Monte Carlo (MCMC) technique. The above problems can then be solved in turn i.e. the exploration of parameter space can overcome runs that become trapped in local minima (i); initial priors can prevent unrealistic solutions (ii) and the reversal of components (iii); model comparison tests can help determine the most probable morphology (iv); and finally the outcome of the MCMC gives a proper description of the parameter uncertainties (v).
2.1.4
Chapter outline
This chapter introduces the use of an adaptive Bayesian MCMC algorithm and with emphasis on the methods, features and uses of such a technique. We demonstrate the feasibility of statistical inference based on obtaining galaxy morphology. A detailed exploration of the influence of the prior distribution, and explicit examples of model comparisons between single Sérsic and two-component bulge+disc models are also presented. The chapter is organised as follows. In Section 2.2, we describe the basic formalism of the inference methodology with an overview of Bayesian statistics and model generation. In Section 2.3 we give an overview of Bayesian statistics and a description of the likelihood and priors used for the new algorithm. We then describe the intricacies of the algorithm as well as how we achieve robust outcomes with convergence tests in Section 2.4. How we compare different model fits is discussed in Section 2.5. Finally, in Section 1.6 we discuss and summarise the algorithm presented.
2.2
Inference Methology
In this section, we describe the main attributes and implementation details of a new 2D galaxy photometric decomposition software code. For writing convenience the algorithm has been titled
PHI
(2D PHotometric decompositions using Bayesian Inference). The version ofPHI
used in this thesis is implemented using theInteractive Data Visulisation(IDL) software language. The flow chart forPHI
is as follows:1.
PHI
reads in either a list or single FITS file for the image(s) with their corresponding point spread function (PSF) and a map containing the sigma values for each pixel which corresponds to the error on the image.2. The user can supply initial guesses to the parameters in the model as well as what type of prior for each parameter is desired. However, the need to input initial parameters is not essential for the algorithm to function correctly.
1D Initial guess Data Proposal of parameter set, 𝜃 Likelihood + Priors Marginal 𝜃 Model counts Instrumentation effects Photometric model MCMC Integration
Figure 2.1: The control flow of the 2D photometric decomposition MCMC algorithm. 3.
PHI
then simulates the galaxy image with the user chosen models for each component. It thenuses a fast fourier transfrom to convolve the model with the PSF image. The PSF can either be in a functional form, or user provided.
4. The likelihood and then the posterior probability are calculated for the current set of parame- ters. These values are fed into the MCMC engine of the code which is based in four phases of a MCMC algorithm. See Section 2.3 for further information.
5. Steps 3-4 are iterated as necessary until the full posterior has been mapped or the user defined iteration maximum is reached.
Figure 2.1 shows a summary flow chart of the above list. Each step will now be covered in detail.
2.2.1
2D-Photometric model functions and pixel sampling
As galaxies can be considered to be primarily two-component systems we will assume to first order that the observed surface brightness distribution of galaxies is composed by the sum of a bulge and a disc. For this chapter the definitions of the components are purely photometric, for example, the
2.2. Inference Methology
bulge is described to be the excess of light over the inner extrapolation of an exponential disc. In this Thesis, elliptical galaxies are considered to be single component systems which have been commonly described in a similar fashion to bulges.
