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\title[Gravitational Waves from the early-universe]
{Gravitational Wave production from MHD Turbulence in the Early-Universe}
\subtitle{Helsinki Institute of Physics Seminar \\
\vspace{-11mm}
\hspace{8.5cm}
\includegraphics[width=.12\textwidth]{HIP_logo} \\
\vspace{-15mm}
\hspace{-8.5cm}
\includegraphics[width=.12\textwidth]{HIP_logo}}
\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
\vspace{2mm}
Laboratory for Atmospheric and Space Physics (LASP) \\
\vspace{2mm}
PhD Visiting fellow at Nordita
}
\date{December 3, 2019}


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\includegraphics[width=.3\textwidth]{nordita_logo} \\
{\tiny
\vspace{4mm}
A. Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.} 
{\bf 114}, 130. arXiv:1807.05479 (2019) \\
\vspace{-1mm}
A. Roper Pol {\em et al., submitted to Phys. Rev. Lett.}
arXiv:1903.08585 (2019) }
\end{flushleft}
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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
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%\begin{frame}
%\frametitle{Introduction and Motivation}
%\begin{itemize}
%\item \visible<1-> { 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}
%\item GW radiation as a probe of early universe physics}
%\item \visible<2-> {Possibility of GWs detection with
%	\begin{itemize}
%	\item \red{\bf Space mission LISA}
%    \item Pulsar Timing Arrays (PTA)
%	\item $B$-mode of CMB polarization (inflation)}
%%	\visible<2-3> {	
%\visible<3->{
%\begin{center}
%\includegraphics[width=.5\textwidth]{lisa_pta}
%\end{center}}
%	\end{itemize}	
%\end{itemize}
%\end{frame}

