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Cosmic microwave background (CMB) temperature and polarization observations indicate that in the best-fit $\Lambda$ Cold Dark Matter model of the Universe, the local geometry is consistent with at most a small amount of positive or negative curvature, i.e., $\vert\Omega_K\vert\ll1$. However, whether the geometry is flat ($E^3$), positively curved ($S^3$) or negatively curved ($H^3$), there are many possible topologies. Among the topologies of $S^3$ geometry, the lens spaces $L(p,q)$, where $p$ and $q$ ($p>1$ and $0

The standard cosmological model, which assumes statistical isotropy and parity invariance, predicts the absence of correlations between even-parity and odd-parity observables of the cosmic microwave background (CMB). Contrary to these predictions, large-angle CMB temperature anomalies generically involve correlations between even-$\ell$ and odd-$\ell$ angular power spectrum $C_\ell$, while recent analyses of CMB polarization have revealed non-zero equal-$\ell$ $EB$ correlations. These findings challenge the conventional understanding, suggesting deviations from statistical isotropy, violations of parity, or both. Cosmic topology, which involves changing only the boundary conditions of space relative to standard cosmology, offers a compelling framework to potentially account for such parity-violating observations. Topology inherently breaks statistical isotropy, and can also break homogeneity and parity, providing a natural paradigm for explaining observations of parity-breaking observables without the need to add parity violation to the underlying microphysics. Our investigation delves into the harmonic space implications of topology for CMB correlations, using as an illustrative example $EB$ correlations generated by tensor perturbations under both parity-preserving and parity-violating scenarios. Consequently, these findings not only challenge the foundational assumptions of the standard cosmological model but also open new avenues for exploring the topological structure of the Universe through CMB observations., Comment: 20 pages, 4 figures. v2: version published in JCAP

Non-trivial spatial topology of the Universe may give rise to potentially measurable signatures in the cosmic microwave background. We explore different machine learning approaches to classify harmonic-space realizations of the microwave background in the test case of Euclidean $E_1$ topology (the 3-torus) with a cubic fundamental domain of a size scale significantly smaller than the diameter of the last scattering surface. This is the first step toward developing a machine learning approach to classification of cosmic topology and likelihood-free inference of topological parameters. Different machine learning approaches are capable of classifying the harmonic-space realizations with accuracy greater than 99% if the topology scale is half of the diameter of the last-scattering surface and orientation of the topology is known. For distinguishing random rotations of these sky realizations from realizations of the covering space, the extreme gradient boosting classifier algorithm performs best with an accuracy of 88%. Slightly lower accuracies of 83% to 87% are obtained with the random forest classifier along with one- and two-dimensional convolutional neural networks. The techniques presented here can also accurately classify non-rotated cubic $E_1$ topology realizations with a topology scale slightly larger than the diameter of the last-scattering surface, if enough training data are provided. While information compressing methods like most machine learning approaches cannot exceed the statistical power of a likelihood-based approach that captures all available information, they potentially offer a computationally cheaper alternative. A principle challenge appears to be accounting for arbitrary orientations of a given topology, although this is also a significant hurdle for likelihood-based approaches., Comment: 34 pages, 12 figures, 4 tables

The chameleon is a theorised scalar field that couples to matter and possess a screening mechanism, which weakens observational constraints from experiments performed in regions of higher matter density. One consequence of this screening mechanism is that the force induced by the field is dependent on the shape of the source mass (a property that distinguishes it from gravity). Therefore an optimal shape must exist for which the chameleon force is maximised. Such a shape would allow experiments to improve their sensitivity by simply changing the shape of the source mass. In this work we use a combination of genetic algorithms and the chameleon solving software SELCIE to find shapes that optimise the force at a single point in an idealised experimental environment. We note that the method we used is easily customised, and so could be used to optimise a more realistic experiment involving particle trajectories or the force acting on an extended body. We find the shapes outputted by the genetic algorithm possess common characteristics, such as a preference for smaller source masses, and that the largest fifth forces are produced by small `umbrella'-like shapes with a thickness such that the source is unscreened but the field reaches its minimum inside the source. This remains the optimal shape even as we change the chameleon potential, and the distance from the source, and across a wide range of chameleon parameters. We find that by optimising the shape in this way the fifth force can be increased by $2.45$ times when compared to a sphere, centred at the origin, of the same volume and mass., Comment: 28 pages, 17 figures, The SELCIE code is available at: https://github.com/C-Briddon/SELCIE

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies $E_{1}-E_{6}$, $E_{11}$, $E_{12}$, $E_{16}$, and $E_{18}$ that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys., Comment: 79 pages, 9 figures. v3: version published in JCAP

