Numerical Simulations of Early Universe Sources of Gravitational Waves

28 Jul 2025, 09:00 → 15 Aug 2025, 19:00 Europe/Stockholm

Albano 3: 4203 - SU Conference Lunch Room (48 seats) (Albano Building 3)

Alberto Roper Pol (University of Geneva)

Venue

Nordita, Stockholm, Sweden

Nordita conference center, House 3, room 4205

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Scope

We are happy to announce the three-week program on numerical simulations of early Universe sources of gravitational waves at Nordita, Stockholm, that will take place from July 28 to August 15 of 2025.

The main objectives of the program are to review the different possible sources contributing to a cosmological background of gravitational waves, the state-of-the-art numerical codes that are available and used by the community, to discuss numerical techniques for these studies, and to discuss what are the priorities that we should address as a community in the coming years.

For this purpose, the program is divided into three weeks, covering the following potential sources of GWs in the early Universe: 

Week 1 (Jul 28-Aug 1): Inflation, (p)reheating, and primordial black holes

Week 2 (Aug 4-8): First order phase transitions and primordial turbulence

Week 3 (Aug 11-15): Topological defects: cosmic strings, domain walls, and others

Registration is now open and will close on March 15th.

To submit an abstract for a contributed talk, please provide the details while registering and then send an abstract in the call for abstracts section by the deadline, March 15th.

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Confirmed Invited speakers

• Pierre Auclair (Louvain University)
• Jorge Baeza-Ballesteros (DESY)
• Jose Juan Blanco-Pillado (University of the Basque Country)
• Axel Brandenburg (Nordita)
• Malte Buschmann (University of Amsterdam)
• Angelo Caravano (IAP, Paris)
• Jose Ricardo Correia (University of Helsinki)
• Emanuela Dimastrogiovanni (University of Groningen)
• Matteo Fasiello (IFT)
• Marco Gorghetto (DESY)
• Mark Hindmarsh (University of Helsinki)
• Oksana Iarygina (Nordita)
• Ryusuke Jinno (Kobe University)
• Tina Kahniashvili (Carnegie Mellon University)
• Marek Lewicki (University of Warsaw)
• Joanes Lizarraga (University of the Basque Country)
• Swagat Mishra (University of Nottingham)
• Ilia Musco (University of Nova Gorica)
• Gerasimos Rigopoulos (Newcastle University)
• Henrique Rubira (LMU/Cambridge)
• Philipp Schicho (University of Geneva)
• Lara Sousa (University of Porto)
• Francisco Torrenti (University of Barcelona)
• Tanmay Vachaspati (Arizona State University)
• Jorinde van de Vis (CERN)
• Masahide Yamaguchi (Institute for Basic Science, Daejeon)

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Accommodation

We will provide accommodation in Stockholm for some accepted applicants, prioritizing early-career researchers and based on travel fund needs. Please indicate if you would like Nordita to cover your accommodation in the registration form and we will confirm accommodation and travel support shortly after the registration deadline.

We have reserved some apartments at Biz Apartment Gärdet at a reduced price. If you would like to reserve one of these apartments and cover the costs on your own, please indicate so in the registration form.

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Application/Registration

Registration is open and will close on March 15th. Registrants will receive an on-site/remote participation confirmation from the organizers shortly after this date.

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Organizers

Chiara Caprini (CERN and University of Geneva)

Amelia Drew (ICTP, Trieste)

Daniel Figueroa (IFIC, Valencia)

Alberto Roper Pol (University of Geneva)

David Weir (University of Helsinki)

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Code of conduct

All participants are expected to uphold our Code of Conduct, which aims to foster a harassment-free environment where everyone feels comfortable engaging in both scientific and social activities.

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Publications acknowledging this program:

1. I. Rogachevskii, N. Kleeorin, "Alpha effect in density-stratified turbulence with large-scale shear in astrophysical clouds and discs,"  arXiv:2509.03248

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Sponsored by:

Code_of_Conduct_Nordita.pdf

Monday 28 July

1 Welcome and registration

2 Studying Inhomogeneous Inflation with Numerical Relativity

Cosmic inflation is the leading paradigm for describing the early universe, addressing fundamental issues such as the horizon and flatness problems. However, a key unresolved question is the nature of its initial conditions. In this talk, I will explore how numerical relativity helps study inflationary spacetimes with inhomogeneous initial conditions, particularly in the presence of strong gravitational effects from large inhomogeneities. Full numerical simulations allow us to map out the phase space of initial conditions that lead to sufficient duration of slow roll inflation versus those that do not. The results strongly depend on the inflationary model, with a rule of thumb that the models with near- or super-Planckian characteristic scales are more robust to matter and geometric inhomogeneities than those with sub-Planckian scales. We mainly focus on the study of α-attractor models and our simulation results allow us to establish a lower bound on the tensor-to-scalar ratio r.

