\newcommand{\BoldVec}[1]{\mathchoice%
  {\mbox{\boldmath $\displaystyle     #1$}}%
  {\mbox{\boldmath $\textstyle        #1$}}%
  {\mbox{\boldmath $\scriptstyle      #1$}}%
  {\mbox{\boldmath $\scriptscriptstyle#1$}}%
}

\def\half{{\textstyle{1\over2}}}
\def\sixth{{\textstyle{1\over6}}}
\def\fourth{{\textstyle{1\over4}}}
\def\eigth{{\textstyle{1\over8}}}
\def\oneovertwelve{{\textstyle{1\over12}}}
\def\twothird{{\textstyle{2\over3}}}
\def\threehalf{{\textstyle{3\over2}}}
\newcommand{\emf}{\mbox{\boldmath ${\cal E}$} {}}
\newcommand{\EEM}{{\cal E}_{\rm M}}
\newcommand{\HH}{{\cal H}}
\newcommand{\TR}{T_{\rm R}}
\newcommand{\TeV}{{\rm \, TeV}}
\newcommand{\GeV}{{\rm \, GeV}}
\newcommand{\MeV}{{\rm \, MeV}}
\newcommand{\Mpc}{{\rm \, Mpc}}
\newcommand{\J}{{\rm \, J}}
\newcommand{\m}{{\rm \, m}}
\newcommand{\s}{{\rm \, s}}
\newcommand{\Hz}{{\rm \, Hz}}
\newcommand{\Gauss}{{\rm \, G}}
\newcommand{\K}{{\rm \, K}}
\newcommand{\kb}{k_{\rm b}}
\newcommand{\kphy}{k_{\rm phys}}
\newcommand{\Tg}{T_{\rm g}}
\newcommand{\Mpl}{M_{\rm pl}}
\newcommand{\GammaR}{\Gamma_{\rm R}}
\newcommand{\Bphy}{B_{\rm phys}}
\newcommand{\OmM}{\Omega_{\rm M}}
\newcommand{\OmGW}{\Omega_{\rm GW}}
\newcommand{\OmT}{\Omega_{\rm T}}
\newcommand{\EM}{E_{\rm M}}
\newcommand{\ET}{E_{\rm T}}
\newcommand{\EGW}{E_{\rm GW}}
\newcommand{\EEGW}{\mathcal{E}_{\rm GW}}
\newcommand{\bk}{\bar{k}}
\newcommand{\bB}{\bar{B}}
\newcommand{\kH}{k_{\rm H}}
\newcommand{\kf}{k_\ast}
\newcommand{\EErad}{{\cal E}_{\rm rad}}
\newcommand{\kB}{k_{\rm B}}
\newcommand{\gS}{g_{\rm S}}
\newcommand{\tmax}{t_{\rm max}}
\newcommand{\xx}{\BoldVec{x}{}}
\newcommand{\kk}{\BoldVec{k}{}}
\newcommand{\pp}{\BoldVec{p}{}}
\newcommand{\ff}{\BoldVec{f}{}}
\newcommand{\BB}{\BoldVec{B}{}}
\newcommand{\ee}{\BoldVec{e}{}}
\newcommand{\dd}{{\rm \, d}} 
\newcommand{\hc}{h_{\rm c}}
\newcommand{\hij}{h_{ij}}
\newcommand{\thij}{\tilde{h}_{ij}}
\newcommand{\hthij}{\hat{\tilde{h}}_{ij}}
\newcommand{\Tij}{T_{ij}}
\newcommand{\tPij}{\tilde{\Pi}_{ij}}
\newcommand{\tTij}{\tilde{T}_{ij}}
\newcommand{\htTij}{\hat{\tilde{T}}_{ij}}
\newcommand{\TT}{\tilde{T}_+}
\newcommand{\Tx}{\tilde{T}_\times}
\newcommand{\hT}{\tilde{h}_+}
\newcommand{\hx}{\tilde{h}_\times}
\newcommand{\nab}{\BoldVec{\nabla} {}}
\newcommand{\Ci}{{\rm Ci}}
\newcommand{\Si}{{\rm Si}}
\newcommand{\ts}{t_{\rm s}}
\newcommand{\ks}{k_{\rm s}}
\newcommand{\tini}{t_{\rm ini}}
\newcommand{\tfin}{t_{\rm fin}}
\newcommand{\map}{\rightarrow}
\newcommand{\mmap}{\longrightarrow}
\newcommand{\Map}{\Rightarrow}
\newcommand{\MMap}{\Longrightarrow}
\newcommand{\alphaM}{\alpha_{\rm M}}
\newcommand{\alphaT}{\alpha_{\rm T}}
\newcommand{\alphaK}{\alpha_{\rm K}}
\newcommand{\alphaB}{\alpha_{\rm B}}
\newcommand{\cT}{c_{\rm T}}
\newcommand{\WKB}{{\rm WKB}}
\newcommand{\fpeak}{f_{\rm peak}}
\newcommand{\kpeak}{k_{\rm peak}}
\renewcommand{\L}{{\cal L}}

