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We report on a study of axion string networks using fixed-grid simulations of up to $16384$ points per side. The length of string can be characterised in terms of standard dimensionless parameters $\zeta_\text{w}$ and $\zeta_\text{r}$, the length density measured in the cosmic rest frame and the string rest frame, scaled with the cosmic time. The motion of the string can be characterised by the root-mean-square (RMS) velocity of the string. Starting from a range of initial length densities and velocities, we analyse the string network in the standard scaling framework and find evolution towards a fixed point with estimated values $\hat{\zeta}_{\text{w},*} = 1.220(57)$ and $\hat{\zeta}_{\text{r},*} = 1.491(93)$. The two measures are related by the RMS velocity, which we estimate to be $\hat{v}_{*} = 0.5705(93)$. The length density is consistent with previous measurements, while the velocity is about 5% lower. For simulations starting from low enough density, the length density parameters $\zeta_\text{w}$ and $\zeta_\text{r}$ remain below their fixed point values throughout, while growing slowly, giving rise to an impression of approximately logarithmic increase with time. This has been proposed as the true long-term behaviour. We find that the growth tends to slow down as the values of $\zeta_\text{w}$ and $\zeta_\text{r}$ identified as fixed points are approached. In the case of $\zeta_\text{r}$, the growth stops for simulations which started close to the fixed point length density. The difference between $\zeta_\text{w}$ and $\zeta_\text{r}$ can be understood to result from the continuing velocity evolution. Our results indicate that the growth of $\zeta_\text{w}$ is a transient appearing at low densities and while the velocity is converging. This highlights the importance of studying the string density and the velocity together, and the preparation of initial conditions., Comment: 19 pp, 18 figures, typos corrected and link to data added

First order phase transitions in the very early universe are a prediction of many extensions of the Standard Model of particle physics and could provide the departure from equilibrium needed for a dynamical explanation of the baryon asymmetry of the Universe. They could also produce gravitational waves of a frequency observable by future space-based detectors such as the Laser Interferometer Space Antenna (LISA). All calculations of the gravitational wave power spectrum rely on a relativistic version of the classical nucleation theory of Cahn-Hilliard and Langer, due to Coleman and Linde. The high purity and precise control of pressure and temperature achievable in the laboratory made the first-order A to B transition of superfluid $^3$He an ideal for test of classical nucleation theory. As Leggett and others have noted the theory fails dramatically. The lifetime of the metastable A phase is measurable, typically of order minutes to hours, far faster than classical nucleation theory predicts. If the nucleation of B phase from the supercooled A phase is due to a new, rapid intrinsic mechanism that would have implications for first-order cosmological phase transitions as well as predictions for gravitational wave (GW) production in the early universe. Here we discuss studies of the AB phase transition dynamics in $^3$He, both experimental and theoretical, and show how the computational technology for cosmological phase transition can be used to simulate the dynamics of the A-B transition, support the experimental investigations of the A-B transition in the QUEST-DMC collaboration with the goal of identifying and quantifying the mechanism(s) responsible for nucleation of stable phases in ultra-pure metastable quantum phases., Comment: 34 pp, 4 figures. Based on a talk given at Quantum Fluids and Solids 2023, Manchester, U.K, on behalf of the QUEST-DMC collaboration

We perform a detailed comparison of the dynamics of cosmic string loops obtained in cosmological field theory simulations with their expected motion according to the Nambu-Goto action. We demonstrate that these loops follow the trajectories predicted within the NG effective theory except in regions of high curvature where energy is emitted from the loop in the form of massive radiation. This energy loss continues for all the loops studied in this simulation until they self-intersect or become small enough that they annihilate and disappear well before they complete a single oscillation. We comment on the relevance of this investigation to the interpretation of the results from cosmological field theory simulations as well as their extrapolation to a cosmological context., Comment: 17 pages + appendix, 8 figures. Published version

