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\title[Generation of Gravitational Waves]
{Generation of Gravitational Waves due to Magnetohydrodynamic Turbulence in the Early Universe}
\subtitle{Comprehensive Exam}
\author[Alberto Roper Pol]{
Alberto Roper Pol \\
\textit{Research Advisor: }Axel Brandenburg
}
\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{ \today}


\begin{document}

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\titlepage % Print the title page as the first slide
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\frametitle{Overview} % Table of contents slide, comment this block out to remove it
\tableofcontents % Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
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%------------------------------------------------
\section{Introduction and Motivation} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
%------------------------------------------------

% \subsection{Subsection Example} % A subsection can be created just before a set of slides with a common theme to further break down your presentation into chunks

\begin{frame}
\frametitle{Introduction and Motivation}
\begin{itemize}
\item \visible<1-> { Generation of cosmological GWs during the EWPT
\item Source of information from earlier times than recombination epoch }
\item \visible<2-> { MHD sources of GWs: 
	\begin{itemize}
	\item Kinetic turbulence from phase transition bubbles nucleation
	\item Magnetic turbulence from primordial magnetic fields
	\end{itemize} }
\item \visible<3-> {Evolution of primordial magnetic fields
	\begin{itemize}
	\item Comparion to Fermi blazar observations
	\item Helical magnetic fields require first order phase transitions
	\end{itemize} }
\item \visible<4-> {Possibility of GWs detection with:
	\begin{itemize}
	\item Space mission LISA
	\item Polarization of CMB
	\item Pulsar Timing Arrays (PTA)
	\end{itemize}	}
\item \visible<5-> {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}

\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{L. M. Widrow. Origin of galactic and extragalactic magnetic fields. 
\textit{Reviews of Modern Physics},
74(3):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{A. M. Taylor, I. Vovk, and A. Neronov. 
Extragalactic magnetic fields constraints from simultaneous GeV--TeV observations of blazars. 
\textit{ Astronomy \& Astrophysics}, 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{Magnetohydrodynamic turbulence}
%------------------------------------------------

\begin{frame}
\frametitle{EWPT plasma}
\begin{block}{Characterization}
\begin{itemize}
\item \visible<1-> { Quark-gluon plasma }
\item \visible<2-> {  Charge-neutral, electrically conducting fluid }
\item \visible<3-> { Relativistic magnetohydrodynamic (MHD) equations }
\item \visible<4-> { Ultrarelativistic equation of state}
\end{itemize}
\end{block}
\visible<4-> {
\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}
Relevant quantities for MHD equations are:
\begin{block}{Stress-energy tensor $T^{\mu \nu}$}
$T^{\mu \nu} = ( p + \rho) U^{\mu} U^{\nu} + p g^{\mu \nu} + \frac{1}{4 \pi} \left( F^{\mu \sigma} F^{\nu}_\sigma - \frac{1}{4} g^{\mu \nu} F_{\lambda \sigma} F^{\lambda \sigma} \right) $
\end{block}
\begin{block}{Faraday tensor}
$F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$
\end{block}
\begin{block}{Faraday tensor in terms of electromagnetic fields}
$ F_{0i} = E_i/c, \quad F_{i0} = - E_i/c, \quad F_{ij} = -\varepsilon_{ijk} B_k $
\end{block}
\begin{block}{4D vectors}
$U^{\mu} = \left( \gamma, \gamma u^i\right), \quad A^{\mu} = \left( \phi, A^i \right)$
\end{block}
\footnotetext{Natural units are used, i.e., $c = 1$}
\end{frame}

\begin{frame}
\frametitle{MHD equations}
\begin{block}{Fluid conservation laws}
\centering
$T^{\mu \nu}_{;\nu} = 0$
\end{block}
\begin{block}{Equation of state}
\centering
$p = \rho /3 $
\end{block}
\begin{block}{Maxwell equations}
\centering
$ F^{\nu \mu}_{;\nu } = \mu_0 J^{\nu} , \quad 
\varepsilon^{\mu \nu \lambda \gamma} F_{\lambda \gamma} = 0$
\end{block}
\begin{block}{Generalized Ohm's law}
\centering
$ \pmb{E} + \pmb{u} \times \pmb{B} = \pmb{J}/\sigma $
\end{block}
\end{frame}

