1 Addendum to: Eur. Phys. J. C https://doi.org/10.1140/epjc/s10052-022-10194-3

2 The extended model

Recently, we show, the author showed that Einstein–Cartan–Brans–Dicke (ECBD) gravity can suppress the dynamical torsion in inflationary phases [1]; in the same way, Paul et al. [2] have shown that this result is valid in the case of dynamical torsion in the bulk of a five-dimensional (5D) spacetime. As we have done in most of my papers [6], in the present paper, we investigate the dynamics of torsion in the case of chiral dynamos in ECBD. To this end we might consider the same Lagrangian

$$\begin{aligned} \mathcal{{L}}_{{\text {ECBDM}}}=\int d^{4}x\sqrt{-g}\left( -{\phi }R +{\omega }\frac{\partial ^{\mu }\phi \partial _{\mu }\phi }{\phi }-\frac{\phi ^{3}F^{2}}{4}+ J^{\mu }A_{\mu }\right) \end{aligned}$$
(1)

where \({\mu }=0,1,2,3\) and \(F^{2}=F_{{\mu }{\nu }}F^{{\mu }{\nu }}\) is the electromagnetic field tensor squared, and ECBDM is the Einstein–Cartan–Brans–Dicke–Maxwell, where we add the electromagnetic Lagrangian without the axial anomaly term. Note that, torsion is not coupled with the EM fields. In this section we couple scalar fields to torsion via non-minimal coupling [10]. R in this Lagrangian is the Ricci–Cartan scalar. The new EM terms introduced here now allow us to introduce the chiral dynamo problem. The chiral total current is

$$\begin{aligned} {{\textbf {J}}}={\sigma }[{{\textbf {E}}}+{{\textbf {v}}}{\times }{{\textbf {B}}}] +{{\mu }_{5}}{{\textbf {B}}} \end{aligned}$$
(2)

The chiral current is superposed on the electric current, where \({\sigma }={\eta }^{-1}\), where \({\sigma }\) is the electrical conductivity and \({\eta }\) is the electrical resistivity. Now,

$$\begin{aligned} \ddot{\phi }+3H\dot{\phi }=\frac{4{\pi }}{{\omega }}\left( {\rho }-3p-\frac{8{\pi }{\sigma }^{2}}{\phi }\right) \end{aligned}$$
(3)

where \({\phi }\) is the inflaton field and \(H=\frac{\dot{a}}{a}\) is the Hubble expansion, while a is the cosmic scale factor. The Friedmann–Robertson–Walker (FRW) metric is given by

$$\begin{aligned} ds^{2}=dt^{2}-a^{2}(t)\left[ dx^{2}+dy^{2}+dz^{2}\right] \end{aligned}$$
(4)

Units are used where, the gravitational constant \(G=1\), and the Ricci–Cartan scalar R is given by

$$\begin{aligned} R=[{\dot{S}}_{0}-6{S_{0}}^{2}+12H^{2}] \end{aligned}$$
(5)

Now let us obtain the first field equations from the Euler-Lagrange (EL)

$$\begin{aligned} \frac{d}{dt}\frac{{\partial }\sqrt{g}\mathcal{{L}}_{{\text {ECBDM}}}}{{\partial }{\dot{\phi }}} -\frac{{\partial }\sqrt{g}\mathcal{{L}}_{{\text {ECBDM}}}}{{\partial }{\phi }}=0 \end{aligned}$$
(6)

Here, the interaction term between the scalar inflatons and EM invariant \(F^{2}\) is of the Ratra type [11]. Thus, by computing the partial derivatives of the Lagrangian (1) and substituting them into the EL equation, one obtains

$$\begin{aligned} R= \frac{3{\phi }^{2}}{4}F^{2}+2{\omega }^{2}{\beta }^{2}+3{\omega } \left( 3H\frac{\dot{\phi }}{\phi }+\frac{\ddot{\phi }}{\phi }\right) \end{aligned}$$
(7)

This is the well-known Rainich already unified field equation [13], as can be seen by taking the inflaton as constant. Let us now obtain the remaining equations in the form of EL equations as

$$\begin{aligned} \frac{d}{dt}\frac{{\partial }\sqrt{g}\mathcal{{L}}_{{\text {ECBDM}}}}{{\partial }{\dot{S}}_{0}}-\frac{{\partial }\sqrt{g} \mathcal{{L}}_{{\text {ECBDM}}}}{{\partial }S_{0}}=0 \end{aligned}$$
(8)

where a is the cosmic scale and \(H=\frac{{\dot{a}}}{a}\) is the Hubble function. Then, this equation yields

$$\begin{aligned} {\dot{\phi }}+3H{\phi }=-2S_{0} \end{aligned}$$
(9)

where \(S_{0}\) is the zero component of the axial torsion vector. This equation tells us that if torsion is constant, its solution yields a decaying inflaton field in cosmic time t, which depends upon torsion. The stronger the torsion, the faster the decay of the inflaton field. Now we consider the case of the EL equation for the cosmic scale a. This yields

$$\begin{aligned} \mathcal{{L}}_{{\text {ECBDM}}}+8H^{2}(1-\dot{\phi })-8{\phi }\left[ H^{2}+\frac{\dot{a}}{a}\right] =0 \end{aligned}$$
(10)