\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 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}
\item GW radiation as a probe of early universe physics
\item Possibility of GWs detection with
	\begin{itemize}
	\item \red{\bf Space mission LISA}
    \item Pulsar Timing Arrays (PTA)
	\item $B$-mode of CMB polarization (inflation)
%	\visible<2-3> {	
%\includegraphics[width=.5\textwidth]{lisa_pta}}
	\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{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}
\item GW radiation as a probe of early universe physics
\item Possibility of GWs detection with
	\begin{itemize}
	\item \red{\bf Space mission LISA}
    \item Pulsar Timing Arrays (PTA)
	\item $B$-mode of CMB polarization (inflation)
	\end{itemize}	
\item  Magnetohydrodynamic (MHD) sources of GWs: 
	\begin{itemize}
	\item Hydrodynamic turbulence from phase transition bubbles nucleation
	\item Magnetic turbulence from 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{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 $\gamma_{ij} = a^2 \delta_{ij}$
\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}
$$ \Bigl(\partial_t^2  - \red{\xcancel{a''\over a}} - c^2 \nabla^2 \Bigr) 
h_{ij} = \frac{16 \pi G}{\red{\bf a} c^2} T_{ij}^{\rm TT} $$
\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 (2019)}
\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}
\begin{block}{Contributions to the stress-energy tensor}
$$T^{\mu \nu} = \bigl(p/c^2 + \rho\bigr) U^{\mu} U^{\nu} + p g^{\mu \nu} + 
F^{\mu \gamma} F^{\nu}_{\ \, \gamma} - \frac{1}{4} g^{\mu \nu} 
F_{\lambda \gamma} F^{\lambda \gamma}$$
\begin{columns}[c]
\column{.5\textwidth}
\begin{itemize}
\item From fluid motions \\
$ T_{ij} = \left( 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}
\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}
$$\frac{\partial \ln \rho}{\partial t} = - \frac{4}{3} \left( \pmb{\nabla} \cdot \pmb{u} + \pmb{u} \cdot \pmb{\nabla} \ln \rho \right) + \frac{1}{\rho} \left[ \pmb{u} \cdot \left( \pmb{J} \times \pmb{B} \right) + \eta \pmb{J}^2 \right) $$
{\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 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{Traceless and transverse projection}
\begin{itemize}
\item CFL condition for stability:
$$\delta_t \leq C_{\rm CFL} \delta x/U_{\rm eff},$$
$U_{\rm eff} = |\pmb{u}| + (c_{\rm s}^2 + v_{\rm A}^2)^{1/2}$, $c_{\rm s}^2 = 1/3$, 
$v_{\rm A}^2 = B^2/\rho$. 
\vspace{5mm}
%\item For computational reasons, we solve GWs equation 
%sourced by the stress-energy tensor\footnote{\tiny A. 
%Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.}, 
%DOI:10.1080/03091929.2019.1653460, arXiv:1807.05479 (2019)}
%\red{$$ \left( \partial_t^2  - c^2 \nabla^2 \right) 
%h_{ij} = \frac{16 \pi G}{a c^2} T_{ij} $$}
\item Projection of $T_{ij}^{\rm TT}$ requires non-local Fourier 
transform $\tilde{T}_{ij}$: %(computationally expensive): 
\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{Linear polarization modes $h^+$ and $h^{\times}$}
\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}{$h^+$ 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{Solution 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.} 
%{\bf 114}, 130. arXiv:1807.05479 (2019)}
%\red{$$ \left( \partial_t^2  - \nabla^2 \right) 
%h_{ij} = 6 T_{ij}/t $$}
%\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{Solving the GW equation}
\begin{itemize}
\item Compute Fourier transform of stress-energy tensor $\tilde{T}_{ij}$
\item Project into TT gauge $\tilde{T}_{ij}^{\rm TT}$
\item Compute $\tilde{T}^+$ and $\tilde{T}^\times$ modes
\item Discretize time using $\delta t$ from MHD simulations
\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 (2019)}\\
{\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{Gravitational waves energy density}
{\large \bf GWs energy density:\\}
{\footnotesize
$$\Omega_{\rm GW} (t) = {\cal E}_{\rm GW}/{\cal E}_{\rm rad}^*, \quad 
{\cal E}_{\rm rad}^* = \frac{3H_\ast^2 c^2}{8 \pi G}$$
$$ \Omega_{\rm GW} (t) = \int_{-\infty}^\infty 
\Omega_{\rm GW} (k, t) {\rm \, d} \ln k $$
$$\red{\bf \Omega_{\rm GW} (k, t)} = \frac{k}{6 H_\ast^2} 
\int_{4 \pi} \left(\left| \dot{\tilde{h}}^{\rm phys}_+ 
\right|^2 + \left| \dot{\tilde{h}}^{\rm phys}_\times \right|^2 \right)
k^2 {\rm \, d} \Omega_k$$}
{\large \bf 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
$$ \Xi_{\rm GW} (t) = \int_{-\infty}^\infty 
\Xi_{\rm GW} (k, t) {\rm \, d} \ln k $$
$$\red{\bf \Xi_{\rm GW} (k, t)} = \frac{k}{6 H_\ast^2} 
\int_{4 \pi} 2 {\rm Im} \left( \dot{\tilde{h}}^{\rm phys}_+ 
\dot{\tilde{h}}^{{\rm phys}, \ast}_\times \right)
k^2 {\rm \, d} \Omega_k$$}
{\footnotesize
$$H_\ast \approx 2.066 \cdot 10^{-11} \ {\rm s}^{-1} \bigg({
T_\ast \over 100 \ {\rm GeV}}\bigg)^2 \bigg({g_\ast \over 100}\bigg)^{1/2}$$}
\end{frame}

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

%\begin{frame}
%\frametitle{Numerical accuracy\footnote{\tiny A. Roper Pol {\em et al., Geophys. Astrophys. Fluid Dyn.}, 
%DOI:10.1080/03091929.2019.1653460, arXiv:1807.05479 (2019)}}
%\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 seemed to 
%be affected
%\end{itemize}
%\end{columns}
%\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence\footnotemark[7]}
\begin{block}{Initial conditions\footnotemark[8]}
\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[7]{A. Roper Pol, {\em et al.} 
arXiv:1903.08585}
\footnotetext[8]{A. Brandenburg, {\em et al.} 
{\em Phys. Rev. D} (2017)}
\end{frame}

