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\title[Gravitational Waves from the early-universe]
{Generation of Gravitational Waves due to Magnetohydrodynamic
Turbulence in the Early Universe}
\subtitle{PhD Final Examination}
\author[Alberto Roper Pol]{
Alberto Roper Pol (PhD candidate) \\
\textit{Faculty Advisor: }Brian Argrow\\
{\em Research Advisor:} Axel Brandenburg \\ 
{\em Collaborators: }Tina Kahniashvili, Arthur Kosowsky \& Sayan Mandal
}
\vspace{-10mm}
\institute[University of Colorado] 
{
University of Colorado at Boulder \\
\vspace{2mm}
Laboratory for Atmospheric and Space Physics (LASP) \\
}
\date{May 8, 2020}


\begin{document}

\begin{frame}
\titlepage
\vspace{-15mm}
\begin{flushright}
\includegraphics[width=.3\textwidth]{lasp}
\end{flushright}
\vspace{-15mm}
\begin{flushleft}
{\tiny
\vspace{6mm}
A. Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.} 
{\bf 114}, 130. arXiv:1807.05479 (2020) \\
\vspace{-1mm}
A. Roper Pol {\em et al., submitted to Phys. Rev. D}
arXiv:1903.08585 (2020) }
\end{flushleft}
\end{frame}

\begin{frame}
\frametitle{Overview}
\tableofcontents 
\end{frame}


%----------------------------------------------------------------------------------------
%	PRESENTATION SLIDES
%----------------------------------------------------------------------------------------

% % % % % % % % % % % % % % % % % % % % % % % % % % % %
\section{Introduction and Motivation} % % % % % % % % %
% % % % % % % % % % % % % % % % % % % % % % % % % % % %

\begin{frame}
\frametitle{Introduction and Motivation}
\begin{itemize}
\item Generation of cosmological 
gravitational waves (GWs) during phase transitions and inflation
\begin{itemize}
\item \red{\bf Electroweak phase transition} $\sim 100$ GeV
\item Quantum chromodynamic (QCD) phase transition $\sim 100$ MeV
\item Inflation
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Introduction and Motivation}
\centering
\includegraphics[width=.65\textwidth]{chronology_universe}
\end{frame}

\begin{frame}
\frametitle{Introduction and Motivation}
\begin{itemize}
\item \light{Generation of cosmological
gravitational waves (GWs) during phase transitions and inflation}
\begin{itemize}
\item \light{Electroweak phase transition $\sim 100$ GeV}
\item \light{Quantum chromodynamic (QCD) phase transition $\sim 100$ MeV}
\item \light{Inflation}
\end{itemize}
\item GW radiation as a probe of early universe physics
\item Possibility of GWs detection with
	\begin{itemize}
	\item \red{\bf Space-based GW detector LISA}
    \item Pulsar Timing Arrays (PTA)
	\item $B$-mode of CMB polarization
	\end{itemize}	
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Introduction and Motivation}
\includegraphics[width=\textwidth]{lisa_etal}
\end{frame}

\begin{frame}
\frametitle{Introduction and Motivation}
\begin{columns}[c]
\column{0.5\textwidth}
\begin{block}{LISA}
\begin{itemize}
\item Laser Interferometer Space Antenna (LISA) 
is a space--based GW detector
\item LISA is planned for 2034
\item LISA was approved in 2017 as one of the main 
research missions of ESA
\item LISA is composed by three spacecrafts in a 
distance of 2.5M km
\end{itemize}
\end{block}
\column{0.5\textwidth}
\begin{figure}[H]
\includegraphics[width=\textwidth]{LISA-waves}
\caption{Artist's impression of LISA from Wikipedia}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Orbit of LISA}
\begin{center}
\includegraphics[width=.7\textwidth]{lisa_orbit}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Introduction and Motivation}
\begin{itemize}
\item \light{Generation of cosmological
gravitational waves (GWs) during phase transitions and inflation}
\begin{itemize}
\item \light{Electroweak phase transition $\sim 100$ GeV}
\item \light{Quantum chromodynamic (QCD) phase transition $\sim 100$ MeV}
\item \light{Inflation}
\end{itemize}
\item \light{GW radiation as a probe of early universe physics}
\item \light{Possibility of GWs detection with}
\begin{itemize}
\item \light{Space-based GW detector LISA}
\item \light{Pulsar Timing Arrays (PTA)}
\item \light{$B$-mode of CMB polarization}
\end{itemize}
\item  Magnetohydrodynamic (MHD) sources of GWs: 
\begin{itemize}
\item Hydrodynamic turbulence from phase transition bubbles nucleation
\item Primordial magnetic fields
\end{itemize}
\item Numerical simulations using {\sc Pencil Code} to solve:
\begin{itemize}
\item Relativistic MHD equations
\item Gravitational waves equation
\end{itemize}
\end{itemize}
\end{frame}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\section{Evidence of primordial magnetic fields}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % 

