Systematic effects in strong lensing source reconstruction methods

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Systematic effects in strong lensing source reconstruction methods

Vicent Cipri` a Albero Blanquer - s3807487 Supervisor: Prof.dr.Leon Koopmans Co-supervisor: dr.Giorgos Vernardos

April - July 2019

Kapteyn Astonomical Institute

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Abstract

Gravitational lens systems are used nowadays to investigate the distribution of matter in the universe. Reconstructions of these systems using statistical methods bring with them difficulties that need to be studied. The objective of this report is to find systematic effects that emerge from the methods used for the reconstruction of a source in strong gravitational lenses. Mock lensed systems are created using high-resolution images of galaxies as sources. An adaptive grid is used to reconstruct the brightness of the lensed source. The analysis is focused on the role of the Point Spread Function (PSF) and its impact on the reconstruction, and on different types of regularization (curvature and covariance) for different levels of regularization. The use of an incorrect PSF for the reconstruction is shown as decisive, and the curvature regularization is preferred for the optimal value of the regularization parameter.

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Acknowledgements

I want to thank Prof. Dr. Leon Koopmans for helping me from the beginning until the end. For his valued guidance week after week and for making me improve during this project. Also, I want to thank Dr. Giorgos Vernardos for every new version of the code, for the small talks in which the project advanced so much, and for making me feel comfortable in this research environment that was new for me. Fianlly, I want to thank Laura, Mila and Vicent for always supporting me.

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Introduction

1.1 Context of gravitational lensing

The phenomenon of gravitational lensing merges knowledge and technological advances. Therefore, its discovery, study, and observation need to be shortly explained from a historical point of view.

1.1.1 Short History

A study on the bending of a ray of light caused by gravity appeared for the first time in the ”Berliner Astronomisches Jahrbuch auf das Jahr 1804”. Johann Soldner studied its effects in astronomy and gave the value α = 0.84 arcsec for the deflection angle of a light ray close to the solar limb [28]. Nevertheless, it was not until one century later, that an appropriate framework for the study of this deflection was developed.

That framework was provided by the General Theory of Relativity, presented by Albert Einstein in November 1915 [12, 15, 13, 14]. Einstein itself was the first to calculate the correct deflection angle [13]. His prediction for the Sun yielded a value of α = 1.74 arcsec. Arthur Eddington and his group measured this value within an error of 20% during the solar eclipse of 29 May 1919. This experiment was one of the first confirmations of Einstein’s General Theory of Relativity.

During the following decades, this topic was rarely studied, as this phenomenon was only expected due to star, and therefore unobservable at the time.

Also, the probability of observing it was very small. One of the most important advances in the theory was the idea of the ”Einstein-Chwolson ring”[16, 10].

Some years after, in 1937[31, 32], Fritz Zwicky showed that the phenomenon of a gravitational lens is much more likely to be seen in a galaxy (”extragalactic nebulae”) than in a star.

Theoretical studies about the usefulness of gravitational lenses were done during the next decades. Finally, this field experienced a boost when, after the discovery of quasars in the 1960s, Dennis Walsh, Bob Carswell, and Ray

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Figure 1.1: Radio image taken with e-MERLIN[1] of the Double Quasar, first observed by D. Walsh, B. Carswell, and R. Weymann [27]. Image from [18].

Weymann observed a double image of a quasar created by a massive foreground cluster galaxy in 1979 [27]. A new image of this named Double Quasar can be seen in Figure 1.1.

1.1.2 Applications

Nowadays, gravitational lenses are used to study the universe on all scales. They can be used to measure the Hubble constant and the scale factor, which means studying the expansion of the universe[23]. They are also used to constrain cosmological parameters (volume, matter density, density fluctuations, etc.) in the study of clusters, black holes, dark matter, galaxies and stars, and even to detect small masses or perturbations in them . One of the main applications of gravitational lens systems is the study of the distribution of matter in the Universe, including dark matter. Strong lenses, due to their magnification, also enable astronomers to see galaxies that would be too distant otherwise.

Hence, the field of gravitational lensing is expected to help answer some of the major open questions about the universe, or, at least, to provide us more understanding on how the universe works. A short review of some of the applications of strong lensing can be found in [21].

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Model types and requirements

As highly distorted images of the source comprise the data, statistical methods are needed to compute reconstructions. As a consequence, research in gravitational lensing has become strongly dependent on advanced computer modeling.

Nevertheless, no standard package exists for this modeling. Individual investiga- tors (or groups of them) develop their own gravitational lens modeling software.

This allows more flexibility to develop novel algorithms and methods to model these complex systems [22].

In general, modeling of gravitational lenses can be separated into a “forward” problem, where the image of the lensed source is predicted from a source model, and the“inverse” (or “reverse”) problem, where the source is reconstructed from the lensed images. Besides, we can divide models in parametric and non-parametric ones. Parametric models are commonly used in the “forward” problem, where the source and a lensing mass describe the observed lensed images (i.e. the data) with a few free parameters. Non-parametric models are commonly used in “inverse” problems. One of the largest challenges here is that many models can often fit the observed data almost equally well. It is important to not get confused by the name “non-parametric”, as these models actually require a large number of parameters. With this technique, the reconstruction of the brightness distribution of the source is achieved using a grid of pixels [22]. Some “rules” that an ideal source inversion algorithm should follow are [22]:

1. Limited assumptions about the mass or luminosity distributions.

2. Limited dependence on prior information 3. Ensure physical solutions

1.2 Objectives

The study of strong gravitational lenses leads to difficulties as to how to reconstruct the source that has been lensed, or how to describe the lens potential of a galaxy. As a consequence, numerous different approaches and modeling codes have been developed to solve these issues.

During my bachelor research project, I use a code developed by dr. Giorgos Vernardos and Prof.dr. Leon Koopmans. Using this code, I study a range of systematic effects that can occur during the process of the source reconstruction of a known source. During this process, we assume that the lens mass distribution is known, in order to separate the effects of the source reconstruction from the lens mass modeling, and only study the former.

First, in order to understand the basic features of the code, I test some char- acteristics of the gravitational lensing theory. At the same time, I assess the effects of the data configuration (i.e. the shape of the images) in the goodness of the reconstruction achieved. In the same vein, I study more precise techni- cal aspects and possible problems arising during the reconstructions. Special

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emphasis will be placed in the role of the Point Spread Function (PSF) and the type of source regularization, which is a prior ensuring smoothness in the reconstruction of the source.

1.3 Overview of the project

In Chapter 2, the statistical framework that forms the basis of our reconstruction program is explained. In Section 2.1, the methodology followed during my tests is explained. In Section 2.2, I explain the basics of the Bayesian statistics used to compare models.

