charge density uniformly distributed throughout space, while possessing no electric or magnetic fields, thus allowing for homogeneous and isotropic Robertson-Walker solutions and evading most of the aforementioned limits. In particular, for constant photon mass and assuming charge conservation, the electromagnetic energy-momentum tensor in this theory behaves, for homogeneous fields, as that of a perfect fluid with equation of state $p_{P}=\rho_{P}$ (see Zeldovich ) so that the energy density in a charged-dominated universe would scale as $a^{-6}$. However, this scaling suggests that in order to avoid large contribution in the early universe, the potential contribution of charge in the present universe should be tiny. Extensions of these ideas in which the photon mass can depend on time were considered in Woodard1 ; Woodard2 where a model for cosmic acceleration was proposed.

In this work we re-examine the cosmology of a charged universe in Maxwell electrodynamics (with two propagating physical modes) from a different perspective. The well-known local and covariant formulation of quantum electrodynamics contains two fundamental ingredients: on one hand the dynamics, which is provided by Maxwell equations and, on the other, the constraints, which allow to eliminate the unphysical degrees of freedom, and are given in the canonical formalism by the Gupta-Bleuler GB1 ; GB2 subsidiary conditions.

[TABLE]

where $(\nabla_{\mu}A^{\mu})^{(-)}$ is the negative frequency part of the operator $\nabla_{\mu}A^{\mu}$ and $|\Phi\rangle$ denotes a physical state. However, this formulation of quantum electrodynamics present certain difficulties in the charged sector which are known since the seventies Kulish ; Ferrari . In particular Maison and Zwanziger MZ proved a general result that states that there is no localized charged state in covariant QED which satisfies the above subsidiary conditions. In other words, either we abandon locality in the description of the charged states or if we insist in a local description of charges we must assume that all charged particles are produced from the decay of neutral states.

A possible way out of this limitation of covariant QED is the modification of the subsidiary condition in the infrared. An explicit implementation of these ideas have been presented by Zwanziger in Z (see also Haller ; Z2 ) and amounts to the introduction of an additional classical conserved current which is generated by the quantum current. In momentum space, this current has only support in the infrared, i.e. below the cutoff frequency, and cancels the effects of the quantum charges on very large scales. This property suggests that the cosmology of charged universes could exhibit important differences in this kind of modified electrodynamics, and, in particular this opens the possibility of having isotropic charged solutions without including additional polarizations for the photon field.

Even though Zwanziger model Z is relatively old, its cosmological implications had not been analyzed so far. The aim of the present work is precisely to evaluate the cosmological viability of that model in the context of charged cosmologies and dark energy models. We will find that the use of the modified formalism generates new terms in the electromagnetic energy-momentum tensor which are not present in standard QED and whose dominant contribution at late-times is a cosmological constant term. By imposing such terms to be compatible with current observations we will set upper limits on the charge asymmetry of the universe in this scenario

The paper is organized as follows: in Section II we review the infrared problems of covariant QED and their implications in the definition of charged states. In Section III, we present the Zwanziger subsidiary conditions and obtain the consistency condition for the classical current. In Section IV, we derive the equations of motion and energy-momentum tensors for the different components. Section V is devoted to the energy density and pressure of the scalar field. In Section VI, we calculate the induced electromagnetic energy density when different boundary conditions are imposed on the classical $b$ field and obtain the limits on the charge asymmetry. Finally in Section VII we present the main conclusions of the work.

II The infrared problem of perturbative QED

Let us start by reviewing the long-standing infrared problem of QED and its connection with the definition of charged particles.

In the standard description of scattering processes in perturbative field theory, the assumption is made that in the initial and final asymptotic regions the interactions can be switched off so that (in Minkowski space-time) the fields can be expanded in plane waves solutions with the corresponding creation and annihilation free operators. In the interaction picture, these free field solutions can then be used to construct the corresponding interacting solution using the Green function for the interaction term. Even though this method can be straightforwardly applied in some toy models, in the case of unbroken gauge theories such as QCD or QED it exhibits important difficulties. Thus, in the QCD case, low-energy confinement prevents the definition of asymptotically free quark states. In the case of QED the difficulty is less obvious since electrons are not confined, however Fadeev and Kulish Kulish showed that the masslessness of the photon implies that the electromagnetic interaction does not decay sufficiently fast at long distances so as to neglect it in the asymptotic regions. This residual interaction, implies that asymptotic charged fields can no longer be described as plane waves but they appear ”dressed” by an electromagnetic field. This in practice prevents the definition of local charged states in covariant QED.