\frame{\tableofcontents[currentsection]}

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

There are different astrophysical evidences that indicate the presence of a large scale 
coherent magnetic field.\footnote{\tiny
L. M. Widrow\\ \hspace{7mm} 
\textit{Rev. of Mod. Phys.},
{\bf 74} 775--823 (2002)}

\begin{block}{Fermi blazar observations}
\begin{itemize}
\item Gamma rays from blazars ($\sim $1 TeV) interact with extragalactic background light
\item Generation of electron - positron beam
\item Observed power removal from gamma-ray beam
\end{itemize}
\end{block}
\end{frame}

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

% UNCOMMENT TO SHOW FEYNMAN DIAGRAM
\begin{figure}[H]
\includegraphics[width=0.6\textwidth]{feynman_diagram}
%\caption{Obtained from
%\url{https://physics.stackexchange.com/questions/109979/feynman-diagram-for-annihilation}}
\end{figure} 

\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\footnotemark\
\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}
%}
\footnotetext{\tiny A. M. Taylor, I. Vovk, and A. Neronov\\
\hspace{7mm} \textit{ Astron. \& Astrophys.}, {\bf 529} A144, (2011)}
\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 first order phase transition:
	\begin{itemize}
	\item \alert{Electroweak phase transition (EWPT)} $t \sim 10^{-12}$ s
	\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{Magnetohydrodynamics}  % % % % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % 

\frame{\tableofcontents[currentsection]}

\begin{frame}
\frametitle{MHD description}
Right after the electroweak phase transition we can model
the plasma using continuum MHD
\vspace{4mm}
\begin{itemize}
\item Quark-gluon plasma
\item Charge-neutral, electrically conducting fluid
\item Relativistic magnetohydrodynamic (MHD) equations
\item Ultrarelativistic equation of state
$$ p = \rho c^2/3$$
\item Friedmann--Lema\^itre--Robertson--Walker model
$$g_{\mu \nu} = {\rm diag} \{ -1, a^2, a^2, a^2\}$$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Stress--energy tensor}
\begin{block}{Contributions to the stress-energy tensor}
$$T^{\mu \nu} = \bigl(\red{p/c^2} + \rho\bigr) U^{\mu} U^{\nu} + p g^{\mu \nu} + 
\blue{F^{\mu \gamma} F^{\nu}_{\ \, \gamma} - \frac{1}{4} g^{\mu \nu} 
F_{\lambda \gamma} F^{\lambda \gamma}},$$
\begin{columns}[c]
\column{.5\textwidth}
\begin{itemize}
\item From fluid motions \\
$ T_{ij} = \left( \red{p/c^2} + \rho \right) \red{\gamma^2} u_i u_j + p \delta_{ij} $ \\
Relativistic equation of state: $ p = \rho c^2/3$
\end{itemize}
\column{.5\textwidth}
\begin{itemize}
\vspace{-8.5mm}
\item From magnetic fields: \\
$ T_{ij} = \blue{- B_i B_j + \delta_{ij} B^2/2}$
\end{itemize}
\end{columns}
\vspace{4mm}
\begin{itemize}
\item 4--velocity $U^\mu = \red{\gamma}(c, u^i)$
\item 4--potential $\blue{A^\mu = (\phi/c, A^i)}$
\item 4--current $\blue{J^\mu = (c \rho_{\rm e}, J^i)}$
\item Faraday tensor $\blue{F^{\mu \nu} = \partial^\mu A^\nu -
\partial^\nu A^\mu}$
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\begin{block}{Conservation laws}
$$T^{\mu \nu}_{\ \ \ ; \nu} = 0$$
\end{block}
\frametitle{MHD equations}
{\small Relativistic MHD equations are reduced 
to\footnote{\tiny A. Brandenburg, K. Enqvist, and P. 
Olesen, {\em Phys. Rev. D} 
{\bf 54}, 1291 (1996)}
\begin{block}{MHD equations}
$$\frac{\partial \ln \rho}{\partial t} = - \red{\frac{4}{3}}
\left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) +
\red{\frac{1}{\rho c^2} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta \pmb{J}^2 \right]} $$
{\footnotesize $$\frac{D \pmb{u}}{Dt} = 
\red{\frac{1}{3} {\bf u} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right)} -
\red{\frac{\pmb{u}}{\rho c^2} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta J^2 \right]}
- \red{\frac{1}{4}} c^2 \pmb{\nabla} \ln \rho +  \red{\frac{3}{4 \blue{\rho}}} \blue{\pmb{J} \times \pmb{B}} +
\frac{2}{\rho} \pmb{\nabla} \cdot \left( \rho \nu \pmb{S} \right) $$}
\end{block}
for a flat expanding universe with comoving and normalized $p = a^4 p_{\rm phys}, 
\rho = a^4 \rho_{\rm phys}, B_i = a^2 B_{i, _{\rm phys}}, u_i$, and 
conformal time $t$.}
\end{frame}

