Route to nonlocal cosmology

An analytic approach to phenomenological models inspired by cubic string field theory is introduced and applied to some examples. We study a class of actions for a minimally coupled, homogeneous scalar field whose energy density contains infinitely many time derivatives. These nonlocal systems are systematically localized and an algorithm to find cosmological solutions of the dynamical equations is provided. Our formalism is able to define the nonlocal field in regions of the parameter space which are inaccessible by standard methods. Also, problems related to nonlocality are reinterpreted under a novel perspective and naturally overcome. We consider phenomenological models living on a Friedmann-Robertson-Walker background with power-law scale factor, both in four dimensions and on a high-energy braneworld. The quest for solutions unravels general features of nonlocal dynamics indicating several future directions of investigation.

Figure 1

The function $\psi (r,t)$ calculated via the series expansion and the eigenstates method. Here, $\theta =1$ (RS braneworld) and $\nu =-3/2$. The meaning of each dashed and dotted curve is explained in the text. In the upper panel ($r/\alpha =1$), the solid thin curve is ${\psi }_1$ with $C_1=e^{i\pi (\nu -\theta /4)}$, while the solid thick curve is ${\psi }_2$ with $C_2=1$. In the lower panel ($r/\alpha =-1$), the solid thin curve is ${\psi }_1$ with $C_1=1$, while the solid thick curve is ${\psi }_3$ with $C_3=1$.

Figure 2

The function $\psi (r,t)$ calculated via the series expansion and the eigenstates method. Here, $\theta =-1$ (GB braneworld) and $\nu =-3/2$. The meaning of each dashed and dotted curve is explained in the text. In the upper panel ($r/\alpha =1$), the solid thin curve is ${\psi }_1$ with $C_1=e^{i\pi (\nu -\theta /4)}$, while the solid thick curve is ${\psi }_2$ with $C_2=1$. In the lower panel ($r/\alpha =-1$), the solid thin curve is ${\psi }_1$ with $C_1=1$, while the solid thick curve is ${\psi }_3$ with $C_3=1$.

Figure 3

The 4D function $\psi (r,t)$ calculated via the series expansion and the eigenstates method. The dashed curve is the local solution $\varphi =\ln t$, while the dotted curves are the series-type functions at different level truncations. The solid curves are (increasing thickness) ${\psi }_1$, ${\psi }_2$, and ${\psi }_3$, respectively. Here, $\nu =-3.9$, $r=0.26$, $\alpha =-1$.