To represent the radial distribution of the stellar light i.ethe surface brightness distribution, of each component, various mathematical functions are used. Over the years, observers have changed their opinion on what model function best describes which component. Today, there is a consensus on the function that best describes spheroidal objects such as the ordinary elliptical galaxy and the bulges of disc galaxies. This function is Sérsic’s 1968 generalization of de Vaucouleurs’ 1948; 1956
R1/4 model to give the R1/n surface density profile. The Sérsic profile which describes how the projected surface-intensityI varies with the projected radiusRhas the form,
I(R) =Ie exp¦−bn R Re
1/n
−1©, (2.3)
whereIeis the intensity at the effective radiusRethat encloses half of the total light from the model (Ciotti (1991); Caon et al. (1993)). bn is defined in terms of the third parameter n, the Sérsic
index, which describes the concentration of the light profile. When n = 1 the model follows an exponential surface-intensity profile and n = 4 reproduces the de Vaucouleurs’ model; thus the Sérsic profile can describe the main body of observed spheroidal objects. The term bn is estimated to bebn≈1.9992n−0.3271 within the range 0.5<n<10 (Capaccioli 1989). The exact value of
bn can be obtained by solving the complete gamma functionΓ(2n) =2γ(2n,bn), whereγ(2n,bn)is the incomplete gamma function (Ciotti, 1991). A detailed review of Sérsic’s profile plus associated quantities has been provided by Graham & Driver (2005).
Galaxies in the Universe can be broadly distinguished into two groups; spheroids and discs. It was discussed above that spheroidal objects can be described by the Sérsic profile, with observed objects spanning a range of concentrations and sizes. Galaxies with large-scale stellar discs are termed ‘disc galaxies’ and commonly have centrally located stellar distributions (i.ethe bulge) that appears as an excess from the relative inward extrapolation of the outer exponential disc light. The exponential extent of the radial distribution of the starlight from the disc component of disc galaxies has been known for sometime (de Vaucouleurs, 1956; Freeman, 1970), with the intensityIchanging withRin the form
I(R) =I0 exp¦− R
h
©
whereI0is the central intensity andhis the e-folding disc scale length. As mentioned above, a value forn=1 for the Sérsic profile will also achieve an exponential model (Graham & Driver, 2005).
For this work the components of galaxies are characterised by elliptical and concentric isophotes with constant (and likely different) ellipticity ε = 1−q (whereq is the ratio between the semi- minor and semi-major axis of the ellipse) and position angleθPAin degrees counter-clockwise from
the vertical axis of the image. The image function deals with a converted ellipticity of the form
q=b/a(=1−ε)and the position angle as the angle relative to the image x-axis PA(=θPA+90 deg)
The isophotes are centred on (x0, y0) with the projected radius given by
r =xp2+ y 2 p q2 (2.5)
wherexpandypare coordinates in the reference frame centred on the image-function centre(x0,y0)
and rotated to its position angle:
xp= (x−x0)cos(PA) + (y−y0)sin(PA) yp=−(x−x0)sin(PA) + (y−y0)cos(PA)
(2.6)
This projected radius is then used to compute the surface brightness distributions as discussed above.
Most published references state the need to oversample the central pixels of the image. This is due to the sharp gradient in flux between adjacent pixels in the centre. Oversampling a fixed region or the entire image can be highly inefficient. GALFIT oversample pixels based on their distance from the centre of the image with the exception of Nuker profile Peng et al. (2002) while PROFIT uses the gradient along the minor axis of the profile and utilises a multilayered oversampling approach Robotham et al. (2017). Figure 2.2 and Fig.2.3 show how increasing the Sérsic index increases the gradient of the profile in the central regions.
PHI
adapts the oversampling region by calculating the Euclidean norm of the gradient vector at every pixel of the image using the adjacent pixels. Where the gradient is over a tolerance specified by the user the pixel is subsequently flagged. Flagged pixels are divided into smaller grids (10×10 sub-pixel grids) with each sub-pixel becoming redefined.2.2. Inference Methology
Figure 2.2: A range of two-component models with the intensity along the y-axis and the radius normalised with the effective radius of the Sérsic profile on the x-axis. The black solid line is a Sérsic profile withn=1. Each frame has a different value of the ratioRe/hwith exponential component designated with the coloured lines. The colour of the line represents both the B/T shown in the larger colour bar on the right and theB/Dwithin one effective radius shown by the individual colour bars to the right of each frame.
Figure 2.3: A range of two-component orientations. See the caption to Fig. 2.2. The solid line shows a Sérsic profile withn=4.