The Einstein field equations of general relativity constrain the local curvature at every point in spacetime, but say nothing about the global topology of the Universe. Cosmic microwave background anisotropies have proven to be the most powerful probe of non-trivial topology since, within $\Lambda$CDM, these anisotropies have well-characterized statistical properties, the signal is principally from a thin spherical shell centered on the observer (the last scattering surface), and space-based observations nearly cover the full sky. The most generic signature of cosmic topology in the microwave background is pairs of circles with matching temperature and polarization patterns. No such circle pairs have been seen above noise in the WMAP or Planck temperature data, implying that the shortest non-contractible loop around the Universe through our location is longer than 98.5% of the comoving diameter of the last scattering surface. We translate this generic constraint into limits on the parameters that characterize manifolds with each of the nine possible non-trivial orientable Euclidean topologies, and provide a code which computes these constraints. In all but the simplest cases, the shortest non-contractible loop in the space can avoid us, and be shorter than the diameter of the last scattering surface by a factor ranging from 2 to at least 6. This result implies that a broader range of manifolds is observationally allowed than widely appreciated. Probing these manifolds will require more subtle statistical signatures than matched circles, such as off-diagonal correlations of harmonic coefficients., Comment: 22 pages, 7 figures. v3: Corrected Figs. 3, 4, and 5 as published in Erratum

The shortest distance around the Universe through us is unlikely to be much larger than the horizon diameter if microwave background anomalies are due to cosmic topology. We show that observational constraints from the lack of matched temperature circles in the microwave background leave many possibilities for such topologies. We evaluate the detectability of microwave background multipole correlations for sample cases. Searches for topology signatures in observational data over the large space of possible topologies pose a formidable computational challenge., Comment: 7 pages, 4 figures. v4: version published in PRL

A key goal in cosmology in the upcoming decade will be to form a better understanding of the accelerated expansion of the Universe. Upcoming surveys, such as the Vera C. Rubin Observatory's 10-year Legacy Survey of Space and Time (LSST), Euclid and the Square Killometer Array (SKA) will deliver key datasets required to tackle this and other puzzles in contemporary cosmology. With this data, constraints of unprecedented power will be put on different models of dark energy and modified gravity. In this context it is crucial to understand how screening mechanisms, which hide the deviations of these theories from the predictions of general relativity in local experiments, affect structure formation. In this work we approach this problem by using a combination of analytic and numerical methods to describe chameleon screening in the context of cosmic voids. We apply a finite element method (FEM) code, SELCIE, to solve the chameleon equation of motion for a number of void profiles derived from observational data and simulations. The obtained results indicate a complex relationship between the properties of cosmic voids and the size of the chameleon acceleration of a test particle. We find that the fifth force on a test particle in a void is primarily related to the depth and the inner density gradient of the void. For realistic void profiles, the obtained chameleon-to-Newtonian acceleration ratios range between $a_{\phi}/a_{\rm Newt} \approx 10^{-6} - 10^{-5}$. However, it should be noted that in unusually deep voids with large inner density gradients, the acceleration ratios can be significantly higher. Similarly, other chameleon models, such as $f(R)$ Hu-Sawicki theory allow for significantly higher acceleration ratios. Given these results, we also discuss the optimal density profiles for detecting the fifth force in the upcoming observational surveys., Comment: 19 pages, 9 figures. The SELCIE code is available at: https://github.com/C-Briddon/SELCIE. Submitted to JCAP

The chameleon model is a modified gravity theory that introduces an additional scalar field that couples to matter through a conformal coupling. This `chameleon field' possesses a screening mechanism through a nonlinear self-interaction term which allows the field to affect cosmological observables in diffuse environments whilst still being consistent with current local experimental constraints. Due to the self-interaction term the equations of motion of the field are nonlinear and therefore difficult to solve analytically. The analytic solutions that do exist in the literature are either approximate solutions and or only apply to highly symmetric systems. In this work we introduce the software package SELCIE (https://github.com/C-Briddon/SELCIE.git). This package equips the user with tools to construct an arbitrary system of mass distributions and then to calculate the corresponding solution to the chameleon field equation. It accomplishes this by using the finite element method and either the Picard or Newton nonlinear solving methods. We compared the results produced by SELCIE with analytic results from the literature including discrete and continuous density distributions. We found strong (sub-percentage) agreement between the solutions calculated by SELCIE and the analytic solutions., Comment: 25 pages, 9 figures, 1 table. SELCIE code available at https://github.com/C-Briddon/SELCIE.git