Speaker: Panagiotis Giannadakis (King's College London)

Early Universe Workshop Inflation-1.pdf

giannadakis.mp4

11:00 Coffee break

3 Lectures on Inflation. Part I

Speaker: Matteo Fasiello

12:30 Lunch break

4 Nordita Niels Bohr Colloquium: Numerical Simulations of Early Universe Sources of Gravitational Waves

Gravitational wave (GW) astronomy is emerging as an exciting new field, offering unprecedented opportunities for breakthroughs in beyond the standard model physics and early Universe cosmology. The key point is to note that early Universe dynamics operates at energies unreachable by any terrestrial means, creating GW backgrounds that redshift down to detectable frequencies today. A detection of any such background can therefore probe energies far above those accessible to particle colliders, shedding light on fundamental physics questions, such as the state of the early Universe, the baryon asymmetry of the Universe, the nature of the dark matter, or whether exotic objects like primordial black holes or cosmic strings exist. Out of the effort to detect GW backgrounds over a wide range of frequencies, a detection program including a large variety of experiments is emerging, including pulsar timing array (PTA) observations, space-based GW detectors (e.g. LISA), or next-generation ground-based detectors (e.g. ET or CE). PTA collaborations have just announced the first evidence for a GW background at nHz frequencies. Although a signal from supermassive black hole binaries is naturally expected at those frequencies, cosmological backgrounds also represent a viable explanation. In order to demonstrate that a potential detection can only be explained by a cosmological signal, an accurate modelling of the different GW backgrounds from the early Universe is of paramount importance. Early Universe GW sources are inherently characterised by nonlinear dynamics and, hence, their study requires conducting the use of high-performance computing. I will give a biased review on recent advances on the study of nonlinear dynamics of early Universe physics that are required to provide a precise characterization of the resulting GW background from the early Universe.

Nordita Niels Bohr Colloquium: https://indico.fysik.su.se/event/9274/

Speaker: Alberto Roper Pol (University of Geneva)

nordita_bohr_colloquium_RoperPol.pdf

Tuesday 29 July

5 Towards Preheating after Inflation: Inflation Fragmentation, Oscillon Formation and Decay (online)

The transition from cosmic inflation to the hot Big Bang, known as reheating, remains a key open question in cosmology. During its early stage, called preheating, the inflaton field decays explosively via parametric resonance into lighter bosonic offspring fields. However, when these external couplings are weak, strong self-interaction (cohesive force) causes the oscillating inflaton condensate to fragment, forming extremely long-lived scalar-field lumps known as oscillons. We investigate the conditions for oscillon formation during preheating, particularly in the presence of external couplings, within the class of inflationary potentials favored by the latest CMB observations. Using high-resolution (3+1)-dimensional lattice simulations on CosmoLattice platform, we systematically map the parameter space that supports oscillon formation. Our results suggest that preheating may have proceeded through both oscillon decay and the conventional decay of the inflaton condensate, offering new insights into the early reheating dynamics.

Speaker: Swagat Saurav Mishra (University of Nottingham, UK)

Formation and decay of oscillons after inflation in the presence of an external coupling. Part I. Lattice simulations

mishra.mp4

SSM_Oscillons_Nordita_2025.pdf

6 What is the maximum temperature ever reached in the universe?

Gravitational waves are naturally sourced by hydrodynamical fluctuations in a thermal medium, as the one that filled the universe before recombination. Since the corresponding gravitational wave spectrum is expected to show rapid growth at high frequencies, f ∼ 1...1000 Hz, unprecedented prospects to detect these signals may be offered by the proposed Einstein Telescope. While the spectral shape is well understood, the peak amplitude is set by the plasma temperature at emission. We present a model-independent numerical method to estimate an upper bound for the maximal temperature reached after inflation and discuss the validity of the formalism across possible scale hierarchies.