Let us then review in detail how the problem arises in the standard manifestly covariant formulation of QED in the Lorentz gauge. Here we will closely follow the analysis in MZ .

The equations of motion in Minkowski space-time read Itzykson :

[TABLE]

where $J^{\nu}$ is the conserved current. In order to recover the classical Maxwell equation, the Lorentz condition $\partial_{\mu}A^{\mu}=0$ should be imposed. As is well known Itzykson , this cannot be done at the operator level but only in the weak sense given by the Gupta-Bleuler subsidiary conditions which in fact defines the physical Fock space of the theory:

[TABLE]

where $(\partial_{\mu}A^{\mu})^{(-)}$ is the negative frequency part of the operator $\partial_{\mu}A^{\mu}$ and $|\Phi\rangle$ denotes a physical state.

Rewriting equations (2) as

[TABLE]

and decomposing the external current as $J_{\mu}(x)=J_{\mu+}(x)+J_{\mu-}(x)$ with $J_{\mu\pm}=\theta(\pm\,x^{0})J_{\mu}(x)$, the general interacting solution $A_{\mu}(x)$ can be written in terms of the free solutions $A_{\mu}^{f}(x)$ satisfying $\Box A_{\mu}^{f}=0$ as

[TABLE]

where the retarded and advance propagators are

[TABLE]

and the free field can be expanded in plane-wave solutions as

[TABLE]

where $\omega=|\vec{k}|$, and $a_{\mu}$, $a_{\mu}^{\dagger}$ denote the free annihilation and creation operators satisfying

[TABLE]

From (5) we can write

[TABLE]

with

[TABLE]

Thus, substituting back (9) in (3), we get the Gupta-Bleuler subsidiary conditions in Fourier space

[TABLE]

where $a_{0}(\vec{k})$ and $a_{\parallel}(\vec{k})=\frac{k^{i}}{\omega}a_{i}(\vec{k})$ denote the temporal and longitudinal annihilation operators of the free fields respectively and $\rho(\vec{k})$ is the charge density operator in Fourier space, i.e

[TABLE]

If $\rho(\vec{k})$ is a smooth function, then

[TABLE]

with $Q$ the total charge. It is now possible to obtain solutions of the subsidiary condition (11) for the physical states $|\Phi\rangle$ in the form

[TABLE]

with

[TABLE]

where $a_{\perp}^{\dagger}(\vec{k})$ denotes the two transverse creation operators and ${\cal F}$ is an analytical function. Thus, the norm squared of a physical state is given by

[TABLE]

Notice that since $\rho(\vec{k})$ is a smooth function with $\rho(0)=Q/(2\pi)^{3/2}$, the above integral is infrared divergent for $Q\neq 0$. This means that the Gupta-Blueler condition has no Fock space solution with finite norm in the charged sector MZ .

One of the possible solutions to this problem is the modification of the subsidiary condition in the infrared. In particular it can be seen MZ that a modified condition given by:

[TABLE]

where $Q$ is the charge operator and $f_{c}(\omega)$ is a cut-off function such that $f_{c}(0)=(2\pi)^{-3/2}$ and $f_{c}(\omega)=0$ for $\omega>\omega_{0}$, defines a non-empty Fock space of physical states. Indeed, the new term can be seen as a classical current which screens the effect of the quantum charges on large scales thus allowing for finite norm states in (16) and in this way avoids the infrared problem. Notice that this expression implies that the subsidiary conditions are only modified below an arbitrarily low frequency $\omega_{0}$, so that standard QED is recovered on small scales. In next section we will describe in detail the implementation of this modified electrodynamics in the Zwanziger model Z .