We comment on a claim that axion strings show a long-term logarithmic increase in the number of Hubble lengths per Hubble volume [arXiv:2007.04990], thereby violating the standard "scaling" expectation of an O(1) constant. We demonstrate that the string density data presented in [arXiv:2007.04990] are consistent with standard scaling, at a string density consistent with that obtained by us [arXiv:1908.03522, arXiv:2102.07723] and other groups. A transient slow growth in Hubble lengths per Hubble volume towards its constant scaling value is explained by standard network modelling [arXiv:2102.07723]., Comment: 10 pages, 4 figures. Revised version: interpretation error corrected, improved discussion, global fit added

Using 3+1 classical lattice simulations, we follow the symmetry breaking pattern and subsequent non-linear evolution of a spectator field non-minimally coupled to gravity when the post-inflationary dynamics is given in terms of a stiff equation-of-state parameter. We find that the gradient energy density immediately after the transition represents a non-negligible fraction of the total energy budget, steadily growing to equal the kinetic counterpart. This behaviour is reflected on the evolution of the associated equation-of-state parameter, which approaches a universal value $1/3$, independently of the shape of non-linear interactions. Combined with kination, this observation allows for the generic onset of radiation domination for arbitrary self-interacting potentials, significantly extending previous results in the literature. The produced spectrum at that time is, however, non-thermal, precluding the naive extraction of thermodynamical quantities like temperature. Potential identifications of the spectator field with the Standard Model Higgs are also discussed., Comment: 26 pages, 12 figures, 1 appendix. Minor revisions, updated references, matches published version. Computer generated movies of the simulation are available at https://youtube.com/playlist?list=PLl1K9-81ct6yHh6boyTiZAfNv_zlwux7-

We study the approach to scaling in axion string networks in the radiation era, through measuring the root-mean-square velocity $v$ as well as the scaled mean string separation $x$. We find good evidence for a fixed point in the phase-space analysis in the variables $(x,v)$, providing a strong indication that standard scaling is taking place. We show that the approach to scaling can be well described by a two parameter velocity-one-scale (VOS) model, and show that the values of the parameters are insensitive to the initial state of the network. The string length has also been commonly expressed in terms of a dimensionless string length density $\zeta$, proportional to the number of Hubble lengths of string per Hubble volume. In simulations with initial conditions far from the fixed point $\zeta$ is still evolving after half a light-crossing time, which has been interpreted in the literature as a long-term logarithmic growth. We show that all our simulations, even those starting far from the fixed point, are accounted for by a VOS model with an asymptote of $\zeta_*=1.20\pm0.09$ (calculated from the string length in the cosmic rest frame) and $v_* = 0.609\pm 0.014$., Comment: 14 pages, 9 figures. v2: Minor changes, matches published version

Semilocal strings -- a particular limit of electroweak strings -- are an interesting example of a stable non-topological defect whose properties resemble those of their topological cousins, the Abrikosov-Nielsen-Olesen vortices. There is, however, one important difference: a network of semilocal strings will contain segments. These are 'dumbbells' whose ends behave almost like global monopoles that are strongly attracted to one another. While closed loops of string will eventually shrink and disappear, the segments can either shrink or grow, and a cosmological network of semilocal strings will reach a scaling regime. We discuss attempts to find a "thermodynamic" description of the cosmological evolution and scaling of a network of semilocal strings, by analogy with well-known descriptions for cosmic strings and for monopoles. We propose a model for the time evolution of an overall lengthscale and typical velocity for the network as well as for its segments, and some supporting (preliminary) numerical evidence., Comment: 11 pages, 3 figures, in `Topological Avatars of New Physics'