Speaker: Simona Procacci (U. Geneva)

procacci.mp4

SProcacci_Nordita_25.pdf

11:00 Coffee break

7 Lectures on Inflation. Part II

Speaker: Matteo Fasiello

fasiello_lecture2.mp4

12:30 Lunch break

8 Stochastic Inflation in (Numerical) General Relativity

I provide a formulation of Stochastic Inflation in full general relativity that goes beyond the slow-roll and separate universe approximations. All degrees of freedom are included in the dynamics and all gradient terms are kept. The stochastic source terms are gauge invariant and defined in terms of the only dynamical scalar degree of freedom in single-field inflation, the Mukhanov-Sasaki variable. The equations can therefore be formulated in any gauge and enable a well-posed implementation for 3+1 numerical relativity simulations with stochastic noise. Some preliminary examples are discussed, including an ultra slow roll potential and a stochastic resonance model.

Speaker: Gerasimos Rigopoulos

rigopoulos.mp4

Wednesday 30 July

9 Backreaction and cosmic butterflies: non-perturbative insights into the small-scale physics of inflation

The small-scale physics of inflation can leave unique observational signatures in the gravitational wave background and may also generate primordial black holes as a dark matter candidate. These phenomena often involve a significant enhancement of inflationary fluctuations, potentially leading to the breakdown of standard perturbation theory. In this talk, I will discuss how lattice simulations provide a crucial tool for addressing these challenges. Focusing on single-field inflationary models with deviations from slow-roll, I will demonstrate how nonlinear physics can significantly alter the inflationary dynamics and their predictions, even in regimes where perturbation theory is typically assumed valid. This highlights the indispensable role of lattice simulations in probing inflation with current and future gravitational wave experiments.

Speaker: Angelo Caravano

Caravano.pdf

10 Primordial gravitational waves from from fully relativistic inflation simulations

Speaker: Paul Shellard

shellard.mp4

11:00 Coffee break

11 Lectures on Inflation. Part III

Speaker: Matteo Fasiello

fasiello_lecture3.mp4

12:30 Lunch break

12 GW background signals from Inflation: lattice calculation

Axion inflation remains one of the most compelling scenarios for early-universe dynamics, offering distinctive observational signatures. In this talk, I will present our recent work on the strong backreaction regime in abelian axion inflation, using fully self-consistent lattice simulations to capture the non-linear dynamics. Particular attention will be given to possible extensions to more general inflationary potentials. I will also present our first steps toward modelling the resulting gravitational wave signal and discuss the prospects for future observational detection.

Speaker: Joanes Lizarraga

lizarraga.mp4

NumericalSimulations_JoanesLizarraga.pdf

Thursday 31 July

13 Gravitational waves from axion inflation with non-Abelian gauge fields

Currently, the search for primordial gravitational waves is largely focused on detecting the parity-odd polarization pattern in the Cosmic Microwave Background—the B-modes. Accurately interpreting B-mode measurements depends heavily on understanding their production mechanisms. A particularly compelling scenario involves gravitational wave generation through the interaction of axion with gauge fields. I will discuss recent advances in axion inflation incorporating non-Abelian gauge fields, highlighting primordial gravitational wave background signatures and implications for primordial magnetogenesis.

Speaker: Oksana Iarygina (Stockholm University, Nordita)

iarygina.mp4

NorditaGWs_Oksana.pdf

14 Stochastic simulation of reheating and/or warm inflation

The late stage of the reheating process may be captured by a two-component approach, in which a self-interacting plasma has already attained local equilibrium, while the inflaton field is still far from equilibrium. This should be particularly suitable if the plasma contains non-Abelian gauge bosons, which are believed to equilibrate fast. We describe the foundations of such an approach, which can in principle be studied both in a linear and non-linear regime. Recent progress towards a gauge-invariant numerical implementation of the linear regime is summarized, and steps towards determining the curvature and tensor power spectra are outlined.

Speaker: Mikko Laine

laine.mp4

nordita25.pdf

11:00 Coffee break

15 Simulations of inflationary magnetogenesis and gravitational waves

A significant fraction of the observable stochastic gravitational wave background can come from the early universe. The spectral shape reveals information about the nature of the early universe and its magnetic fields. The relic gravitational wave spectrum is particularly sensitive to the time-dependence of the source and therefore reveals information about the generation mechanism of magnetic fields. Gravitational wave production is more effective on large length scales, making inflationary magnetogenesis an obvious candidate. Parity-violating processes such as axion inflation and in principle also the chiral magnetic effect directly imprint their helicity onto the circular polarization spectrum of gravitational waves. In my talk, I will present the results of three-dimensional numerical simulations of various generation mechanisms and how their time-dependence shapes the resulting gravitational wave field both spectrally and in real space.