\begin{frame}
\frametitle{MHD equations}
Relativistic MHD equations are reduced to\footnote{A. Brandenburg, K. Enqvist, and P. Olesen. 
Large-scale magnetic fields from hydromagnetic turbulence in the very early universe. 
\textit{Physical Review 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}
\end{frame}

\begin{frame}
\frametitle{MHD energy density}
\small
\begin{columns}[t]
\column{.5\textwidth}
\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}
\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: }
$$ \frac{1}{2} k \left| H_i \right|  \leq E_i, \quad i = \text{K}, \text{M} $$
\footnotetext[4]{Lorentz-Heaviside units are used}
\normalsize
\end{frame}

\begin{frame}
\frametitle{Initial conditions}
\begin{block}{Fractional helicity $\sigma_i$}
\begin{columns}[t]
\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}
$$ \varepsilon_{\text{K}} = \frac{k H_{\text{K}}(k, t)}{2 E_{\text{K}}(k, t)} = \frac{2 \sigma_{\text{K}}}{1 + \sigma_{\text{K}}^2} $$
\end{columns}
\normalsize
\end{block}

\begin{block}{Ratio of energy densities}
$$ Q_* = \frac{B_*}{\sqrt{\rho_*} u_*}$$
Three cases:
\begin{itemize}
\item $Q_* \gg 1$:  Magnetically dominated 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 subdominated or kinetically dominated turbulence
$\left( E_{\text{K}} \propto k^{\alpha_{\text{K}}} \right.$, $\left. \alpha_{\text{K}} \geq 2 \right)$ 
\end{itemize}
\end{block}
\footnotetext[5]{$t_*$ is the time of generation, where the initial conditions are defined}
\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}{Inverse cascade condition}
$$ s_i = \frac{\partial \ln E_i}{\partial \ln t} = \left( \alpha_i - \beta_i \right) q_i > 0 $$
\end{block}

Since $\mathcal{E}_i = \int E_i dk$, we have the relation
$$p_i = \left( 1 + \beta_i \right) q_i$$
\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_* = 1$ } \includegraphics[width=.8\textwidth]{MHD_results/B} \\
{\tiny $ Q_* = 0.1$ } \includegraphics[width=.8\textwidth]{MHD_results/C}
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\large \textbf{Case 2.} Initial non-helical ($\sigma_{\text{M}} = 0$) white noise spectrum ($\alpha_i = 2$) \\}
\centering
{\tiny $ Q_* = 1$ } \includegraphics[width=.84\textwidth]{MHD_results/D} \\
{\tiny $ Q_* = 1$ } \includegraphics[width=.82\textwidth]{MHD_results/E} \\
{\tiny $ Q_* \rightarrow \infty$ } \includegraphics[width=.8\textwidth]{MHD_results/F}
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\large \textbf{Case 3.} Initial helical ($\sigma_{\text{M}} = 0.3$) white noise spectrum ($\alpha_i = 2$) \\}
\centering
{\tiny $ Q_* \rightarrow \infty$ } \includegraphics[width=1\textwidth]{MHD_results/G} \\
\end{frame}

\begin{frame}
\frametitle{Numerical simulations}
{\normalsize \textbf{Case 4.} Initial fully kinetic helical ($\sigma_{\text{K}} = 1$) Batchelor spectrum ($\alpha_{\text{K}} = 4$) \\}
\centering
\includegraphics[width=.9\textwidth]{MHD_results/H} \\
\includegraphics[width=.9\textwidth]{MHD_results/I} \\
\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
\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 
$$ \left< \pmb{\omega} \cdot \pmb{u} \right> + \left< \pmb{A} \cdot \pmb{B} \right> $$
\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}
\includegraphics[width = .95\textwidth]{MHD_results/B_corrl}
\end{frame}