III Zwanziger subsidiary conditions

Let us consider a simple renormalizable scalar electrodynamics theory minimally coupled to gravity. The Lagrangian density of this model can be written as Z

[TABLE]

where $D_{\mu}=\partial_{\mu}+iqA_{\mu}$, and $q$ is the $U(1)$ charge of the scalar field. The fundamental fields appearing here are $A_{\mu}$, $\varphi$ and $g_{\mu\nu}$, and the action is

[TABLE]

with $\mathfrak{g}\equiv|\det(g_{\mu\nu})|$.

The corresponding equations of motion are

[TABLE]

and

[TABLE]

where

[TABLE]

is the $U(1)$ conserved current, i.e. such that $\nabla_{\mu}J^{\mu}=0$. We will see later how current conservation can be derived from (21), as one would expect. Notice that although the action in (20) is invariant under the restricted gauge transformations, $A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\Lambda$ with transformation parameter satisfying $\Box\Lambda=0$, the scalar sector is fully gauge invariant, so that current conservation is preserved.

Taking the divergence of (20) and taking into account the conservation of the current, we get

[TABLE]

Following Z , we define the physical states $|\Phi\rangle$ as those given by the following modification of the Gupta-Bleuler condition:

[TABLE]

where $b(x)$ is a real c-number solution of the wave equation

[TABLE]

which can be separated into its positive and negative frequency parts as $b(x)=b^{(+)}(x)+b^{(-)}(x)$ and that in Fourier space would generate the cut-off function $f_{c}(\omega)$ in (17).

Then it can be shown that if $b(x)$ satisfies:

[TABLE]

with $\Sigma$ a constant-time hypersurface, $g_{\Sigma}$ the metric on $\Sigma$, $d\Sigma^{\mu}$ the future-oriented volume element on $\Sigma$ and $q$ the charge of the state $|\Phi\rangle$ then $\langle\Phi|\Phi\rangle\geq 0$, i.e. states satisfying (24) have non-negative norm and such subspace is invariant under the action of observables ${\cal O}$ i.e. $\langle{\cal O}\Phi|{\cal O}\Phi\rangle\geq 0$. Notice that unlike Z we are working with an arbitrary $\lambda$.

These conditions imply that for the expectation value we get

[TABLE]

so that the classical Maxwell equations are modified with the introduction of an additional classical current and can be written as Z

[TABLE]

Notice that even though the subsidiary condition are modified, the gauge invariance of the scalar sector preserves the Ward identities of the theory, so that we do not expect any uncompensated production of temporal and longitudinal photons in the theory. As a matter of fact, as shown Z , all the cross sections formulae of standard QED are recovered in the modified formalism.

IV Charged cosmologies

We will apply the previous formalism in cosmology for the description of a homogeneous and isotropic universe with a uniform charge density. With that purpose we consider a spatially-flat Robertson-Walker metric

[TABLE]

where $a\equiv a(t)$ is the scale factor and hence, $\sqrt{\mathfrak{g}}=a^{3}$.

We would like to find non-trivial solutions in which the matter fields (scalar and electromagnetic fields) are also homogeneous and isotropic. With that purpose we will look for solutions of the classical equation of motion where the vector fields cannot point in any spatial direction, so that $A_{i}=0$. Then, the only remaining component of the field is $A_{0}$. By homogeneity, spatial derivatives of any field must vanish, which gives us $\partial_{i}A_{\mu}=0$ and $\partial_{i}\varphi=0$. With these conditions, current conservation reads

[TABLE]

being $\kappa$ the constant comoving charge density of the scalar field.