In the QCD axion dark matter scenario with post-inflationary Peccei-Quinn symmetry breaking, the number density of axions, and hence the dark matter density, depends on the length of string per unit volume at cosmic time $t$, by convention written $\zeta/t^2$. The expectation has been that the dimensionless parameter $\zeta$ tends to a constant $\zeta_0$, a feature of a string network known as scaling. It has recently been claimed that in larger numerical simulations $\zeta$ shows a logarithmic increase with time, while theoretical modelling suggests an inverse logarithmic correction. Either case would result in a large enhancement of the string density at the QCD transition, and a substantial revision to the axion mass required for the axion to constitute all of the dark matter. With a set of new simulations of global strings we compare the standard scaling (constant-$\zeta$) model to the logarithmic growth and inverse-logarithmic correction models. In the standard scaling model, by fitting to linear growth in the mean string separation $\xi = t/\sqrt{\zeta}$, we find $\zeta_0 = 1.19 \pm 0.20$. We conclude that the apparent corrections to $\zeta$ are artefacts of the initial conditions, rather than a property of the scaling network. The residuals from the constant-$\zeta$ (linear $\xi$) fit also show no evidence for logarithmic growth, restoring confidence that numerical simulations can be simply extrapolated from the Peccei-Quinn symmetry-breaking scale to the QCD scale. Re-analysis of previous work on the axion number density suggests that recent estimates of the axion dark matter mass in the post-inflationary symmetry-breaking scenario we study should be increased by about 50%., Comment: 8 pages including supplemental material; 5 figures. v2 matches published version

Models of symmetry breaking in the early universe can produce networks of cosmic strings threading 't Hooft-Polyakov monopoles. In certain cases there is a larger global symmetry group and the monopoles split into so-called semipoles. These networks are all known as cosmic necklaces. We carry out large-scale field theory simulations of the simplest model containing these objects, confirming that the energy density of networks of cosmic necklaces approaches scaling, i.e. that it remains a constant fraction of the background energy density. The number of monopoles per unit comoving string length is constant, meaning that the density fraction of monopoles decreases with time. Where the necklaces carry semipoles rather than monopoles, we perform the first simulations large enough to demonstrate that they also maintain a constant number per unit comoving string length. We also compare our results to a number of analytical models of cosmic necklaces, finding that none explains our results. We put forward evidence that annihilation of poles on the strings is controlled by a diffusive process, a possibility not considered before. The observational constraints derived in our previous work for necklaces with monopoles can now be safely applied to those with semipoles as well., Comment: 12 pages and 7 figures, updated to match with published version

We present the first CMB power spectra from numerical simulations of the global O(N) linear $\sigma$-model with N = 2,3, which have global strings and monopoles as topological defects. In order to compute the CMB power spectra we compute the unequal time correlators (UETCs) of the energy-momentum tensor, showing that they fall off at high wave number faster than naive estimates based on the geometry of the defects, indicating non-trivial (anti-)correlations between the defects and the surrounding Goldstone boson field. We obtain source functions for Einstein-Boltzmann solvers from the UETCs, using a recent method that improves the modelling at the radiation- matter transition. We show that the interpolation function that mimics the transition is similar to other defect models, but not identical, confirming the non-universality of the interpolation function. The CMB power spectra for global strings and monopoles have the same overall shape as those obtained using the non-linear $\sigma$-model approximation, which is well captured by a large-N calculation. However, the amplitudes are larger than the large-N calculation predict, and in the case of global strings much larger: a factor of 20 at the peak. Finally we compare the CMB power spectra with the latest CMB data to put limits on the allowed contribution to the temperature power spectrum at multipole $\ell$ = 10 of 1.7% for global strings and 2.4% for global monopoles. These limits correspond to symmetry-breaking scales of 2.9x1015 GeV (6.3x1014 GeV with the expected logarithmic scaling of the effective string tension between the simulation time and decoupling) and 6.4x1015 GeV respectively. The bound on global strings is a significant one for the ultra-light axion scenario with axion masses ma 10-28 eV. These upper limits indicate that gravitational wave from global topological defects will not be observable at the GW observatory LISA., Comment: Minor changes, matches published version; 24 pages, 9 figures