\section{Gravitational waves}

\begin{frame}
\small
\frametitle{Gravitational waves equation}
\begin{itemize}
\item GWs equation is obtained by linearization of Einstein field equations: \\
\vspace{2mm}
$ G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} $, \\
where $G^{\mu \nu}$ is the Einstein tensor
and $T^{\mu \nu}$ is the stress-energy tensor
\item Tensor-mode metric perturbations: $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$ \\
\item Background metric tensor $\eta_{\mu \nu}$ is the Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric, valid for 
cosmological applications
\item Gravitational waves equation: \\
$$ \frac{\partial^2 h_{ij}}{\partial t^2} + 2 \mathcal{H} \frac{\partial h_{ij}}{\partial t} - \frac{c^2}{a^2} \nabla^2 h_{ij} = \frac{16 \pi G}{a^2 c^2} T^{\text{TT}}_{ij}, $$
where $\mathcal{H} = a'/a$ is the comoving Hubble parameter, $T_{ij}^{\text{TT}}$ is the traceless and 
transverse projection of $T_{ij}$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves equation}
\begin{block}{Non-expanding universe GWs equation}
\begin{itemize}
\item Assumption: Sub-Hubble scales, i.e., $k t \ll 1$
\item GWs equation:
$$ \frac{\partial^2 h_{ij}}{\partial t^2}  - c^2 \nabla^2 h_{ij} = \frac{16 \pi G}{c^2} T_{ij}^{\text{TT}} $$
\end{itemize}
\end{block}
\begin{block}{Properties}
\begin{itemize}
\item Tensor-mode perturbations are gauge invariant
\item $h^{\mu \nu}$ has only two degrees of freedom: $h^+$, $h^{\times}$
\item The metric tensor is traceless and transverse (TT gauge)
\end{itemize}
\end{block}
\begin{block}{Contributions to the stress-energy tensor}
\begin{columns}[c]
\column{.5\textwidth}
\begin{itemize}
\item From fluid motions \\
$ T_{ij} = \left( p/c^2 + \rho \right) \gamma^2 u_i u_j + p \delta_{ij} $
\end{itemize}
\column{.5\textwidth}
\begin{itemize}
\item From magnetic fields: \\
$ T_{ij} = - B_i B_j + \frac{1}{2} \delta_{ij} B^2 $
\end{itemize}
\end{columns}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Traceless and transverse projection}
\begin{itemize}
\item For computational reasons, we solve GWs equation sourced by the stress-energy tensor
$$ \frac{\partial^2 h_{ij}}{\partial t^2}  - c^2 \nabla^2 h_{ij} = \frac{16 \pi G}{c^2} T_{ij} $$
\item Projection of $h_{ij}^{\text{TT}}$ only at specific times, not at every time-step
\item Projection requires Fourier projection (computationally expensive)
\item TT projection:
$$ \tilde{h}^{\text{TT}}_{ij} = \left( P_{il} P_{jm} - \frac{1}{2} P_{ij} P_{lm} \right) \tilde{h}_{lm} $$
where $P_{ij} = \delta_{ij} - \hat{k}_i \hat{k}_j$
\end{itemize}
\end{frame}

% UNCOMMENT TO SHOW h+ hx GIFs
\begin{frame}
\frametitle{Linear polarization modes $h^+$ and $h^{\times}$}
\transduration<0-20>{5}
\centering
\multiinclude[<+->][format=png, graphics={width=0.4\textwidth}]{GW+/GW+}
\multiinclude[<+->][format=png, graphics={width=0.4\textwidth}]{GWx/GWx}
\begin{columns}[c]
\column{.5\textwidth}
\hspace{5mm} $h^+$ mode of polarization
\column{.5\textwidth}
\hspace{5mm} $h^{\times}$ mode of polarization
\end{columns}
\end{frame}

