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\title[Generation of Gravitational Waves]
{Gravitational Wave Spectra from Primordial 
Magnetohydrodynamic Turbulence (additional)}
\subtitle{Nordita program on ``Gravitational Waves from the Early Universe'' \\ 
Aug. 26 -- Sep. 20 2019}
\author[Alberto Roper Pol]{
Alberto Roper Pol (PhD candidate) \\
\textit{Research Advisor: }Axel Brandenburg \\ 
{\em Collaborators: }Tina Kahniashvili, Arthur Kosowsky, Sayan Mandal
}
\institute[University of Colorado] 
% Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{
University of Colorado at Boulder \\
\medskip
Laboratory for Atmospheric and Space Physics (LASP)
}
\date{August 26, 2019}


\begin{document}

\begin{frame}
\titlepage % Print the title page as the first slide
\end{frame}

\begin{frame}
\frametitle{1D Beltrami field}
\begin{itemize} 
\item Magnetic field: $\pmb{B}(x) = B_0 \left(0,  \sin k_0 x, \cos k_0 x \right)$,
such that $\pmb{B} \cdot \left( \pmb{\nabla} \times \pmb{B} \right) = \text{sgn} \left(k_0\right)$
\item Stress-energy tensor: 
$ T_{ij}  = \frac{1}{2}B_0^2 \left( \begin{array}{c c c} 
0 & 0 & 0 \\ 0 & - \cos 2k_0 x & \sin 2k_0 x \\ 0 & \sin 2k_0 x & \cos 2k_0 x
\end{array} \right) $ \\
\vspace{2mm}
which is already TT with two independent modes:
$$ T^+ = -\frac{1}{2} B_0^2 \cos 2k_0 x, \quad T^{\times} = -\frac{1}{2} B_0^2 \sin 2k_0 x $$
\item Tensor-mode perturbations: 
{\small
$$ h^+(x, t) = -\frac{2 \pi G}{ c^4 k_0^2} B_0^2 \cos 2k_0x (1 - \cos 2k_0c t) $$
$$h^{\times}(x, t) = - \frac{2 \pi G}{ c^4 k_0^2} B_0^2 \sin 2k_0x (1 - \cos 2k_0c t) $$ } 
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{1D Beltrami field}
\begin{itemize}
\item GWs energy density:
$$ E_{\rm GW} (k, t) = \frac{\pi G}{c^4 k_0^2}
\mathcal{E}_{\text{M}}^2 \left[ 
\delta (k - 2k_0) + \delta(k + 2k_0) \right] $$ 
$$ H_{\rm GW} (k, t) = \frac{\pi G}{c^4 k_0^2}
\mathcal{E}_{\text{M}}^2 \left[ 
\delta (k - 2k_0) - \delta(k + 2k_0) \right] $$ 
\item Polarization degree:
$$\mathcal{P}(2k_0) = {\rm sgn} \left( k_0 \right) $$
\item We see that polarization degree is $\pm 1$, 
with the same sign as the sign of the magnetic 
helicity
\end{itemize}
\end{frame}

\section*{Evidence of primordial magnetic fields}

\begin{frame}
\frametitle{Evidence of primordial magnetic fields}

There are different evidence that indicate the presence of a large scale 
coherent magnetic field

\begin{block}{Fermi blazar observations\footnotemark}
\begin{itemize}
\item Gamma rays from blazars interact with 
extragalactic background light
\item Generation of electron - positron beam
\item Observed power removal from gamma-ray beam
\end{itemize}
\end{block}
\footnotetext{A. M. Taylor, I. Vovk, and A. Neronov, 
{\em Astronomy \& Astrophysics,} 529:A144 (2011)}
\end{frame}

\begin{frame}
\frametitle{Evidence of primordial magnetic fields}
\visible<1-> {
\begin{block}{Solution}
\begin{itemize}
\item Large scale (intergalactic) magnetic fields could deviate the electron-positron from beam in opposite directions
\item Recombination does not happen leading to lose of energy
\item Strength $\sim 10^{-16}$ G, scale $\sim 100 $ kpc
\end{itemize}
\end{block}
}
\visible<2> {
\begin{block}{Origin}
Intergalactic magnetic fields could have been originated from: 
\begin{itemize}
\item Astrophysical or
\item \alert{Cosmological } seed fields
\end{itemize}
subsequently amplified during structure formation
\end{block}
}
\end{frame}

\begin{frame}
\frametitle{Evidence of primordial magnetic fields}
\begin{block}{Helicity}
\begin{itemize}
\item Magnetic helicity is observed in present astrophysical objects
\item Fractional magnetic helicity is required in cosmological seed fields
\item Primordial helical magnetic fields require a
cosmological first order phase transition:
	\begin{itemize}
	\item \alert{Electroweak phase transition 
	(EWPT),} $t \sim 10^{-12}$ s, $\sim 250$ GeV
	\item Quantum chromodynamics (QCD) phase 
	transtion, $t \sim 10^{-6}$ s
	\end{itemize}
\end{itemize}
\end{block}
\begin{definition}[Magnetic Helicity]
$ \mathcal{H} = \left< \pmb{B} \cdot \left( \pmb{\nabla} \times \right)^{-1} \pmb{B} \right>  = \left< \pmb{A} \cdot \pmb{B} \right> $
\end{definition}
\end{frame}

