Optimal inflationary potentials

Inflation is a highly favored theory for the early Universe. It is compatible with current observations of the cosmic microwave background and large scale structure and is a driver in the quest to detect primordial gravitational waves. It is also, given the current quality of the data, highly underdetermined with a large number of candidate implementations. We use a new method in symbolic regression to generate all possible simple scalar field potentials for one of two possible basis sets of operators. Treating these as single-field, slow-roll inflationary models we then score them with an information-theoretic metric (“minimum description length”) that quantifies their efficiency in compressing the information in current data. We explore two possible priors on the parameter space of potentials, one related to the functions’ structural complexity and one that uses a Katz back-off language model to prefer functions that may be theoretically motivated. This enables us to identify the inflaton potentials that optimally balance simplicity with accuracy at explaining current data, which may subsequently find theoretical motivation. Our exploratory study opens the door to extraction of fundamental physics directly from data, and may be augmented with more refined theoretical priors in the quest for a complete understanding of the early Universe.

Figure 1

Pareto front of inflationary potentials found with ESR when compared to the data for the two basis sets. We show the best-fitting functions according to the change in the description length, $L(D)$, (red) and the likelihood, $\mathcal{L}$, (blue) relative to the corresponding minima. More accurate functions appear at lower $|\mathrm{\Delta }\log (L)|$, while overall superior functions appear at lower $L(D)$. For the solid line we use the $k\log (n)$ term in the description length to penalize model complexity, while for the dashed line we instead use the Katz language model.

Figure 2

Inflationary trajectories of the potentials with the minimum description length for different analyses: (a) The MDL function for both basis sets A and B using the $k\log (n)$ function prior, (b) The MDL potential for basis set A with the language model prior, and (c) The MDL expression for basis set B with the language model. For context, in (d) we give the corresponding plot for the Starobinsky or Higgs inflation, with ${\theta }_0$ optimized to fit $A_S$, and in (e) we give the potential which is at the “knee” of the Pareto front [Eq. (22)]. The shaded region shows the range of $\varphi$ during which inflation occurs, where the inflaton rolls from the region of high to low potential. (a) $V(\varphi )=e^{-e^{e^{e^{\varphi }}}}$, (b) $V(\varphi )={\theta }_0({\theta }_1+\log {(\varphi )}^2)$, (c) $V(\varphi )={\theta }_0{\varphi }^{\frac{{\theta }_1}{\varphi }}$, (d) $V(\varphi )={\theta }_0{(1-e^{-\sqrt{2/3\varphi }})}^2$, (e) $V(\varphi )={\theta }_0{e}^{-e^{\varphi }}$.

Figure 3

Variation of the predicted tensor-to-scalar ratio, $r$, with complexity. For both sets of basis operators, the lowest achievable $r$ decreases with the complexity of the potential. We also plot the prediction of the MDL potentials in red, where the result using the $k\log (n)$ prior is solid and with the language model is dashed. The blue line shows the predictions of the potentials which maximize the likelihood at a given complexity.

Figure 4

The predicted tensor-to-scalar ratio and spectral index for the ten best models in each analysis (Tables 1, 2, 3, 4). The 68% and 95% CL regions from Eqs. (16) and (17) are shown in blue. We color the points by the change in description length relative to the optimal model for each analysis, such that darker points indicate better potentials.