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

\begin{frame}
\frametitle{Gravitational waves energy density}
{\large \bf GWs energy density:}
$$ \mathcal{E}_{\text{GW}} = \frac{c^2}{32 \pi G} \left< \left| \dot{\tilde{h}}^+ \right|^2 + \left| \dot{\tilde{h}}^{\times} \right|^2 \right>_k =  \int_0^{\infty} E_{\text{GW}} dk $$
{\large \bf Antisymmetric GWs energy density:}
$$ \mathcal{H}_{\text{GW}} = \frac{c^2}{32 \pi G} \left< 2 \text{Im} \left( \dot{\tilde{h}}^+ \dot{\tilde{h}}^{\times} \right) \right>_k = \int_0^{\infty} H_{\text{GW}} dk $$
{\large \bf Polarization degree: }
$$ \mathcal{P}(k) = \frac{H_{\text{GW}}(k)}{E_{\text{GW}}(k)} $$
{\large \bf Spectral amplitude $h_c$: }
$$ h_{\text{rms}}^2 = \left< \left| \tilde{h}^{+} \right|^2 + \left| \tilde{h}^{\times}\right|^2 \right>_k = \int_{0^+}^{\infty} h_c^2 d \log k $$
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\begin{block}{Initial conditions}
\begin{itemize}
\item Fully helical stochastic magnetic field
\item Batchelor spectrum, i.e., $E_{\text{M}}(t_*) \propto k^4$ for small $k$
\item Total energy density at $t_*$ is $\sim $ 10\% to the radiation energy density\footnotemark[6] 
\item Different positions of the spectral peak $k_{\text{M}}$, 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 Duration of runs is $\sim $1 -- 5 days
\end{itemize}
\end{block}
\footnotetext[6]{limit from the Big Bang Nucleosynthesis bound}
\end{frame}

% UNCOMMENT FOR GWs RESULTS
\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
{\bf Positive initial helicity} \\
\begin{center}
\includegraphics[width = 0.32\textwidth]{GW_results/M1152c3} 
\includegraphics[width = 0.32\textwidth]{GW_results/M1152g3} 
\includegraphics[width = 0.32\textwidth]{GW_results/M1152f3} 
\end{center}
\vspace{-4mm}
\begin{columns}[t]
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 300$$
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 60$$
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 2$$
\end{columns}
\vspace{3mm}
{\bf Negative initial helicity} \\
\begin{center}
\includegraphics[width = 0.6\textwidth]{GW_results/M1152e3} 
\vspace{-4mm}
\tiny $$ k_{\text{M}}/k_H = 6 $$
\end{center}
\end{frame}

\begin{frame}
\frametitle{Polarization degree}
\begin{itemize}
\item Helical magnetic fields induce circularly polarized GWs
\item Helicity and GWs polarization have same sign
\item Agreement with theoretical prediction \footnotemark[7]
\item Illustrated with a 1D Beltrami field example
\end{itemize}
\footnotetext[7]{T. Kahniashvili, G. Gogoberidze, and B. Ratra. 
Polarized Cosmological Gravitational Waves from Primordial Helical Turbulence. 
\textit{Physical Review Letters}, 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}

\begin{frame}
\frametitle{Linear polarization basis error}
\begin{block}{$e_{ij}^{\times}$ implementation}
\begin{itemize}
\item The test-case used is already TT and has only 2 degrees of freedom
\item If we apply definitions to obtain $T^+$ and $T^{\times}$ modes we get:
$$ T^+ = - \frac{1}{2} \cos 2k_0x, \quad T^{\times} = - \frac{1}{2} i \cos 2k_0 x \neq - \frac{1}{2} \sin 2k_0 x$$
\item Not the expected values and not independent
\item Error arises from $e_{ij}^{\times}(\pmb{k})$ being odd, i.e., $e_{ij}^{\times}(-\pmb{k}) = - e_{ij}^{\times} (\pmb{k})$
\item To solve this issue we have performed a numerical trick:
\begin{itemize}
\item Define half of the $\pmb{k}$ to be considered positive directions. This decision is arbitrary
\item Compute $e_{ij}^{\times} (\pmb{k})$ for only positive directions
\item Assign $e_{ij}^{\times} (-\pmb{k}) = e_{ij}^{\times} (\pmb{k})$ for negative directions
\end{itemize}
\item This leads to correct solution for 1D Beltrami field and to for more complex problems
\end{itemize}
\end{block}
\end{frame}

