Cosmological parameter estimation using particle swarm optimization

Constraining theoretical models, which are represented by a set of parameters, using observational data is an important exercise in cosmology. In Bayesian framework this is done by finding the probability distribution of parameters which best fits to the observational data using sampling based methods like Markov chain Monte Carlo (MCMC). It has been argued that MCMC may not be the best option in certain problems in which the target function (likelihood) poses local maxima or have very high dimensionality. Apart from this, there may be examples in which we are mainly interested to find the point in the parameter space at which the probability distribution has the largest value. In this situation the problem of parameter estimation becomes an optimization problem. In the present work we show that particle swarm optimization (PSO), which is an artificial intelligence inspired population based search procedure, can also be used for cosmological parameter estimation. Using PSO we were able to recover the best-fit $\Lambda$ cold dark matter (LCDM) model parameters from the WMAP seven year data without using any prior guess value or any other property of the probability distribution of parameters like standard deviation, as is common in MCMC. We also report the results of an exercise in which we consider a binned primordial power spectrum (to increase the dimensionality of problem) and find that a power spectrum with features gives lower chi square than the standard power law. Since PSO does not sample the likelihood surface in a fair way, we follow a fitting procedure to find the spread of likelihood function around the best-fit point.

Figure 1

In this figure the red line shows a Markov chain which has been obtained from a typical run of COSMOMC and the green line shows the trajectory of a PSO particle, along the same dimension i.e. ${\Omega }_b{h}^2$. The Markov chain as well as a PSO trajectory can begin anywhere in the range and progressively move towards the best-fit location. However, in the case of PSO the particle approaches towards the best-fit location (Gbest) in an oscillatory manor with successively decreasing amplitude, which is not the case for a Markov chain since its step size does not vary much. Only after a sufficient number of PSO steps the particle positions and the Markov chain converge. Since there are more number of points for the Markov chain as compared to the PSO, we use $x$-scale such that we have five Markov points for every PSO point.

Figure 2

In this figure the red and the green points show the distribution of the positions of PSO particles and samples from a Markov chain, respectively, in the same plane. From the figure it can be noticed that in the initial stage the scatter of PSO particles is very large (see Fig. 1 also), however, close to the convergence all particles get confined in a very compact region. The distribution of the sample points in the case of Markov chain is much more symmetric than in PSO. We suspect that this is due to the different role played by the stochastic variables (random numbers) in PSO as compared to that in the Markov chains. The nonsymmetric distribution makes PSO less favorable if we want to find the shape of the likelihood close to the best-fit values (in order to report errors) in comparison to the Markov chain.

Figure 3

The solid line in the above figure shows the change in the fitness function $-2\log \mathcal{L}$ as PSO steps progress, and the dashed line shows the value for the WMAP seven year data. From this figure it can be noticed that in the beginning improvement in the value of the fitness function is quite rapid, but after some time it saturates, primarily because once the particles reach close to the global maximum as given by the Gbest their velocities drop.

Figure 4

In each panel of the above figure we show the location of the best-fit points (location of Gbest) and the average location of PSO particles, by the solid and dot-dashed lines, respectively. We also show the best-fit values given by the WMAP team by dashed lines. From the above figure it can be noticed that as PSO progresses the average location of PSO particles and the location of the Gbest converge, which can be used as a robust check. For most cases the best-fit values obtained by PSO match well with standard $\Lambda$ cold dark matter (LCDM) model values, but there are some differences also (see Table ).

Figure 5

The red, black and blue lines in the above figure represent the best-fit angular power spectrum recovered from PSO, standard LCDM power spectrum and the binned power spectrum of WMAP seven year data, respectively. Note that the PSO best-fit angular power spectrum is very close to that provided by the WMAP team. The small difference (see Table ) between the PSO best-fit parameters and the WMAP best-fit parameters leads to a difference of 7 in ${\chi }_{\mathrm{eff}}^2$ (it is smaller by 7 for PSO).

Figure 6

Panels in this figure show the two-dimensional contour plots for various pairs of the six cosmological parameters ${\Omega }_b{h}^2$, ${\Omega }_c{h}^2$, ${\Omega }_{\Lambda }$, $n_s$, $A_S$ and $\tau$. We have considered a grid of size $24\times 24$ for our exercise considered the $3\sigma$ region, (where $\sigma$ is the error computed from the fitting) around the best-fit point. In the case when we do not have any idea about the $\sigma$ we can give any other trial value also. The contours in the figure (from inside) are for $\Delta {\chi }_{\mathrm{eff}}^2=2$, 4, 6, 8 and 10. Note that the contour plots not only give an idea about the area around the best-fit region, they also clearly demonstrate correlation between various cosmological parameter.

Figure 7

Same as in Fig. 6 but for different combination of cosmological parameters.

Figure 8

This figure shows how the binned primordial power spectrum (20 logarithmic bins over the $k$ range) changes as PSO progresses (line 1 is for the initial PPS and 6 is for the final PPS). The lower and upper values of the power in bins are represented by the dashed line. Starting with a power law PPS we found that a power spectrum which has low power in some bins and high in others fits better than a power law model.

Figure 9

The red, black and blue lines in the above figure represent the best-fit angular power spectrum recovered from PSO, standard LCDM power spectrum and the binned power spectrum of WMAP seven year data, respectively. Note that at low $l$ the angular power spectrum with binned PPS fits better as compared to the standard power law PPS to the observed data (the improvement in $\Delta {\chi }_{\mathrm{eff}}^2$ is around 7).