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\title[Gravitational Waves from the early-universe]
%{Gravitational Wave production from MHD Turbulence in the Early-Universe}
{Generation of Gravitational Waves due to Magnetohydrodynamic
Turbulence in the Early Universe}
\subtitle{Extra slides}
%\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{Faculty Advisor: }Brian Argrow\\
%{\em Research Advisor:} Axel Brandenburg \\ 
%{\em Collaborators: }Tina Kahniashvili, Arthur Kosowsky \& Sayan Mandal
%}
%\vspace{-10mm}
%\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 \\
%\vspace{2mm}
%Ann \& H.J. Smead Aerospace Engineering Sciences \\
%\medskip
%\vspace{2mm}
%Laboratory for Atmospheric and Space Physics (LASP) \\
%\vspace{2mm}
%PhD Visiting fellow at Nordita
%}
\date{May 8, 2020}


\begin{document}

\begin{frame}
\titlepage % Print the title page as the first slide
%\vspace{-15mm}
%\begin{flushright}
%%\includegraphics[width=.3\textwidth]{lasp}
%\end{flushright}
%\vspace{-15mm}
%\begin{flushleft}
%%\includegraphics[width=.1\textwidth]{APSLogo_2color} \\
%{\tiny
%\vspace{6mm}
%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. D}
%arXiv:1903.08585 (2019) }
%\end{flushleft}
\end{frame}

\begin{frame}
\frametitle{MHD energy density}
\small
\begin{columns}[t]
\column{.5\textwidth}
\blue{
\textbf{Kinetic energy density: }
$$ \Omega_{\text{K}}(t) = \frac{1}{2} \left< \rho \pmb{u}^2 \right> = \int E_{\text{K}}(k, t) dk $$ \\
\textbf{Kinetic helicity: }
$$ \Xi_{\text{K}}(t) = \rho \left< \pmb{u} \cdot \pmb{\omega} \right> = \int H_{\text{K}} (k, t) dk $$ \\
\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: }
$$ \Omega_{\text{M}}(t) = \frac{1}{2 } \left< \pmb{B}^2 \right> = \int E_{\text{M}}(k, t) dk $$
\textbf{Magnetic helicity: }
$$ \Xi_{\text{M}}(t) = \left< \pmb{B} \cdot \pmb{A} \right> = \int H_{\text{M}}(k, t) dk $$
\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[1]{
\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}

\begin{frame}
\frametitle{Numerical simulations}
\begin{block}{Numerical scheme used in {\sc Pencil Code}}
\begin{itemize}
\item MHD modular code
\item DNS simulations
\item High--order finite difference method (6th--order by
default)\footnotemark[2]
\item 3rd--order Runge Kutta time integration\footnotemark[3]
\end{itemize}
\end{block}
\footnotetext[2]{A. Brandenburg, {\em The Fluid Mechanics
of Astrophys. and Geophys.}, Vol.9 (2003)}
\footnotetext[3]{J. H. Williamson, {\em J. Comp. Phys.} {\bf 35},
48--56 (1980)}
\end{frame}

\begin{frame}
\frametitle{Time integration for GW equation\footnotemark[4]}
\centering
\includegraphics[width=.95\textwidth]{williamson}
\footnotetext[4]{\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{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}{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}
\end{frame}

\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
$$\Xi_{\rm M, K} = \int_{-\infty}^\infty 
\Xi_{\rm M, K} (k, t) {\rm \, d} \ln k $$
$$\red{\bf \Xi_{\rm M} (k, t)} = k
\int_{4 \pi} \left(\tilde{\bf B} \cdot \tilde{\bf A}^\ast\right)
k^2 {\rm \, d} \Omega_k, \quad
\red{\bf \Xi_{\rm K} (k, t)} = k
\int_{4 \pi} \rho\left(\tilde{\bf u} \cdot
\tilde{\bf \omega}^\ast\right) k^2 {\rm \, d} \Omega_k$$}
\end{frame}