\begin{frame}
\frametitle{MHD equations}
Electromagnetic fields are obtained from Faraday tensor as
{\scriptsize
$${\bf B} = {\bf \nabla} \times {\bf A}, \quad
{\bf E} = - {\bf \nabla} \phi - \frac{\partial
{\bf A}}{\partial t}$$}
\vspace{-4mm}
\begin{block}{Generalized Ohm's law}
$${\bf E} = \eta {\bf J} - {\bf u}
\times {\bf B}$$
\end{block}
\begin{block}{Maxwell equations}
\begin{columns}[c]
\column{.5\textwidth}
$${\bf \nabla} \cdot {\bf E} =
\rho_{\rm e} c^2,$$
$${\bf \nabla} \times {\bf B} =
{\bf J} + \red{\xcancel{\frac{1}{c^2} \frac{\partial {\bf E}}
{\partial t}}}$$
\column{.5\textwidth}
$$\nabla \cdot {\bf B} = 0$$
$$\frac{\partial {\bf B}}{\partial t} = -
{\bf \nabla} \times {\bf E}$$
\end{columns}
\end{block}
Maxwell equations + Ohm's law combined:
$$\blue{\frac{\partial \pmb{B}}{\partial t} = \pmb{\nabla} \times \left( \pmb{u} \times \pmb{B} -
\eta \pmb{J} \right)}$$
\end{frame}
%
% SLIDES BELOW MOVED TO EXTRA SLIDES
%
%\begin{frame}
%\frametitle{MHD energy density}
%\small
%\begin{columns}[t]
%\column{.5\textwidth}
%\blue{
%\textbf{Kinetic energy density: }
%$$ \mathcal{E}_{\text{K}}(t) = \frac{1}{2} \left< \rho \pmb{u}^2 \right> = \int E_{\text{K}}(k, t) dk $$ \\
%\textbf{Kinetic helicity: }
%$$ \mathcal{H}_{\text{K}}(t) = \rho \left< \pmb{u} \cdot \pmb{\omega} \right> = \int H_{\text{K}} (k, t) dk $$ \\
%\textbf{Kinetic correlation length: }
%$$ \xi_{\text{K}}(t) =  \frac{\int dk k^{-1} E_{\text{K}}}{\mathcal{E}_{\text{K}}} $$
%}
%\column{.5\textwidth}
%\red{
%\textbf{Magnetic energy density: }
%$$ \mathcal{E}_{\text{M}}(t) = \frac{1}{2 } \left< \pmb{B}^2 \right> = \int E_{\text{M}}(k, t) dk $$
%\textbf{Magnetic helicity: }
%$$ \mathcal{H}_{\text{M}}(t) = \left< \pmb{B} \cdot \pmb{A} \right> = \int H_{\text{M}}(k, t) dk $$
%\textbf{Magnetic correlation length: }
%$$ \xi_{\text{M}}(t) =  \frac{\int dk k^{-1} E_{\text{M}}}{\mathcal{E}_{\text{M}}} $$
%}
%\end{columns}
%\vspace{2mm}
%\textbf{Realizability condition: }
%$$ \frac{1}{2} k \left| H_{\rm M}
%\right| \Rightarrow \green{\xi_{\rm M} \geq
%\frac{{\cal H}_{\rm M}}{2{\cal E}_{\rm M}}}$$
%\normalsize
%\end{frame}
%
%\begin{frame}
%\frametitle{Initial conditions}
%\small
%\begin{block}{Fractional helicity $\sigma_{\rm M}$
%and peak frequency $k_{\rm M}$}
%%\begin{columns}[t]
%%%\small
%%\column{.5\textwidth} 
%%\vspace{-8mm}
%%$$\varepsilon_{\text{M}} =
%%\frac{k H_{\text{M}}(k, t)}{2 E_{\text{M}}(k, t)} =
%%\frac{2 \sigma_{\text{M}}}{1 + \sigma_{\text{M}}^2}$$
%%\column{.5\textwidth}
%\centering
%\includegraphics[width=.8\textwidth]{km}
%%\end{columns}
%\end{block}
%
%\begin{block}{Ratio of energy densities}
%$$ Q_* = \frac{B_*}{\sqrt{\rho_*} u_*}, \quad
%\text{where $t_*$ is the time of generation}$$
%Three cases:
%\begin{itemize}
%\item $Q_* \gg 1$:  Magnetically dominant turbulence
%$\left( E_{\text{M}} \propto k^{\alpha_{\text{M}}} \right.$,
%$\left. \alpha_{\text{M}} \geq 4 \right)$ 
%\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}
%\normalsize
%\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 ${\cal E}_i = \int E_i dk$, 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}
%\vspace{3mm}
%\includegraphics[width=.7\textwidth]{inverse_cascade}
%\end{center}
%\end{frame}
%
%\begin{frame}
%\frametitle{Numerical simulations\footnotetext{
%\scriptsize A. Brandenburg, {\em et al.} 
%{\em Phys. Rev. D} {\bf 96}, 123528 (2017)}}
%{\bf Case 1.} Initial non--helical ($\sigma_{\rm M} = 0$)
%Batchelor spectrum ($\alpha_i = 4$)
%\vspace{3mm}
%\begin{center}
%\includegraphics[width=.8\textwidth]{case1}
%\end{center}
%\end{frame}
%
%\begin{frame}
%\frametitle{Numerical simulations}
%{\bf Case 2.} Initial helical ($\sigma_{\rm M} = 0.3$)
%white noise spectrum ($\alpha_i = 2$)
%\begin{center}
%\includegraphics[width=\textwidth]{case2}
%\end{center}
%{\bf Realizability condition:}
%$$\frac{1}{2}k|H_{\rm M} \leq E_M \Rightarrow
%\green{\xi_{\rm M} \geq \frac{{\cal H}_{\rm M}}
%{2 {\cal E}_{\rm M}} = \xi_{\rm M}^{\rm min}}$$
%\end{frame}
%
%\begin{frame}
%\frametitle{Numerical simulations}
%{\bf Case 3.} Initial fully helical ($\sigma_{\rm K} = 0$)
%Batchelor kinetic spectrum ($\alpha_{\rm K} = 4$)
%\vspace{3mm}
%\begin{center}
%\includegraphics[width=.9\textwidth]{case3}
%\end{center}
%\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 inetically dominated turbulence, $pq$ diagram tends to
%$\beta = 4$ and conservation
%of Loitsiansky integral dominates
%\end{itemize}
%\end{block}
%\begin{itemize}
%\tiny
%\item Total helicity 
%$$ \langle \pmb{\omega} \cdot \pmb{u}\rangle +
%\langle \pmb{J} \cdot \pmb{B} \rangle $$
%\item Saffman integral
%$$ \mathcal{S} = \int \langle\pmb{u}(\pmb{x}, t) \cdot
%\pmb{u}(\pmb{x} + \pmb{r}, t) \rangle d\pmb{r} $$
%\item Loitsianski integral
%$$ \mathcal{L} = \int r^2 \langle \pmb{u}(\pmb{x}, t)
%\cdot \pmb{u}(\pmb{x} + \pmb{r}, t) \rangle d\pmb{r} $$
%\end{itemize}
%\end{frame}

