Precision cosmology with Padé rational approximations: Theoretical predictions versus observational limits

We propose a novel approach for parametrizing the luminosity distance, based on the use of rational “Padé” approximations. This new technique extends standard Taylor treatments, overcoming possible convergence issues at high redshifts plaguing standard cosmography. Indeed, we show that Padé expansions enable us to confidently use data over a larger interval with respect to the usual Taylor series. To show this property in detail, we propose several Padé expansions and we compare these approximations with cosmic data, thus obtaining cosmographic bounds from the observable Universe for all cases. In particular, we fit Padé luminosity distances with observational data from different uncorrelated surveys. We employ Union 2.1 supernova data, baryonic acoustic oscillation data, Hubble Space Telescope measurements and differential age data. In so doing, we also demonstrate that the use of Padé approximants can improve the analyses carried out by introducing cosmographic auxiliary variables, i.e., a standard technique usually employed in cosmography in order to overcome the divergence problem. Moreover, for any drawback related to standard cosmography, we emphasize possible resolutions in the framework of Padé approximants. In particular, we investigate how to reduce systematics, how to overcome the degeneracy between cosmological coefficients, how to treat divergences and so forth. As a result, we show that cosmic bounds are actually refined through the use of Padé treatments and the thus derived best values of the cosmographic parameters show slight departures from the standard cosmological paradigm. Although all our results are perfectly consistent with the $\mathrm{\Lambda }\mathrm{CDM}$ model, evolving dark energy components different from a pure cosmological constant are not definitively ruled out. Finally, we use our outcomes to reconstruct the effective Universe’s. equation of state, constraining the dark energy term in a model-independent way.

Figure 1

Analytical curves for the luminosity distance of the $\mathrm{\Lambda }\mathrm{CDM}$ model compared to its Taylor and Padé approximations. As we see, the Taylor polynomials $T3$, $T4$ and $T5$ tend to quickly diverge outside the region $z\le 2$. At the same time, not all the Padé approximants give good approximations of this model. For example, $P_{11}$, $P_{13}$ and $P_{23}$ give spurious singularities when used to approximate the $\mathrm{\Lambda }\mathrm{CDM}$ model. We will see how to avoid this problem in the numerical analysis. On the contrary, $P_{21}$, $P_{22}$ and $P_{32}$ give excellent approximations to the $d_L$ derived from assuming $\mathrm{\Lambda }\mathrm{CDM}$. Additional comments have been reported in the text.

Figure 2

Analytical curves for the luminosity distance of the $\omega \mathrm{CDM}$ model compared to its Taylor and Padé approximations. As we see, the Taylor polynomials $T3$, $T4$ and $T5$ tend to quickly diverge outside the region $z\le 2$. At the same time, not all the Padé approximants give good approximations of this model. For example, $P_{11}$, $P_{13}$, $P_{23}$ and $P_{41}$ give spurious singularities when used to approximate the $\omega \mathrm{CDM}$ model. We will see how to avoid this problem before our numerical analysis. On the contrary, $P_{21}$, $P_{22}$ and $P_{32}$ give excellent approximations to the $d_L$ derived from assuming $\omega \mathrm{CDM}$. Additional comments have been reported in the text.

Figure 3

The Padé approximants used for the different $\theta$. The approximations enclosed in the triangle give conclusive results.

Figure 4

One-dimensional marginalized posteriors for the parameter set $\mathcal{A}$. $SC$ stands for standard cosmography.

Figure 5

One-dimensional marginalized posteriors for the parameter set $\mathcal{B}$. $SC$ stands for standard cosmography.

Figure 6

One-dimensional marginalized posteriors for the parameter set $\mathcal{C}$. $SC$ stands for standard cosmography.

Figure 7

Marginalized posterior constraints for the parameter set $\mathcal{A}$ using $P_{21}$.

Figure 8

Marginalized posterior constraints for the parameter set $\mathcal{B}$ using $P_{31}$.

Figure 9

Marginalized posterior constraints for parameter set $\mathcal{C}$ using $P_{23}$.