Speaker: Axel Brandenburg (Stockholm University, Nordita)

brandenburg.mp4

Brandenburg.pdf

12:30 Lunch break

16 Gravitational wave background anisotropies as a probe of the early universe

Speaker: Emanuela Dimastrogiovanni

dimastrogiovanni.mp4

slides_Nordita_2025.pdf

Friday 1 August

17 Equation of state during (p)reheating and its observational implications

I will present a complete characterization of the equation of state from the end of inflation until perturbative reheating, when an inflaton with quadratic potential is coupled to a daughter field through both trilinear and scale-free interactions. By simulating the dynamics in 2+1-dimensional lattices, we track the evolution of the equation of state for up to 10 e-folds of expansion, with the later evolution being resolved through a Boltzmann approach. Our results show that, despite the daughter field experiencing an initial tachyonic excitation, the equation of state never reaches w=1/3 before perturbative reheating, independently of the coupling strengths. I will discuss the implications of our results for the GW spectrum from preheating observed today, as well as for theoretical predictions of inflationary CMB observables.

Speaker: Francisco Torrenti

Nordita-talk-EoS.pdf

torrenti.mp4

11:00 Coffee break

18 Numerical Simulations of Primordial Black Holes

I will present a mathematical and numerical investigation on primordial black hole formation from the collapse of adiabatic cosmological perturbations, describing into details the mechanism of gravitational collapse. For a radiation dominated fluid this is described by the mechanism of critical collapse. Starting from a time independent curvature profile, specified on the super horizon regime using the gradient expansion approach, I study primordial black hole formation from adiabatic cosmological perturbations. This allows to compute the threshold and the mass distribution of primordial black holes consistently with the power spectrum of cosmological perturbations obtained from inflation. This scenario can be generalised during the quark-hadron (QCD) phase transition, when the equation of state is characterised by a softening, reducing significantly the pressure gradients around the solar mass scale. This enhances the formation of primordial black holes, in the mass range of binary systems merging into a bigger black hole, emitting gravitational waves observed by LIGO/Virgo. This gives a possible interesting explanation for the signals of the LVK catalog falling into the lower mass gap, which are difficult to be explained by black holes formed from stellar collapse.

Speaker: ILIA Musco (Sapienza University of Rome, INFN)

Musco PBHs Nordita.pdf

12:30 Lunch break

Panel discussion week 1

Convener: Daniel Figueroa

Monday 4 August

19 Supercooled cosmological phase transitions and their gravitational wave signals

We are currently witnessing the dawn of a new era in astrophysics and cosmology, started by the LIGO/Virgo observations of Gravitational Waves (GWs). Recently, also the detection of a stochastic background of GWs at very low frequencies was announced by the Pulsar Timing Array collaborations. In this talk, I will discuss how GW signals are produced in very strong cosmological phase transitions and examine the possible implications of current data for this source as well as the prospects for detection in the upcoming next generation of experiments.

Speaker: Marek Lewicki

lewicki.mp4

20 Bubble Nucleation and Gravitational Waves from Strongly Coupled QFT's

Speaker: Nicklas Ramberg (SISSA Trieste)

ramberg1.mp4

11:00 Coffee break

21 Perturbative cosmological phase transitions in a broad temperature range

Cosmological phase transitions, particularly the electroweak one, continue to draw attention due to their potential to generate a stochastic gravitational wave background and to provide a possible mechanism for baryogenesis.

In this talk, I will discuss the perturbative description of such transitions, focusing on recent developments in high-temperature effective field theory (EFT) relevant to transition thermodynamics. Key aspects include the automated construction of the high-temperature EFT, the identification of the effective transition scale for nucleation, and the incorporation of the final perturbative order of soft fluctuations in the effective potential. Ultimately, by examining the structure of higher-dimensional operators in the EFT, we gain an appreciation for the limitations of the high-temperature expansion, particularly in describing the strongest transitions.

Confronted with these limitations, I will conclude by outlining old and new strategies to systematically extend perturbative control beyond the high-temperature regime, enabling descriptions valid across a broader temperature range.

Speaker: Philipp Schicho

nordita.25.pdf

schicho.mp4

12:30 Lunch break

22 Origin of chiral magnetic effect, production of turbulence and generation of large-scale magnetic fields

In the standard model of particle physics, the chiral anomaly can occur in relativistic plasmas and plays an important role in the early Universe, proto-neutron stars, heavy-ion collisions, and quantum materials. It gives rise to a chiral magnetic effect if the number densities of left- and right-handed electrically charged fermions are
unequal. At high energies, the dynamics of a plasma with charged fermions can be described in terms of chiral magnetohydrodynamics.
We show that a chiral magnetic effect can result just from spatial fluctuations of the chemical potential, causing a chiral dynamo instability, magnetically driven turbulence, and ultimately a
generation of large-scale magnetic field through the magnetic alpha effect. This have consequences for the dynamics of certain high-energy plasmas, such as the early Universe. We discuss how the chiral magnetic effect can be a source for gravitational waves.

