Since cosmic acceleration was first discovered in 1998 by Riess et al., physicists have predicted the existence of dark energy to explain the accelerating expansion of the universe. Meanwhile, dark energy theory can be used to explain the cosmic microwave background (CMB) anisotropies very well. The research of dark energy mainly focuses on two aspects included physical models and its properties concerning whether or not dark energy density evolves with time. The physical models are usually proposed from the physical nature of dark energy density and pressure, the other is independent of physical models and can be verified by reconstructing the equation of state w(z) for dark energy. The reconstruction of the equation of state includes parametric and non-parametric methods. The parametric model is still not found which has evident superiority compared to the ΛCDM model, so we adopt non-parametric method to reconstruct the equation of state of dark energy.
We have presented a new, nonparametric technique to reconstruct the equation of state. In order to avoid instability of the derivative for the functional data, we linearize the luminosity-distance integral formula in near-flat space by employing numerical integral based on Lagrange interpolation, and proposing a method of combining principal component analysis (PCA) and biased estimation on the basis of ridge regression analysis to reconstruct the regression parameters. We also present a principal component selection criterion to better distinguish between ΛCDM and w(z) ≠-1 models. In order to test the validity of the method, we construct hypothetical the luminosity-distance data sampled from various equations of the state, which can be used to reconstruct w(z). The results are shown in Fig.1, which indicates the method can be used to determine the most probable behavior of w(z) and to infer how likely a target trajectory is given the current data. Thus it can be used to accept or reject classes of the ΛCDM model.
We applied this method for the dark energy equation of state. When only using Type Ia Supernova (SNIa) JLA sample, the results indicate that the equation-of-state has not obviously deviated from w=-1, or it may be consistent with ΛCDM. In the future, we hope to observe more high-redshift SNIa data. It will be more convenient to reconstruct the dark energy equation of state.