The tests are separated into two sets. The first set, discussed in Chapter 3, is centered on understanding how the code works. This is done by testing different aspects of the theory of gravitational lensing, observe how it is able to recreate lensed images, and by removing some parts of the data to assess how the reconstruction works. These tests are done varying parameters that involve both the source and the lens potential. The parameters that I test are the source position in Section 3.2, the Einstein radius in Section 3.3, the axis ratio in Section 3.4, the position angle of the lens in Section 3.5, and the relative position or distance between the source and the lens in Section 3.6. Lastly, I test how changes in the mask affect the reconstruction of the source in Section 3.7. Every test is self-contained. An overall discussion is presented in Section 3.8.

The main part of the project focuses on issues that arise when trying to achieve the best source reconstruction. For this purpose, the first test involves the pixel resolution on the image plane in Section 4.1. The following step tests how using an incorrect Point Spread Function (PSF) for the inverse problem affects the source reconstruction. This is done in Section 4.2. Different values of the regularization value λ and different types of regularization (curvature and covariance) are tested in Section 4.3. Related to the resolution, in Section 4.4, I study the effect of having smoother or more structured data in the reconstruction of the system. Our last step is studying the effect of higher resolution of the adaptive-grid on the source plane (Section 4.5). A final conclusion and a description of possible future work is done in Chapter 5.

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Chapter 2

Methodology

We focus on the source inversion problem, as mentioned in Chapter 1. The technique that is used for our research is based on Bayesian strong gravitational- lens modeling on an adaptive-grid for the reconstruction of the source [25].

In this context, we differentiate between point-like images and extended sources. When studying point-like images, one often focuses on differential time delays, positions, and fluxes of the images [11], providing constraints on the lens potential and its derivatives at discrete positions [20]. When studying extended sources, the objects of study are usually arcs or rings [11]. This project is focused on the modeling of extended sources. An example of extended source is shown in Figure 2.1.

Images of gravitational lens systems, obtained with earth or space-based telescopes, are often described by a set of pixels with a given surface brightness.

Since the number of pixels in general is very large, many free parameters are needed to achieve a reasonable reconstruction of the shape of the source.

To start the modeling, the pixel values are stacked into a vector. The image data vector is called d and the vector of unknown source brightness values is called s. Based on the conservation of surface brightness during the lensing (see e.g. Congdon [11], 4.6.1.), a set of linear equations relating the pixels of the image and the source can be expressed as:

d = BLs. (2.1)

The derivation of this expression can be seen in Warren and Dye (2003) [30].

The matrix L is the lensing operator, which can be Lij = 1 if pixel i in the image plane relates to pixel j from the source or Lij = 0 when those pixels are not related. They can also take fractional values via interpolation. The matrix B is called the blurring operator, which describes the Point Spread Function (PSF) produced by the observations (in particular telescope optics and atmosphere). The photons are spread from their actual direction to where the CCD device receives them. The blurring conserves the flux but does not conserve the surface brightness. During imaging, we introduce this blurring

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Figure 2.1: Image of the Cosmic Horseshoe (SDSS J114833.14+193003.2). The lens galaxy and the source galaxy present lens alignment, generating a lens image which is almost a ring of light. This system was discovered in 2007[26].

The image is taken with HST[2].

effect by using a generated PSF¹. This is studied in one experimental section later on.

If the data of the lensed images (after galaxy-light subtraction) obtained from observations is represented by d^obs with a diagonal covariance matrix Cd, the goodness of fit can be defined as[20]:

χ²= (BLs − d^obs)^TC⁻¹_d (BLs − d^obs). (2.2) An additional feature comes from the level of freedom that the model obtains due to the pixelation of the the source. When the number of free parameters is too large, this can lead to an unrealistic reconstruction of the source. This effect will increase with the noise of the data. To control this, an additional regularization term can be added to the goodness of fit to penalize, if not eliminate, this effect. This additional term can be seen as prior information based on our experience about the surface brightness distribution of the objects that are being lensed. In our case, we use a quadratic regularization term R = s^TH^THs.

The matrix H will define the type of regularization. Usually, the type or regularization is chosen to generate smoother source solution, in better agreement with observations. Besides, its relative strength will be regulated by the regularization parameter λ. With the addition of this term, the discrepancy between

1PSF generation is done using the TinyTim software [9]. http://tinytim.stsci.edu/cgi- bin/tinytimweb.cgi

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Figure 2.2: Schematic overview of the non-linear source reconstruction method.

A subset of the image pixels given by the positions x^s_i on the image plane (filled circles) is cast back to the source plane on y^s_i through the lens equation 2.5.

These form the vertices of an adaptive grid on the source plane. The open circles are cast to the source plane to positions y^d_i. The surface brightness at the empty circles is computed by a linear superposition of the surface brightness at the three triangle vertices that enclose it. [25]

the model and the data can be expressed with a penalty function:

P = (BLs − d^obs)^TC⁻¹_d (BLs − d^obs) + λs^TH^tHs = χ²+ λR. (2.3) We can use a low value of λ to allow a model solution that is more structured, or we can use a high value of λ in order to favor a smoother, usually more reasonable, source. The selection of the value for the regularization parameter is discussed further later in the thesis. I will also compare two different types of regularization: curvature and covariance.

For a given value for λ, the optimal solution for the source can be found by solving the equation ∇sP = 0 for the source-solution vector s. This is equivalent to finding the solution of:

(L^TC⁻¹_d L + λH^TH)s = L^TC⁻¹_d d^obs. (2.4) Having introduced the statistical framework, the next step is to describe how the source grid is constructed. For our reconstructions, a grid-based lensing technique is used. To create the source grid, the data pixels are divided using a spacing of 3x3 pixels as a default. The visual schematics of the image and source plane grids are shown in Figure 2.2.

On the left part of the figure, representing the image plane, two different grids can be differentiated, with filled and open circles. The filled circles mark the positions x^s_i. The vectors d and s defined above characterize the surface brightness distribution on a set of the spatial points x^d_i and y^d_i which are, respectively, in the lens and in the source plane. These points define the grid

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that we use in our reconstruction. Using the lens equation,

y^d_i = x^d_i − ∇ψ(x^d_i), (2.5) these points are cast back to the appropriate positions y^s_i in the source plane, which define the vertices of the adaptive grid in the reconstruction. In the lens equation 2.5, x^d_i is the spatial position of the image plane pixel, with surface brightness d_i (the i^thelement of the vector d). ψ(x^d_i) represents the lensing potential (a more detailed explanation of it can be found in Vegetti and Koopmans, 2009 [25]).

The surface brightnesses at the vertices of the triangles forming the adaptive grid is linearly interpolated to compute the brightness at the open circles that they enclose. In this way, the brightness for all the points in the source plane are computed. As it is been shown, the generated grid for the reconstruction on the source plane is completely constructed from a subset of pixels in the image plane and changes as the lens mass model changes, becoming in this way fully adaptive.

A mask is applied in order to ensure the reconstruction is largely limited to the lensed images and not the surrounding noise-dominated pixels. The mask is auto-generated by the code. The values that it takes are 1 for the pixels those comprise the image on the grid, and 0 for that outside (with some extra space to warrant that no information is lost). For the source, the mask takes value 0 only in the pixels where no constraints from the data are found.