Authors: I. Rogachevskii , J. Schober, A. Brandenburg

Speaker: Igor Rogachevskii (Ben-Gurion University of the Negev)

Chiral-MHD-Rogachevskii (1).pdf

Tuesday 5 August

23 The bubble wall velocity in first order phase transitions

Cosmological first order phase transitions are a promising source of gravitational waves, and a possible explanation of the baryon asymmetry and dark matter abundance. Predicting the phenomenological consequences of such phase transitions requires knowledge of the expansion velocity of the bubbles formed in a phase transition. In this talk, I will summarize the computation of the wall velocity and present WallGo, a software package for the computation of bubble wall velocities. WallGo is the first publicly available code that computes the matrix elements, collision integrals Boltzmann equation and scalar field equation of motion for user-defined models. I will present results obtained with WallGo, and discuss estimates of the theoretical uncertainty of the bubble wall velocity.

Speaker: Jorinde van de Vis

TalkJvdVis.pdf

vandevis.mp4

24 Simulating cosmic bubbles on the lattice and in the lab

Bubble nucleation plays a pivotal role in many models of particle physics and the early Universe, and is a promising potential source of cosmological gravitational waves. However, we lack a satisfying theoretical understanding of this process, with existing approaches working only in imaginary (Euclidean) time, and relying on assumptions that have yet to be empirically tested. A promising route forward is to use cold-atom systems which undergo first-order phase transitions that are analogous to vacuum decay. In this talk, I will present recent theoretical work to understand this analogy using semiclassical lattice simulations, and will discuss possibilities and challenges for realising these analogues in the laboratory.

Speaker: Alex Jenkins (University of Cambridge)

jenkins.mp4

simulating-cosmic-bubbles-alex-jenkins.key

simulating-cosmic-bubbles-alex-jenkins.pdf

11:00 Coffee break

25 Curvature perturbation from first-order phase transitions & A positive-definite formulation of tunneling

This talk is consists of two parts. In the first part I will talk about curvature perturbation produced in very strong first-order phase transitions. I will first explain super-hubble curvature perturbation based on 2311.16222, and then move on to hubble-scale curvature perturbation based on 2503.01962. In the second part I change the topic completely and talk about a new type of formulation of quantum tunneling that features positive-definite action based on my ongoing work.

Speaker: Ryusuke Jinno (DESY)

jinno.mp4

12:30 Lunch break

26 Primordial Magnetic Fields: A Source of Relic Gravitational Waves

Primordial magnetic fields (PMFs) are a unique probe of the early Universe and a source of relic gravitational waves. In this talk, I will present recent advances in our understanding of PMFs, focusing on three interconnected topics. First, I will discuss the evolution of PMFs within the framework of chiral magnetohydrodynamics (MHD), highlighting the emergence of mean-field dynamos. Second, I will examine current observational constraints on PMFs in cosmic voids, and explore the potential contamination from galactic dipole fields that may affect the interpretation of lower bounds. Finally, I will present new models for the late-time evolution of PMFs during cosmic structure formation, with implications for both theory and observation. These insights help clarify how cosmic magnetism may leave detectable imprints in upcoming gravitational wave observations.

Speaker: Jennifer Schober (University of Bonn)

schober.mp4

Wednesday 6 August

27 Cold baryogenesis revisited

The matter-antimatter asymmetry of the Universe represents one of the main open questions in particle physics and cosmology. In this talk, we will present a novel realization of cold baryogenesis (a mechanism involving the formation and decay of topological defects associated with the gauge group of the Standard Model known as SU(2) textures) that relies on the out-of-equilibrium dynamics during a strong first order electroweak phase transition. By performing extensive lattice simulations of the Higgs doublet and gauge field dynamics, we evaluate the related Chern-Simons number production as well as the rate of baryon number violation, as a function of the parameters of the phase transition and the shape of the Higgs potential. We finally provide an estimate for the total baryon asymmetry generated this way.

Speaker: Simone Blasi (DESY Hamburg)

blasi.mp4

talk Stockholm.pdf

11:00 Coffee break

28 Simulations and modelling of gravitational waves from first order phase transitions

I review the modelling of acoustic production of gravitational waves at first order phase transitions, and discuss its evolution in the light of recent developments in numerical simulation and theory, which are entering the regime of strong transitions and large bubbles where signals are loudest.

