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Centro de Astrofísica da Universidade do Porto

CAUP Researchers: Amélia M. M. Leite, André S. Nunes, Carlos J. A. P. Martins
Team at CAUP: Galaxies and Observational Cosmology
Other Researchers: Anastasios Avgoustidis (UK), Jon Urrestilla (SP/UK)

Analytic Models for the Evolution of Semilocal String Networks,
Physical Review D, Volume 84, pp. 06504 (2011)

Scaling properties of domain wall networks,
Physical Review D, Volume 84, 103523 (2011)

Topological defects unavoidably form at cosmological phase transitions; their study is therefore a key part of any attempt to understand the very early universe. The type and properties of the defect networks will depend on details of the phase transitions and, in the cases where networks are stable, they can survive until the present day, becoming a kind of cosmic fossil: a relic of a much earlier phase in the universe’s history. Being highly non-linear objects, the study of topological defect networks necessarily requires a combination of analytic and numerical work. Analytic models aim to encapsulate the key evolution mechanisms, but so far have only been derived for the simplest (and somewhat idealized) models. Numerical simulation can tackle the more complex (and realistic) models, but requires considerable computing resources. Moreover, the two approaches are necessarily linked: analytic models rely on model parameters whose values can only be determined by comparison with simulations, leading to the so-called calibration of the model. All these techniques are used in these papers. In the first, available analitic models for defect evolution were extended and applied to semilocal string networks. This confirmed the numericaly-derived hint that linear scaling evolution is the attractor solution for a broad range of model parameters. The second paper focused on high-performance numerical simulations for the evolution of domain networks. By carrying out the world’s largest simulations to date for this type of defect, it was possible to confirm that the linear (scale-invariant) evolution of the network is also an attractor in this case. This suggests that previous hints of deviations from this behavior may have been due to the limited dynamical range of those simulations. The results of these simulations were used to provide the first rigorous calibration for the two free parameters of the velocity-dependent one-scale model for domain walls.

Figure 1 | The evolution of the domain wall density (ρw = A/V) and velocity (γ2v2) as a function of time, for the 3D simulation with 5123 boxes, corresponding to different cosmological epochs: radiation era (λ = 1/2), matter era (λ = 2/3) and a faster expansion epoch with λ = 4/5. The plotted curves are the average of 25 different simulations with different (random) initial conditions. After an initial transient phase (which is due to the initial conditions of the simulations) density is inversely proportional to time while velocity is constant. Both of these are the hallmarks of scale-invariant evolution.

Although such cosmological defect networks are yet to be observed, they exist and have been studied in a range of condensed matter systems (including liquid crystals, superconductors and superfluids). We are now seeing a reborn interest in topological defect theories, thanks to the fact that it was shown that they arise naturally from a much wider class of models than previously expected and also thanks to new theoretical developments which showed topological defects could be potentially observable.

Despite considerable recent developments, many kinds of topological defects are still poorly understood. This is particularly the case for defects in models with extra dimensions, in which case it has been recently realized that the realistic models tend to be more complex than the cases that have been studied for several decades. One such example is that of semilocal string networks, arising from supersymmetric grand unified theory inflationary models. Even for simpler cosmological defects, like wall networks, the resolutions used in previous studies are still too low to allow accurate calibrations, and may have produced misleading results. Thus, much work, both analytical and numerical, is still needed in the area of topological defects. Both these issues were addresses in the papers here highlighted.

For the study of semilocal string networks, the team extended simpler, already existing analytical models, and developed the formalism which allowed them to determine, for the first time, that a linear scaling evolution is the attractor solution for these networks for a broad range of model parameters. This had been hinted before, by numeric simulations, but that suspicion could not be generalized due to the fact that the results obtained from simulations are limited by their dynamical ranges. Indeed, the comparison of this analytical model with previous numerical work yielded good results, although testing with a greater range of simulation parameters is still needed.

Regarding domain walls, some simulations had already been performed but only with lower resolutions. This led to results which were somewhat unexpected, namely some hints of deviation from the linear scaling as the attractor solution arose from them. However, with the new, high-resolution simulations now performed, no deviations from the linear scaling solution were found, in accordance to what happens with many other kinds of topological defects. The higher-resolution simulations also allowed the researchers to determine the two parameters needed for the calibration of the equivalent to the velocity-dependent one-scale model for domain walls networks. This model, which is the only one currently available for describing wall networks in an analytical way, has two free parameters that can only be measured from simulations. Thus, this first accurate calibration of these parameters (cw = 0.5 ± 0.2 and kw = 1.1 ± 0.3) is crucial for the research being done in the field of domain walls networks.

The team at CAUP contributed the analytical work and the model calibrations in both papers, and also the numerical studies of the domain walls, while the numerical studies of the semilocal networks were done by external collaborators. The domain wall simulations, which are the world’s largest to date, also required the in situ development and optimization of a new parallel code that exploits the most recent advances in high-performance computing.

It is also worth noting that the first authors of both these papers were, at the time, undergraduate students at the Faculty of Science of the University of Porto, who developed their work in the context of scholarships supervised by the CAUP researcher Carlos J. A. P. Martins.

Institute of Astrophysics and Space Sciences

Institute of Astrophysics and Space Sciences (IA) is a new but long anticipated research infrastructure with a national dimension. It embodies a bold but feasible vision for the development of Astronomy, Astrophysics and Space Sciences in Portugal, taking full advantage and fully realizing the potential created by the national membership of the European Space Agency (ESA) and the European Southern Observatory (ESO). IA resulted from the merging the two most prominent research units in the field in Portugal: the Centre for Astrophysics of the University of Porto (CAUP) and the Center for Astronomy and Astrophysics of the University of Lisbon (CAAUL). It currently hosts more than two-thirds of all active researchers working in Space Sciences in Portugal, and is responsible for an even greater fraction of the national productivity in international ISI journals in the area of Space Sciences. This is the scientific area with the highest relative impact factor (1.65 times above the international average) and the field with the highest average number of citations per article for Portugal.

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