Chameleon gravity is an example of a model that gives rise to interesting phenomenology on cosmological scales while simultaneously possessing a screening mechanism, allowing it to avoid solar system constraints. Such models result in non-linear field equations, which can be solved analytically only in simple highly symmetric systems. In this work we study the equation of motion of a scalar-tensor theory with chameleon screening using the finite element method. More specifically, we solve the field equation for spherical and triaxial NFW cluster-sized halos. This allows a detailed investigation of the relationship between the NFW concentration and the virial mass parameters and the magnitude of the chameleon acceleration, as measured at the virial radius. In addition, we investigate the effects on the chameleon acceleration due to halo triaxiality. We focus on the parameter space regions that are still allowed by the observational constraints. We find that given our dataset, the largest allowed value for the chameleon-to-NFW acceleration ratio at the virial radius is $\sim 10^{-7}$. This result strongly indicates that the chameleon models that are still allowed by the observational constraints would not lead to any measurable effects on galaxy cluster scales. Nonetheless, we also find that there is a direct relationship between the NFW potential and the chameleon-to-NFW acceleration ratio at the virial radius. Similarly, there is a direct (yet a much more complicated) relationship between the NFW concentration, the virial mass and the acceleration ratios at the virial radius. Finally, we find that triaxiality introduces extra directional effects on the acceleration measurements. These effects in combination could potentially be used in future observational searches for fifth forces., Comment: 32 pages, 21 figures, 1 table, submitted to JCAP

This thesis introduces a set of methods for testing models of modified gravity using galaxy clusters. In particular, a technique for constraining models with a chameleon screening is introduced. In addition, the outlined technique is expanded to test a wider class of models, such as the theory of emergent gravity. Finally, the first part of the thesis is concluded by adapting the mentioned tests for model independent constraints. The obtained results indicate that galaxy clusters can be used to obtain some of the most powerful constraints on cosmological scales. The second part of the thesis is dedicated to the topic of cosmological emulators. More specifically, a technique of emulating cosmological N-body simulation output data based on machine learning is introduced. Generative adversarial networks (GANs) are used to emulate dark matter-only as well as hydrodynamical simulation data. In addition, N-body modified gravity simulations are explored as well. The presented investigation of the GAN algorithm shows that such emulators offer a powerful, fast and efficient way of producing simulation output data with different cosmological parameters. The power spectrum analysis indicates a 1-20% difference between the training and the generated data depending on the dataset used and whether Gaussian smoothing is applied or not., Comment: PhD thesis, University of Portsmouth, November 2020

Generative adversarial networks (GANs) have been recently applied as a novel emulation technique for large scale structure simulations. Recent results show that GANs can be used as a fast, efficient and computationally cheap emulator for producing novel weak lensing convergence maps as well as cosmic web data in 2-D and 3-D. However, like any algorithm, the GAN approach comes with a set of limitations, such as an unstable training procedure and the inherent randomness of the produced outputs. In this work we employ a number of techniques commonly used in the machine learning literature to address the mentioned limitations. In particular, we train a GAN to produce both weak lensing convergence maps and dark matter overdensity field data for multiple redshifts, cosmological parameters and modified gravity models. In addition, we train a GAN using the newest Illustris data to emulate dark matter, gas and internal energy distribution data simultaneously. Finally, we apply the technique of latent space interpolation to control which outputs the algorithm produces. Our results indicate a 1-20% difference between the power spectra of the GAN-produced and the training data samples depending on the dataset used and whether Gaussian smoothing was applied. Finally, recent research on generative models suggests that such algorithms can be treated as mappings from a lower-dimensional input (latent) space to a higher dimensional (data) manifold. We explore such a theoretical description as a tool for better understanding the latent space interpolation procedure., Comment: 20 pages, 23 figures, 3 tables

Verlinde's theory of Emergent Gravity (EG) describes gravity as an emergent phenomenon rather than a fundamental force. Applying this reasoning in de Sitter space leads to gravity behaving differently on galaxy and galaxy cluster scales; this excess gravity might offer an alternative to dark matter. Here we test these ideas using the data from the Coma cluster and from 58 stacked galaxy clusters. The X-ray surface brightness measurements of the clusters at $0.1 < z < 1.2$ along with the weak lensing data are used to test the theory. We find that the simultaneous EG fits of the X-ray and weak lensing datasets are significantly worse than those provided by General Relativity (with cold dark matter). For the Coma cluster, the predictions from Emergent Gravity and General Relativity agree in the range of 250 - 700 kpc, while at around 1 Mpc scales, EG total mass predictions are larger by a factor of 2. For the cluster stack the predictions are only in good agreement at around the 1 - 2 Mpc scales, while for $r \gtrsim 10$ Mpc EG is in strong tension with the data. According to the Bayesian information criterion analysis, GR is preferred in all tested datasets; however, we also discuss possible modifications of EG that greatly relax the tension with the data., Comment: 19 pages, 5 figures, 5 tables, accepted for publication on JCAP