\begin{frame}
\frametitle{Numerical results for decaying MHD turbulence}
\includegraphics[width=\textwidth]{pol_ini2}
\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 issue}
\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{First--order electroweak phase transition}
\begin{itemize}
\item Electroweak phase transition: symmetry of massless electroweak
bosons is broken
\item Production of W$^+$, W$^-$ and Z$^0$ mass bosons (weak force),
and photons $\gamma$ (EM)
\item Standard Model predicts a smooth crossover at a critical temperature
of $159.5 \pm 1.5$ GeV for $m_{\rm H} = 125$ GeV\footnote[5]{K. Kajantie,
M. Laine, K. Rummukainen, M. Shaposhnikov, \\ \hspace{5mm}
{\em Nucl. Phys.}
{\bf 466}, 189--258 (1996)}
\end{itemize}
\begin{center}
\includegraphics[width=0.6\textwidth]{SM_phase_diagram}
\end{center}
\end{frame}

\begin{frame}
\frametitle{First--order electroweak phase transition}
\begin{columns}[t]
\column{.5\textwidth}
\centering
\includegraphics[width=\textwidth]{potential}
\column{.5\textwidth}
\centering
\includegraphics[width=.7\textwidth]{bubbles}\\
From D. Cutting and collaborators
\end{columns}

\vspace{4mm}

The scale of these bubbles is expected to be
a hundredth of the Hubble scale $\rightarrow$
$k_{\rm K} \approx 100$ \footnote[6]{M.S. Turner,
E.J. Weinberg, L. M. Widrow,\\ \hspace{5mm} {\em Phys. Rev. D},
{\bf 46}, 2384 (1992)}
\end{frame}

\begin{frame}
\frametitle{First--order electroweak phase transition}
\begin{itemize}
\item SM extensions: singlet or triplet scalar extensions,
two Higgs doublet, higgs portals, composite Higgs, scale invariant
scalar extensions, SUSY models, low scale higher dimension operators.
\item BSM additions are motivated by electroweak baryogenesis
(Sakharov conditions)
\begin{enumerate}
\item C and CP violation: particles scattering off the bubble walls,
producing asymmetries
\item Baryon number violation: sphaleron transitions in front
of the wall are biased to produce more baryons than antibaryons
\item Out of equilibrium: the bubble walls disturb the
symmetric-phase equilibrium state
\end{enumerate}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{EWPT magnetogenesis}
\begin{itemize}
\item CP violation terms are added to the Lagrangian
\item EWPT should produce
magnetic fields from perturbations
on the Higgs expectation value\footnote[7]{T. Vachaspati,
{\em Phys. Lett. B} {\bf 265}, 258--261 (1991)}
\item It can produce magnetic fields up to 10\% of the
radiation energy density\footnote[8]{Y. Zhang, T. Vachaspati,
F. Ferrer, {\em Phys. Rev. D},
{\bf 100}, 083006 (2019)}
\item It cannot be larger than 10\% due to the BBN observations
\footnote[9]{T. Kahniashvili, A. G. Tevzadze, B. Ratra,
\\ \hspace{5mm}{\em Astrophys. J.} {\bf 726}, 78 (2011)}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Forcing terms}
We use two different fourcings:\\
\begin{columns}[t]
\column{.5\textwidth}
\blue{Acoustic forcing on velocity fields:}
\begin{itemize}
\item Described in Mee \& Brandenburg (2006)
\item Forcing from Gaussian potential
$${\bf f} = {\bf \nabla} \phi$$
$$\phi = N \exp([{\bf x} - {\bf x_f}]^2/R^2)$$
$$N = k_{\rm K}/(2\pi)$$
\item Monochromatic forcing into momentum equation
\end{itemize}
\column{.5\textwidth}
\red{
Helical forcing on magnetic fields:}
\begin{itemize}
\item From Brandenburg (2001)
\item Fully helical forcing term (in Fourier space)

\includegraphics[width=.7\textwidth]{fk}

\item Monochromatic forcing into induction
equation
\end{itemize}
\end{columns}
\end{frame}



\end{document}