Estimate of a non-helical electroweak primordial bootstrap field today

TL;DR

This paper estimates the current strength of a primordial non-helical magnetic field originating from the electroweak phase transition, finding it consistent with observational lower limits and providing a comparison with helical field scenarios.

Contribution

The study provides a novel estimate of the present-day magnitude of a non-helical primordial magnetic field from the electroweak epoch, including its correlation scale and comparison with helical cases.

Findings

Primordial field strength today is about 10^{-14} G at 2 kpc scale.

Initial field at electroweak transition was 10^{23}-10^{24} G.

Results align with observational lower limits on intergalactic magnetic fields.

Abstract

We estimate the magnitude today of the primordial magnetic field originating at the electroweak phase transition. We find that the field, which at the electroweak phase transition is originally of order $10^{23}-10^{24}$ G correlated over the Hubble scale, today is of order $10^{-14}$ G at a scale of order 2 kpc. This result is consistent with the lower limit on the strength of intergalactic magnetic fields obtained by Neronov and Vovk from observations of TeV blazars. The field is non-helical. We compare our results with the helical case discussed by Field and Carroll.

Equations

$$\mathcal{E}={\mathcal{E}}_1+{\mathcal{E}}_2+{\mathcal{E}}_3+{\mathcal{E}}_4+{\mathcal{E}}_5-\frac{g^2{\varphi }_0^4}{8{\sin }^2\theta }+\frac{g{\varphi }_0^2}{2\sin \theta }B-\frac{g{\varphi }_0^2}{2\cos \theta }Z-\frac{g}{2\cos \theta }{\partial }_j({ϵ}_{ij}{Z}_j{\varphi }^2),$$

$${\mathcal{E}}_1=|(D_1+i{D}_2)W{|}^2,$$

$${\mathcal{E}}_2=\frac{1}{2}{\left(B-\frac{g}{2\sin \theta }{\varphi }_0^2-2g\sin \theta |W{|}^2\right)}^2,$$

$${\mathcal{E}}_3=\frac{1}{2}{\left(Z-\frac{g}{2\cos \theta }({\varphi }^2-{\varphi }_0^2)-2g\cos \theta |W{|}^2\right)}^2,$$

$${\mathcal{E}}_4={\left(\frac{g\varphi }{2\cos \theta }{Z}_i+{ϵ}_{ij}{\partial }_j\varphi \right)}^2,$$

$${\mathcal{E}}_5=\left(\lambda -\frac{g\text{texttwosuperior}}{8{\cos }^2\theta }\right)({\varphi }^2-{\varphi }_0^2)\text{texttwosuperior}.$$

$$D_i={\partial }_i-ig(A_i\sin \theta +Z_i\cos \theta ),$$

$$B=\frac{m_W^2}{e}+2e|W{|}^2.$$

$$B\ge \frac{m_W^2}{e}\approx 3\times 10^{23}\mathrm{G}.$$

$$B\le \frac{m_H^2}{e}\approx 7.64\times 10^{23}\mathrm{G},$$

$$1.82\times 10^{-2}\ge {\Omega }_B^{\mathrm{EW}}\ge 2.8\times 10^{-3}$$

$${\mathcal{E}}_{\mathrm{one}\mathrm{cell}}=\frac{m_W^2}{e}{\int }_{\mathrm{one}\mathrm{cell}}B{d}^{2}x-\frac{m_W^4}{2e\text{texttwosuperior}}A=2\pi \frac{m_W^2}{e^2}-\frac{m_W^4}{2e^2}A,{\int }_{\mathrm{one}\mathrm{cell}}B{d}^{2}x=\frac{2\pi }{e},$$

$${\partial }_t\mathbf{v}+(v\nabla )\mathbf{v}=-\nabla p+(\nabla \times \mathbf{B})\times \mathbf{B}+\nu {\nabla }^2\mathbf{v},{\partial }_t\mathbf{B}=\nabla \times (\mathbf{v}\times \mathbf{B})+\eta {\nabla }^2\mathbf{B},$$

