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\date{\today,~ $ $Revision: 1.10 $ $}
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\maketitle
Primordial gravitational waves (GWs), observable as stochastic background
with the planned Laser Interferometer Space Antenna (LISA), could be
produced by magnetic fields generated at the time of the electroweak
phase transition.
There is the possibility that they might be helical \cite{Vac01,DiasGil08}.
Such fields would have decayed more slowly than nonhelical ones, and would
thus have a chance to survive until the present time \cite{BKMRPTV17} to
explain the lower limits on the magnetic field strength inferred through
the non-observation of secondary cascade photons with FERMI \cite{NV10}.
Such fields could be generated during the electroweak phase transition and
would undergo forward and inverse cascading during the entire radiation
era until recombination \cite{BEO96}.
The inverse cascade manifests itself through a shift in the peak of the
magnetic energy spectrum \cite{CHB01,BJ04}.
The forward cascade, on the other hand, would manifest itself through
the development of a steeper slope of the normalized magnetic helicity
\cite{BS05},
so that the magnetic field is no longer fully helical at high wavenumbers,
if the initial magnetic field was fully helical at all wavenumbers.
This change would become evident if the development of the new slope
takes a sufficient amount of time.
Details of this process, including its duration, could manifest themselves
in the circular polarization of the resulting GW signal.
However, the magnetic field will gradually decay and its ability to
contribute to the resulting GW production diminishes.
This happens rather rapidly.
Therefore, whether or not it is detectable in the GW signal depends on
how rapidly the magnetic field starts to decay.
%In this Letter, we use direct numerical simulations of primordial magnetic
Here, we use direct numerical simulations of primordial magnetic
fields together with the resulting GW production to investigate the
effect of the time-dependence of the source on the GW signal.
We use the {\sc Pencil Code} \cite{PC} for these calculations, which
is well suited for primordial MHD simulations and which comes with a GW
solver readily available for our purposes \cite{RPBKKM18}.
All simulations have a resolution of $1024^3$ meshpoints.
Earlier analytic work showed that the degree of circular polarization
changes with wave number if the two spectral slopes are different from
each other, but it stays independent of $k$ if the spectra have the
same slope \cite{KGR05,KK15}.
Those analytic calculations made use of certain approximations, but
recent numerical work \cite{RPMBKK19} showed that the predictions from
the analytical model regarding the degree of polarization are surprisingly
well reproduced by the numerical simulations.
As we have already mentioned, the decay of the magnetic field causes a
decline of the turbulent driving of GWs.
This decline is further enhanced by the expansion of the universe,
although this effect is small if the turbulence decays within a Hubble
time.
Mathematically, this decline is caused by an increase of the scale factor
$a$ that enters in the denominator of the stress term in the GW equation.
Using conformal time, $t=\int dt_{\rm phys}/a(t_{\rm phys})$, and comoving
strain, $h=ah_{\rm phys}$, where $t_{\rm phys}$ and $h_{\rm phys}$ are
physical time and strain, the linearized GW equation reads
\EQ
\left(\partial_t^2-\nabla^2\right)h_\lambda=6T_\lambda/t
\quad\mbox{(for $t\ge1$)},
\EN
where $\lambda=+$ or $\times$ denote the plus or cross polarization
modes, which are the two independent tensor modes compatible with the
Einstein equations, and the $T_\lambda$ are the sums of Reynolds and
Maxwell stresses, projected onto the $\lambda=+$ or $\times$ modes.
For a fully helical magnetic field, the magnetic energy, and thus
also the magnetic stress decay like $T_\lambda\sim(t-1)^{-2/3}$;
see \cite{BM99} for a corresponding result in ordinary MHD, and
Refs.~\cite{BKMRPTV17,BK17} for applications in the cosmological context.
For $t\ll2$, this effect is clearly more important than that of the
cosmological expansion, which will nevertheless be retained in our
simulations.
The authors of Ref.~\cite{RPMBKK20} considered two types of simulations;
(i) one where the initial turbulence spectrum is {\em given} and (ii)
one where the magnetic energy spectrum is {\em driven} by the injection
of an electromotive force.
Their direct numerical simulations showed that when the initial turbulence
spectrum is given, the degree of circular polarization is independent
of the wave number for $k>2k_*$, where $k_*$ is the wave number of the
energy-carrying scale of the turbulence, i.e., where the magnetic energy
spectrum peaks.
On the other hand, when the magnetic energy spectrum is driven, the degree
of circular polarization declines to zero with increasing value of $k$.