%%% SLIDES ABOVE MOVED TO EXTRA SLIDES

\begin{frame}
\frametitle{Evolution of magnetic strength and correlation length}
\includegraphics[width = .95\textwidth]{MHD_results/B_corrl}
\end{frame}

% % % % % % % % % % % % % % % % % % % % % % % %
\section{Gravitational waves} % % % % % % % % %
% % % % % % % % % % % % % % % % % % % % % % % %

\frame{\tableofcontents[currentsection]}

\begin{frame}
\frametitle{Gravitational waves equation}
\begin{block}{\bf GWs equation for an expanding 
flat Universe}
\begin{itemize}
\item Assumptions: isotropic and homogeneous Universe
\item Friedmann--Lema\^itre--Robertson--Walker (FLRW) 
metric $\gamma_{ij} = a^2 \delta_{ij}$
\item Tensor-mode perturbations above the FLRW model:
$$g_{ij} = a^2 \left( \delta_{ij} + 
h_{ij}^{\rm phys} \right)$$
\item GWs equation is\footnotemark
\vspace{-4mm}
$$ \Bigl(\partial_t^2  - \red{\xcancel{a''\over a}} - c^2 \nabla^2 \Bigr)
% \Bigl(\partial_t^2 - c^2 \nabla^2 \Bigr)
h_{ij} = \frac{16 \pi G}{\red{\bf a} c^2} T_{ij}^{\rm TT} $$
\vspace{-4mm}
\begin{itemize}
\item $h_{ij}$ are rescaled $h_{ij} = a 
h_{ij}^{\rm phys}$
\item Comoving spatial coordinates $\nabla = a \nabla^{\rm phys}$ 
\item Conformal time d$t = a$ d$t^{\rm phys}$
\item Comoving stress-energy tensor components 
$T_{ij} = a^4 T_{ij}^{\rm phys}$
\item Radiation-dominated epoch such that 
$a'' = 0$ 
\end{itemize}
\end{itemize}
\end{block}
\footnotetext{\tiny L. P. Grishchuk, {\em 
Sov. Phys. JETP}, 40, 409-415 (1974)}
\end{frame}

\begin{frame}
\begin{block}{Normalized GW equation\footnotemark}
\vspace{5mm}
\blue{\bf
$$\Bigl(\partial_t^2 - \nabla^2 \Bigr) h_{ij} = 6 T_{ij}^{\rm TT}/t$$}
\vspace{2mm}
\end{block}
\begin{block}{Properties}
\begin{itemize}
\item All variables are normalized and non-dimensional
\item Conformal time is normalized with $t_\ast$
\item Comoving coordinates are normalized with $c/H_\ast$
\item Stress-energy tensor is normalized with ${\cal E}_{\rm rad}^\ast = 
3 H_\ast^2 c^2/(8 \pi G)$
\item Scale factor is $a_\ast = 1$, such that $a = t$
\end{itemize}
\end{block}
\footnotetext{\tiny A. 
Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.} 
{\bf 114}, 130.\\ \hspace{7mm} arXiv:1807.05479 (2020)}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves equation}
\begin{block}{Properties}
\begin{itemize}
\item Tensor-mode perturbations are gauge invariant
\item $h_{ij}$ has only two degrees of freedom: $h^+$, $h^{\times}$
\item The metric tensor is traceless and transverse (TT gauge)
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Linear polarization modes $+$ and $\times$}
\begin{block}{Linear polarization basis (defined in Fourier space)}
$$ e_{ij}^+ = \left( \pmb{e}_1 \times \pmb{e}_1 - \pmb{e}_2 \times \pmb{e}_2\right)_{ij} $$
$$ e_{ij}^{\times} = \left(\pmb{e}_1 \times \pmb{e}_2 + \pmb{e}_2 \times \pmb{e}_1 \right)_{ij} $$
\end{block}
\begin{block}{Orthogonality property}
$ e_{ij}^A e_{ij}^B = 2 \delta_{AB} $, where $A, B = +, \times$
\end{block}
\begin{block}{$+$ and $\times$ modes}
$$ \tilde{h}^+ = \frac{1}{2} e_{ij}^+ \tilde{h}^{\text{TT}}_{ij},
\quad \quad  \tilde{T}^+ = \frac{1}{2} e_{ij}^+ \tilde{T}^{\text{TT}}_{ij}$$
$$ \tilde{h}^{\times} = \frac{1}{2} e_{ij}^{\times} \tilde{h}^{\text{TT}}_{ij},
\quad \quad  \tilde{T}^\times = \frac{1}{2} e_{ij}^\times \tilde{T}^{\text{TT}}_{ij}$$
\end{block}
\end{frame}