$$\mathcal{E}(k,t)=\frac{2\pi k^2}{(2\pi {)}^3}\int d^{3}y{e}^{i\mathbf{ky}}<B(\mathbf{x},t)B(\mathbf{x}+\mathbf{y},t)>$$

$$\mathcal{E}(k,t)=\sqrt{\frac{t_0}{t}}\mathcal{E}\left(k\sqrt{\frac{t}{t_0}},t_0\right).$$

$$\frac{1}{2}{B}_{\mathrm{rms}}^2\equiv \frac{1}{2}<\mathbf{B}(\mathbf{x},t{)}^2>={\int }_0^{\infty }dk\mathcal{E}(k,t).$$

$$d{\tau }^2=d{t}^2-a(t{)}^{2}d{\mathbf{x}}^2=a(t{)}^2[(d{t}^{\ast }{)}^2-d{\mathbf{x}}^2]$$

$$t^{\ast }=\int \frac{dt}{a(t)},$$

$$\mathcal{E}(k,t_{\mathrm{today}})=\frac{a(t_{\mathrm{EW}}{)}^4}{a(t_{\mathrm{today}}{)}^4}\sqrt{\frac{t_{\mathrm{EW}}^{\ast }}{t_{\mathrm{today}}^{\ast }}}\mathcal{E}\left(k\sqrt{\frac{t_{\mathrm{today}}^{\ast }}{t_{\mathrm{EW}}^{\ast }}},t_{\mathrm{EW}}\right).$$

$$t_{\mathrm{EW}}\approx 6\times 10^{-12}\sec .$$

$$\lambda (t)=<1/k>,<1/k>={\int }_0^{\infty }\mathcal{E}(k,t)\frac{dk}{k}/{\int }_0^{\infty }\mathcal{E}(k,t)dk.$$

$$\lambda (t_{\mathrm{today}})=\frac{a(t_{\mathrm{today}})}{a(t_{\mathrm{EW}})}\sqrt{\frac{t_{\mathrm{today}}^{\ast }}{t_{\mathrm{EW}}^{\ast }}}\lambda (t_{\mathrm{EW}}).$$

$$B_{\mathrm{rms}}(t_{\mathrm{today}})=\frac{a(t_{\mathrm{EW}}{)}^2}{a(t_{\mathrm{today}}{)}^2}\sqrt{\frac{t_{\mathrm{EW}}^{\ast }}{t_{\mathrm{today}}^{\ast }}}{B}_{\mathrm{rms}}(t_{\mathrm{EW}}).$$

$$\mathrm{amplification}\mathrm{factor}=\sqrt{\frac{t_{\mathrm{today}}^{\ast }}{t_{\mathrm{EW}}^{\ast }}}\approx 5.7\times 10^6.$$

$$t_{\mathrm{EQ}}=21.6\times 10^{22}{t}_{\mathrm{EW}},$$

$$\lambda (t_{\mathrm{today}}=13.8\mathrm{Gyr})\approx 1.8\mathrm{kpc},$$

$$\lambda (t_{\mathrm{EW}})=1/H_{\mathrm{EW}}\approx 0.4\mathrm{cm}$$

$$B_{\mathrm{rms}}(13.8\mathrm{Gyr})\approx (1\mathrm{to}2.55)\times 10^{-14}\mathrm{G}.$$

$$\mathcal{E}(k,t_{\mathrm{EW}}\approx 6\times 10^{-12}\sec )\approx 14.45\times 10^{67}\times \mathcal{E}(1.76\times k\times 10^{-7},t_{\mathrm{today}}\approx 13.8\mathrm{Gyr}),$$

$$\lambda (t_{\mathrm{EW}})=f_{\lambda }/H_{\mathrm{EW}}\approx 0.4f_{\lambda }\mathrm{cm}\mathrm{and}{B}_{\mathrm{rms}}(t_{\mathrm{EW}})\approx 2\times 10^{25}{f}_B\mathrm{G}.$$

Full text

1 Introduction

It is well known that the standard electroweak theory is in agreement with experiments. It is therefore of interest to investigate cosmological consequences of this theory and compare these to observations.