\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}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\begin{itemize}
\item Observe smoother structures in the box as $k_{\text{M}}$ is smaller
\item We can observe phase shifted structures between $h^+$ and $h^{\times}$
\item Indication of helicity
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Numerical results for hydrodynamic forced turbulence}
\begin{block}{Physical conditions}
\begin{itemize}
\item Forcing hydrodynamic term from phase transition bubble nucleation
\item Two cases considered:
\begin{itemize}
\item Rotational turbulence: very small vorticity, i.e., $\pmb{\nabla} \times \pmb{u} \ll 1$
\item Acoustic turbulence: very small divergence, i.e., $\pmb{\nabla} \cdot \pmb{u} \ll 1$
\end{itemize}
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Numerical results for hydrodynamic forced turbulence}
{\bf Rotational turbulence} $\pmb{\nabla} \times \pmb{u} \approx 0$ \\
\begin{center}
\includegraphics[width = 0.6\textwidth]{GW_results/RT1152b} \\
\end{center}
{\bf Acoustic turbulence} $\pmb{\nabla} \cdot \pmb{u} \approx 0$ \\
\begin{center}
\includegraphics[width = 0.6\textwidth]{GW_results/AT1152c}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Numerical results for hydrodynamic forced turbulence}
\begin{itemize}
\item Much stronger signal found for acoustic turbulence than for rotational turbulence
\item Result in agreement with literature\footnotemark[8]
\item For rotational turbulence, GWs polarization is negative and very small
\item For acoustic turbulence, GWs polarization is zero, as expected for incompressible flow fields
\end{itemize}
\footnotetext[8]{M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir. 
Numerical simulations of acoustically generated gravitational waves at a first order phase transition. 
\textit{Physical Review D}, 92(12):123009, 2015}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
\begin{columns}[c]
\column{0.5\textwidth}
\begin{block}{LISA}
\begin{itemize}
\item Laser Interferometer Space Antenna (LISA) is a space based GWs 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 $\sim $2M km
\end{itemize}
\end{block}
\column{0.5\textwidth}
\begin{figure}[H]
\includegraphics[width=1\textwidth]{LISA-waves}
\caption{Artist's impression of LISA from Wikipedia}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
\transduration<0-29>{.5}
\multiinclude[<+->][format=png, graphics={width=\textwidth}]{LISA_GW/LISA_GW}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
{\footnotesize
\begin{itemize}
\item LISA sensitivity is usually expressed as $h^2 \Omega_{\text{GW}}$
\item $\Omega_{\text{GW}}$ is the ratio of GWs energy density to critical energy density
\item Critical energy density is
$$ \mathcal{E}_{\text{crit}} = \frac{3 H_0^2 c^2 }{ 8 \pi G} $$
\item Current Hubble parameter is usually expressed as
$$ H_0 = 100 h \  \text{km s}^{-1} \text{Mpc}^{-1} $$
where $h$ represents the uncertainties in the actual value of $H_0$
\item We consider two different LISA configurations \footnotemark[9]:
\begin{itemize}
\item 4-link configuration with $2 \times 10^9$ m arm length after 5 years of duration
\item 6-link configuration with $6 \times 10^9$ m arm length after 5 years of duration
\end{itemize}
\end{itemize} }
{\footnotesize \footnotetext[9]{C. Caprini, M. Hindmarsh, S. Huber, T. Konstandin, J. Kozaczuk, et al. 
Science with the space-based interferometer eLISA. II: gravitational waves from cosmological phase transitions. 
\textit{Journal of Cosmology and Astroparticle Physics}, 2016(04):001–001, 2016} }
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
{\bf Decaying MHD turbulence with initial + fully helical magnetic field: }
\begin{center}
\includegraphics[width = 0.32\textwidth]{GW_results/M1152c3_LISA} 
\includegraphics[width = 0.32\textwidth]{GW_results/M1152g3_LISA} 
\includegraphics[width = 0.32\textwidth]{GW_results/M1152f3_LISA} 
\end{center}
\vspace{-8mm}
\begin{columns}[t]
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 300$$
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 60$$
\column{.33\textwidth}
\tiny $$ k_{\text{M}}/k_H = 2$$
\end{columns}
\vspace{3mm}