\begin{frame}
\frametitle{Numerical considerations}
\begin{itemize}
\item CFL condition for stability:
$$\delta_t \leq C_{\rm CFL} \delta x/U_{\rm eff},$$
$U_{\rm eff} = |\pmb{u}| + (c_{\rm s}^2 + v_{\rm A}^2)^{1/2}$,
$c_{\rm s}^2 = c^2/3$, 
$v_{\rm A}^2 = B^2/\rho$. 
\vspace{5mm}
\item Projection of $T_{ij}^{\rm TT}$ requires non-local Fourier 
transform $\tilde{T}_{ij}$:
\red{
$$ \tilde{T}^{\rm TT}_{ij} = \left( P_{il} P_{jm} -
\frac{1}{2} P_{ij} P_{lm} \right) \tilde{T}_{lm} $$}
where $P_{ij} = \delta_{ij} - \hat{k}_i \hat{k}_j$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Method 1}
\begin{itemize}
\item Solve the GWs equation 
sourced by the stress-energy tensor\footnote{\tiny A. 
Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.} \\
\hspace{7mm} {\bf 114}, 130. arXiv:1807.05479 (2020)}
\red{$$ \left( \partial_t^2  - \nabla^2 \right) 
h_{ij} = 6 T_{ij}/t $$}
\item Evolution of 6 components in physical space
\item Project $h_{ij}^{\rm TT}$ only when we are interested in 
spectra
\red{
$$ \tilde{h}^{\rm TT}_{ij} = \left( P_{il} P_{jm} -
\frac{1}{2} P_{ij} P_{lm} \right) \tilde{h}_{lm} $$}
\item Compute $\tilde{h}^+$, $\tilde{h}^\times$ modes
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves energy density}
{\large \bf GWs energy density:\\}
{\scriptsize
$$\Omega_{\rm GW} (t) = {\cal E}_{\rm GW}/{\cal E}_{\rm rad}^*, \quad 
{\cal E}_{\rm rad}^* = \frac{3H_\ast^2 c^2}{8 \pi G}$$
$$ \Omega_{\rm GW} (t) = \int_{-\infty}^\infty 
\Omega_{\rm GW} (k, t) {\rm \, d} \ln k $$
$$\red{\bf \Omega_{\rm GW} (k, t)} = \frac{k}{6 H_\ast^2} 
\int_{4 \pi} \left(\left| \dot{\tilde{h}}^{\rm phys}_+ 
\right|^2 + \left| \dot{\tilde{h}}^{\rm phys}_\times \right|^2 \right)
k^2 {\rm \, d} \Omega_k$$}


{\large \bf Magnetic/kinetic energy density:\\}
\scriptsize{
$$\Omega_{\rm M, K} (t) = \int_{-\infty}^\infty 
\Omega_{\rm M, K} (k, t) {\rm \, d} \ln k $$
$$\red{\bf \Omega_{\rm M} (k, t)} = \frac{k}{2}
\int_{4 \pi} \left(\tilde{\bf B} \cdot \tilde{\bf B}^\ast\right)
k^2 {\rm \, d} \Omega_k, \quad
\red{\bf \Omega_{\rm K} (k, t)} = \frac{k}{2}
\int_{4 \pi} \rho\left(\tilde{\bf u} \cdot
\tilde{\bf u}^\ast\right) k^2 {\rm \, d} \Omega_k
$$
}
%}
%{\large \bf Antisymmetric GWs energy density:}
%\small
%$$ \mathcal{H}_{\text{GW}} (t) = \int_0^\infty 
%H_{\rm GW} (k, t) {\rm \, d} k = \frac{c^2}{16 \pi G} 
%\int_0^\infty \int_{4 \pi} 2 {\rm Im} \left(
%\tilde{\dot{h}}^+ \tilde{\dot{h}}^\times \right) 
%k^2 {\rm \, d} \Omega_k {\rm \, d} k$$
%{\large \bf Polarization degree: }
%$$ \red{\mathcal{P}(k) = \frac{H_{\rm GW}(k)}
%{E_{\rm GW}(k)}} \in [-1, 1]$$
%{\footnotesize
%$$ \Xi_{\rm GW} (t) = \int_{-\infty}^\infty 
%\Xi_{\rm GW} (k, t) {\rm \, d} \ln k $$
%$$\red{\bf \Xi_{\rm GW} (k, t)} = \frac{k}{6 H_\ast^2} 
%\int_{4 \pi} 2 {\rm Im} \left( \dot{\tilde{h}}^{\rm phys}_+ 
%\dot{\tilde{h}}^{{\rm phys}, \ast}_\times \right)
%k^2 {\rm \, d} \Omega_k$$}
\end{frame}

