ner_tags: O O O O O O O O O O O O O O O O O O B-Model I-Model O B-Model O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Model I-Model O B-Model I-Model I-Model O B-Tag O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O B-Software O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O
tokens: show the regions which contain 95% of the models. The central blue line is the median line for Planck TT+lowP+BSH. Here Planck indicates Planck TT+lowP. One might wonder whether it is reasonable to stop at first order in . We have therefore tested a generic expansion in powers of the scale factor up to order : We find that all parameters are very stable when allowing higher order polynomials; the parameters are weakly constrained and going from (the linear case) to (quadratic case) to (cubic expansion) does not improve the goodness of fit and stays compatible with , which indicates that a linear parameterization is sufficient. 5.1.2. 1-parameter varying <italic>w</italic> A simple example of a varying model that can be written in terms of one extra parameter only (instead of ) was proposed in Gott Slepian (2011), Slepian et al. (2014) , motivated in connection to a DE minimally-coupled scalar field, slowly rolling down a potential , analogous to the one predicted in chaotic inflation ( Linde 1983 ). More generally, one can fully characterize the background by expanding a varying equation of state , where: at first order in , which is then the only extra parameter. Marginalized posterior contours in the plane – are shown in Fig. 6 . The tightest constraints come from the combination Planck TT+lowP+lensing+BSH that gives at 68% confidence level, which slightly improves constraints found by Aubourg et al. (2015) . Fig. 6 Marginalized posterior contours in the – plane are shown for 1-parameter varying models (see Sect. 5.1.2 ) for different data combinations. Here Planck indicates Planck TT+lowP. 5.1.3. Principal Component Analysis on <italic>w(z)</italic> A complementary way to measure the evolution of the equation of state, which is better able to model rapid variations, proceeds by choosing in fixed bins in redshift and by performing a principal component analysis to uncorrelate the constraints. We consider different bins in and assume that has a constant value in each of them. We then smooth the transition from one bin to the other such that: with , a smoothing scale , and a binning . We have tested also a larger number of bins (up to ) and have found no improvement in the goodness of fit. The constraints on the vector of values that can assume in each bin is difficult to interpret, due to the correlations between bins. To uncorrelate the bins, we perform a principal component analysis ( Huterer Starkman 2003 ; Huterer Cooray 2005 ; Said et al. 2013 ). We first run COSMOMC ( Lewis Bridle 2002 ) on the original binning values ; then extract the covariance matrix that refers to the parameters we want to constrain: where is the vector of parameters and is its transpose. We calculate the Fisher matrix, , and diagonalize it, , where is diagonal and is the orthogonal matrix whose columns are the eigenvectors of the Fisher matrix. We then define (e.g., Huterer Cooray 2005 ) and normalize this such that its rows sum up to unity; this matrix can be