Direct Reconstruction of Dark Energy

An important issue in cosmology is reconstructing the effective dark energy equation of state directly from observations. With so few physically motivated models, future dark energy studies cannot only be based on constraining a dark energy parameter space. We present a new nonparametric method which can accurately reconstruct a wide variety of dark energy behavior with no prior assumptions about it. It is simple, quick and relatively accurate, and involves no expensive explorations of parameter space. The technique uses principal component analysis and a combination of information criteria to identify real features in the data, and tailors the fitting functions to pick up trends and smooth over noise. We find that we can constrain a large variety of $w(z)$ models to within 10%–20% at redshifts $z\lesssim 1$ using just SNAP-quality data.

Figure 1

Extracting $w=-1$. The first 10 eigenfunctions for $\Lambda$ are shown (left). We take two separate reconstructions, using $M=4$ and $M=5$ ($N=10$), and generate the errors on ${\mu }_{\mathrm{resid}}$ via a Monte Carlo calculation (with 100 samples each for clarity). The two reconstructions are combined with equal weight to give the $1\mathrm{\text{−}}\sigma$ errors. The choice of number of eigenfunctions is not yet favorable; beyond $z\sim 1.2$ (not shown) the reconstruction is hopeless for these eigenfunctions.

Figure 2

Reconstruction of $w=-1$ using the CIC criteria, Eq. (4), with $s=0.2$. All [$N$, $M$] values satisfying Eq. (4) are simulated through Monte Carlo calculations using 200 runs each (left); the results are bundled into a single probability distribution (middle) from which we infer $1\mathrm{\text{−}}\sigma$ errors at each redshift, shown left. For this example, the CIC selects for $M=2$, 3 for each $N$. We show two probability distributions for $N=10$ (middle), as well as the combination for all $N$, $M$ used. It is at the stage of bundling up the different $N$, $M$ eigenmodes into one probability distribution where the method becomes nonparametric. The combination of different eigenmodes leads to slight non-Gaussianity of the distribution. In this example, $s$ can be increased all the way up to 1 which shrinks the errors even further, and allows tight constraints on $w$ over the full range of $z$ (right).

Figure 3

Reconstruction of evolving dark energy. With a choice of $s=0.2$ and $\kappa =5$ we can reconstruct $w$ to $z\sim 1$ with $\sim 10\%–20\%$ accuracy.

Figure 4

Reconstruction of $w$ using the constitution SN Ia.