The Sakharov conditions

Perhaps the most convincing motivation for requiring an explanation for the BAU is the fact that inflation, a cornerstone of modern cosmology, will wash out any initial BAU. Before inflation was proposed, however, Sakharov proposed that any explanation of the BAU must satisfy three conditions, now famously known as the 'Sakharov conditions'. These conditions are:

• violation of baryon number conservation;
• violation of C and CP;
• departure from equilibrium.

The conditions are intuitively obvious as they basically say in crude terms that you need a process that allows you to create baryons, treat particles differently to anti-particles ¹ and the future different to the past, and you need the process to change something in the Universe [1]. Even though the last condition is the most obvious it actually has a well-known loophole if CPT invariance is violated in a theory [2]. This can be seen from the proof of the third condition. Consider the equilibrium average of the baryon asymmetry B using standard thermal quantum mechanics ² ,

$$\begin{eqnarray}\begin{array}{ccl}{\rm{\langle }}B{\rm{\rangle }} & = & {\rm{Tr}}\left[{{\rm{e}}}^{-\beta { \mathcal H }}B\right]\\ & = & {\rm{Tr}}\left[{({\rm{CPT}})({\rm{CPT}})}^{-1}{{\rm{e}}}^{-\beta { \mathcal H }}B\right]\\ & = & {\rm{Tr}}\left[{{\rm{e}}}^{-\beta { \mathcal H }}{({\rm{CPT}})}^{-1}B({\rm{CPT}})\right]\\ & = & -\,{\rm{Tr}}\left[{{\rm{e}}}^{-\beta { \mathcal H }}B\right].\end{array}\end{eqnarray} \tag{ 2.1 }$$

The proof is quite elementary but does reveal that in asserting that baryon asymmetry production requires a departure from equilibrium you are implicitly assuming CPT invariance. This, however is a safe assumption in the vast majority of models including any this author is aware of that use the electroweak baryogenesis mechanism. Despite these conditions being intuitively obvious they are useful for the organization of this book, which is structured around these conditions. Remarkably one can, in principle, satisfy all three conditions using Standard Model particle content. However, the precise values of parameters within the Standard Model rule out it being sufficient to explain the BAU.