\begin{frame}
\frametitle{Numerical accuracy\footnote{\tiny A. 
Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.} \\
\hspace{7mm} {\bf 114}, 130. arXiv:1807.05479 (2020)}}
\begin{columns}[t]
\column{.6\textwidth}
\includegraphics[width=1.05\textwidth]{accuracy}
\column{.4\textwidth}
\vspace{-60mm}
\begin{itemize}
\item CFL condition is not enough for GW solution 
to be numerically accurate
\item $c \delta t/\delta x \sim 0.05 \ll 1$
\item Higher resolution is required
\item Hydromagnetic turbulence does not seem to 
be affected
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Method 2}
\begin{itemize}
\item Compute Fourier transform of stress-energy tensor $\tilde{T}_{ij}$
\item Project into TT gauge $\tilde{T}_{ij}^{\rm TT} = 
\left(P_{il} P_{jm} - \frac{1}{2} P_{ij} P_{lm}\right) \tilde{T}_{lm}^{\rm TT}$
\item Compute $\tilde{T}^+$ and $\tilde{T}^\times$ modes
\item Discretize time using $\delta t$ from MHD simulations (CFL condition)
\item Assume $\tilde{T}^{+, \times}/t$ to be constant between 
subsequent timesteps \\ (robust as $\delta t \rightarrow 0$)
\item GW equation solved analytically between subsequent timesteps in 
Fourier space\footnote{\tiny A. 
Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.}, 
{\bf 114}, 130 \\ \hspace{7mm} arXiv:1807.05479 (2020)}\\
{\footnotesize
$$\left( \begin{array}{c}
\omega \tilde{h} - 6 \omega^{-1} \tilde{T}/t \\ 
\tilde{h}'
\end{array} \right)^{t + \delta t}_{+, \times} = 
\left(\begin{array}{c c}
\cos \omega \delta t 	& \sin \omega \delta t \\ 
- \sin \omega \delta t & \cos \omega \delta t
\end{array} \right) \left( \begin{array}{c}
\omega \tilde{h} - 6 \omega^{-1} \tilde{T}/t \\ 
\tilde{h}'
\end{array}\right)^t_{+, \times}$$}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Frequency of oscillations of GWs vs MHD waves}
\includegraphics[width=\textwidth]{frequency_scales}
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence\footnotemark}
\begin{block}{Initial conditions}
\begin{itemize}
\item Fully helical stochastic magnetic field
\item Batchelor spectrum, i.e., $E_{\rm M} \propto k^4$ for small $k$
\item Kolmogorov spectrum for inertial range, i.e., $E_{\rm M} \propto k^{-5/3}$
\item Total energy density at $t_*$ is $\sim $ 10\% to the radiation energy density
\item Spectral peak at $k_{\rm M} = 100 \cdot 2 \pi$, normalized with $k_H = 1/(cH)$ 
\end{itemize}
\end{block}
\begin{block}{Numerical parameters}
\begin{itemize}
\item 1152$^3$ mesh gridpoints
\item 1152 processors
\item Wall-clock time of runs is $\sim$\ 1 -- 5 days
\end{itemize}
\end{block}
\footnotetext[9]{{\scriptsize A. Brandenburg, {\em et al.} 
{\em Phys. Rev. D} {\bf 96}, 123528 (2017),\\
\hspace{5mm} A. Roper Pol, {\em et al.} 
arXiv:1903.08585 (2020)}}
\end{frame}

\begin{frame}
\frametitle{Initial magnetic spectra}
\begin{center}
{\footnotesize $ k_{\rm M} = 15 $} \\
\includegraphics[width=\textwidth]{EM_ET}
\end{center}
\vspace{-8mm}
\begin{columns}
\column{.5\textwidth}
{\scriptsize
$$ E_{\rm M} (k) = \frac{1}{2} \int_{4\pi} \Bigl( \tilde{\bf B} (k) \cdot
\tilde{\bf B}^* (k) \Bigr) k^2 d \Omega$$
$$\Omega_{\rm M} = \int_0^\infty E_{\rm M} (k) dk$$}
\column{.5\textwidth}
\vspace{5mm}
{\scriptsize
$$ E_{\rm T} (k) = \int_{4\pi} \Bigl( \tilde{\bf T}_{ij}
(k) \tilde{\bf T}_{ij}^* (k) \Bigr) k^2 d \Omega$$}
{\scriptsize
$$ E_{\rm T} (k) = \frac{1}{2} \int_{4\pi} \Bigl( \tilde{\bf B}^2
(k) \tilde{\bf B}^{2, \ast} (k) \Bigr) k^2 d \Omega$$
$$\Omega_{\rm T}  = \int_0^\infty E_{\rm T} (k) dk$$}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\includegraphics[width = .9\textwidth]{M1152e_exp6k4}
\end{frame}

