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Once observations have been obtained, the analysis and interpretation of the data are the next steps. The result of an analysis can be evident, such as the clear (by-eye) detection of a known molecular feature in a spectrum. Yet, in most cases one wants to answer more complicated questions like “what is the abundance of molecule X (with errorbars)”, “what is the upper abundance limit for an undetected molecule?” or “what is the vertical temperature profile of an atmosphere which gives rise to the observation?”. The answers to such questions lie in some sort of comparison of the observational data with some sort of model, the choice of which can influence the results and the implications of which need to be kept in mind during the analysis. In this section I want to summarize the tools which are available for drawing such inferences, also known as ’retrievals’, and concentrate mostly on methods using the Markov Chain Monte Carlo (MCMC) or Nested Sampling approach.

Self-consistent models and their data – model comparison

Self-consistent models typically calculate atmospheric observables such as emission or trans- mission spectra by solving for the atmospheric structure in a physically consistent fashion. This means that input parameters are physically meaningful quantities such as atmospheric elemental abundances, planetary mass, radius and luminosity, host star radius and temper- ature, and the orbital distance between the planet and star. The models then solve for the atmospheric temperature and abundance structure, as well as the planet’s radiation field, by enforcing physical concepts such as radiative-convective equilibrium and chemical equi- librium (see sections3.6and2.3.1, respectively). The result are the “correct” planetary ob- servables, “correct” in the sense that they fulfill all the physical constraints that make up the model. Such models are highly useful to explore the influence of parameters as, say, the host star spectral type on planetary atmospheric structures and spectra. They allow to explore the atmosphere’s physical behavior if the physical assumptions entering into the model are justified. The petitCODE which I constructed as part of my PhD project is such a self-consistent code (see Chapter3), and the example of varying the host star spectral type is described in Chapter4, Section4.5.

For quantitative data evaluation self-consistent models are of limited use, however: they are usually computationally expensive, as they often try to satisfy the underlying physical constraints in an iterative fashion. Thus, it can be cumbersome to carry out a 2_minimiza- tion, for example, because many model realizations may have to be calculated. This problem can be circumvented by calculating, and interpolating in model grids, but only if the grid range and dimensionality is not too large. Results of my code were used for such an ap- proach, in combination with an MCMC framework, seeSamland et al.(2017). Alternatively, only a small set of models is calculated, and a first order, more qualitative characterization can be carried out by exploring which model realization is in closest agreement with the data. This has been done in, e.g.,Fortney et al.(2005);Burrows et al.(2005,2007);Fischer

28 Chapter 1. Introduction

et al.(2016), and also I contributed to such studies (Mancini et al. 2016a,b;Southworth et al. 2017;Mancini et al. 2017). The second disadvantage is more fundamental: If there is an im- portant physical phenomenon which is not covered by the self-consistent model, then it will not be able to find a meaningful best-fit to the data.

Parametrized models and bayesian retrievals

A solution around the numerically costly self-consistent models is the construction of pa- rameterized models. These models have the molecular abundances and temperature values of the atmospheric layers as free parameters. For computational convenience most retrieval codes use a few-parameter function to describe the atmospheric temperature profile, and use this function’s parameters as free parameters instead. In this case, the need for fulfilling certain physical constraints is neglected, and the spectra of many model realizations can be calculated quickly.

There exist multiple forms of retrieval approaches. One is, for example, the so-called Op- timal Estimation method, which is essentially a 2_{minimization between the model and the} data, but this method assumes a Gaussian distribution of the retrieved parameters, and thus their uncertainty estimate may be inaccurate (see, e.g.,Line et al. 2013b, and the references therein).