Summary of the procedure: A source model is defined on a finite irreg- ular adaptive grid. This source and the observed data are written as a vector.

Then, a lensing operator is constructed using the lens potential. When apply- ing this to the source model, it gives a lensed source in the image plane that is compared with the data after a PSF convolution. Hence, the data model is created from the source with the lensing and the blurring operators. The goodness of fit χ²is defined, with the addition of a quadratic regularization function to penalize unrealistic models (based on ”experience”), with the regularization parameter λ to weight this term with respect to the goodness of fit. The penalty function is optimized to find the solution for the source model, for a given (i.e.

fixed) lens potential.

2.1 Methodology of the experiments

Even though the experiments are separated into two different test sets, the analysis is similar in both of them. My tests consist in assessing the outcome of the relevant steps in the reconstruction and try to identify ways to improve these or avoid failures in the source reconstruction.

The code works with “JSON”² files from which it reads the data that are processed afterward. Hence, the changes that I study are written in these JSON

2JSON file definition: file that stores simple data structures and objects in JavaScript Object Notation (JSON) format, which is a standard data interchange format.[6]

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Figure 2.3: M83 galaxy as used for doing the tests. Squared image with pixel values for the brightness from 0 to 1. [7]

files. Two different files are used, one for the forward problem and one for the inverse problem. The JSON file for the forward and inverse problems are explained in Appendices 6 and 7, respectively.

The test sets use two different galaxies. The first is Messier 83 (M83), shown in Figure 2.3, which is a barred spiral galaxy. The second galaxy is NGC2623, which is a merger galaxy, and can be seen in Figure 2.4. The main point for choosing these galaxies is that they represent more realistic sources than the very regular shapes that are often used for these simulations. Brightness in both images is scaled to a range from 0 to 1.

2.2 Model comparison

For a thorough study of the statistics that yields to our model comparison, see Vegetti and Koopmans, 2009 [25]. Here, I explain the relevant parts that I use in this work.

Bayesian statistics: Bayesian statistics is a framework that has been shown to be powerful to analyze over model parameters and, as in our case, to compare different models objectively. A good introduction to the required basics is given in section 7.4.5. of the book Astrophysical Applications of Gravi-

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Figure 2.4: NGC2623 galaxy as used for doing the tests. Squared image with pixel values for the brightness from 0 to 1. [8]

tational Lensing, from the XXIV Canary Islands Winter School of Astrophysics [20]. Here, I introduce the necessary concepts to make our comparison between models, when applicable.

If we consider a set of data d and a model of parameters q, we can express their joint probability distribution, using conditional and marginal distributions, as

p(d, q) = p(q|d)p(d) = p(d|q)p(q) (2.6) from where we can deduce simply the Bayes’s Theorem, which is the base of all Bayesian statistics:

p(q|d) = p(d|q)p(q)

p(d) . (2.7)

In this equation, p(d|q) is the probability of the data given the model, also called Likelihood (L(d|q)); p(q) is the prior probability distribution for q; p(q|d) is the posterior probability distribution for the model given the data; and p(d) is the evidence, henceforward called Z. The evidence tells us the probability of a model to generate the data, including the effect of the prior.

Bayes factor: In order to compare models, the Bayes factor is used. It is supposed that there is no a priori knowledge for both models. I consider this because both models are going to be the same except for one change (parameter, psf, etc.). Therefore, the Bayes factor is the ratio of the evidence of both models.

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It can be defined as:

f = Z(M₁) Z(M2)

P (M₁)

P (M2) ≈ Z(M₁)

Z(M2) (2.8)

In equation 2.8, P are the priors and Z are the evidence for each model. Using the value of f to compare between two models, the comparison method that I use rely on the one provided by Kass and Raftery (1995)[19], which is based on Jeffrey’s scale (1961) [17]. Table 2.1 shows this scale which, even though it is only qualitative, can help us to decide between models.

log10f Relation between models 0 to 1/2 Not worth more than a bare mention

1/2 to 1 Substantial

1 to 2 Strong

> 2 Decisive

Table 2.1: Kass and Raftery [19] version of Jeffreys’ scale [17] for a qualitative indication of the meaning of the Bayes factor f .

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Chapter 3

Experimental:

Understanding the code

Having introduced the statistical method that we are using, some small trials are done to understand how the program works. Physical aspects of gravitational lens theory (source position, lens axis ratio, etc.) are tested. The outcome is used to localize systematic effects in source reconstruction. The effect of different masks on the same data is used to understand the capability of the reconstruction method. Because of the use of a different mask for each trial, every data set is different from each other. Hence, comparison between models using the evidence can not be done. Comparison of the models with real gravitational lens systems is made sometimes to show the ability of the program to reproduce real results.

3.1 Default model

To study the differences that changing one parameter produces, a default model is needed. The values used for these specific forward and inverse problems are shown in Appendices 6 and 7, respectively. With these values, the mock lens and the reconstruction that is achieved can be seen in Figure 3.1.

This first plot shows the final result of the methods explained in Chapter 2.

At the bottom-left of the image, the true source is shown. This is a squared image of a galaxy, which, in this case, is Messier 83 (M83). The pixel brightness is rescaled to range values between 0 and 1. Above the true source, there is the data generated from it using a specific lens model, which shown in Appendix 6 for this case. Next to the true source, the reconstruction of the source in the adaptive grid is shown. Above the reconstruction, the lens image model that the reconstructed source generates. The last column shows the difference between the data and the data model, namely the residual; and lastly the errors are shown (calculated using eq. B22 from Suyu et al, 2006 [24]).

The source reconstruction reproduces the general contour and the position

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Figure 3.1: Outcome of the program for source reconstruction with default values. True Source: a squared image of the source with pixel values between 0 and 1. Data: lens image produced by the true source with a defined lens model.

Reconstruction: source reconstruction in an adaptive-grid. Model: lens image produced by source reconstruction. Residual: the difference between data and model. Errors: errors in source reconstruction.

of the true source. However, all the details are lost. For instance, the “substructure” at the bottom-right of the source has disappeared. Hence, the reconstruction using default values is not really good. For the residuals, our goal is to only see the noise. This would imply the residuals looking homogeneous, meaning that the data model does not exceedingly diverge from the actual data.

In the residuals from Figure 3.1 it can be seen that the top image of the lensed source shows both positive and negative residuals significantly exceeding the noise level. These structures reveal that the model needs to be improved to achieve a good source reconstruction. Chapter 4 is more focused on this.

3.2 Source position

The position of the source is defined by x0 and y0, representing distance in arcsec along horizontal and vertical axes, respectively. The default model shown in Figure 3.1 uses source position values x0 = −0.1 arcsec and y0 = 0 arcsec.

As a result, after using the lens equation, four images of the source (the top one is double) have been formed.