% Box simulations commented out (moved to extra slides)
%
%\begin{frame}
%\frametitle{Numerical results for decaying MHD turbulence}
%Box results for positive initial helicity:
%\begin{center}
%\includegraphics[width = 0.32\textwidth]{GW_results/M1152c3_hhT} 
%\includegraphics[width = 0.32\textwidth]{GW_results/M1152g3_hhT} 
%\includegraphics[width = 0.32\textwidth]{GW_results/M1152f3_hhT} \\
%\includegraphics[width = 0.32\textwidth]{GW_results/M1152c3_hhX}
%\includegraphics[width = 0.32\textwidth]{GW_results/M1152g3_hhX}
%\includegraphics[width = 0.32\textwidth]{GW_results/M1152f3_hhX}
%\vspace{-4mm}
%\end{center}
%\begin{columns}[c]
%\column{.32\textwidth} \hspace{3mm} \tiny $k_{\text{M}}/k_H = 300$
%\column{.32\textwidth} \hspace{3mm} \tiny $k_{\text{M}}/k_H = 60$
%\column{.32\textwidth} \hspace{3mm} \tiny $k_{\text{M}}/k_H = 2$
%\end{columns}
%\end{frame}
%
% Box simulations commented out (moved to extra slides)
%
% Definition of polarization moved to extra slides
%
%\begin{frame}
%\frametitle{Antisymmetric and helical spectra}
%{\large \bf Antisymmetric GWs energy density:}
%{\scriptsize
%$$ \Xi_{\rm GW} (t) = \int_{-\infty}^\infty 
%\Xi_{\rm GW} (k, t) {\rm \, d} \ln k $$
%$$\red{\bf \Xi_{\rm GW} (k, t)} = \frac{k}{6 H_\ast^2} 
%\int_{4 \pi} 2 {\rm Im} \left( \dot{\tilde{h}}^{\rm phys}_+ 
%\dot{\tilde{h}}^{{\rm phys}, \ast}_\times \right)
%k^2 {\rm \, d} \Omega_k$$}
%
%{\large \bf Polarization degree: }
%{\scriptsize
%$$ \red{\mathcal{P}(k) = \frac{\Xi_{\rm GW}(k)}
%{\Omega_{\rm GW}(k)}} \in [-1, 1]$$}
%
%{\large \bf Magnetic/kinetic helical density: }
%{\scriptsize
%$$\Xi_{\rm M} = \int_{-\infty}^\infty 
%\Xi_{\rm M, K} (k, t) {\rm \, d} \ln k $$
%$$\red{\bf \Xi_{\rm M} (k, t)} = \frac{k}{2}
%\int_{4 \pi} \left(\tilde{\bf B} \cdot \tilde{\bf A}^\ast\right)
%k^2 {\rm \, d} \Omega_k, \quad
%\red{\bf \Omega_{\rm K} (k, t)} = \frac{k}{2}
%\int_{4 \pi} \rho\left(\tilde{\bf u} \cdot
%\tilde{\bf \omega}^\ast\right) k^2 {\rm \, d} \Omega_k$$}
%\end{frame}
%
% Definition of polarization moved to extra slides

\begin{frame}
\frametitle{Comments}
\begin{itemize}
\item \red{\bf Novel $k^0$ scaling in the subinertial
range}
\end{itemize}
\begin{block}{Polarization degree}
\begin{itemize}
\item Helical magnetic fields induce circularly 
polarized GWs
\item Helicity and GWs polarization have same sign
\item Agreement with analytical 
prediction\footnotemark
\item Illustrated with a 1D Beltrami field example
\end{itemize}
\end{block}
%\begin{block}{Box simulations}
%\begin{itemize}
%\item For smaller $k_{\rm M}$, $h^+$ and 
%$h^\times$ field structures become smoother
%\item $h^+$ and $h^\times$ appear shifted, which 
%indicates presence of helicity
%\end{itemize}
%\end{block}
\footnotetext{
T. Kahniashvili {\em et al., Phys. Rev. Lett.,}\\
\hspace{7mm} {\bf 95} (15):151301 (2005)}
\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:
%$$ \tilde{\mathcal{E}}_{\text{GW}} = \frac{\pi G}{c^4 k_0^2	} \mathcal{E}_{\text{M}}^2 \left[ 
%\delta (k - 2k_0) + \delta(k + 2k_0) \right] $$ 
%$$ \tilde{\mathcal{H}}_{\text{GW}} = \frac{\pi G}{c^4 k_0^2	} \mathcal{E}_{\text{M}}^2 \left[ 
%\delta (k - 2k_0) - \delta(k + 2k_0) \right] $$ 
%\item Polarization degree:
%$$\mathcal{P}(2k_0) = \text{sgn} \left( k_0 \right) $$
%\item We see that polarization degree is $\pm 1$, with the same sign as the sign of the magnetic helicity
%\end{itemize}
%\end{frame}
%
% Second slide about Beltrami field moved to extra slides