For the spatial hypersurface of constant $t$, we have $\sqrt{g_{\Sigma}}=a^{3}(t)$ and $d\Sigma^{\mu}=d^{3}x(1,0,0,0)$, so that the consistency condition (26) reads

[TABLE]

with $V$ a given comoving volume. In the cosmological context it is natural to impose that $b$ is a homogeneous field, i.e. $b=b(t)$ which means that in Fourier space $b(k)$ only has contribution from the zero mode, i.e. the corresponding cutoff frequency would be essentially $k_{0}=0$. In other words, the modification of the subsidiary condition would only affect the zero mode electromagnetic fields. For the rest of states, the standard Gupta-Bleuler condition is recovered. Notice that since the tip of the light-cone $\omega_{0}=0$ is Lorentz invariant, we do not expect any modification in the subsidiary condition in a boosted frame. If we further assume that the consistency condition is valid for an arbitrary cosmological volume $V$, then we finally obtain

[TABLE]

which implies that the right-hand side of (28) vanishes, i.e. even though we have a net charge density, the presence of the new current cancels its effects on cosmological scales. This means a vanishing Faraday tensor on large scales, so that it is possible to get exact homogeneous and isotropic Robertson-Walker solutions. This de-electrification of the electric current, which is decoupled from the electromagnetic fields, is different from the degravitation mechanism Dvali of the cosmological constant from gravity. Although both cases resort to infrared modifications of the theory, in the gravitational case it is the dynamics rather than the subsidiary conditions what is modified in order to absorb the vacuum energy contribution.

On the other hand, notice that introducing (30) into the Maxwell equations (20) and taking into account the isotropy and homogeneity of $A_{\mu}$, we can write a $\varphi$-independent equation of motion for the classical electromagnetic field:

[TABLE]

which is compatible with (32) and can be rewritten as:

[TABLE]

being $H\equiv\dot{a}/a$ the Hubble parameter.

The scalar field can be written in terms of a modulus and a phase: $\varphi=f\,e^{i\theta}$. Introducing this expression in (21) gives, after splitting the resulting equation in its real and imaginary parts,

[TABLE]

and

[TABLE]

In order to simplify this expression, we write the zero component of the current density in terms of $f$ and $\theta$ as

[TABLE]

in such a way that (35) becomes

[TABLE]

Now, we can write (36) in a different way as

[TABLE]

Thus, comparison with (37) shows us that this equation of motion is completely equivalent to current conservation, as was mentioned before.

The stress-energy tensor is obtained by the variation of the action with respect to the metric as

[TABLE]

We get the complete stress-energy tensor as a sum of the contributions from the scalar and electromagnetic fields, $T^{\mu\nu}=T^{\mu\nu}_{\varphi}+T^{\mu\nu}_{A}$. For our model, we get its symmetrized components as

[TABLE]

and

[TABLE]

Although gauge covariant derivatives appear in (41), we see that in our case, using (37), $T^{\mu\nu}_{\varphi}$ can be written in terms of the modulus of the scalar field $f$ only. As a matter of fact, the only non-vanishing components of (41) and (42) are the energy densities and pressures, obtained as $\rho_{(\alpha)}={{T_{(\alpha)}}^{0}}_{0}$ and $p_{(\alpha)}=-{{T_{(\alpha)}}^{i}}_{i}$, where $(\alpha)$ stands for $\varphi$ or $A$. Thus we get

[TABLE]

and

[TABLE]

where we have made use of (34) to get the standard Coulomb interaction $\kappa A_{0}a^{-3}$ term. Notice the difference with respect to the standard electromagnetic energy density in QED since, in addition to the Coulomb term, in the Zwanziger model a new extra contribution is present which is proportional to $b^{2}$ according to (27).

V Energy and pressure of the scalar field

In order to obtain the scaling behaviour of the energy density of the scalar field, we will particularize the scalar field potential to the simplest case corresponding to a mass term,

[TABLE]

We will solve (38) numerically, so we rewrite it here in a dimensionless form. To do so, we define a dimensionless time as $\tau\equiv mt$ and define $\tau$ valued fields and parameters as the barred ones:

[TABLE]

In the following, we shall omit the $\tau$ dependence, which is obvious in barred quantities.