Another interesting case is when x0 = −0.1 arcsec and y0 = −0.1 arcsec.

This can be seen in Figure 3.2. This case can be directly compared with the real system which is shown in Figure 3.3. In this real system, a quasar is lensed

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Figure 3.2: Mock lens and reconstruction using the values x₀= −0.1arcsec and y₀= −0.1arcsec for the position of the source.

by a foreground galaxy, creating a four image distribution similar to that on our model.

With values x₀= y₀= 0.0 arcsec for the source position, four symmetrically distributed images of the source are expected because of the symmetry of this particular system. A real example matching this configuration case can be seen in Figure 3.5.

For both cases, the lensed image of the source is similar to a real gravitational lens system. Hence, the systems generated in the code are able to generate lens systems that appear realistic.

On the source reconstruction for the first case (x0= y0= −0.1 arcsec, Figure 3.2), the final result does not really differ from the default model. Nevertheless, for the second case (x0= y0= 0.0 arcsec, Figure 3.4), the source reconstruction shows some structure in the bottom-right part. I relate this improvement with having four separated images in the data. As in the previous trials, part of the images were merged, part of the information was lost or difficult to extract.

This shows that higher definition data conduce to better reconstructions. The latter model is going to be used for the next trials in this Chapter 3.

3.3 Einstein radius

Symmetric ring-like images of the source occur when the source lies exactly behind the lens, as in Figure 3.4. The Einstein radius is then defined as the angular radius of the ring-like image. A derivation of its expression can be found in Wambsganss (2001)[29]. From the theory, increasing (decreasing) the

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Figure 3.3: RXJ1131-1231. Quasar lensed by a foreground galaxy (in the middle of the picture) that creates four images of the background quasar, smearing three of them into a bright arc in the left side of the image.[4]

Figure 3.4: Mock lens and reconstruction with x0= y0= 0.0 arcsec

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Figure 3.5: HE0435-1223. The foreground galaxy (in the middle of the picture) creates four almost evenly distributed images of the distant quasar around it.[3]

Einstein radius also means increasing (decreasing) the mass of the lens.

The default model uses an Einstein radius with a value of b = 0.9 arcsec.

The first trial on this Section uses Einstein radius value of b = 0.2 arcsec, again with source position x0 = y0 = 0.0 arcsec. This model can be seen in Figure 3.6. The second test uses, with the same source position, a value of b = 1.3 arcsec, and can be seen in Figure 3.7. A worse reconstruction is expected for the smallest value of b, as it implies more mixed source images.

We can infer from the image that the distance between the images in Figure 3.6 is actually around 0.2 arcsec. The data (and the model) are more compact, with no homogeneous residuals. The reconstruction is similar or even poorer than in previous tests. This is expected, as all the information has to be ex- tracted from compact images blurred by the PSF, with correspondent loss of details.

With an Einstein radius value of b = 1.3 arcsec, the four source images in Figure 3.7 are seen separately. From the graph, we can infer that the distance between the images is actually around 1.3 arcsec. The residuals are still not homogeneous. Nevertheless, as the images are separated from each other, the amount of “lost” information decreases and the reconstruction is better. This model shows again the small structure at the bottom-right of the M83 galaxy.

3.4 Axis ratio

The axis ratio for the lens potential takes values from 0 to 1, being 1 for a circle and 0 for the maximum ellipticity. In the default model, the axis ratio of the lens potential was q = 0.8. As in the default model the value is close to 1, the images are distributed almost symmetrically. When that value decreases, it becomes easier to distinguish the phenomenon called “Einstein cross”. The name is received directly from the distribution of the generated images, as they are symmetrical two by two. The next test shows a gravitational lens system

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Figure 3.6: Mock lens and reconstruction with b = 0.2 arcsec

Figure 3.7: Mock lens and reconstruction with b = 1.3 arcsec

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Figure 3.8: Mock lens and reconstruction with axis-ratio q = 0.5

using a source position of x0 = y0 = 0.0 arcsec with an axis ratio value of q = 0.5. This is shown in Figure 3.8.

It is clearly seen that the images are distributed elliptically, as expected.

The residuals show some structure again. The source reconstruction shows only a circular shape without details. This is probably occurring because of the small auto-generated mask used for this test. However, the effects of the mask are discussed further later in Section 3.7.

3.5 Position angle

The position angle defines the angular difference between the Cartesian axis and the axis in which the lens potential is constructed. Our default model uses an angle of pa = −45^o. This explains why all our images are rotated by exactly this angle.

We also carry out a test with pa = 0^o. To notice this effect, this model can be compared with the one in Figure 3.4, as both models have x0 = y0 = 0.0 arcsec. The test using pa = 0^o can be seen in the Figure 3.9.

This effect happens because of the elliptical shape of the potential. Hence, it does not have circular symmetry, and this effect arises. The residual and source reconstruction does not show any improvement. Hence, the position angle does not play an important role in source reconstructions.

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Figure 3.9: Mock lens and reconstruction with pa = 0

3.6 Dimensions of the source

I use a larger extend in arcsec for the True Source image to simulate the effect of the source being closer to the lens. Hence, a larger (smaller) extend of the source is related to smaller (larger) redshifts. The expected outcome of a larger True Source this is a larger image of the lensed source. In the default model, the dimensions of the source image are height = width = 0.5 arcsec. In this trial, the dimensions are set as the double: height = width = 1.0 arcsec. For this test, the source position values are set again to x0= −0.1 arcsec and y0= 0 arcsec.

We can see its effects on Figure 3.10.

As it was expected, the images of the lensed source are this time larger.

The residuals still show more structure than only noise, so the data and the model do not match completely. Nevertheless, the source reconstruction shows more details than in the default model on Figure 3.1. This improvement on the reconstruction of the source happens now with a larger True Source. This creates more extended lensed images from which more details can be discerned.

It all brings the program to produce a better source reconstruction.

3.7 Reconstruction without the full image

Here I study the ability to reconstruct the source without the whole lensed image. I change the mask¹ to emulate this effect.

First of all, the mock lens is generated. By doing some changes in the values

1The changes in the mask are done manually, using the website created by Giorgos Vernar- dos: gerlumph.swin.edu.au/tools/mask creator/

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Figure 3.10: Mock lens and reconstruction with height = width = 1.0 arcsec.

of the parameters for both the forward and inverse problem, I achieve a relatively good source reconstruction (using all the source images given by the data). I maintain the starting model always the same. After this, I apply different masks to the inverse problem. This means that everything remains the same during the tests except for the mask that is applied to the data.

I do not use only M83 but also NGC2623 as source model. Thus, reconstructions for different shapes are compared. I chose this galaxy because of its peculiar shape, as it has a regular shape in the middle and two thin arms. This set of tests is done in a slightly different way for each galaxy.

As the lens potential is known, the reconstruction should be still good for fewer images. The relevant point is to check if the code can reconstruct one source even without all its lensed images. With the same data set but using different images of the lensed source, the reconstructions might somehow differ.