But it is easy to show that

$$\begin{aligned} \dot{H}+H^{2}= \frac{{\ddot{a}}}{a} \end{aligned}$$
(11)

Substitution of expression (11) into (10) yields

$$\begin{aligned} {\phi }^{-1}\mathcal{{L}}_{{\text {ECBDM}}}+8H^{2}\left( 1-\frac{{{\phi }}}{\phi }\right) -[2H^{2}+\dot{H}]=0 \end{aligned}$$
(12)

Making use of expression (9) and only considering Hubble parameter H up to \(\mathcal{{O}}(H^{2})\), we obtain

$$\begin{aligned} {\phi }^{-1}\mathcal{{L}}_{{\text {ECBDM}}}+8H^{2}(1+2S_{0})-[2H^{2}+\dot{H}]=0 \end{aligned}$$
(13)

Taking the ECBD Lagrangian in the form

$$\begin{aligned} {\phi }^{-1}\mathcal{{L}}_{{\text {ECBDM}}}\approx {12H^{2} -{\omega }\left( \frac{\dot{{\phi }}}{\phi }\right) ^{2}} \end{aligned}$$
(14)

Note that from this equation that if we constrain ECBDM cosmology to the case where expansion rebounds, universe expansion stops, (H = 0) and torsion is constant, this assumption can be used as a boundary condition to determine the BD parameter \({\omega }\). The last expression reduces to

$$\begin{aligned} {\omega }\left( \frac{\dot{\phi }}{\phi }\right) ^{2}+{S^{2}}_{0}=0 \end{aligned}$$
(15)

Assuming this specific boundary condition where inflation turns into deflation, one obtains from the last expression that

$$\begin{aligned} S^{0}={\sqrt{{\omega }}}t^{-1} \end{aligned}$$
(16)

from previous equations. Then, torsion is non-constant, and substitution of the last expression squared into (14) yields the result

$$\begin{aligned} \left( \frac{\dot{\phi }}{\phi }\right) ={S}_{0}={{\sqrt{{\omega }}}}t^{-1} \end{aligned}$$
(17)

This expression shall be very useful to solve chiral dynamo equation in the next section. Note that expression (16) above is quite important, due to the fact that torsion is determined in terms of the BD parameter \({\omega }\). This agrees with the status quo that general relativity is a torsionless equation, since GR is characterized by \(({\omega }=0)\).

3 Helical chiral dynamos in Einstein–Cartan–Brans–Dicke–Maxwell gravity

In this section we shall finally derive the chiral dynamo equation in the case of the ECBDM model of the previous section. We recall that though we do not start from the coupling of torsion to EM fields; the magnetic field may be computed in terms of torsion, as we shall see in this section. Here in differential forms notation \(F=dA\) where \(A=(A_{i}dx^{i})\) is the magnetic potential four-dimensional one form, where no torsion is present. Variation of the four-dimensional potential in the above Lagrangian is

$$\begin{aligned} {\partial }_{i}(a^{3}{\phi }^{3}F^{ij})= a^{3}{\phi }^{3}{J_{5+c}}^{j} \end{aligned}$$
(18)

where electric current \(J_{c}\) is given by

$$\begin{aligned} {{\textbf {J}}}_{(5+c)}={\mu }_{5}{{\textbf {B}}}+{\sigma }({{\textbf {E}}} +{{\textbf {v}}}{\times }{{\textbf {B}}}) \end{aligned}$$
(19)

First current is the chiral, and the second is the normal electric current. Substitution of this current into the expression (19) yields

$$\begin{aligned} ({\lambda }^{2}-{\lambda }{\mu }_{5})\left[ 3\left( \frac{\dot{\phi }}{\phi }+H\right) -{\sigma }\right] {\nabla }{\times }{{\textbf {E}}}={\sigma }{\nabla }{\times } [{\textbf {v}}{\times }{} {\textbf {B}}] \end{aligned}$$
(20)

which from from the Bianchi identity

$$\begin{aligned} {\partial }_{[i}F_{jk]}=0 \end{aligned}$$
(21)

we obtain the Faraday effect

$$\begin{aligned} {\nabla }{\times }{{\textbf {E}}=-{\partial }_{t}{} {\textbf {B}}} \end{aligned}$$
(22)

