% Equations
\newcommand{\be}[1]{\begin{equation} #1 \end{equation}}
\newcommand{\ba}[1]{\begin{align} #1 \end{align}}
\newcommand{\bes}[1]{\begin{equation}\begin{split} #1 \end{split}\end{equation}}
\newcommand{\bsa}[2]{\begin{subequations}\label{#1}\begin{align} #2 \end{align}\end{subequations}}

% Basic Math
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\dabs}[1]{\| #1 \|}
\newcommand{\tr}[1]{\mathrm{Tr}\left( #1\right)}
\newcommand{\half}{\frac{1}{2}} 
\newcommand{\ord}[2][ ]{O\left( #2 \right)^{#1}}
\newcommand{\ex}[1]{\ \mathrm{exp} \left[ #1 \right]}
\renewcommand{\exp}[1]{\ \mathrm{exp} \left[ #1 \right]}
\newcommand{\pint}[1]{\mathcal{D} #1 \ }
\newcommand{\Max}[1]{\mathrm{max}\left\{ #1 \right\}}
\newcommand{\Min}[1]{\mathrm{min}\left\{ #1 \right\}}
\newcommand{\im}{\mathrm{Im} \, }
\newcommand{\re}{\mathrm{Re} \, }
\newcommand{\const}{\, \mathrm{const.}}
\newcommand{\few}{\, \mathrm{few} \, }
\newcommand{\ud}{\mathrm{d}}

% Vector Calc
\newcommand{\curl}[1]{{\bm \nabla} \times {{\bf #1}}}
\newcommand{\divg}[1]{{\bm \nabla} \cdot {{\bf #1}}}
\newcommand{\grad}{{\bm \nabla}}

\newcommand{\condprob}[2]{\mathbb{P} \left( #1 | #2 \right)}
\newcommand{\prob}[1]{\mathbb{P}\left(#1 \right)}
\newcommand{\factorial}[1]{#1!}
\newcommand{\deval}[2]{\left. #1 \right|_{#2}}

\DeclareSymbolFont{bbold}{U}{bbold}{m}{n}
\DeclareSymbolFontAlphabet{\mathbbold}{bbold}
\newcommand{\bbone}{{\huge \mathbbold{1}}}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\wt}[1]{\widetilde{#1}}

% Quantum Mech bra-ket notation
\newcommand{\bra}[1]{\bigl<#1\bigr|}
\newcommand{\ket}[1]{\bigl|#1\bigr>}
\newcommand{\amp}[2]{\bigl< #1 \bigl| #2\bigr>}
\newcommand{\ampl}[2]{\bigl< #1 \bigl| #2\bigr>}
\newcommand{\expval}[3]{\bigl< #1 \bigr| #2 \bigl| #3 \bigr>}

% Groups
\newcommand{\SU}[1]{\mathrm{SU}(#1)}
\newcommand{\U}[1]{\mathrm{U}(#1)}
\newcommand{\SO}[1]{\mathrm{SO}(#1)}
\newcommand{\Sp}[1]{\mathrm{Sp}(#1)}
\newcommand{\su}[1]{\mathfrak{su}(#1)}
\newcommand{\uAlg}[1]{\mathfrak{u}(#1)}
\newcommand{\so}[1]{\mathrak{so}(#1)}
\newcommand{\spAlg}[1]{\mathrak{sp}(#1)}

% References
\newcommand{\Eref}[1]{Eq.~(\ref{#1})}
\newcommand{\eref}[1]{eq.~(\ref{#1})}
\newcommand{\Erefs}[2]{Eqs.~(\ref{#1})~and~(\ref{#2})}
\newcommand{\erefs}[2]{eqs.~(\ref{#1})~and~(\ref{#2})}
\newcommand{\Fref}[1]{Fig.~\ref{#1}}
\newcommand{\fref}[1]{fig.~\ref{#1}}
\newcommand{\Sref}[1]{Sec.~\ref{#1}}
\newcommand{\sref}[1]{sec.~\ref{#1}}
\newcommand{\Aref}[1]{App.~\ref{#1}}
\newcommand{\aref}[1]{app.~\ref{#1}}
\newcommand{\Tref}[1]{Tab.~\ref{#1}}
\newcommand{\tref}[1]{tab.~\ref{#1}}
\newcommand{\Rref}[1]{Ref.~\cite{#1}}
\newcommand{\rref}[1]{ref.~\cite{#1}}
\newcommand{\Rrefs}[1]{Refs.~\cite{#1}}
\newcommand{\rrefs}[1]{refs.~\cite{#1}}
\newcommand{\pref}[1]{(\ref{#1})}