With this knowledge, some prior information on the lens potential and the source galaxy could be inferred, and better reconstructions may be achieved.

As the applied mask is changed, a comparison between the evidence of two models cannot be done. Hence, the analysis will be qualitative.

3.7.1 M83

I create mock data with four images of the galaxy M83 by centering the true source with the lens potential. I eliminate one by one the images (selecting them randomly). I run the reconstruction using four, three, two and only one of the images at the same time. This process is shown in Figure 3.11. As the residuals do not change significantly, only the model and the reconstruction are shown for each case.

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Figure 3.11: Changes in the mask for M83

All four source reconstructions are quite similar. Between the first reconstruction (four images) and the second (three images), the only difference is at the edges of the reconstructed image. Then, the source reconstruction is not affected. In the third test (two images), these effects start to be more noticeable, but it is almost not perceived inside of the source. Finally, in the last reconstruction, even though the shape of the reconstructed source seems to remain exactly the same, the body of the source is full of voids that the program is not able to fill.

The volume of data decreases with each test. Hence, source reconstruction misses enough data from which to create the whole adaptive-grid. The reconstructed source matches the model and corresponds to the real galaxy where there is data but, at the same time, it is not complete.

The reconstruction is reasonably good until there is only one image left.

This is happening because the lens potential is completely fixed. Nevertheless, the code is able to generate reconstructions with only part of the data. The capability for doing it when adding additional uncertainties has to be studied as a part of a future project.

3.7.2 NGC2623

The lensed image contains two images from the source. This is obtained by moving the position of the true source, as we did in Section 3.2. After this, I vary some parameters to achieve again a reasonably good reconstruction of the source.

In the first place, I mask the image of one of the arms and run a reconstruction. The next trial is done without both images of the same arm. The obtained source reconstructions are shown in Figure 3.12. For the second set of tests, I run two more reconstructions with only one of the two source images at each time, see Figure 3.13.

Again, if only one image is masked, the reconstruction of this part is done properly. Nevertheless, the details in this part of the data become important.

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Figure 3.12: Reconstructions with changes in the mask of one arm of NGC2623

Figure 3.13: Reconstructions with changes in the mask of the images of NGC2623

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In this case, the mask applied to the bottom image should be bigger in order to reconstruct the whole arm. The curvature of the arm changes a bit, and some voids in the structure can be noticed.

From the second part, we conclude the same. Besides, when the masked image is closer to the True Source, the source reconstruction tends to fail less.

As the distortion caused by the lens is weaker, the lensed source is closer to the true source and the reconstruction is easier to compute.

3.8 Discussion

In this first experimental part, different effects that take place in gravitational lenses have been tested. The code offers flexibility to reproduce real gravitational lens systems. The spatial distribution of the data has been shown to be important to the quality of the source reconstruction. When the data is sufficiently spaced or detailed, the source reconstruction shows more details.

Contrarily, less detailed data or smaller data sets lead to poorer source reconstructions. Finally, when the lens potential is known a priori, the code is able to reconstruct the source with only partial information about the image.

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Chapter 4

Experimental: Studying the code

The main part of the bachelor research project is to study the code to find systematic effects in the reconstruction. I focus on the effects of using an incorrect PSF and on different types of regularization. The resolution of the image, different levels of resolution in the source, and the adaptive grid are also analyzed.

The tests are done, with some exceptions, for both galaxies M83 and NGC2623.

Bayesian statistics is used to analyze the results where possible.

4.1 Image resolution

The parameters for the forward and the inverse problem are kept the same as in the default model in Figure 3.1, but the pixel resolution of our image is increased. For observations, this means using an instrument with better resolution. The default model has pixx= pixy = 70 pixels, with pixx and pixy

being the number of pixels in directions x and y, respectively. The next test has values pixx= pixy = 110 pixels. As less information is lost between the pixels when the image is created, an improvement is expected in both the model and the reconstruction. This test is shown in Figure 4.1.

The residuals are a bit closer to a noise than in the default model. However, the improvement is not very large, and the reconstruction is still poor, with an important lack of details in the model of the lensed images. To improve the image resolution will help to achieve better reconstructions, but it is not a decisive factor. Even though, this is an improvement which is reasonable and will help us to better see the effect of other changes in the reconstruction.

Consequently, the next sections inside of this chapter use a pixel resolution of 110x110 pixels.

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Figure 4.1: Mock lens and reconstruction with pixx= pixy= 110.

4.2 Point Spread Function (PSF)

In this section, the role that the Point Spread Function (PSF) plays in the process of reconstruction of the lensed source is analyzed. In many instances, we do not know which is the precise PSF for a specific problem. To study the impact of using an slightly different PSF, I create one mock lens and then reconstruct the source using different PSFs. The test are carried out using only NGC2623.

I generate a mock lens image using a PSF from a blackbody with a temperature of 10000K¹. For the first case, the same PSF is used also for the inverse problem. This is the case which is going to be used as a reference model. This reconstruction is shown on Figure 4.2.

Next, I use different PSFs for the inverse problem. To study whether small deviations form the true PSF matters, I use PSFs with a Blackbody temperate at 5000K and at 8000K. Also, other PSFs that present greater differences are tested: 10000K rotated 90^o and power law F (λ) = λⁱ, with spectral index i = 3. It is supposed that the reconstructions using a PSF closer to the original one will have a logarithm of the Bayes factor closer to 0 than the models created with a PSF completely different from the original.

Individual values of the evidence and the Bayes factor for each model can be seen in Table 4.1.

The reconstruction done with the PSF from a Blackbody at 10000K rotated

1Generated using Tiny Tim [9] with parameters: Camera WFC3 UVIS channel, Chip 2, Pixel position 2048 1024, Filter F390W, Spectrum type variable. http://tinytim.stsci.edu/cgi- bin/tinytimweb.cgi

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Figure 4.2: Same PSF for forward and inverse problem (Blackbody at 10000K)

PSF ln(evidence) ln(Z) log(Bayes factor) log10f

Blackbody 10000K 10305 0

Blackbody 8000K 10299 2.6

Blackbody 5000K 10278 11.7

Blackbody 10000K rotated 90^o clockwise 9479 >100

F (λ) = λ³ 10274 13.5

Table 4.1: Different PSF used to reconstruct the source. The PSF used to create the mock image was from a Blackbody at 10000K.

90^oclockwise gives the worst source reconstruction. It is shown in Figure 4.3.

Analyzing the Bayes factor shown in Table 4.1, the source reconstruction done with the same PSF as the forward problem is decisively preferred with respect to the others. The greatest difference is given by the PSF from Black- body at 10000K rotated 90^o. It is also seen that the models using a PSF from a Blackbody spectrum with different temperature show higher evidence.

The PSFs based on a Blackbody spectrum are created from the same distribution, but with different temperature. Hence, the generated PSFs are closer between them. When rotating the PSF, its lack of symmetry resulted in a shift of the reconstructed source. This effect was induced by different smearing for different directions.