Now, the equation of motion (38) with the mass potential can be written as

[TABLE]

where the prime means $\tau$ derivative, and we have defined the dimensionless constant

[TABLE]

In order to simplify the numerical evaluation of (47) we define

[TABLE]

so that the equation of motion becomes

[TABLE]

where the damping term $\bar{g}^{\prime}$ does not appear. We will solve for different cosmological eras assuming $\bar{a}\propto\tau^{n}$, so that we get the final equation

[TABLE]

Thus for radiation dominated era with $n=1/2$, we get

[TABLE]

Solving these equations numerically we get, for a wide range of initial conditions an oscillatory behaviour for the field around a constant value at late times when $m\gg H$, as showed in Fig.1, both for $\bar{g}$ and $\bar{g}^{\prime}$. This translates into an oscillatory behaviour for the field $\bar{f}$ with an amplitude that decays with the scale factor as $a^{-3/2}$.

If we introduce this behaviour in the scalar field energy density (43a), with the mass potential (45), we will find that it decays at late time as

[TABLE]

so that the scalar field has a matter like behaviour as expected Turner . This means that provided that initial conditions ensure that it is subdominant with respect to the total matter density it will be a subdominant component at all times.

VI Energy and pressure of the electromagnetic field

VI.1 Setting the background

In order to study analytically the behaviour of both the energy density and pressure in (44b) we should solve the equation of motion for $A_{0}(a)$, given by (34). Changing time derivatives to $a$-derivatives, and integrating twice and once respectively from some initial value $a_{i}$ we can get, after some manipulations,

[TABLE]

where we have defined the primitive functions

[TABLE]

and $H(a)\equiv H_{0}E(a)$. These primitives are evaluated in $a$ except if they have some subscript that indicates a particular constant scale factor.

In order to obtain explicit expressions for the integrals, we will consider the standard cosmological behaviour with an initial inflationary phase followed by a reheating phase connecting with the radiation era of standard $\Lambda$CDM cosmology. We will assume for simplicity that the energy density scales as matter during the reheating phase KolbTurner . This background scheme is depicted in Figure 2.

We consider a quasi- de Sitter inflationary phase with almost constant Hubble parameter $H_{I}$ which can be estimated from the Friedmann equation as

[TABLE]

where $M_{p}^{2}=1/(8\pi G)$ and $V_{I}^{1/4}$ is the scale of inflation. On the other hand we have, $a_{b}=e^{-\mathcal{N}}a_{e}$, where $\mathcal{N}$ is the total number of inflation e-folds. In addition, since we are assumming the reheating era to be matter dominated, where $\rho\propto a^{-3}$, it is possible to write

[TABLE]

Taking $\rho(a_{e})=V_{I}$ and that $\rho(a_{RH})=\frac{\pi^{2}}{30}g_{*}T_{RH}^{4}$, one can approximate

[TABLE]

where we have ignored numerical factors of order one. Finally, we need $a_{RH}$. Assuming adiabatic expansion after reheating we have

[TABLE]

where $T_{eq}\simeq 0.83$ eV and $a_{eq}=2.8\times 10^{-4}$, the standard values for the temperature and scale factor at matter-radiation equality KolbTurner . Thus we see that the details of the inflationary and reheating phases are encapsulated in the three parameters $T_{RH}$, $V_{I}$ and $\mathcal{N}$.

Thus, up to order-one numerical factors, the final function to integrate is

[TABLE]

with

[TABLE]

VI.2 Evolution of the energy density

Now, we can study how the energy density of the electromagnetic part evolves in the background described by (60) and (61c). This would depend on (a) the initial time where the charge density appears, and (b) on the boundary conditions that we set for both $A_{0}$ and $\nabla_{\mu}A^{\mu}$.

If we introduce the fields into the energy density, grouping terms adequately, we get

[TABLE]

Let us study the evolution of the various energy density terms in different situations.

VI.2.1 Instantaneous charge density generation

Let us first consider the case in which charge and electromagnetic fields vanish initially and at some $a=a_{i}$ a net charge density is generated, so that we take $A_{0}|_{i}=0$ and $\nabla_{\mu}A^{\mu}|_{i}=0$ in (62) and obtain

[TABLE]

The value of $a_{i}$ will be determined by the charge generation mechanism. Thus, as mentioned before a charge density could be generated, for example, during inflation due to some fluctuation of the charged scalar field scalar1 ; scalar2 ; scalar3 , or at a phase transition gauge1 ; gauge2 ; gauge3 well inside the radiation era.

