Can we see Lorentz-violating vector fields in the CMB?

We investigate the perturbation theory of a fixed-norm, timelike Lorentz-violating vector field. After consistently quantizing the vector field to put constraints on its parameters, we compute the primordial spectra of perturbations generated by inflation in the presence of this vector field. We find that its perturbations are sourced by the perturbations of the inflaton; without the inflaton perturbation the vector field perturbations decay away leaving no primordial spectra of perturbations. Since the inflaton perturbation does not have a spin-$1$ component, the vector field generically does not generate any spin-$1$ “vector-type” perturbations. Nevertheless, it will modify the amplitude of both the spin-$0$ “scalar-type” and spin-$2$ “tensor-type” perturbation spectra, leading to violations of the inflationary consistency relationship.

Figure 1

Figure showing the allowed values for ${\beta }_i$. The arrows indicate extension to infinity. The slope of the wedge makes a $45°$ angle with the ${\beta }_2-{\beta }_3$ plane. An example of a valid choice of parameters is ${\beta }_1=1$, ${\beta }_2=2.5$, ${\beta }_3=-3$.