This problem with the PSF could be solved by setting free the parameters for the lens potential. By doing that, we would recover a reconstruction closer to reality. Nevertheless, with that recovery, we would solve an error in the PSF with another error in the parameters for the lens potential. The use of an incorrect PSF generates errors that are critical for a good reconstruction of the source.

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Figure 4.3: PSF used: Forward problem, Blackbody at 10000K; Inverse problem, Blackbody at 10000K rotated 90^oclockwise.

During the next sections, the same PSF is used for the forward and for the inverse problem. By doing this, other effects in the source reconstruction can be seen.

4.3 Regularization

A study of the relevance of the type of regularization is realized. First, a data set is created and maintained invariable through the tests. Different values of the regularization parameter λ are tested. Regarding its type, the most common regularizations are the zeroth order, the gradient, and the curvature [25]. I study two cases for the regularization matrix: curvature and covariance. The curvature regularization calculates derivatives between the triangles of the reconstruction in the source plane (further description in [25]). The covariance regularization uses an exponential e^−r/σ^s to correlate positions, with r the distance between pixels and σ_s the correlation length in arcsec. I use a value of σ_s= 1.

The experiment is done using as a True Source the galaxy M83 and the merger galaxy NGC2623. As the best type of regularization depends on the source distribution [24], the study of the regularization needs to be realized separately for M83 and for NGC2623. However, the preparation for both can be explained together.

I modify the level or regularization by adjusting the value of the regularization parameter, which starts with a value of 100 and decreases by a factor of 100 every test until reaching 0.0001. For some set of tests I add one more source reconstruction with a different value of λ. The comparison between models is

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going to be done using the Bayes factor explained in Section 2.2.

The regularization works as a sort of prior which makes the reconstructed source smoother. Hence, it is expected to find more realistic and smoother sources when using higher values of the regularization parameter λ. At the same time, the model needs to fit the data. This usually occurs for lower values of λ. These two opposite directions lead to think about an intermediate point where the reconstruction fits properly the data while being realistic. By doing tests with λ in a range from 0.0001 to 100, I expect to have enough information to study that ”intermediate point” of regularization needed to obtain a better- reconstructed source.

Testing the curvature and the covariance matrix for the regularization with similar λ parameters can show us how important is the correlation between pixels depending on the distance. When every pixel is largely correlated with its close surrounding pixels, more structure is expected in the reconstructed source. When the correlation comprises faraway pixels, reconstructed sources are expected to be smoother.

4.3.1 M83

For the study of the regularization with galaxy M83, the results are shown in Table 4.2 and Table 4.3.

λ values regularisation ln(Evidence) log(Bayes factor) log₁₀(f )

100 -14114 -24550 >100

1 -2464 3069 >100

0.01 -520 9009 >100

0.0001 -124 9434 0

Table 4.2: M83 - curvature. Regularization value, natural logarithm of the evidence, and the logarithm (base 10) of the Bayes factor for reconstructions of M83 with curvature matrix regularization and different values of the regularization parameter λ.

λ values regularisation ln(Evidence) log(Bayes factor) log₁₀(f )

100 -2166 3748 >100

1 -343 7683 28

0.1 -133 7747 0

0.01 -31 7370 >100

0.0001 -6 6258 >100

Table 4.3: M83-covariance. Regularization value, natrual logarithm of the evidence, and the logarithm (base 10) of the Bayes factor for reconstructions of M83 with covariance matrix regularization and different values of the regularization value λ.

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The plots for the reconstructions, the model and the residuals of the model are shown in Figure 4.4 for the regularization using the curvature matrix and in Figure 4.5 for the regularization using the covariance matrix.

Starting with the curvature, we can see in Table 4.2 the Bayes factor for the different values of λ. Using Kass and Raftery scale as shown in Section 2.2, the comparison is made with respect to the test with λ = 0.0001, as it is the one with the highest evidence. The analysis shows that this model is decisively supported when compared to any other. However, it can be seen that the evidence for each test increases when lowering the parameter λ. The optimum value for λ cannot be found as the evidence never decreases in our set of tests.

Anyway, the difference in the evidence value between models decreases as the evidence increases. Hence, an optimum value for the regularization parameter is expected to be somewhere below λ = 0.0001. We can match this analysis with the plot of the source reconstruction. The model with λ = 100, the least preferred model, gives a reconstruction which is too smooth, losing almost every detail in the reconstruction. Also, the residual is significantly far from the noise level. When lowering the regularization parameter, more details are found in the source reconstruction, and the residual starts to look like noise. Nevertheless, it seems to happen that between the tests with the lowest λ, the shape in the residual does not diminish as much as between the tests with the highest λ. This is correlated to the rate of change of the evidence and shows a limitation in the goodness of the source reconstruction when only changing the regularization parameter.

Analyzing the case with the covariance regularization, the model with the highest evidence has now λ = 0.1. We find smaller differences between the evidence value of each model. Comparison between models is done using Kass and Raftery scale as in Table 2.1, always with respect the model λ = 0.1. All the models are decisively discarded against the reference model λ = 0.1. The maximum value for λ is unknown, but we can assess that is between 1 and 0.01, and probably close to 0.1. Analyzing the plot, the source reconstruction always reproduces the shape of the source, including its sub-structure. The residuals become similar to noise but, again, there is always some shape that does not disappear.

To compare different regularizations, it is seen that they behave in a different way. The curvature regularization changes its evidence much more than the covariance one. The former is decisively preferred for the optimum λ value of both. Extending this, taking the evidence for the covariance with λ = 0.1 (lnZ = 7747), and for the curvature with λ = 0.0001 (lnZ = 9434), the logarithm of the Bayes factor is > 100, which is decisive in favour of the curvature.

The covariance regularization shows its maximum evidence value for a higher regularization parameter than the curvature. The correlation length σsfor the covariance is 1 arcsec. Hence, its effect is smoother than in the curvature, which correlates only the pixels next to each other. As a consequence, the change in λ affects much more the curvature than the covariance. Besides, this allows the curvature to reach a better reconstruction (only for its optimum λ value) than the covariance regularization (also for its optimum λ value). The residual, for

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M83 curvature

Figure 4.4: Reconstruction with curvature regularization matrix for labelled values of regularization parameter λ.

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M83 covariance

Figure 4.5: Reconstruction with covariance regularization matrix for labelled values of regularization parameter λ.

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both models, is never given only by noise.

4.3.2 NGC2623

The results for the galaxy NGC2623 are shown in tables 4.4 and 4.5.

In addition, the plots for the reconstruction, the model, and the residuals for this case are shown in Appendix 8. This is done because the tendency is the same as in the case of M83, and plotting it here would offer no new information.