Based on the model of Ref.~\cite{KGR05}, this can be understood as a
consequence of the fact that with a given initial spectrum, the magnetic
field is fully helical at all wave numbers, and only at late times a
decline of the degree of circular polarization of GW could be expected
when the current helicity cascade of the magnetic field gets established
and $k\HM(k)$ becomes steeper than $\EM(k)$ for $k>2k_*$.
To test this idea in detail, we must consider different time dependencies
of the magnetic field.
This cannot be done in a fully selfconsistent model, where the decay
law is always fixed.
To model a slow-down in the decay, we could ``modify'' the MHD equations.
As an example, let us mention here that, if the induction equation
is dominated by the Hall effect, the magnetic energy decay becomes
proportional to $\sim t^{-2/5}$.
However, to have a more controlled experiment, we now consider solutions
of the GW equation where the source term on the right-hand side is scaled
by a time-dependent factor $F(t)$, i.e.,
\EQ
T_\lambda(\xx,t)\to F(t)\,T_\lambda(\xx,t).
\EN
We consider two possibilities.
In models of type~A, we compensate or even overcompensate for the decay
of the magnetic field.
Note, however, that we preserve the natural temporal fluctuations and
the intrinsic changes of the spectra, including changes in there peak
due to the forward cascade and inverse cascades, respectively.
To an excellent approximation, the decay of a helical field can be
modeled as
\EQ
\EEM=\EEMz/[1+(t-1)/\tau]^{2/3}.
\EN
For the fully helical simulation of Ref.~\cite{RPMBKK20}, which is the
fiducial model for the experiments presented here, we find $\EEMz=0.0031$
and $\tau=0.053$.
In that case, we choose $F(t)=[1+(t-1)/\tau]^n$ and vary the value of $n$.
For $n=2/3$, the overall amplitude of the effective stress stays
constant, while for $n>2/3$, the decay is overcompensated, so we expect
an accelerated growth of the GW energy.
As we will see below, when $n=2/3$, the effect on the growth of GW energy
is very weak, because the resulting GW amplitude depends decisively on
the temporal variation of the source at later times.
To boost the effective temporal frequency of the signal, we also
consider a model where we give regular instantaneous ``kicks'' to
the amplitude of the stress.
This corresponds to a staircase profile for the scaling factor $F(t)$
in front of the stress term, i.e.,
\EQ
F(t)=1+\epsilon\sum_{m=1}^\infty m\theta(t-m\tau),
\EN
where $\epsilon$ determines the slope of the increase of $F(t)$ and $\tau$
(here $\tau=0.05$) is the interval of kicks.
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_Ft}
\end{center}\caption[]{
Normalized GW and magnetic energies versus time for models of type~A
(red) and $B$ (blue).
The black solid and dashed lines refer to GW and magnetic energies
for $F(t)=1$.
}\label{pcomp_Ft}\end{figure}
In \Fig{pcomp_Ft}, we present results for the energy evolution
for different values of $n$ for models of type~A and for different
combinations of $\epsilon$ and $\tau$ for models of type~B.
In all cases, there is a statistically monotonous increase of $\EGW$.
We emphasize that the GW energies obtained for models with $F(t)\gg1$
reach obviously unrealistically large values, which is necessary to
show that the magnetic influence very quickly becomes negligible in
all realistic situations.
Next, we consider for different times magnetic and GW energy spectra,
$\EM(k)$ and $\EGW(k)$, respectively.
They are normalized such that $\int\EM(k)\,dk=\EEM$ and
$\int\EGW(k)\,dk=\EEGW$.
Occasionally, we also use the GW energy for logarithmic wavenumber
interval, $\Omega_{\rm GW}(k)\equiv k\EGW(k)$.
We see that at early times, the spectra of $2\EM(k)$ and $k\HM(k)$ coincide,
as required by the realizability condition for a fully helical magnetic
field.
At late times, however, the current helicity, whose spectrum is
proportional to $k^2\HM(k)$, displays a forward cascade with a slope
proportional to $k^{-5/3}$, which implies a $k^{-8/3}$ spectrum for
$\EGW(k)$; see Ref.~\cite{RPMBKK19}.
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{power_frac_M1152e_exp6k4_sig1d}
\end{center}\caption[]{
Magnetic energy spectra (upper panel) and fractional magnetic helicity
spectra (lower panel) at $t=1.01$, $1.1$, $1.2$, ..., $1.6$, maked by
colors ranging from red to green and blue, for the fully helical run
of Ref.~\cite{RPMBKK20}.
}\label{power_frac_M1152e_exp6k4_sig1d}\end{figure}
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_Ft_M1152e_exp6k4_sig1d}
\end{center}\caption[]{
GW energy (upper panel) and degree of polarization (lower panel)
at the same times as in \Fig{power_frac_M1152e_exp6k4_sig1d}
for a model of type~A with $n=2$.
