Machine learning cosmic inflation

We present a machine learning approach, based on the genetic algorithms (GA), that can be used to reconstruct the inflationary potential directly from cosmological data. We create a pipeline consisting of the GA, a primordial code and a Boltzmann code used to calculate the theoretical predictions, and cosmic microwave background (CMB) data. As a proof of concept, we apply our methodology to the Planck CMB data and explore the functional space of single-field inflationary potentials in a nonparametric, yet analytical way. We show that the algorithm easily improves upon the vanilla model of quadratic inflation and proposes slow-roll potentials better suited to the data, while we confirm the robustness of the Starobinsky inflation model (and other small-field models). Moreover, using unbinned CMB data, we perform a first concrete application of the GA by searching for oscillatory features in the potential in an agnostic way, and find very significant improvements upon the best featureless potentials, $\mathrm{\Delta }{\chi }^2<-20$. These encouraging preliminary results motivate the search for resonant features in the primordial power spectrum with a multimodal distribution of frequencies. We stress that our pipeline is modular and can easily be extended to other CMB data sets and inflationary scenarios, like multifield inflation or theories with higher-order derivatives.

Figure 1

Flowchart of a typical run of a genetic algorithm (from Ref. [39]).

Figure 2

A flowchart briefly summarizing our methodology: First the GA creates a set of random functions ${\mathrm{GA}}_i(\varphi )$ of the scalar field $\varphi$, where $i\in \{1,\dots ,N\}$ and $N$ is the population size. Each of these functions is then combined with the prior in order to create a potential $V_i(\varphi )$, that is subsequently fed to the “primordial module” that calculates the inflationary predictions. The latter are passed on to a Boltzmann code, in our case CLASS, in order to calculate the theoretical CMB predictions of each potential, which are in turn passed to the Planck likelihood clik (including the temperature, polarization and lensing spectra) that returns the log-likelihood, i.e., the ${\chi }^2$, for each potential. In our analysis the primordial package used is the one included in CLASS, but any other one could also be used. Finally, this process is repeated hundreds of times to ensure convergence.

Figure 3

Typical runs of the GA showing the minimum ${\chi }^2$ in the population as a function of the generation. Individual colored lines correspond to independent runs with different random seeds for the initial conditions. Dashed lines correspond to the run with $\alpha =0.4$ and solid lines to the one with $\alpha =1$. The black dashed line represents the ${\chi }^2$ of the prior, shown for comparison. Left panel: Quadratic inflation model as a prior, and polynomials chosen as a grammar. The GA easily improves upon the prior, although some independent runs perform better than other ones. Larger steps in functional space, for $\alpha =1$, help the GA to find a better best-fit. Right panel: Starobinsky inflation model as prior, and polynomials chosen as a grammar. The GA hardly improves on the prior which is already well-fitted to the data, as can be seen from the modest value of the $\mathrm{\Delta }{\chi }^2$. Changing $\alpha$ does not help, showing the robustness of the prior with respect to the data. In both panels and for all cases, the GA converges to a minimum.

Figure 4

Inflationary potentials reconstructed by the GA at the last generation (blue lines), including the best-fit function (orange line), and with the prior shown for comparison (green line). The upper panels show the total potential constrained by data, the lower panels show the same potential but rescaled by the prior and zoomed around unity. Left panels: Quadratic inflation model as a prior, polynomials chosen as a grammar, and $\alpha =1$. Right panels: Starobinsky inflation model as prior, polynomials chosen as a grammar, and $\alpha =1$. As described in the text, the scatter of the potentials (blue lines) around the best-fit is a proxy for the error estimate of our reconstructions: here we only show the 68.3% best of all the functions in the last generation (across all seeds), after ordering them in terms of their ${\chi }^2$.

Figure 5

The evolution of the population fitness as a function of the generations for the Starobinsky prior with fast trigonometric grammar and the unbinned CMB data, and for $\alpha =1$. Individual colored lines correspond to independent runs with different random seeds for the initial conditions, and the black dashed line represents the ${\chi }^2$ of the prior, shown for comparison.

Figure 6

As in Fig. 4, but for the Planck unbinned data using the Starobinsky inflation model as a prior, a fast trigonometric grammar and $\alpha =1$. Upper panel: The total potential, as constrained by the data, deviations from the prior being invisible by eye. Lower panel: The same potential, but rescaled by the prior and zoomed around unity. We recall that blue lines correspond to the 68.3% best functions in the last generation across all seeds and therefore represent the error scatter. The structure of this error scatter unveils a clear pattern for these fast oscillatory functions.

Figure 7

Left panel: The dimensionless primordial scalar power spectrum $[k^3/(2{\pi }^2)]P_{\zeta }(k)$ as a function of the wave number $k$ for the best-fit candidate found by the GA with the Starobinsky inflation model as a prior, fast trigonometric grammar, $\alpha =1$ and using Planck unbinned data (orange line). As can be seen, the oscillations in $\varphi$-space from the potential are induced on the power spectrum in $\log k$-space. The power spectrum from the prior (blue line) is shown for comparison. Right panel: The relative variation of the best-fit power spectrum with respect to the prior, given by Eq. (12).

Figure 8

Upper panel: Fourier transform of the best-fit potential giving the multimodal distribution of frequencies in field space. Here in the abscissa we have represented the inverse of these frequencies, i.e., the wavelengths in $M_{\mathrm{Pl}}$ units, which could be interpreted as axion-decay constants $f$. Lower panel: Fourier transform of the primordial scalar power spectrum resulting from the best-fit potential, giving the distribution of frequencies in $\log k$ space, often denoted as ${\omega }_{\log }/H$ in the literature.

Figure 9

Distribution of the axion-decay constants $f$ from the 68.3% best of all functions at the last generation, across all random seeds. We used a constant binning width in log-space of ${\log }_{10}(\mathrm{\Delta }f/M_{\mathrm{Pl}})=0.05$. The structure of the histogram shows that specific values are more often encountered than other ones.

Figure 10

The difference between the GA best-fit of the unbinned data with the fast trigonometric functions and the Starobinsky prior for the $TT$, $TE$, and $EE$ spectra. The black points with light-red error bars correspond to the binned Planck 18 data, while the light-gray points for $\ell >30$ correspond to the unbinned ones. Note that for $\ell <30$, binned and unbinned data are the same. Upper panel: The full range of the data. The GA does not address the low-$\ell$ anomaly in the CMB. Lower panel: Zoom in the high-$\ell$ region, where the GA systematically improves the fit to CMB residuals. As can be seen in both regimes, the GA is never overfitting the unbinned data (which are the ones used in the analysis), but rather improving the overall fit. This is further confirmed in Appendix.

Figure 11

Distribution of the ${\chi }^2$ values given by the simplified expression of Eq. (a1) for 10,000 realizations, where randomly 20% of the unbinned Planck CMB data points have been removed. We show the $TT$, $TE$, $EE$, and $TT+TE+EE$ combinations and as can be seen, the GA consistently outperforms the prior, thus reinforcing our conclusion that no overfitting is taking place according to the cross-validation criterion.