λ values regularisation ln(Evidence) log(Bayes factor) log10(f )

100 -14569 -36989 >100

1 -3821 4850 >100

0.01 -559 11991 >100

0.001 -287 12648 42.1

0.0001 -130 12745 0

Table 4.4: NGC2623 - curvature. Regularization value, natural logarithm of the evidence and the logarithm (base 10) of the Bayes factor for reconstructions of NGC2623 with curvature matrix regulariation and different values of the regularization value λ.

λ values regularisation ln(Evidence) log(Bayes factor) log10(f )

100 -2417 6692 >100

1 -355 10857 72.1

0.1 -126 11023 0

0.01 -35 10764 >100

0.0001 -12 9909 >100

Table 4.5: NGC2623-covariance. Regularization value, natural logarithm of the evidence and the logarithm (base 10) of the Bayes factor for reconstructions of NGC2623 with covariance matrix regularization and different values of the regularization value λ.

The behavior of these reconstructions is the same as for the M83. With curvature regularization, the maximum achieved for the evidence is for λ = 0.0001. This model is always decisively preferred when Bayesian comparison is done. The difference in evidence values between two lower λ values is smaller than the difference for higher pairs of λ. Hence, a maximum is expected close to the value of λ = 0.0001. The residual behavior is also the same.

For the covariance regularization, the maximum evidence falls again in λ = 0.1. The optimum value is expected to be close to this one. This model is always decisively preferred when studying the Bayes factor. The change between the values of the evidence is again smaller than for the curvature.

For both cases, the residuals seem to reach a decreasing limit when lowering λ. If we compare the best models in both cases: curvature with λ = 0.0001 has

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lnZ = 12745; covariance with λ = 0.1 has lnZ = 11023. The Bayes factor gives

>> 2 with a decisive preference for the curvature with λ = 0.0001.

Discussion

As a general discussion for this Section, it seems that lower values of λ fit better the data. However, if the value is lowered excessively, the reconstructed source starts to over-fit the data. Also, the noise starts to be absorbed in the reconstruction and the source starts looking too structured and unrealistic. Smoother models driven by higher regularization parameters are the worst among the tested, with lack of details and shaped residuals. The covariance regularization seems to work better than the curvature for many values of λ.

Nevertheless, the curvature reaches a better model if optimization of λ is done.

From this last point, it can be inferred that a greater correlation between close pixels is preferred over correlation with distant ones for the general case.

Hence, lower values of λ should be preferred, but maximization of its value would be the main goal for future using of the code. Also, different values for the correlation between pixels in the covariant matrix should be tested, as different regularization matrix which penalizes the correlation between long- distance pixels.

4.4 Levels of blur

For the next sections, the models that are used for the reconstructions are the galaxy M83 with covariance matrix and λ = 0.1 and the galaxy NGC2623 also with covariance matrix and λ = 0.1.

Smearing consists on averaging the brightness value of a certain amount of pixels throughout the whole image and at the same time. Depending on this amount, the smearing is stronger of weaker.

I study the effects of the blurring in the source reconstruction, with emphasis on the effects on the reconstruction of delicate parts of the source like substruc- tures or arms. I also study the behavior of the residual. The models studied here combine a relatively structured source with a progressively higher level of blur. In order to have less influence on the particular shape of one source, the experiment is made, again, for the galaxy M83 and the merger galaxy NGC2623.

The particularity offered by M83 is the substructure at the right-bottom part.

In the case of NGC2623, its two arms are the main focus where we expect the biggest change on the source reconstruction with different levels of blur.

The procedure starts by using the models which were found to be the best when using the covariance matrix, without blur. Images of the true source with increasing levels of blurriness are created using a Gaussian blur filter. To create the filter, I give a radius that works as the standard deviation. By increasing the radius, we achieve more blurriness in the image. I chose slightly different values for the radius for each case.² A summary of the values is given in Table

2Both images have the same pixel density dpi=96. The M83 image galaxy has 550x550

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4.6.

M83 4 8 12 16 mm

NGC2623 5 10 15 20 mm

Table 4.6: Radius values for the Gaussian blur filter applied to each galaxy image.

The relevant results that are obtained from these blurriness values are shown in Table 4.7 and Figure 4.6 for galaxy M83. Table 4.8 and Figure 4.7 show the results for merger galaxy NGC2623.

blur radius (mm) - 4 8 12 16

ln(evidence) 7683 8640 9377 9752 9994

Table 4.7: M83. Radius of the blurring filter and the natural logarithm of the evidence that comes form the source reconstruction.

blur radius (mm) - 5 10 15 20

ln(evidence) 10857 12007 13204 13944 14540

Table 4.8: NGC2623. Radius of the blurring filter and the natural logarithm of the evidence that comes form the source reconstruction.

The evidence increases for more blurred images. This effect happens because, when the blurring in the true image increases, the data become smoother and easier to fit. Nevertheless, source reconstructions from a blurred source lose more details than those without blur. The evidence cannot be compared between models to look for the best reconstruction, as the true image is changed when the smearing filter is applied. However, we can see that it is easier to fit smooth data than structured data.

Analyzing the images, both groups of reconstructions show the same trend.

In the most blurred case, both galaxies lose part of its shape. The left arm of NGC2623 become shorter and the substructure of M83 almost disappears. With only the first level of the filter applied, the source only becomes a bit smoother.

When the images are more blurred, which means that we have smoother data, the residuals look more like noise. For example, the last reconstruction would not be accepted for more than the general shape. Nevertheless, its residuals are the ones that show a model with less discrepancy with the data, being almost only noise.

Hence, we find the opposite behavior from the one in Section 4.3. The main difference is that, here, the data is blurred from the beginning. This is as if the light from the source arrives at the lens already smeared. On the

pixels∼146x146 mm², and the NGC2623 one, 700x700pixels∼185x185 mm². I use the same ratio between the radius and the size of the picture in both cases.

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M83 levels of blurriness

Figure 4.6: Reconstruction for galaxy M83 with different levels of blurriness.

NGC2623 levels of blurriness

Figure 4.7: Reconstruction for galaxy NGC2623 with different levels of blurriness.

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Figure 4.8: Reconstruction and residuals of M83 and NGC2623 with spacing = 2

contrary, in Section 4.3 the smoothness came only for the reconstruction of the source. Hence, in the latter case, the data is not smeared. Smeared data produce smoother source reconstructions, and can be used for the first approach to weirdly structured sets of data.

4.5 Adaptive grid settings

As is seen in Section 4.3, the residual are never close enough to a noise signal.

It is reduced when the reconstructions of the source are structured, but there is always a remaining shape. To achieve noise-residuals, the next step that can be studied is the spacing that creates the adaptive-grid where the reconstruction is generated (see Section 2). The currently used spacing value is 3. As the tendency until now shows that more structured sources usually give better reconstructions, it is worth to test if lowering the value of this parameter can help to achieve better reconstructions. To do this, I choose again the reconstructions made, for both galaxies, with covariance matrix regularization and λ = 0.1. The obtained reconstructions and the residuals obtained for both cases are shown in 4.8.

