Self-interacting $p$-forms and the shear-free anisotropic cosmological model

Abstract

The cosmological principle may require more than observations demand: what is needed is isotropic expansion without vorticity. Extended Friedmann-Lemaitre-Robertson-Walker (eFLRW) cosmologies realise this with a non-vanishing anisotropic spatial curvature that is balanced by an anisotropic stress while the shear vanishes. Thorsrud and others investigated such scenarios for free differential $p$-forms. Here we extend the analysis to non-Abelian $p$-forms ($p=0,\dots,4$) with self interactions. The balance requires the anisotropic stress to decay as $a^{-2}$, $a$ being the cosmological scale factor, which we combine with the gauge-field continuity equation. The general results indicate that, in the presence of a potential, fine-tuning of the same is needed in order to balance the curvature. Notably, this is not the case for a $3$-form field $\underline{H}^a$ with a potential term $V(\underline{H}^2)=A+B\,\underline{H}^2$, where $A$ and $B$ are constants: in Bianchi class A, an uncharged 3-form balances the curvature with this potential without any tuning, and a charged one does so for a suitably chosen connection. This is the only candidate we have found where the potential does not require tuning. It is also an interesting candidate as it can be interpreted as a cosmological constant in combination with a dark-matter sector mimicking curvature.