% Units
\newcommand{\cgs}{\ \mathrm{cgs}}
\newcommand{\SI}{\ \mathrm{SI}}
\newcommand{\erg}{\ \mathrm{erg}}
\newcommand{\EeV}{\ \mathrm{EeV}}
\newcommand{\PeV}{\ \mathrm{PeV}}
\newcommand{\TeV}{\ \mathrm{TeV}}
\newcommand{\GeV}{\ \mathrm{GeV}}
\newcommand{\MeV}{\ \mathrm{MeV}}
\newcommand{\keV}{\ \mathrm{keV}}
\newcommand{\eV}{\ \mathrm{eV}}
\newcommand{\meV}{\ \mathrm{meV}}
\newcommand{\mueV}{\ \mu\mathrm{eV}}
\newcommand{\THz}{\ \mathrm{THz}}
\newcommand{\GHz}{\ \mathrm{GHz}}
\newcommand{\MHz}{\ \mathrm{MHz}}
\newcommand{\kHz}{\ \mathrm{kHz}}
\newcommand{\Hz}{\ \mathrm{Hz}}
\newcommand{\mHz}{\ \mathrm{mHz}}
\newcommand{\muHz}{\ \mu \mathrm{Hz}}
\newcommand{\nHz}{\ \mathrm{nHz}}
\newcommand{\Gpc}{\ \mathrm{Gpc}}
\newcommand{\Mpc}{\ \mathrm{Mpc}}
\newcommand{\kpc}{\ \mathrm{kpc}}
\newcommand{\pc}{\ \mathrm{pc}}
\newcommand{\Gly}{\ \mathrm{Gly}}
\newcommand{\ly}{\ \mathrm{ly}}
\newcommand{\km}{\ \mathrm{km}}
\newcommand{\cm}{\ \mathrm{cm}}
\newcommand{\mm}{\ \mathrm{mm}}
\newcommand{\mum}{\ \mu \mathrm{m}}
\newcommand{\nm}{\ \mathrm{nm}}
\newcommand{\fm}{\ \mathrm{fm}}
\newcommand{\kg}{\ \mathrm{kg}}
\newcommand{\gram}{\ \mathrm{g}}
\newcommand{\fb}{\ \mathrm{fb}}
\newcommand{\pb}{\ \mathrm{pb}}
\newcommand{\Gyr}{\ \mathrm{Gyr}}
\newcommand{\yr}{\ \mathrm{yr}}
\newcommand{\Kel}{\ \mathrm{K}}
\newcommand{\mK}{\ \mathrm{mK}}
\newcommand{\degr}{\ \mathrm{deg}}
\newcommand{\arcsec}{\ \mathrm{arcsec}}
\newcommand{\arcmin}{\ \mathrm{arcsec}}
\newcommand{\sr}{\ \mathrm{sr}}
\newcommand{\sqdeg}{\ \mathrm{sqdeg}}
\newcommand{\Msun}{\ M_{\odot}}
\newcommand{\Mearth}{\ M_{\oplus}}
\newcommand{\Jy}{\ \mathrm{Jy}}
\newcommand{\Gauss}{\ \mathrm{G}}
\newcommand{\Tesla}{\ \mathrm{T}}
\newcommand{\Mpl}{M_{\rm pl}}
\newcommand{\Rsun}{\ R_{\odot}}
\newcommand{\Rearth}{\ R_{\oplus}}
\newcommand{\Lsun}{\ L_{\odot}}
\newcommand{\Joule}{\ \mathrm{J}}
\newcommand{\Watt}{\ \mathrm{W}}
\newcommand{\kel}{\ \mathrm{K}}

% Punctuation
\newcommand{\per}{\;.}
\newcommand{\com}{\;,}
\newcommand{\hc}{\mathrm{h.c.}}
\newcommand{\cc}{\mathrm{c.c.}}
\newcommand{\ie}{{\it i.e.}}
\newcommand{\eg}{{\it e.g.}}

% General Relativity
%\newcommand{\ChristoffelSymbol}[3]{\tensor{\Gamma}{^{#1}_{#2 #3}}}
\newcommand{\ChristoffelSymbol}[3]{\Gamma^{#1}_{#2 #3}}

% Formatting
\newcommand{\draftnote}[1]{{\bf \color{red} #1}}
\newcommand{\nn}{\nonumber \\}