%------------------------------------------------
\section*{Magnetohydrodynamic turbulence}
%------------------------------------------------

\begin{frame}
\frametitle{EWPT plasma}
\begin{block}{Characterization}
\begin{itemize}
\item Quark-gluon plasma 
\item  Charge-neutral, electrically conducting fluid 
\item  Relativistic magnetohydrodynamic (MHD) 
equations
\item  Ultrarelativistic equation of state
\end{itemize}
\end{block}
\begin{block}{Ultrarelativistic equation of state}
 $ p = \rho c_s^2, \quad c_s^2 = c^2/3$
\end{block}
\end{frame}


\begin{frame}
\frametitle{MHD equations}
Relativistic MHD equations are reduced 
to\footnote{A. Brandenburg, K. Enqvist, and P. 
Olesen, {\em Phys. Rev. D}, 
54(2):1291–1300, 1996.}
\begin{block}{Continuity equation}
$\frac{\partial \ln \rho}{\partial t} = - \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) + \frac{1}{\rho} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta \pmb{J}^2 \right) $
\end{block}
\begin{block}{Momentum equation}
\small $ \frac{D \pmb{u}}{Dt} = \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) - \frac{\pmb{u}}{p} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta J^2 \right] - \frac{1}{4} \pmb{\nabla} \ln \rho + \frac{3}{4 \rho} \pmb{J} \times \pmb{B} + \frac{2}{\rho} \pmb{\nabla} \cdot \left( \rho \nu \pmb{S} \right) $
\end{block}
\begin{block}{Induction equation}
$ \frac{\partial \pmb{B}}{\partial t} = \pmb{\nabla} \times \left( \pmb{u} \times \pmb{B} - \eta \pmb{J} \right) $ 
\end{block}
for a flat expanding universe with comoving magnitudes
\end{frame}

\begin{frame}
\frametitle{MHD energy density}
\small
\begin{columns}[t]
\column{.5\textwidth}
\blue{\textbf{Kinetic energy density: }
$$ \mathcal{E}_{\text{K}}(t) = \frac{1}{2} 
\left< \rho \pmb{u}^2 \right> = 
\int E_{\text{K}}(k, t) {\rm \, d}k $$ \\
\textbf{Kinetic helicity: }
$$ \mathcal{H}_{\text{K}}(t) = \rho 
\left< \pmb{\omega} \cdot \pmb{u} \right> 
= \int H_{\text{K}} (k, t) {\rm \, d}k $$ \\
\textbf{Kinetic correlation length: }
$$ \xi_{\text{K}}(t) =  \frac{\int {\rm \, d}k \ 
k^{-1} E_{\text{K}}}{\mathcal{E}_{\text{K}}} $$}

\column{.5\textwidth}
\red{\textbf{Magnetic energy density: }
$$ \mathcal{E}_{\text{M}}(t) = \frac{1}{2 } \left< 
\pmb{B}^2 \right> = \int E_{\text{M}}(k, t) 
{\rm \, d}k $$
\textbf{Magnetic helicity: }
$$ \mathcal{H}_{\text{M}}(t) = \left< \pmb{B} \cdot 
\pmb{A} \right> = \int H_{\text{M}}(k, t) 
{\rm \, d} k $$
\textbf{Magnetic correlation length: }
$$ \xi_{\text{M}}(t) =  \frac{\int {\rm \, d} 
k \ k^{-1} E_{\text{M}}}{\mathcal{E}_{\text{M}}} $$}
\end{columns}
\vspace{2mm}
\textbf{Realizability condition: }
$$ \frac{1}{2} k \left| H_{\rm M} \right|  
\leq E_{\rm M} \Rightarrow \green{
\xi_{\rm M} \geq \frac{{\cal H}_{\rm M}}{2 
{\cal E}_{\rm M}}}$$
\let\thefootnote\relax\footnotetext{
Lorentz-Heaviside units are used}
\normalsize
\end{frame}