After some algebra we obtain the chiral dynamo equation

$$\begin{aligned} {\partial }_{t}{} {\textbf {B}}=-[{\eta }({\lambda }^{2}-{\lambda }{\mu }_{5}) [1+3{\eta }(H+{\omega }t^{-1})-i{{\textbf {v}}}.{{\textbf {k}}}]{\textbf {B}} \end{aligned}$$
(23)

where we have used Eq. (17) of the previous section. Since in the early universe the conductivity is very high and the ohmic resistivity is quite low, we may keep only terms up to first order in resistivity and drop terms like \(\mathcal{{O}}({\eta }^{2})\) and compute second-order contributions of the ohmic resistivity or diffusivity by the end of this section, where we compare both results. Under this assumption chiral dynamo equation reduces to

$$\begin{aligned} {\partial }_{t}{} {\textbf {B}}=-{\eta }({\lambda }^{2}-{\lambda }{\mu }_{5}){{\textbf {B}}} \end{aligned}$$
(24)

We now assume that chirality dominates over the relation between the magnetic helicity parameter and the chiral chemical potential, such as \({\mu }_{5}\ge {\lambda }\). This allows us to express the last chiral dynamo equation in the form

$$\begin{aligned} {\partial }_{t}{} {\textbf {B}}={\eta }{\lambda }{\mu }_{5}{{\textbf {B}}} \end{aligned}$$
(25)

and then now it is much easier to find the solution of this simple dynamo equation. This solution can be approximated by

$$\begin{aligned} {{\textbf {B}}}\approx {{{\textbf {B}}}_{{\text {seed}}}({\eta }{\lambda }{\mu }_{5}t)} \end{aligned}$$
(26)

By making use of the \({\lambda }={L^{-1}}_{B}\), where \(L_{B}\) is the coherent length of the magnetic field and taking \(L_{B}=1pc\) as in the Googol et al. primordial magnetogenesis torsionless paper [15], since \(t=10^{18}s\) and the \({B^{{\text {seed}}}}_{{\text {ECBDM}}}= 10^{13}G\) at today’s axion field, we would need a magnetic field in the present universe of the order of \(B_{{\text {ECBDM}}}= 10^{-34}G\). We now use the expression for the dynamo solution of the order of \(\mathcal{{O}}({\eta }^{2})\), which yields a solution as

$$\begin{aligned} {{\textbf {B}}}\approx {{{\textbf {B}}}_{{\text {seed}}}({\eta }^{2}{\lambda }{\mu }_{5}t)} \end{aligned}$$
(27)

This immediately yields \(B_{{\text {ECBDM}}}{chiral}= 10^{-13}{\omega }G\) as a helical magnetic field in the present universe. The \({\omega }\) factor which is sometimes used in BD gravity as \({\omega }=\frac{1}{6}\) will not appreciably change our results here; since we would obtain a \(10^{-14}G\) for the axion magnetic field. We have used the following data in this computation: a seed field from the Biermann battery mechanism as \(B_{Bierm}=10^{30}G\), a chiral chemical potential as \({\mu }_{5}=10\), as a value used by Shober et al. [18] to investigate chiral dynamos in torsionless spacetime. By substituting these data into expression (27), one obtains the estimate \({B^{{\text {inflaton}}}}_{{\text {ECBDM}}}=10^{-13}\) Gauss, which is exactly the result obtained by Miniati et al. [23] from 20pc coherent length. Though we apparently do not see the presence of torsion in this chiral dynamo equation, it is actually present in the relation between H and \(S_{0}\) above.

4 Conclusions

In this addendum we have investigated the magnetogenesis in the case of the ECBDM model for cosmology. One obtains in the present universe a magnetic field of \(10^{-13}\) Gauss from a second order in the ohmic diffusivity in chiral dynamo equation. With GUT seed fields of the order of \(10^{41}G\) obtained by Berera in the early universe, one would be able to obtain, instead an axionic magnetic field in the present universe of \(10^{-13}\). Using our solution, a magnetic field of \(B_{{\text {ECBD}}}({\eta }^{2})=10^{-3}\) Gauss, which is a field found at the core of some galaxies is found. We note that the ECBDM universe seems to be a promising model to improved results for magnetic fields from torsion, which are able to seed galactic dynamos at a reasonable coherence length. It is also important to note that, here we have not address back reaction into the metric, which would require a metric of non-Friedmannian nature, because we are not using the axial anomaly term E.B, and most of the magnetic fields obtained are very weak to back-react on the isotropic homogeneous universe considered. Of course, solutions of several Lagrangians can be used in the near future to test the model dependence of the electrodynamics which we were using to further investigate magnetogenesis in Riemann–Cartan spacetime [24]. We were told very recently that Bamba et al. [25] have investigated helical magnetogenesis with a reheating phase and high-order curvature baryogenesis without torsion. It would be interesting if we tried to add torsion couplings even non-minimally with magnetogenesis along their lines. This work may appear in the near future. More on GR magnetogenesis may be found in the last reference [26]. Actually, Bamba et al. [28] have investigated the non-helical magnetic fields in the reheating phase in higher-order curvature coupling. This is an interesting subject to work and to review in order to perform extensions to modified gravity magnetogenesis.