In this note we therefore estimate the cosmological evolution of the bootstrap magnetic field generated at the electroweak phase transition [2] and compare our results to some observations. This field is originaly of order $10^{23}-10^{24}$ G over the Hubble scale. During the evolution of the universe it reduces to a field of order $10^{-14}$ G today. This magnitude is consistent with the lower limit $B\geq 3\times 10^{-16}$ G obtained by Neronov and Vovk [1] from observations of TeV blazars. This bound depends on the magnetic correlation length $\lambda$ and is lifted as $\lambda^{-1/2}$ if $\lambda$ is much smaller than of the order a megaparsec. In our case the correlation length is of order 2 kpc, which corresponds to a lower limit of order $\sim 10^{-15}$. Being a lower limit this is perfectly consistent with our result $2.55\times 10^{-14}$.

It should be mentioned that long time ago Vachaspati found the first electroweak candidate for a primordial magnetic field [3]. His solution differs fundamentally from ours.

The contents of this note is the following:

In Section 2 we discuss the electroweak solution originaly due to Ambjørn and the author [4]-[6]. In section 3 we introduce the self-similarity found by Zrake [7] and the author [8] and discuss how this relation can be used to estimate the magnetic field and the correlation length today of the bootstrap field originating at the electroweak phase transition. In section 4 we compare with the helical case, in partilar with the maximally helical case discussed in detail by Field and Carroll [9]. Section 5 cocludes the note.

2 The electroweak bootstrap solution

The magnetic bootstrap solution of the electrowak theory is simplest in the Bogomolnyi case where the Higgs mass equals the $Z$ mass. We shall therefore mainly exhibit this special case. The standard electroweak static energy can be written [4]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Here we took $W_{2}$=$iW_{1}\equiv iW$, $B$ is the magnetic field and $Z$ is the corresponding $Z-$field. Also

[TABLE]

where $A_{i}$ are the potentials for $B$ and $Z_{i}$ are the corresponding potentials for the $Z$ field.

The Bogomolnyi method then consists in taking first ${\cal E}_{5}=0$, which implies unrealistically that the Higgs mass equals the $Z$ mass. Then we minimize the energy by taking the positive terms ${\cal E}_{i},i=1,...,4$ to vanish. We refer to [4] for a discussion of all these equations of motion. Here we just mention ${\cal E}_{2}=0$, i.e.

[TABLE]

We mention that if the Higgs mass is not equal to the $Z$ mass there will be higher order terms in powers of $|W|$ on the right hand side.

The solution arizing from ${\cal E}_{i}=0,i=1,...,4$ is a bootstrap solution from the following point of view:

1)The magnetic field is generated like in a solenoid by a current consisting of charged $W$ vector meson fields. The action of thes fields is anti-screening [4]-[6].

2)The existence of the charged vector bosons is due to the magnetic field, which is unstable [10] above the threshold $m_{W}^{2}/e$ unless the $W$ mesons are present.

Eq.(8) implies that the magnetic field exceeds the threshold

[TABLE]

It can be shown [5] that in the realistic case where the Higgs mass is not equal to the $Z$ mass the upper limit including contributions from the $|W|^{2}-$term is

[TABLE]

where $m_{H}$ is the Higgs mass. Compared to the total energy density at the electroweak (EW) phase transition111We use [9] that $\rho(t_{\rm EW})\approx 1.6\times 10^{49}~{}{\rm G}^{2}$. this gives

[TABLE]

As discussed in the literature the solution of the equation of motion associated with Eq. (8) is a lattice of vortices [4]. This has been shown numerically [6] as well as mathematically [11]-[15]. This solution represents a collective phenomenon in the sense that Eq. (8) does not have a single vortex solution, only a lattice of an infinite number of vortices as solution. Some other properties of Eq. (8) have recently been discussed in refs. [16]-[18].