\begin{frame}
\frametitle{Initial conditions}
\begin{block}{Fractional helicity 
$\sigma_{\rm M}$ and peak frequency $k_{\rm M}$}
\begin{columns}
\small
\column{.5\textwidth} 
$$ \varepsilon_{\text{M}} = \frac{k H_{\text{M}}
(k, t)}{2 E_{\text{M}}(k, t)} = 
\frac{2 \sigma_{\text{M}}}
{1 + \sigma_{\text{M}}^2}$$
\column{.5\textwidth}
\includegraphics[width=0.6\textwidth]{scheme}
\end{columns}
\normalsize
\end{block}
\begin{block}{Ratio of energy densities}
\begin{center}
$Q_* = B_*\left/\left(\sqrt{\rho_*} u_*
\right)\right., \quad $ where $t_\ast$ 
indicates the time of generation
\end{center}
Three cases:
\begin{itemize}
\item $Q_* \gg 1$:  Magnetically dominant 
turbulence $\left( E_{\rm M} \propto 
k^{\alpha_{\rm M}} \right.$, $\left. 
\alpha_{\rm M} \geq 4 \right)$ 
\item $Q_* \sim 1$: Equipartition case
\item $Q_* \ll 1$: Magnetically subdominant or 
kinetically dominant turbulence
$\left( E_{\text{K}} \propto k^{\alpha_{\text{K}}} 
\right.$, $\left. \alpha_{\text{K}} \geq 2 \right)$ 
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Spectral parameters}
\begin{block}{Parameters $\alpha_i$, $\beta_i$, $q_i$, $p_i$}
$$E_i(k, t) \propto k^{\alpha_i}$$
$$E_i(k, t) = \xi_i^{-\beta_i} \phi_i \left( k \xi_i \right)$$
$$\xi_i (t) \propto t^{q_i}$$
$$\mathcal{E}_i(t) \propto t^{-p_i}$$
\end{block}
\begin{block}{$pq$ diagram}
Since ${\cal E}_i = \int E_i {\rm \, d} k$, we have 
the relation
$$p_i = (1 + \beta_i) q_i$$
Straight lines for constant $\beta$
\end{block}
\end{frame}

\begin{frame}
\begin{block}{Inverse cascade condition}
$$ s_i = \frac{\partial \ln E_i}{\partial \ln t} = 
\left( \alpha_i - \beta_i \right) q_i > 0 $$
\end{block}
\begin{center}
\includegraphics[width=0.6\textwidth]{inverse_cascade}
\end{center}
\end{frame}
%
%% UNCOMMENT TO SHOW RESULTS
\begin{frame}
\frametitle{Numerical simulations}
{\large \textbf{Case 1.} Initial non-helical 
($\sigma_{\text{M}} = 0$) Batcehlor spectrum 
($\alpha_i = 4$) \\}
\centering
{\tiny $ Q_* = 10$ } 
\includegraphics[width=.8\textwidth]{MHD_results/A} \\
{\tiny $ Q_* = 0.1$ } 
\includegraphics[width=.8\textwidth]{MHD_results/C}
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\large \textbf{Case 2.} Initial helical 
($\sigma_{\text{M}} = 0.3$) white noise spectrum 
($\alpha_i = 2$) \\}
\begin{center}
{\tiny $ Q_* \rightarrow \infty$ } \includegraphics[width=1\textwidth]{MHD_results/G} \\
\end{center}
{\large \bf Realizabilty condition:}
$$\frac{1}{2} k |H_{\rm M}| \leq E_{\rm M} \Rightarrow
\green{\xi_{\rm M} \geq \frac{{\cal H}_{\rm M}}
{2 {\cal E}_{\rm M}} = \xi_{\rm M}^{\rm min}}$$
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\normalsize \textbf{Case 3.} Initial fully helical 
($\sigma_{\text{K}} = 1$) Batchelor spectrum 
($\alpha_{\text{K}} = 4$) \\}
\vspace{2mm}
\centering
{\tiny $\sigma_{\rm M} = 0$}
\includegraphics[width=.9\textwidth]{MHD_results/I} \\
{\tiny $\sigma_{\rm M} = -1$}
\includegraphics[width=.9\textwidth]{MHD_results/J}
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
\begin{block}{Conclusions}
\begin{itemize}
\item Inverse cascade can be predicted with the 
condition $s_i = \left( \alpha_i - \beta_i \right) 
q_i > 0$
\item With initial helicity, conservation of total 
helicity dominates ($\beta = 0$)
\item With no initial helicity, $pq$ diagram tends 
to $\beta = 2$ and conservation of Saffman integral 
dominates
\item With kinetically dominant turbulence, 
$pq$ diagram tends to $\beta = 4$ and conservation
of Loitsiansky integral dominates
\end{itemize}
\end{block}
\begin{itemize}
\tiny
\item Total helicity 
$$ \left< \pmb{\omega} \cdot \pmb{u} \right> + 
\left< \pmb{J} \cdot \pmb{B} \right> $$
\item Saffman integral
$$ \mathcal{S} = \int \left< \pmb{u}(\pmb{x}, t) \cdot 
\pmb{u}(\pmb{x} + \pmb{r}, t) \right> d\pmb{r} $$
\item Loitsianski integral
$$ \mathcal{L} = \int r^2 \left< \pmb{u}(\pmb{x}, t) 
\cdot \pmb{u}(\pmb{x} + \pmb{r}, t) \right> 
d\pmb{r} $$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Evolution of magnetic strength and correlation length\footnote{A. Brandenburg {\em et 
al. Phys. Rev. D,} 96, 123528 (2017)}}
\includegraphics[width = .9\textwidth]{MHD_results/B_corrl}
\end{frame}

\end{document}