In these images, it can be seen how the residuals actually diminish until almost the noise level. Only some smooth shape can be noticed. Besides, the reconstruction program is able to reproduce with great detail the true sources.

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Chapter 5

Conclusions

Systematic effects arising during the reconstruction of a lensed source have been studied. Different possibilities for avoiding possible errors and improving source reconstruction methods have been found. As a result of the tests that have been done, can be concluded:

• The models that we study with the code are similar to those that we see in reality.

• Compressed or small images of the source on the data give worse source reconstructions because of the loss of information.

• With a known potential, we are able to reconstruct the source with in- complete, but sufficient, data.

• Higher resolution gives better reconstructions.

• Using the correct PSF for each case is decisive to achieve a proper reconstruction. This will be even more important when adding to the code the reconstruction of the lens potential.

• The value of the regularization parameter λ needs to be optimized for every type of regularization in order to obtain the best reconstruction.

• In the cases of the reconstruction of galaxies M83 and NGC2623, with optimized value of the regularization parameter λ, curvature regularization gives a better result than the optimum for covariance regularization.

For no-optimal values, covariance regularization works better. The selection of the type of regularization is an important step for every source reconstruction.

• Smaller spacing used during the adaptive grid source reconstruction represents an important improvement.

• Smooth data is easier to fit for a reconstruction than detailed data.

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As future work, some of the topics that can further studied are:

• The influence of using an incorrect PSF for the source reconstruction on a model using free parameters to describe the lens potential.

• A major understanding of the influence of the correlation between pixels in the regularization matrix and its possible optimization.

• The optimization of the regularization parameter λ as one powerful tool to obtain better reconstructions.

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Chapter 6

Appendix Forward Problem

Table 6.1 includes the parameters defined in the “.json” file used for the “forward” problem that are relevant to this bachelor project. Shown values are the ones used for the default model.

General - Image plane

pixx= 70 pixel pixy = 70 pixel width=3.5 arcsec height=3.5 arcsec Source: light - fromfits

x0= 0.1 arcsec x0= 0.0 arcsec Ni = 550 pixel Nj= 550 pixel width=0.5 arcsec height=0.5 arcsec

Lens: mass: SIE

b = 0.9 arcsec q = 0.8 pa = −45^o

x₀= 0.0 arcsec y₀= 0.0 arcsec Noise: uniform

sn = 40

PSF : fromfits

pix_x= 74 pix pix_y= 74 pix Mask : automask smear=0.04 arcsec threshold=0.1

Table 6.1: “Forward” problem: relevant parameters for this bachelor project with values for the generation of the mock lens image. Used in “.json” format.

General - Image plane: observation with an intrument. i) pix_x, pix_y: image width and height in pixels. ii) Width and height: image width and height in arcsec.

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Source: brightness distribution, image from a “.fits”¹ file [5]. i) x0, y0: source center abscissa and ordinate in arcsec, respectively. ii) Ni, Nj: pixel dimensions of the source image (Ni= Nj). iii) Width and height: size of the image in arcsec.

Lens: distribution of mass, Singular Isothermal Ellipsoid. i) b: Einstein radius in arcsec. ii) q: axis ratio (minor/major, major is always on the x-axis).

iii) pa: position angle of the major axis (degrees, east of north). iv) x₀, y₀: lens center abscissa and ordinate in arcsec.

Noise: Uniform noise. i) sn: signal to noise ratio at the maximum lensed image brightness

PSF: “.fits” file containing the Point Spread Function. i) pixx, pixy: Width and height of the PSF in pixels, respectively.

Mask: Produce a mask automatically. i) Smear factor: the resulting image brightness is smeared by a 2-d Gaussian filter of this size (in arcsec). ii) Threshold: pixels above this threshold value in brightness after smearing are accepted.

1Data format used for the transport, analysis, and archival storage of scientific data sets.

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Chapter 7

Appendix Inverse Problem

Table 7.1 includes the parameters defined in the “json” file used for the “inverse”

problem that are relevant to this bachelor project. The values are the ones used for the default model.¹ This part of the code takes the “fits” files of the image of the source, the mask, and the PSF. It also takes the noise data. Hence, the Image plane, noise, PSF, mask and the lens parameters remain as in the “forward”

problem (see Appendix 6) by default, but they can be changed manually.

Adaptive grid spacing = 3 Regularization curvature λ = 1

covariance λ = 1 sdesv = 1

Table 7.1: “Inverse” problem: relevant parameters with values for the reconstruction of the source. Used in “.json” format.

Adaptive grid: i) Spacing: division of the data pixels from the image plane to create the source reconstruction adaptive grid, every 3x3 pixels in our case.

Regularization: type of regularization. i) curvature with regularization parameter λ. ii) covariance with regularization parameter λ and standard deviation sdesv.

1The default model uses curvature regularization with λ = 1. The covariance is added here for completeness

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Chapter 8

Appendix Plots

The figures from Subsection 4.3.2 are shown here.

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NGC2623 curvature

Figure 8.1: Reconstruction of NGC2623 with curvature regularization matrix for labelled values of the regularization parameter λ.

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NGC2623 covariance

Figure 8.2: Reconstruction of NGC2623 with covariance regularization matrix for labelled values of the regularization parameter λ.

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Bibliography

[1] e-merlin, enhanced multi element remotely linked interferometer network, http://www.e-merlin.ac.uk/.

[2] Esa/hubble, nasa, https://apod.nasa.gov/apod/ap111221.html.

[3] Esa/hubble, nasa, suyu et al., lensed quasar he0435-1223;

http://sci.esa.int/jump.cfm?oid=58733.

[4] Esa/hubble, nasa, suyu et al., lensed quasar rxj1131-1231;

https://www.spacetelescope.org/images/heic1702d/.

[5] Fits file - flexible image transport system, nasa/gsfc;

https://fits.gsfc.nasa.gov/.

[6] .json file extension, https://fileinfo.com/extension/json.

[7] M83 from simbad database / nasa/ipac extragalactic database ; http://simbad.u-strasbg.fr/simbad/sim-id?ident=m83.

[8] Ngc2623 from esa/hubble nasa; https://apod.nasa.gov/apod/ap180110.html.

[9] Tinytim software from space telescope science institute(stsci), http://www.stsci.edu/software/tinytim/.

[10] O. Chwolson. “uber eine mogliche form fiktiver doppelsterne”. Astron.

Nachr., 221:329, 1924.

[11] Keeton C.R. Congdon, A.B. Principles of Gravitational Lensing, Light Deflection as a Probe of Astrophysics and Cosmology. Sringer Praxis Books, 2018.

[12] A. Einstein. ”grundgedanken der allgemeinen relativit¨atstheorie und an- wendung dieser theorie in der astronomie”, ”fundamental ideas of the general theory of relativity and the application of this theory in astronomy”. Preussische Akademie der Wissenschaften, Sitzungsberichte,, page 315, 1915 (part 1).

Systematic effects in strong lensing source reconstruction methods