\def\aB{a_{\rm B}}
\def\mg{m_{\rm g}}
\def\bT{b_{\rm T}}
\def\cs{c_{\rm s}}
\def\crit{{\rm crit}}
\def\cT{c_{\rm T}}
\def\CMB{{\rm CMB}}
\def\crit{{\rm crit}}
\def\klim{k_{\rm lim}}
\def\klimap{k_{{\rm lim}, \, a''}}
\def\klimaM{k_{{\rm lim}, \, \alpM}}
\def\klimaMp{k_{{\rm lim}, \, \alpM'}}
\def\EQ{{\rm EQ}}
\def\EW{{\rm EW}}
\def\DE{{\rm DE}}
\def\RD{{\rm RD}}
\def\MD{{\rm MD}}
\def\LD{{\Lambda\rm D}}
\def\ini{{\rm ini}}
\def\incr{{\rm incr}}
\def\IR{{\rm IR}}
\def\fin{{\rm fin}}
\def\eff{{\rm eff}}
\def\mat{{\rm mat}}
\def\GR{{\rm GR}}
\def\GW{{\rm GW}}
\def\gS{g_{\rm S}}
\def\UV{{\rm UV}}
\def\rad{{\rm rad}}
\def\QCD{{\rm QCD}}
\def\phys{{\rm phys}}
\def\turb{{\rm turb}}
\def\kap{\kappa}
\def\lam{\lambda}
\def\Lam{\Lambda}
\def\alp{\alpha}
\def\del{\delta}
\def\Del{\Delta}
\def\sig{\sigma}
\def\zet{\zeta}
\def\cM{c_{\rm M}}
\def\alpK{\alpha_{\rm K}}
\def\alpB{\alpha_{\rm B}}
\def\alpM{\alpha_{\rm M}}
\def\alpMz{\alpha_{\rm M, 0}}
\def\alpT{\alpha_{\rm T}}
\def\alpTz{\alpha_{\rm T, 0}}
\def\alpMT{\alpha_{\rm M,T}}
\def\betaT{\beta_{\rm T}}
\def\om{\omega}
\def\Om{\Omega}
\def\GN{G_{\rm N}}
\def\aaa{\mathcal{A}}
\def\bbb{\mathcal{B}}
\def\ccc{\mathcal{C}}
\def\ddd{\mathcal{D}}
\def\eee{\mathcal{E}}
\def\fff{\mathcal{F}}
\def\ggg{\mathcal{G}}
\def\hhh{\mathcal{H}}
\def\iii{\mathcal{I}}
\def\jjj{\mathcal{J}}
\def\kkk{\mathcal{K}}
\def\lll{\mathcal{L}}
\def\mmm{\mathcal{M}}
\def\nnn{\mathcal{N}}
\def\ooo{\mathcal{O}}
\def\ppp{\mathcal{P}}
\def\qqq{\mathcal{Q}}
\def\rrr{\mathcal{R}}
\def\sss{\mathcal{S}}
\def\ttt{\mathcal{T}}
\def\uuu{\mathcal{U}}
\def\vvv{\mathcal{V}}
\def\www{\mathcal{W}}
\def\xxx{\mathcal{X}}
\def\yyy{\mathcal{Y}}
\def\zzz{\mathcal{Z}}
\def\munu{{\mu\nu}}
%\def\ij{{ij}}
\def\ij{{{ij}}}
\def\p{\partial}
\def\lan{\langle}
\def\ran{\rangle}