Thus, for charge generation occurring during the inflation era ($a_{i}\ll a_{e}$), we can see that the terms appearing in (63b) behave as shown in Fig. 3. In this plot $\mathcal{N}_{i}\equiv\ln\frac{a_{e}}{a_{i}}=2$, i.e. only two inflation e-folds are needed to reach the asymptotic behaviour. We can see that at the end of inflation, the terms containing $\mathcal{I}$ are two orders of magnitude bigger than those with $\mathcal{F}$ and $\mathcal{G}$. After inflation, the difference remains increasing, and nowadays only the $\mathcal{I}$ terms are relevant. Moreover, we see that $\mathcal{I}-\mathcal{I}_{i}$ remains constant after that, i.e. $\mathcal{I}\simeq\mathcal{I}_{e}$ for $a>a_{e}$, so that the dominant contribution at late times is a cosmological constant component.

Since in inflation one has the analytical expression

[TABLE]

we can see that, at the end of inflation,

[TABLE]

for big enough $\mathcal{N}_{i}$. Then, the energy density today is

[TABLE]

with

[TABLE]

where we have used (61b) and $a_{i}=e^{-\mathcal{N}_{i}}a_{e}$. Thus for $\lambda<0$, it is possible to have a positive cosmological constant today.

We now want to study the range of values of the charge density $\kappa$ which can provide a value for $\rho_{A}$ today consistent with dark energy density measurements. In order to do it, we use that

[TABLE]

Comparing this expression with (66) and (67), we get

[TABLE]

Thus we get for the constant comoving charge density

[TABLE]

so that

[TABLE]

which allows to obtain, assuming individual particles of charge $e=\sqrt{4\pi\alpha}$, the limit on the charge asymmetry as

[TABLE]

for $\mathcal{N}_{i}\in[2,60]$.

Let us now consider the case in which the charge generation takes place well inside the radiation era at a temperature $T=T_{Q}$, corresponding to an initial scale factor

[TABLE]

In Fig. 4 we can see that, again, the $\mathcal{I}$ terms are several orders of magnitude bigger than the $\mathcal{F}$ and $\mathcal{G}$ ones. Moreover, after $a_{eq}$, $\mathcal{I}-\mathcal{I}_{i}$ has again a constant value until today, so that we can approximate the energy density today as the energy density in the radiation-matter equality. We use the analytical solution for $\mathcal{I}$ during a radiation dominated universe,

[TABLE]

in order to write

[TABLE]

and then, following the same steps as in (68-70), one can estimate

[TABLE]

which, gives the limit on the charge asymmetry

[TABLE]

Thus we see that for $|\lambda|={\cal O}(1)$, a tiny charge asymmetry of order $\eta_{Q}\simeq 10^{-43}$ produced at the electroweak phase transition would generate a cosmological constant compatible with observations. The possibility of generating a cosmological constant in the context of modified electrodynamics was considered in the uncharged case in BeltranMaroto1 ; BeltranMaroto2 . Notice that in the uncharged sector with $q=0$ i.e. $\kappa=0$, the only possible homogeneous solution of the consistency condition (26) is a constant $b$ field, which contributes to $\rho_{A}$ in (44a) as a pure cosmological constant.

VI.2.2 Vanishing fields in the asymptotic future

Another type of solutions correspond to those in which charge density has ever been present but the induced electromagnetic field vanish asymptotically in the future as the charge density decrease as $\kappa\propto a^{-3}$, i.e.

[TABLE]

For $a\gg 1$ only the $\Omega_{\Lambda}$ term will survive, and $\mathcal{I}$ vanishes in the limit. Then this yields

[TABLE]

Now, taking the same limit in (54a), one can see that it is automatically satisfied for any $A_{0}|_{i}$ taking the previous result for $\nabla_{\mu}A^{\mu}|_{i}$. By introducing (79) in the energy density (62), we can see that this is equivalent to removing the constant mode, as one would expect. Moreover, the term $\mathcal{F}a^{-6}-\mathcal{I}$ is removed too. The energy density is then

[TABLE]

and the pressure

[TABLE]

As mentioned above, we can safely neglect the term with $A_{0}|_{i}$ which makes the analysis easier. Now, let us define the dimensionless functions

[TABLE]

We solve (55b) and (55c) for the different epochs

Inflation:

[TABLE] 2. 2.