We continue a comprehensive numerical study of semilocal string networks and their cosmological evolution. These can be thought of as hybrid networks comprised of (non-topological) string segments, whose core structure is similar to that of Abelian Higgs vortices, and whose ends have long--range interactions and behaviour similar to that of global monopoles. Our study provides further evidence of a linear scaling regime, already reported in previous studies, for the typical length scale and velocity of the network. We introduce a new algorithm to identify the position of the segment cores. This allows us to determine the length and velocity of each individual segment and follow their evolution in time. We study the statistical distribution of segment lengths and velocities for radiation- and matter-dominated evolution in the regime where the strings are stable. Our segment detection algorithm gives higher length values than previous studies based on indirect detection methods. The statistical distribution shows no evidence of (anti)correlation between the speed and the length of the segments., Comment: 33 pages, 22 figures. v2: Minor changes. Matches published version. 23 pages and 12 figures. Extra figures have been included as Supplemental Material in an Ancillary file

We present an estimation of the average velocity of a network of global monopoles in a cosmological setting using large numerical simulations. In order to obtain the value of the velocity, we improve some already known methods, and present a new one. This new method estimates individual global monopole velocities in a network, by means of detecting each monopole position in the lattice and following the path described by each one of them. Using our new estimate we can settle an open question previously posed in the literature: velocity-dependent one-scale (VOS) models for global monopoles predict two branches of scaling solutions, one with monopoles moving at subluminal speeds and one with monopoles moving at luminal speeds. Previous attempts to estimate monopole velocities had large uncertainties and were not able to settle that question. Our simulations find no evidence of a luminal branch. We also estimate the values of the parameters of the VOS model. With our new method we can also study the microphysics of the complicated dynamics of individual monopoles. Finally we use our large simulation volume to compare the results from the different estimator methods, as well as to asses the validity of the numerical approximations made., Comment: v2: Minor changes. Matches published version. 22 pages, 4 figures. v3: Erratum added that corrects a mistake in a formula and the corresponding tables, without modifying the conclussions

In this work we calibrate two different analytic models of semilocal strings by constraining the values of their free parameters. In order to do so, we use data obtained from the largest and most accurate field theory simulations of semilocal strings to date, and compare several key properties with the predictions of the models. As this is still work in progress, we present some preliminary results together with descriptions of the methodology we are using in the characterisation of semilocal string networks., Comment: 6 pages and 4 figures. Prepared for the Spanish Relativity Meetings ERE2014

We report on a detailed numerical study of the evolution of semilocal string networks, based on the largest and most accurate field theory simulations of these objects to date. We focus on the large-scale network properties, confirming earlier indications (coming from smaller simulations) that linear scaling is the attractor solution for the entire parameter space of initial conditions that we are able to probe. We also provide a brief comparison of our numerical results with the predictions of a previously developed one-scale model for the overall evolution of these networks. Two subsequent papers will discuss in more detail the analytic modeling of the semilocal segment populations as well as optimized numerical diagnostics., Comment: 13 pages, 4 figures; v2: minimal changes, matches version published in PRD

We study field theoretical models for cosmic strings with flat directions in curved space-time. More precisely, we consider minimal models with semilocal, axionic and tachyonic strings, respectively. In flat space-time, the string solutions of these models have a flat direction, i.e., a uniparametric family of configurations with the same energy exists which is associated to a zero mode. We prove that the zero mode survives coupling to gravity, and study the role of the flat direction when coupling the strings to gravity. Even though the total energy of the solution is the same, and thus the global properties of the family of solutions remains unchanged, the energy density, and therefore the gravitational properties, are different. The local structure of the solutions depends strongly on the value of the parameter describing the flat direction; for example, for supermassive strings, the value of the free parameter can determine the size of the universe., Comment: 14 pages, 11 figures; matches version to be published in JHEP