Speaker: Mark Hindmarsh (University of Sussex)

G-Waves 2025 08 Nordita.pdf

hindmarsh.mp4

12:30 Lunch break

29 Langer’s nucleation rate on the lattice

First-order phase transitions in the early universe provide a possible mechanism for producing observable gravitational waves. Predicting the gravitational wave spectrum requires accurate nucleation rate computations, as the rate determines the transition temperature and duration. These computations often rely on Langer's nucleation rate formula, which has long resisted validation in numerical simulations. Here, we present work that, for the first time, demonstrates agreement between Langer's formula and lattice simulations. Our findings clarify the conditions necessary for successful lattice simulations of nucleation and reveal insights into nucleation processes and the limitations of Langer's formula.

Speaker: Joonas Hirvonen (University of Nottingham)

hirvonen.mp4

Thursday 7 August

30 The Higgsless simulations (and PTs to address the H0 tension)

In the first part of the talk, I will present the Higgsless approach to simulate the plasma dynamics and to predict the stochastic GW spectrum from PTs. I will explain how the Higgsless simulations can produce fully nonlinear results. In the second part of the talk I will focus on how phase transitions can address the so-called H0 tension.

Speaker: Henrique Rubira

Nordita_GWsim_25_08_05.pdf

rubira.mp4

31 Gravitational wave templates for cosmological phase transitions with non-linear decay of the fluid motion

A stochastic gravitational-wave (GW) background of cosmological origin is a prime target for present and forthcoming detectors, from pulsar–timing arrays to LISA. In this talk I present new results on GW production during cosmological first-order phase transitions, obtained with the recently developed Higgsless simulation framework. From these simulations we derive a set of GW templates built within a generalized modelling scheme that extends the standard sound-wave description to include damped sources, thereby allowing to capture the saturation of the GW amplitude at long source durations. I will report insights into strong transitions, showcase the first high-resolution Pencil-Code simulation results utilizing the Higgsless approach, and demonstrate how spectra can be generated on the fly with the open-source Python package CosmoGW.

Speaker: Isak Stomberg (DESY)

NORDITA-talk.pdf

stomberg.mp4

11:00 Coffee break

32 What do we learn from pulsar timing data: testing early Universe physical processes

Recent detection of stochastic gravitational wave background by pulsar timing arrays (PTAs) missions opens a new window of testing fundamental physics laws at energy scales far beyond what is reached by particle physics experiments and/or by astrophysical observations;  In my talks I will addresses some puzzles in our understanding of the Universe nature (its past, present day, and future), that motivate to consider extensions of the standard model of cosmology and particle physics; In particular, I will discuss the PTAs data in the context of massive gravity and reconstructing the earliest moments of the Universe expansion.

Speaker: Tina Kahniashvili (Carnegie Mellon University (USA) & Ilia State University (Georgia))

kahniashvili.mp4

NORDITA2025-2.pdf

12:30 Lunch break

33 Relativistic hydrodynamics with CosmoLattice

In this talk we will explore the newly developed hydrodynamic module in CosmoLattice that allows to investigate relativistic fluids in an FLRW background. We first review the hydrodynamic equations in the conservation and non-conservation form, which have been implemented in CosmoLattice in the the relativistic and non-relativistic limit. We may then have a look at the evolution algorithms and the discretisation schemes that are used. As a first application we study the production of gravitational waves from a turbulent plasma in the relativistic and non-relativistic regime. At last we will have an outlook on how this hydrodynamic module will be further developed and connected to the other capabilities of CosmoLattice.

Speaker: Kenneth Marschall

Relativistic_Hydrodynamics_with_CL.pdf

34 Confining Sectors in the Early Universe

I will present some ongoing work on early universe dynamics relevant to a strongly coupled confining sector, in a bottom-up holographical approach. I will focus on two scenarios: during inflation and after inflation. I will discuss the inflationary signals if such a sector is present during inflation, and aspects of a deconfinement to confinement phase transition after reheating.

Speaker: Rashmish Mishra (Harvard University)

Friday 8 August

35 Gravitational Waves from slowly decaying sources in the early Universe

Understanding the Gravitational Wave background from stochastic processes in the early Universe requires studying the unequal time correlator (UETC) of the anisotropic stress tensor of the source. Some of the contributions to the Gravitational Wave spectrum generated in a First-Order Phase Transition can be described assuming a constant-in-time UETC. We show how the constant-in-time model can be used to understand those contributions and, in particular, its application to the Gravitational Wave spectrum from lattice simulations of relativistic hydrodynamic turbulence.