{\bf Forcing hydrodynamic rotational and acoustic turbulence: }
\vspace{-2mm}
\begin{center}
\includegraphics[width = 0.49\textwidth]{GW_results/RT1152b_LISA} 
\includegraphics[width = 0.49\textwidth]{GW_results/AT1152c_LISA} 
\end{center}
\vspace{-6mm}
\begin{columns}[t]
\column{.5\textwidth}
\centering \tiny Rotational turbulence
\column{.5\textwidth}
\centering \tiny Acoustic turbulence
\end{columns}
\end{frame}

\section{Aeroacoustic analogy}

\begin{frame}
\frametitle{Aeroacoustic analogy}
\begin{block}{}
\begin{itemize}
\item Research in progress so no calculations or details are complete
\item Similar equations for GWs and for sound waves, both are sourced by $T_{ij}$ terms
\item Aeroacoustic analogy applicable for GWs
\item Breathing mode $h^b$ of GWs has not been taking into account
\item Breathing mode appears because traceless tensor is not possible when space-time is not Minkowskian
\item Breathing mode could potentially be described as a Lighthill's equation
\item Aeroacoustic analogy could help explain why acoustic turbulence produces stronger GWs signal
and study the dependence of GWs with Mach number
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Timeline of the project}
{\scriptsize 
\begin{block}{General}
\begin{itemize}
\item Research started approximately May 2017
\item Expected thesis defense semester: Summer 2020
\end{itemize}
\end{block}
\begin{block}{Year 1}
\begin{itemize}
\item Numerical simulations of GWs using {\sc Pencil Code}
\item Improve my skills using {\sc Pencil Code}
\item Prepare publications with GWs results
\item Prepare publication in special issue of GAFD Journal about the {\sc Pencil Code} algorithms and implementation
\item Attend 12th International LISA Symposium in Chicago (8 -- 13 July 2018)
\item PhD visiting stay at the Nordic Institute for Theoretical Physics (Nordita) in Stockholm from August to October 2018
with Prof. Axel Brandenburg
\item Attend ''Quantum Anomalies and Chiral Magnetic Phenomena'' scientific program at Nordita
\item Possible academic stay at McWilliams Center for Cosmology at Carnegie Mellon University with Prof. Tina Kahniashvili
\end{itemize}
\end{block} }
\end{frame}

\begin{frame}
\frametitle{Timeline of the project}
{\scriptsize 
\begin{block}{Year 2}
\begin{itemize}
\item Improve my skills using {\sc Pencil Code}
\item Use {\sc Pencil Code} for running other types of numerical simulations
\item Actively contribute to {\sc Pencil Code} 
\item Study effects of expansion on GWs
\item Study $E$ and $B$-polarization modes
\item Study in more detail the breathing mode and the aeroacoustic analogy
\item Explore other GWs detectors as, for example, PTAs
\item Attend Nordita scientific program on gravitational waves on Summer 2019
\item Conduct some teaching practicum with Prof. Brian Argrow
\item Prepare PhD thesis report
\end{itemize}
\end{block} }
\end{frame}

\section{Timeline}


\begin{frame}
\Huge{\centerline{The End}}
\end{frame}

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\end{document}