\newcommand{\VecThreeD}[3]{\left( \begin{array}{c}
#1 \\ #2 \\ #3 \end{array} \right)}
\newcommand{\VecTwoD}[2]{\left( \begin{array}{c}
#1 \\ #2 \end{array} \right)}
\newcommand{\Matrixsym}[6]{\left(\begin{array}{c c c}
#1 & #2 & #3 \\ #2 & #4 & #5 \\ #3 & #5 & #6
\end{array} \right)}
\newcommand{\MatTwoD}[4]{\left(\begin{array}{c c}
#1 & #2 \\ #3 & #4 \end{array} \right)}

\newcommand{\bra}[1]{\langle #1\rangle}
\newcommand{\Bra}[1]{\bigl\langle #1 \bigr\rangle}
\newcommand{\Para}[1]{\bigl[ #1 \bigr]}
\newcommand{\PPara}[1]{\Bigl[ #1 \Bigr]}
\newcommand{\PPPara}[1]{\biggl[ #1 \biggr]}
\newcommand{\PPPPara}[1]{\Biggl[ #1 \Biggr]}
\newcommand{\Pare}[1]{\bigl( #1 \bigr)}
\newcommand{\PPare}[1]{\Bigl( #1 \Bigr)}
\newcommand{\PPPare}[1]{\biggl( #1 \biggr)}
\newcommand{\PPPPare}[1]{\Biggl( #1 \Biggr)}

%\newcommand{\Eq}[1]{Eq.~(\ref{#1})}
%AB: lowercase for JCAP
\newcommand{\Eq}[1]{equation~(\ref{#1})}
\newcommand{\EEq}[1]{Equation~(\ref{#1})}
\newcommand{\Eqs}[2]{equations~(\ref{#1}) and (\ref{#2})}
\newcommand{\Eqss}[2]{Eqs.~(\ref{#1}--\ref{#2})}
%\newcommand{\Fig}[1]{Fig.~(\ref{#1})}
%\newcommand{\FFig}[1]{Figure~(\ref{#1})}
%\newcommand{\Sec}[1]{Sec.~(\ref{#1})}
%\newcommand{\App}[1]{App.~(\ref{#1})}
%\newcommand{\Tab}[1]{Tab.~(\ref{#1})}
%\newcommand{\TTab}[1]{Table~(\ref{#1})}
%AB: no parentheses
%\newcommand{\Fig}[1]{Fig.~\ref{#1}}
%\newcommand{\Fig}[1]{Fig.~\ref{#1}}
\newcommand{\FFig}[1]{Figure~\ref{#1}}
%\newcommand{\Sec}[1]{Sec.~\ref{#1}}
%\newcommand{\App}[1]{App.~\ref{#1}}
%\newcommand{\Tab}[1]{Tab.~\ref{#1}}
%\newcommand{\Tab}[1]{Table~\ref{#1}}
\newcommand{\Fig}[1]{figure~\ref{#1}}
\newcommand{\Sec}[1]{section~\ref{#1}}
\newcommand{\App}[1]{appendix~\ref{#1}}
\newcommand{\Tab}[1]{table~\ref{#1}}

\newcommand{\CC}[1]{\textcolor{blue}{{Chiara: #1}}}

\usepackage{amsfonts}
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}

\newcommand{\lambdabar}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}

\newcommand{\blue}[1]{\textcolor{blue}{#1}}
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\green}[1]{\textcolor{green}{#1}}
\newcommand{\axel}[1]{\textcolor{magenta}{#1}}

\def\boxbegin{\begin{tcolorbox}[breakable,opacityback = 0, opacityframe=0.5, standard jigsaw]}
\def\boxend{\end{tcolorbox}}