Speaker: Antonino Midiri (University of Geneva)

midiri.mp4

Midiri Stockholm.pdf

36 Gravitational wave production: the interplay between vortical and compressional motions.

In the early Universe, during a phase transition, the surrounding plasma is subjected to a fluid motion sourcing gravitational waves. This fluid motion can be composed of compressional and vortical motions. Most of the time, they are considered distinct and studied independently. However, by analyzing the UETC of the anisotropic stresses, a mixed term combining vortical and compressional motions leads to another contribution to gravitational wave production.
In this presentation, the interplay between the two motions will be at the heart of the discussion. This mixed motion will be studied and characterized using numerical simulations from the Pencil Code.

Speaker: Madeline Salome

GWs-Nordita-Madeline_SALOME_compressed.pdf

salome.mp4

37 GWs from generalized fluid perturbations during first-order phase transitions

Given their weak interaction with different degrees of freedom, gravitational waves (GWs) offer fresh opportunities to probe the earliest moments of the Universe, and physics beyond the Standard Model. For instance, a first-order phase transition (FOPT) in the primordial plasma at the electroweak scale could emit a recognizable signal in the frequency range of the upcoming detectors like LISA. During a FOPT, bubbles of the stable phase nucleate, expand, and collide, generating perturbations that result in GWs. Thus, to model the consequent GW spectrum realistically, it is crucial to understand the fluid perturbations of the primordial plasma. Given the high energy and the relativistic speeds involved in strong enough phase transitions, the shape and amplitude of the power spectrum could be significantly affected by nonlinearities and the generated turbulence. While these topics have been generally studied in fluid dynamics for decades, their impact on GW spectra from FOPTs remains unclear. We investigate the production of vorticity and turbulence in the relativistic regime beyond linear theory. Using a semi-analytical approach, we report estimates of the GW spectrum, as well as the time scales and the strength of vorticity production when the initial field is purely compressional, as in the case of sound waves. This analysis helps constrain the applicability of models that approximate the fluid perturbations induced by bubble collisions as a superposition of the longitudinal modes, such as sound waves. Furthermore, it lays the foundation for more detailed numerical studies of the relativistic regime in the future.

Speaker: Deepen Garg (University of Bonn)

garg.mp4

11:00 Coffee break

38 Gravitational Effects on Sound Waves: A Perturbative Approach for Large Bubbles in Cosmological First-Order Phase Transitions

If a cosmological first-order phase transition takes place over the cosmological time scale, gravity must affect the profile of sound waves. To investigate such a regime beyond the self-similarity, we combine a hydrodynamic scheme in the presence of gravity with a fluid computation scheme under energy injection from the bubble wall.
In this talk, I present the results of our (1+1)d hydrodynamic simulations around an expanding bubble in the cosmological background. We find that gravitational effects lead to a thinner fluid shell and reduce the kinetic energy budget, in qualitative agreement with previous discussions on late-time fluid behavior in an expanding universe. Moreover, our simulations reveal the development of sub-structures in the fluid profile for accelerating bubble walls. We also discuss the potential impact of these effects on the broadening of the SGWB spectral plateau.

Speaker: Jun'ya Kume (University of Padova, INFN Padova)

202508_Nordita_Kume.pdf

kume.mp4

39 Generating gravitational wave spectra from phase transitions with realistic equations of state

The Sound Shell Model provides a computationally efficient way of calculating the gravitational wave spectra of first-order cosmological phase transitions, reproducing the results of lattice simulations for intermediate strength transitions. The Sound Shell Model has been encapsulated in the Python-based simulation framework PTtools, which enables easy generation of the gravitational wave spectra. Using PTtools one can simulate hundreds of spectra in a matter of minutes on a laptop. This enables charting the full parameter space.

The vast majority of existing simulations have been based on the bag model equation of state, which assumes the ultrarelativistic sound speed $c_s = \frac{1}{\sqrt{3}}$. As the latest addition, PTtools has been extended to include support for arbitrary equations of state and therefore for a temperature- and phase-dependent sound speed $c_s(T,\phi)$, extending the simulations beyond the ultrarelativistic assumption of the bag model.

In addition to PTtools, we have developed the web-based plotting utility
PTPlot, which makes generating the gravitational wave spectra possible through a web browser. Integrating PTtools with PTPlot enables easy access to the full Sound Shell Model for the research community.