\begin{frame}
\frametitle{Early time evolution of GW energy density spectral slope}
\begin{columns}
\column{.5\textwidth}
\includegraphics[width=\textwidth]{fig3}
\column{.5\textwidth}
\includegraphics[width=\textwidth]{fig4}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Signal at the present time}
\begin{block}{Characteristic amplitude of GWs}
{\scriptsize
$$h_{\rm c}^2 (t) =  \int_{-\infty}^\infty h_{\rm c}^2 (k, t) {\rm \, d} \ln k,
\quad \red{\bf h_{\rm c}^2 (k, t)} =  \int_{4\pi} \left(\left|\tilde{h}_+^{\rm phys}\right|^2 + 
\left|\tilde{h}_\times^{\rm phys}\right|^2 \right) k^2 {\rm \, d} \Omega_k$$}
\end{block}
\begin{block}{GW energy density and characteristic amplitude}
\begin{itemize}
\item Shifting due to the expansion of the universe:
\vspace{2mm}
\begin{itemize}
\item $\Omega_{\rm GW}^0 (k) = a_0^{-4} (H_\ast/H_0)^2 \Omega_{\rm GW} (k, t_{\rm end})$
\vspace{2mm}
\item $h_{\rm c}^0 (k) = a_0^{-1} h_{\rm c} (k, t_{\rm end})$
\vspace{2mm}
\item $f = a_0^{-1} H_\ast k/(2 \pi)$
\end{itemize}
\end{itemize}
\end{block}
\vspace{-4mm}
$$a_0 \approx 1.254 \cdot 10^{15} \left(T_\ast/ 100 {\rm \ GeV} \right) 
\left(g_{\rm S}/100\right)$$
$$H_\ast \approx 2.066 \cdot 10^{-11} \ {\rm s}^{-1}
\left(T_\ast/ 100 {\rm \ GeV} \right)^2
\left(g_\ast/100\right)^{1/2}$$
$$H_0 = 100 h_0 {\rm \ km s}^{-1} {\rm \ Mpc}^{-1}$$
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\begin{columns}
\column{.6\textwidth}
\centering
\includegraphics[width = \textwidth]{fig1}
\column{.4\textwidth}
\includegraphics[width=\textwidth]{tableini}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Time evolution of GW energy density}
\centering
\includegraphics[width=.6\textwidth]{fig5}
\end{frame}

\begin{frame}
\frametitle{Forced turbulence (built-up primordial magnetic fields and 
hydrodynamic turbulence)}
\begin{columns}
\column{.6\textwidth}
\centering
\includegraphics[width=\textwidth]{fig2}
\column{.4\textwidth}
\includegraphics[width=\textwidth]{table}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Forced turbulence (built-up primordial magnetic fields and 
hydrodynamic turbulence)}
\begin{block}{Conclusions}
\begin{itemize}
\item Much stronger signal found for acoustic 
than for rotational turbulence
\item Result in agreement with 
literature\footnotemark
\item In both cases, GWs polarization is zero
\item \red{\bf Novel $k^0$ scaling in the subinertial range}
\item Smooth bump for acoustic runs around the spectral peak
\item Steeper than Kolmogorov GW spectra in the inertial range
\end{itemize}
\end{block}
\footnotetext{M. Hindmarsh {\em et al., Phys. Rev. D,}\\
\hspace{7mm} {\bf 92} 12:123009 (2015)}
\end{frame}

\begin{frame}
\frametitle{Conclusions}
\begin{itemize}
\item We have implemented a module within the open-source {\sc Pencil Code} that allows to 
obtain background stochastic GW spectra from primordial magnetic fields and 
hydrodynamic turbulence.
\item For some of our simulations we obtain a detectable signal by future 
GW detector LISA.
%\item LISA data analysis require models for different contributions to the
%GW background for extraction of single sources.
\item GW equation is normalized such that it can be easily scaled for different 
times within the radiation-dominated epoch.
\item Novel $f$ spectrum obtained for GWs in high frequencies range vs $f^3$ obtained 
from analytical estimates
\item Bubble nucleation and magnetogenesis physics can be coupled to our 
equations for more realistic production analysis.
\end{itemize}
\end{frame}

\begin{frame}
\begin{columns}
\column{.2\textwidth}
\centering
\includegraphics[width=.7\textwidth]{nsf_logo}\\
\vspace{5mm}
\includegraphics[width=1.3\textwidth]{gnsf_logo}
\column{.5\textwidth}
\centering
\Huge{The End \\ Thank You! \\}
\includegraphics[width=.8\textwidth]{question} \\
\blue{\bf \small{alberto.roperpol@colorado.edu}}
\column{.2\textwidth}
\centering
\includegraphics[width=\textwidth]{lasp}\\
\vspace{5mm}
\includegraphics[width=\textwidth]{nordita_logo}\\
\vspace{5mm}
\includegraphics[width=.7\textwidth]{ilia_state}
\end{columns}
\end{frame}

\end{document}

