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\title[Generation of Gravitational Waves]
{Subinertial range of gravitational wave spectra
sourced by hydromagnetic turbulence.}
\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{September 10, 2019}


\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
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%	PRESENTATION SLIDES
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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
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% \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 Generation of cosmological 
gravitational waves (GWs) during the electroweak 
phase transition (EWPT)
\item MHD sources of GWs: 
	\begin{itemize}
	\item Hydrodynamic turbulence from phase transition bubbles nucleation
	\item Generation of primordial magnetic fields
	\end{itemize}
\item Possibility of GWs detection with LISA
\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{Gravitational waves}

\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 $a^2 \delta_{ij}$
\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}
$$ \left(\partial_t^2  - c^2 \nabla^2 \right) 
h_{ij} = \frac{16 \pi G}{a c^2} T_{ij}^{\rm TT} $$
\vspace{-4mm}
\begin{itemize}
\item $h_{ij}$ are rescaled $h_{ij} = a 
h_{ij}^{\rm phys}$
\item Comoving spatial coordinates $\nabla$ and 
conformal time $t$
\item Comoving stress-energy tensor components 
$T_{ij}$
\item Radiation-dominated epoch such that 
$a'' = 0$ 
\end{itemize}
\end{itemize}
\end{block}
\footnotetext{L. P. Grishchuk, {\em 
Sov. Phys. JETP}, 40, 409-415 (1974)}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves equation}
\begin{block}{Normalized GW equation\footnotemark}
\blue{\bf
$$\left(\partial_t^2 - \nabla^2 \right) h_{ij} = 6 T_{ij}^{\rm TT}/t$$}
\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 ${\cal E}_{\rm rad}^\ast$
\item Scale factor is $a_\ast = 1$
\end{itemize}
\end{block}
\footnotetext{\tiny A. 
Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.}, 
DOI:10.1080/03091929.2019.1653460, arXiv:1807.05479 (2019)}
\end{frame}

\begin{frame}
\frametitle{Gravitational waves equation}
\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} $ \\
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} = - B_i B_j + \delta_{ij} B^2/2 $
\end{itemize}
\end{columns}
\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}{MHD equations}
$$\frac{\partial \ln \rho}{\partial t} = - \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) + \frac{1}{\rho} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta \pmb{J}^2 \right) $$
{\footnotesize $$\frac{D \pmb{u}}{Dt} = \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) - \frac{\pmb{u}}{\rho} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta J^2 \right] - \frac{1}{4} \pmb{\nabla} \ln \rho +  \frac{3}{4 \rho} \pmb{J} \times \pmb{B} + \frac{2}{\rho} \pmb{\nabla} \cdot \left( \rho \nu \pmb{S} \right) $$}
$$ \frac{\partial \pmb{B}}{\partial t} = \pmb{\nabla} \times \left( \pmb{u} \times \pmb{B} - \eta \pmb{J} \right) $$
\end{block}
for a flat expanding universe with comoving magnitudes
\end{frame}


\begin{frame}
\frametitle{Gravitational waves energy density}
{\large \bf GWs energy density:\\}
{\footnotesize
$$\Omega_{\rm GW} (t) = {\cal E}_{\rm GW}/{\cal E}_{\rm rad}^*, \quad 
{\cal E}_{\rm rad}^* = \frac{3H_\ast^2 c^2}{8 \pi G}$$
$$ \Omega_{\rm GW} (t) = \int_{-\infty}^\infty 
\Omega_{\rm GW} (k, t) {\rm \, d} \ln k $$
$$\Omega_{\rm GW} (k, t) = \frac{k}{12 H_\ast^2} 
\int_{4 \pi} \left(\dot{\tilde{h}}^{\rm phys}_{ij} 
\dot{\tilde{h}}^{\rm phys, *}_{ij} \right)
k^2 {\rm \, d} \Omega_k$$
$$H_\ast \approx 2.066 \cdot 10^{-11} \ {\rm s}^{-1} \bigg({
T_\ast \over 100 \ {\rm GeV}}\bigg)^2 \bigg({g_\ast \over 100}\bigg)^{1/2}$$}
\end{frame}

\begin{frame}
\frametitle{Characteristic amplitude of gravitational waves}
{\large \bf GWs amplitude:\\}
$$h_{\rm c} (t) =  \int_{-\infty}^\infty h_{\rm c} (k, t) {\rm \, d} \ln k$$
$$h_{\rm c} (k, t) =  \frac{1}{2} \int_{4\pi} \left(\tilde{h}_{ij}^{\rm phys} 
\tilde{h}_{ij}^{\rm phys, *}\right) k^2 {\rm \, d} \Omega_k$$
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence\footnotemark[4]}
\begin{block}{Initial conditions for Set I}
\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\footnotemark[5]
\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[4]{A. Roper Pol, {\em et al.} 
arXiv:1903.08585}
\footnotetext[5]{limit from the Big Bang Nucleosynthesis 
bound}
\end{frame}


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

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\begin{columns}
\column{.6\textwidth}
\centering
\includegraphics[width = \textwidth]{fig1}
\column{.4\textwidth}
\begin{itemize}
{\footnotesize
\item ini1: $k_{\rm M} = 100$, $\Omega_{\rm M} \approx 0.1$
\item ini2: $k_{\rm M} = 100$, $\Omega_{\rm M} \approx 0.01$
\item ini3: $k_{\rm M} = 10$, $\Omega_{\rm M} \approx 0.01$}
\end{itemize}
{\tiny
\vspace{2mm}
We consider two different LISA configurations\footnotemark
\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{columns}
\footnotetext{C. Caprini {\rm et al., JCAP,} 2016(04):
001001 (2016)}
\end{frame}

\begin{frame}
\frametitle{Built-up magnetic fields and acoustic turbulence}
\begin{center}
\includegraphics[width=.6\textwidth]{time_evol_M} \\
\includegraphics[width=.6\textwidth]{time_evol_GW}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Forced turbulence, low resolution (work in progress)}
\centering
\includegraphics[width=.7\textwidth]{fig2}
\end{frame}

\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{General Conclusions}
\begin{itemize}
\item We have implemented a module within the {\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 
mission LISA
\item GW equation is normalized such that it can be easily scaled for different 
moments within the radiation-dominated epoch
\item Bubble nucleation and magnetogenesis physics can be coupled to our 
equations for more realistic production analysis
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Subinertial range Conclusions}
\begin{itemize}
\item Spectra of GWs obtained in subinertial range is shallower than predicted analytically 
$\Omega_{\rm GW} \propto k$, instead of $\Omega_{\rm GW} \propto k^3$.
\item Expected $k^3$ for larger scales than present in our simulations so far.
\item Interesting study on the time evolution of energy injection to large scales 
to study development of flat linear GW spectrum.
\item Effect on detectability \footnote{Baker {\em et al.}, 2019, arXiv:1907.11305}.
\end{itemize}
\begin{columns}
\column{.48\textwidth}
\includegraphics[width=\textwidth]{lisa_pta}
\column{.48\textwidth}
\includegraphics[width=\textwidth]{lisas}
\end{columns}
\end{frame}


\begin{frame}
\begin{center}
\Huge{The End \\ Thank You! \\}
\includegraphics[width=0.5\textwidth]{question}
\end{center}
\end{frame}
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\end{document}