Reheating:

[TABLE] 3. 3.

Radiation

[TABLE] 4. 4.

Matter:

[TABLE] 5. 5.

Dark energy:

[TABLE]

The evolution of the equation of state is shown in Fig. 5. We can see that far from the transition regions i.e. neglecting $\mathcal{G}_{i}$, $w_{A}=-1/3$ in the radiation era, $w_{A}=0$ in the matter era and during the accelerated expansion eras, the behaviour goes as:

[TABLE]

that tends asymptotically to that of a stiff fluid $w_{A}=1$.

In Fig. 6 we show the evolution of the dimensionless $\bar{\rho}_{A}$ compared to $E^{2}(a)$ which follows the scaling of $\rho_{\Lambda CDM}$. As we can see the maximum contribution occurs in the matter dominated era, when the ratio $R=\bar{\rho}/E^{2}$ reaches a maximum value $R_{max}=4.8$. Thus, in order for the charge-induced energy density not to spoil the predictions of standard $\Lambda$CDM, we impose \rho_{A}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}10^{-2}\rho_{\Lambda CDM} which implies

[TABLE]

which, in turn, can be translated into a limit on the charge asymmetry as

[TABLE]

This limit relaxes in several order of magnitude, the present bounds on the charge asymmetry in standard Maxwell electrodynamics mentioned before.

VII Conclusions

We have explored the possibility of constructing homogeneous and isotropic cosmologies with a non-vanishing charge density in the context of modified Maxwell electrodynamics. Unlike previous works which considered theories that include a small photon mass and thus propagate three degrees of freedom, we have limited ourselves to the Zwanziger model of electrodynamics, with two propagating polarizations, but with modified subsidiary conditions. The modification affects only the physical photon Fock space in the infrared.

We show that in the context of this model, the induced classical current counterbalance the effects of the physical (quantum) charges so that the Faraday tensor vanishes on cosmological scales, thus allowing for the construction of exact Robertson-Walker geometries. Depending on the boundary conditions imposed on the classical $b(t)$ field, different scenarios are possible. Thus, if $b(t)$ vanishes at some initial time when the charge density is generated, then the dominant contribution to the electromagnetic energy-momentum tensor is a cosmological constant-like term. Imposing the value of the induced constant to be smaller than the observed one sets stringent limits on the comoving charge density, which translates into limits on the charge asymmetry which can range from \eta_{Q}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}10^{-131} if charges are produced during inflation (for typical inflationary models) to \eta_{Q}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}10^{-43}|\lambda|^{1/2}(100\;\mbox{GeV}/T_{Q}) if the charge density is generated in the radiation era at a temperature $T_{Q}$. In the case in which the $b$ field vanishes asymptotically in the future when the charge density also vanishes, the cosmological constant-like term is absent and the dominant contribution appears as an extra matter density in the matter dominated era. Imposing again compatibility with the observed matter density sets a weaker limit \eta_{Q}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{<}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt\sim}}10^{-28}|\lambda|^{1/2}, several orders of magnitude below the limits in standard Maxwell electrodynamics.

The de-electrification mechanism discussed in this work only takes place on cosmological scales, and, a priori, could not prevent the appearance of effects on smaller scales. However, the small charge densities suggest that such effects could be actually suppressed. Thus for example, in the highest density case, corresponding to $\eta_{Q}\simeq 10^{-28}$, the corresponding density of charged particles today would be $n_{Q}\simeq 10^{-26}$ cm*-3*. Of course density perturbations would induce also charge density perturbations which could be enhanced in high-density objects. The study of the evolution of these charge fluctuations is however beyond the scope of the present work.

acknowledgments

This work has been partially supported by MINECO grant FIS2016-78859-P(AEI/FEDER, UE) and by Red Consolider MultiDark FPA2017-90566-REDC. The research of J.F.S. is supported by U.S. National Science Foundation (Grant PHY-1620661).