% Abbreviations
\newcommand{\LCDM}{$\Lambda${\rm CDM}}
\newcommand{\CL}{\% \ {\rm C.L.}}
\newcommand{\atCL}[1]{\ \text{at #1\% C.L.}}
\newcommand{\rhs}{{\rm r.h.s.} }
\newcommand{\lhs}{{\rm l.h.s.} }
\newcommand{\etal}{et.\,al.}
\newcommand{\SM}{\text{\sc \fontfamily{ptm}\selectfont sm}}
\newcommand{\QCD}{\text{\sc \fontfamily{ptm}\selectfont qcd}}
\newcommand{\EW}{\text{\sc \fontfamily{ptm}\selectfont ew}}
\newcommand{\DM}{\text{\sc \fontfamily{ptm}\selectfont dm}}
\newcommand{\DE}{\text{\sc \fontfamily{ptm}\selectfont de}}
\newcommand{\CC}{\text{\sc \fontfamily{ptm}\selectfont cc}}
\newcommand{\QED}{\text{\sc \fontfamily{ptm}\selectfont qed}}
\newcommand{\GUT}{\text{\sc \fontfamily{ptm}\selectfont gut}}
\newcommand{\PMNS}{\text{\sc \fontfamily{ptm}\selectfont pmns}}
\newcommand{\BE}{\text{\sc \fontfamily{ptm}\selectfont be}}
\newcommand{\FD}{\text{\sc \fontfamily{ptm}\selectfont fd}}

% Misc.
\newcommand{\sigv}{\left< \sigma v \right>}
\newcommand{\Acal}{\mathcal{A}}
\newcommand{\Bcal}{\mathcal{B}}
\newcommand{\Ccal}{\mathcal{C}}
\newcommand{\Dcal}{\mathcal{D}}
\newcommand{\Ecal}{\mathcal{E}}
\newcommand{\Fcal}{\mathcal{F}}
\newcommand{\Gcal}{\mathcal{G}}
\newcommand{\Hcal}{\mathcal{H}}
\newcommand{\Ical}{\mathcal{I}}
\newcommand{\Jcal}{\mathcal{J}}
\newcommand{\Kcal}{\mathcal{K}}
\newcommand{\Lcal}{\mathcal{L}}
\newcommand{\Mcal}{\mathcal{M}}
\newcommand{\Ncal}{\mathcal{N}}
\newcommand{\Ocal}{\mathcal{O}}
\newcommand{\Pcal}{\mathcal{P}}
\newcommand{\Qcal}{\mathcal{Q}}
\newcommand{\Rcal}{\mathcal{R}}
\newcommand{\Scal}{\mathcal{S}}
\newcommand{\Tcal}{\mathcal{T}}
\newcommand{\Ucal}{\mathcal{U}}
\newcommand{\Vcal}{\mathcal{V}}
\newcommand{\Wcal}{\mathcal{W}}
\newcommand{\Xcal}{\mathcal{X}}
\newcommand{\Ycal}{\mathcal{Y}}
\newcommand{\Zcal}{\mathcal{Z}}

\newcommand{\Lscr}{\mathscr{L}}

\newcommand{\Csf}{\textsf{C}}
\newcommand{\Psf}{\textsf{P}}
\newcommand{\CPsf}{\textsf{CP}}
\newcommand{\Tsf}{\textsf{T}}
\newcommand{\Bsf}{\textsf{B}}
\newcommand{\Lsf}{\textsf{L}}

\newcommand{\Cbb}{\mathbb{C}}
\newcommand{\Pbb}{\mathbb{P}}
\newcommand{\Rbb}{\mathbb{R}}
\newcommand{\Zbb}{\mathbb{Z}}

\newcommand{\MSbar}{\overline{\rm MS}}
\newcommand{\DRbar}{\overline{\rm DR}}

\newcommand{\cmark}{\ding{51}} % Requires \usepackage{pifont}
\newcommand{\xmark}{\ding{55}} % Requires \usepackage{pifont}
\newcommand{\qmark}{{\color{blue} \it ?}} % Requires \usepackage{pifont}

\newcommand{\dbar}{\text{\dj}}

\newcommand{\tx}{(t,\xvec)}
\newcommand{\tk}{(t,\kvec)}
\newcommand{\wx}{(\omega,\xvec)}
\newcommand{\wk}{(\omega,\kvec)}
\newcommand{\xt}{(\xvec,t)}
\newcommand{\kt}{(\kvec,t)}
\newcommand{\xw}{(\xvec,\omega)}
\newcommand{\kw}{(\kvec,\omega)}

\newcommand{\pvec}{{\bm p}}
\newcommand{\kvec}{{\bm k}}
\newcommand{\xvec}{{\bm x}}

\newcommand{\vev}[1]{\langle 0 | #1 | 0 \rangle}

\newcommand{\pprime}{{\prime\prime}}