The lattice of flux tubes have a simple expression for the energy in the Bogomolnyi case. Integrating over one lattice cell the expression (1) gives by use of periodicity [4]

[TABLE]

where $A$ is the area of a cell. Here we used that the flux is quantized [4]. Realistically, with the right Higgs mass, the energy becomes more complicated, with quadratic terms in the fields.

At the electroweak phase transition there is a high temperature which will cause the flux tubes to oscillate. This is known also to happen in high temperature superconductors where it has been shown that the Lindemann ratio222The Lindemann ratio is the ration of the fluctuations of the tubes (or molecules) relative to the equilibrium distance. If the ratio exceeds 1/10 a liquid is formed. is so large that a liquid state is formed, where the flux tubes enter a (boiling) spaghetti state [19]. So we assume that a similar phenomenon happens in our case. For us the main point is that the high temperature state is statistically isotropic. We shall use this to compute the subsequent evolution of the magnetic field to obtain its present value.

3 Evolution of the magnetic field

in the expanding universe

We now assume statistical isotropy and the validity of the standard magnetohydrodynamics (MHD) in the form

[TABLE]

where $\nu$ and $\eta$ are time independent constants. From isotropy and scaling invariance of the MHD equations it follows [8] that the energy density

[TABLE]

satisfies

[TABLE]

We also define a mean field $B_{\rm rms}$ by

[TABLE]

We next consider the evolution of the universe with the flat expanding metric

[TABLE]

where $t^{*}$ is the conformal time, i.e.

[TABLE]

where $t$ is the Hubble time. The scaling of the energy density now reads

[TABLE]

This was derived from isotropy and validity of the MHD equations. It may be that these assumptions are only valid slightly after the electroweak transition happening at the time

[TABLE]

In the following we shall assume that this difference is so small that it can be ignored.

The magnetic coherence length $\lambda(t)$ is obtained by integrating over the energy,

[TABLE]

From the scaling (19) we obtain

[TABLE]

Similarly we have from the scaling (19)

[TABLE]

It should be noted that from Eq. (22) there is an amplification of the coherence length in addition to the general relativity scale factor $a(t)$

[TABLE]

Therefore, in terms of the Hubble time $t$ the amplification behaves as either $t^{1/4}$ at early times before matter domination at $t_{\rm EQ}$,

[TABLE]

or it behaves at at later times like $t^{1/6}$.

Numerically we obtain from Eq. (22)

[TABLE]

where we used that the initial coherence length of the field is given by the Hubble radius,

[TABLE]

since the initial solution is not limited in space.

From Eqs. (9), (10), and (23) we have for the rms field

[TABLE]

These results are consistent with the lower limits obtained in [1], according to which the field should exceed $3\times 10^{-16}$G with this bound improving as $\lambda^{-1/2}$ for correlation lengths much smaller than a megaparsec. With our correlation length of order 2 kpc the bound (28) is thus rather close (within a factor $\sim 10$) to the bound of Neronov and Vovk.

The rms field and the coherence length are statistical quantities. A more informative quantity is the energy spectrum. The scaling relation (19) relates the energy densities at different times. So measuring the energy today, one can obain the spectrum at earlier times. For example,

[TABLE]

allows knowledge of the spectrum at the electroweak phase transition. The self-similarity scaling essentially says that the energy spectrum does not change except kinematically.

There is a subtility in using Eq. (19), since it assumes the validity of the MHD equations at all times. These may not be valid exactly at the time of the phase transition, but only slightly later when the creation of the field is finished [20]. Therefore the time $t_{\rm EW}$ is in general only close to the transition time. Numerical examples show that this effect is quite small. For exampel, in the numerical calculations by Zrake [7] after a fraction of an Alfvén time the magnetic energy spectrum relaxes to a self-similar power behavior.