}\label{pcomp_Ft_M1152e_exp6k4_sig1d}\end{figure}
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_Ft_M1152e_exp6k4_sig1e}
\end{center}\caption[]{
Same as \Fig{pcomp_Ft_M1152e_exp6k4_sig1d}, but with $n=5$.
}\label{pcomp_Ft_M1152e_exp6k4_sig1e}\end{figure}
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_Ft_M1152e_exp6k4_sig1f}
\end{center}\caption[]{
Same as \Fig{pcomp_Ft_M1152e_exp6k4_sig1d}, but
for a model of type~B with $\epsilon=1$ and $\tau=0.01$.
}\label{pcomp_Ft_M1152e_exp6k4_sig1f}\end{figure}
For small values of $n$, the GW spectrum displays at all times the same
spectrum for $\Omega_{\rm GW}(k)\equiv\EGW(k)$ and the antisymmetric part,
$\Xi_{\rm GW}(k)$; see Ref.~\cite{CDK04} for details.
The reason is that at late times, the magnetic energy is so weak that
it cannot change the GW spectrum noticeably.
To get an idea of how small the effect is in comparison with the much
larger effect of fluctuations, we now decrease the value of $n$.
For $n=2$, for example, the value of $\Omega_{\rm GW}(k,t)$ grows by
a factor of $\approx30$ compared to the standard case, but
$\Omega_{\rm GW}(k,t)$ shows growth only at $k>2k_*$.
Consequently, the dependence of polarization also only changes at small
$k$; see \Fig{pcomp_Ft_M1152e_exp6k4_sig1d}.
For $n=5$, on the other hand, we see a clear growth at all $k$,
and now ${\cal P}(k)$ also declines rapidly at high $k$; see
\Fig{pcomp_Ft_M1152e_exp6k4_sig1e}.
Finally, we consider models of type~B.
It turns out that now $\EGW(k,t)$ increases preferentially
only at discreet wave numbers, $k=jk_*$, with integers $j\ge1$.
The resulting polarization, ${\cal P}(k)$ increases more smoothly over
the full range of wave numbers.
Our work has illuminated some of the more subtle aspects of turbulent
GW production by time-dependent magnetic fields.
%In this Letter, we have only considered the case of a given initial
Here, we have only considered the case of a given initial
magnetic energy spectrum.
This allowed us to produce a magnetic field that is fully helical at
all wave numbers.
In the other case of a driven initial spectrum, considered in
Ref.~\cite{RPMBKK20}, the fractional magnetic helicity declines
to zero at higher wave numbers.
This also happens in our present models with a given initial spectrum,
but at those later times, the temporal fluctuations associated with the
turbulence are too weak to have a noticeable effect.
To illustrate this more clearly, we have artificially modified the time
dependence of the stress by applying an $F(t)$ function.
Although the details between models of type~A and B are different,
they all have in common that the degree of circular polarization now
decreases with time.
However, our work also shows that, if the initial magnetic spectrum can
indeed be assumed given as fully helical at all wave numbers, the degree
of circular polarization would indeed be 100\%, even at high wave numbers.
%r e f
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\bibitem{DiasGil08}
A. D{\'{\i}}az-Gil, J. Garc{\'{\i}}a-Bellido, M. Garc{\'{\i}}a P{\'e}rez and A. Gonz{\'a}lez-Arroyo\yprl{2008}{100}{241301}
\bibitem{BKMRPTV17}
A. Brandenburg, T. Kahniashvili, S. Mandal, A. Roper Pol, A. G. Tevzadze, and T. Vachaspati\yprd{2017}{96}{123528}
\bibitem{NV10}
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\bibitem{BEO96}
A. Brandenburg, K. Enqvist, and P. Olesen\yprd{1996}{54}{1291}
\bibitem{CHB01}
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\bibitem{BJ04}
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\bibitem{BS05}
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\bibitem{PC}
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DOI:10.5281/zenodo.2315093
\bibitem{RPBKKM18}
A.~Roper Pol, A.~Brandenburg, T.~Kahniashvili, A.~Kosowsky, and S.~Mandal\ygafd{2020}{114}{130}
\bibitem{KGR05}
T. Kahniashvili, G. Gogoberidze, and B. Ratra\yprl{2005}{95}{151301}
\bibitem{KK15}
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\bibitem{RPMBKK19}
A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili, and A. Kosowsky\sprd{2019}
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\bibitem{BK17}
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\bibitem{CDK04}
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\end{thebibliography}
%\appendix
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