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

%% UNCOMMENT FOR GWs RESULTS
%\begin{frame}
%\frametitle{Numerical results for decaying MHD turbulence (old)}
%\begin{columns}
%\column{.5\textwidth}
%{\bf Negative initial helicity}
%\column{.5\textwidth}
%{\bf Positive initial helicity}
%\end{columns}
%\vspace{4mm}
%\begin{columns}
%\column{.5\textwidth}
%\includegraphics[width = \textwidth]{GW_results/M1152e3} 
%\tiny $$ k_{\text{M}}/k_H = 2$$
%\column{.5\textwidth}
%\includegraphics[width = 0.8\textwidth]{GW_results/M1152f3} 
%\vspace{-2mm}
%\tiny $$ k_{\text{M}}/k_H = 6 $$
%\end{columns}
%\vspace{4mm}
%\blue{Blue dots indicate negative helicity spectrum 
%(magnetic fields) and negative antisymmetric 
%spectrum (GW)} \\
%\red{Red dots indicate positive helicity spectrum 
%(magnetic field) and positive antisymmetric 
%spectrum (GW)}
%\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}
\end{columns}
\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{Comments}
%\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.,} 95
%(15):151301 (2005)}
%\end{frame}
%%

\begin{frame}
\frametitle{Forced turbulence (built-up primordial magnetic fields and 
hydrodynamic turbulence)}
\centering
\includegraphics[width=.7\textwidth]{fig2}
\end{frame}

\begin{frame}
\frametitle{Forced turbulence (built-up primordial magnetic fields and 
hydrodynamic turbulence)}
\centering
\includegraphics[width=.7\textwidth]{table}
\end{frame}

%\begin{frame}
%\frametitle{Numerical results for hydrodynamic forced turbulence}
%\begin{block}{Conclusions}
%\begin{itemize}
%\item Much stronger signal found for acoustic 
%turbulence than for rotational turbulence
%\item Result in agreement with 
%literature\footnotemark
%\item In both cases, GWs polarization is zero
%\end{itemize}
%\end{block}
%\footnotetext{M. Hindmarsh {\em et al., Phys. Rev. D,} 
%92(12):123009 (2015)}
%\end{frame}

\begin{frame}
\frametitle{Time evolution of GW energy density}
\centering
\includegraphics[width=.6\textwidth]{fig5}
\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{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=\textwidth]{LISA-waves}
\caption{Artist's impression of LISA from Wikipedia}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
{\footnotesize
\begin{itemize}
\item LISA sensitivity is usually expressed as 
$h_0^2 \Omega_{\text{GW}}$
\item $\Omega_{\rm GW}$ is the ratio of GWs energy 
density to critical energy density
\item Critical energy density is
$$ \mathcal{E}_{\rm crit} = \frac{3 H_0^2 c^2 }
{ 8 \pi G} $$
\item Current Hubble parameter is usually expressed as
$$ H_0 = 100 \, h_0 \  {\rm km \, s}^{-1} 
{\rm Mpc}^{-1} $$
where $h_0$ represents the uncertainties in the 
actual value of $H_0$
%\item We consider two different LISA configurations 
%\footnote{C. Caprini {\rm et al., JCAP,} 2016(04):
%001001 (2016)}
%\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}}
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
\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 $f^0 = a_0^{-1} f$
\end{itemize}
\end{itemize}
\end{block}
$$a_0 \approx 1.254 \cdot 10^{15} \left(T_\ast/ 100 {\rm \ GeV} \right) 
\left(g_{\rm S}/100\right)$$
\end{frame}

\begin{frame}
\frametitle{Detectability with LISA}
\begin{columns}
\column{.5\textwidth}
\includegraphics[width=\textwidth]{fig1}
\column{.5\textwidth}
\includegraphics[width=\textwidth]{fig2}
\end{columns}
\end{frame}

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

%----------------------------------------------------------------------------------------

\end{document}