4 Comparison with a helical case

The bootstrap solution we have considered is non-helical, with ${\bf AB}$=0. Some time ago Field and Carroll [9] estimated how a maximally helical field develops during the evolution of the universe. They assumed an initial behavior at the electroweak phase transition given by

[TABLE]

After maximally helical amplification Field and Carroll obtain

[TABLE]

This should be compared to our results from the last section given in Eqs. (26) and (28), i.e.

[TABLE]

We see that the coherence lengths differ by a factor $\sim$ 10 if $f_{\lambda}=1$, and the rms fields differ by a factor $\sim 10^{4}$ if $f_{B}=1$. The latter case means that all energy at the electroweak phase transition is magnetic. In our case this is far from the case, since from Eq. (11) we see that $\Omega^{\rm EW}_{B}\sim 3\times 10^{-3}$. If we limit ourselves to such an $\Omega_{B}$ then the helical case would be reduced by a factor $\geq\sim$10.

In this connection it should be mentioned that from a dimensional argument it is difficult to get the entire energy at the electroweak phase transition to be magnetic, since (in natural units) the magnetic field has dimension (mass)2, so except if some large numerical factor is present it is impossible to get a result for the magnetic field which is much larger than what we obtained. There are simply not large enough masses to produce an $\Omega_{B}\sim 1$ in the conventional electroweak theory!

The results mentioned above shows that it is not necessary to have helicity in order to obtain a type of inverse cascade. However, the enhancement effect is larger with helicity present. For completeness, we compare the amplification factors in the two cases, namely for the helical case [9]

[TABLE]

and Eq. (24) for the non-helical case with the amplification factor

[TABLE]

Thus the difference in the amplification is a factor of 10.

Since the helical case gives a larger effect there has been many studies of the effects of the chiral anomaly on the evolution of the (hyper-)magnetic field, see refs. [21]-[27]. In the modified MHD equations found in some of these papers there is no simple scale invariance which allows one to derive a self-similar expression for the time development of the energy. However, when a field generated early ”passes” the electroweak phase transition there may be the possibility that this helical field have a further development similar to the magnetic properties discussed in this note, in paticular if the usual MHD equations take over from the modified ones at the transition [22],[26].

We mention that Jedamzik and Sigl [30] have shown, using different methods, that appreciable primordial fields originating from phase transitions is possible, even when viscosity and dissipation is considered. These results as well as the estimate of the rms magnetic field today is in reasonable agreement with our result.

5 Conclusion and discussion

Primordial magnetic fields are generated at the electroweak phase transition. They could act as extremely weak extragalactic seed fields for various astrophysical dynamo effects. It is important for our result that the coherence length is amplified by a factor of order $10^{6}$, as mentioned in Eq. (24). If this had not been the case the coherence length would have been of the order $\sim 10^{-3}$ pc, which would have been without interest cosmologically.

If the magnetic field discussed here is relevant as a cosmological seed field it is of interest from the point of view of principles that then this primordial field is related to the non-Abelian vacuum, which is magnetic in nature. The flux tubes also exist above the critical temperature, but the string (flux tube) tension vanishes. By a non-perturbative phase transition one can gauge transform these fields to the trivial fields [28]-[29]. Thus, in this picture the seed field has been woken up from the non-Abelian vacuum in the electroweak theory by a phase transition.

In our estimate of the value of the magnetic field today we assumed statistical isotropy and the validity of the standard MHD equations (13) after the phase transition. The actual solution (8) of the electroweak theory ceases to be valid after some time, since the magnetic field decreases due to the expansion of the universe and the non-linear dynamics of the MHD equations and hence after some time reaches a value below the threshold $m_{W}^{2}/e$. The magnetic field then becomes electroweak stable and there is no longer any reason [10] for the presence of the $W-$condensate, so these vector bosons can decay according to the standard theory. The magnetic field is then subsequently ruled entirely by the MHD equations (13).