MCMC and Nested Sampling make use of Bayes’ Theorem, P (B|A)P (A) = P (A|B)P (B), where P (A) is the probability that event A is observed and P (A|B) is the conditional prob- ability that A is observed, given that the event B has been observed. Thus, the probability that a set of given free parameters x describes the observed spectral data d is

P (x|d) = P (d|x)P (x)

P (d) , (1.15)

where P (d|x) may simply be calculated from the 2 _{between the model and the data, such} that P (d|x) = e 2_/2

, and P (x) encapsulates any prior knowledge about the parameter dis- tribution, such as the fact that temperatures cannot become negative. It is thus also called ’prior’. Finally, P (d), also called ’model evidence’, is simply the integral of P (d|x)P (x) over all x values. Hence, all the quantities on the right hand side (RHS) of Equation 1.15

are straightforward to calculate. However, if the model describing the atmosphere is highly dimensional (such as 5 parameters describing the molecular abundances and 5 parameters describing the temperature structure, which yields 10 dimensions) getting a good estimate of P (x|d) can become quickly unwieldy, especially because we require the model evidence P (d), which is a 10-dimensional integral! This is where MCMC comes into play, as it allows to efficiently sample P (x|d) using a Monte Carlo approach (also see Section6.2), efficiently meaning that still of the order of 106_{model evaluations need to be carried out. Nonetheless,} one is rewarded with samples of the actual multidimensional parameter distribution func- tions, which allows to assess the parameter uncertainties, but also correlations between the retrieval parameters.

Even though I just said that the calculation of the model evidence is too numerically costly, there exists the method of so-called nested samplingSkilling(2004), which can do this

in a numerically efficient way, by estimating which volume dX = P (x)dx maps to a certain P (d_{|x) value interval. The algorithm goes to increasingly narrower P (d|x) iso-contours of} larger P (d|x) values, which is why it is called ‘nested sampling’. As a by-product, nested sampling generates samples of P (x|d). Because it is so efficient, nested sampling allows to carry out model comparisons, i.e. answering the question if a parametrized model Ma is better than another parametrized model Mb. Here “another model” really means a different model, not just a different value choice of the free parameters of model Ma. Mb may have completely different free parameters. Model comparisons are interesting because they allow to answer the question: “Given the data, which of the two models is the most likely one?”. This helps to prevent preferring model Ma, which may fit the data better than model Mb, but purely because model Ma has so many free parameters that it is essentially able to fit any data well (which is called ’overfitting’). For answering which of the two models is more likely, given the data, we can write, using Bayes’ theorem,

P (M|d) = P (d|M)P (M) P (d) (1.16) which yields P (Ma|d) P (M_b|d) = P (d|Ma)P (Ma) P (d_|Mb)P (Mb) = P (d|Ma) P (d_|Mb) = Bab, (1.17) where it was assumed that we don’t have any prior knowledge regarding which of the two models is more likely, in general. Bab is also called ’Bayes factor’. Now, P (d|Ma) is nothing more than the model evidence, i.e. the norm for which I neglected to write “|M” in Equation1.15. Hence, if a model Mais favored over model Mb it will hold that Bab > 1. During the calculation of the model evidences the nested sampling algorithm automatically draws samples of P (x|d, M), such that the posterior parameter distributions are found at the same time as the preferred model. If more than two models are considered, the best of these models may be found by a pair-wise comparison of all model evidences.

There exist applications of optimal estimation (Line et al. 2012; Barstow et al. 2013), MCMC (Madhusudhan et al. 2011b;Benneke & Seager 2012;Line et al. 2013b), and nested sampling retrieval (Benneke 2015; Waldmann et al. 2015b; Line & Parmentier 2016; Lavie et al. 2016; MacDonald & Madhusudhan 2017) in the literature, with the latter becoming the state-of-the-art because in spite of nested sampling being the most numerically expen- sive method, it allows the direct comparison of different models. The most recent study by (MacDonald & Madhusudhan 2017) is a particularly good example to show the usefulness of nested sampling retrievals: the authors analyzed the transmission spectrum of HD 209458b, for which it was previously unclear if its weak water signals are due to its atmosphere be- ing cloudy, or depleted in water. By carrying out a nested sampling retrieval they found that a third model is the most likely of the three, which mixes a clear and a cloudy trans- mission spectrum, corresponding to a partially cloudy terminator. Consequently the water abundance was slightly decreased, but not as much as in the case of the clear atmosphere.

30 Chapter 1. Introduction