Speaker: Mr Mika Mäki (University of Helsinki)

maki.mp4

Slides

12:30 Lunch break

Panel discussion week 2

Conveners: Alberto Roper Pol (University of Geneva), David Weir (University of Helsinki)

Monday 11 August

40 Lectures on topological defects. Part I: Kinks and domain walls

Speaker: Tanmay Vachaspati (Arizona State University)

topoDefectsWallsNordita2025.pdf

vachaspati_lecture1.mp4

41 Gravitational Signatures of Domain Walls

In this talk, we present the results of simulations that explore the gravitational wave spectra produced by domain walls in a $\mathbf{Z}_2$ model, using the publicly available $CosmoLattice$ code. Focusing on the approach to scaling, we investigate the impact of various initial fluctuations and mass-to-Hubble ratios. We demonstrate that the Velocity-Dependent One-Scale (VOS) model accurately describes the evolution toward scaling after just a few e-folds, irrespective of the initial fluctuation conditions. Using a $2048^3$ grid, we compute the gravitational wave spectra for a domain wall system in an expanding Universe with different equations of state. Additionally, we conduct semi-analytical studies of the unequal time correlator (UTC) to refine our understanding of gravitational wave production and the dynamics of domain walls.

Speaker: Aäron Rase (Vrije Universiteit Brussel)

Domain_Walls_Aaron_Rase_Nordita.pdf

rase.mp4

11:00 Coffee break

42 Stochastic Gravitational Wave Background generated by superconducting cosmic strings

Speaker: Lara Sousa

Talk Nordita 2.pdf

12:30 Lunch break

Tuesday 12 August

43 Lectures on topological defects. Part II: Cosmic strings

Speaker: Tanmay Vachaspati (Arizona State University)

vachaspati_lecture2.mp4

44 Simulating primordial black hole formation from domain wall collapse

A variety of theories beyond the Standard Model predict the existence of topological defects such as domain walls in the early universe. Given certain conditions, it is possible for enclosed domain wall configurations to collapse to primordial black holes, which may contribute to some if not all of the estimated dark matter abundance. However, there are a number of uncertainties regarding the likelihood of black hole formation. One such uncertainty is the degree by which deformations of the domain walls hinders black hole production. By simulating the entire collapse of deformed domain wall configurations using numerical relativity, we attempt to alleviate this uncertainty.

Speaker: Matthew Elley

elley.mp4

11:00 Coffee break

45 The QCD axion mass

The QCD axion arises as the pseudo-Goldstone mode of a spontaneously broken abelian Peccei-Quinn (PQ) symmetry. If the scale of PQ symmetry breaking occurs below the inflationary reheat temperature and the domain wall number is unity, then there is a unique axion mass that gives the observed dark matter (DM) abundance. Computing this mass has been the subject of intensive numerical simulations for decades since the mass prediction informs laboratory experiments. In this talk, I will present the most precise and accurate large-scale simulations to date of the axion-string network leveraging adaptive mesh refinement. Accounting for axion production from strings prior to the QCD phase transition leads us to predict that the axion mass should be approximately ma ∈ (45, 65) µeV. However, we provide preliminary evidence that axions are produced in greater quantities from the string-domain-wall network collapse during the QCD phase transition, potentially increasing the mass prediction to as much as 300 µeV.

Speaker: Malte Buschmann

buschmann.mp4

12:30 Lunch break

46 More axion stars and gravitational waves from strings

Axion strings inevitably produce a contribution to the stochastic gravitational wave background. Combining effective field theory analysis with numerical simulations, we show that the resulting gravitational wave spectrum has logarithmic deviations from a scale invariant form with an amplitude that is significantly enhanced at low frequencies. As a result, a single ultralight axion-like particle with a decay constant larger than 10^14 GeV and any mass between 10^-18 eV and 10^-28 eV leads to an observable gravitational wave spectrum and is compatible with constraints from dark matter overproduction, isocurvature and dark radiation. Since the spectrum extends over a wide range of frequencies, the resulting signal could be detected by multiple experiments. We also comment on the recent possible NANOgrav signal in light of our results.

I will also show that if dark matter consists of QCD axions in the post-inflationary scenario more than ten percent of it efficiently collapses into Bose stars at matter-radiation equality. Such a result is mostly independent of the present uncertainties on the axion mass. This large population of solitons, with asteroid masses and Earth-Moon distance sizes, might plausibly survive until today, with potentially interesting implications for phenomenology and experimental searches.

Speaker: Marco Gorghetto (DESY Hamburg)

gorghetto.mp4

Gorghetto_nordita.pdf

Wednesday 13 August

47 Lectures on topological defects. Part III: Magnetic monopoles

Speaker: Tanmay Vachaspati (Arizona State University)

topoDefectsMonopolesNordita2025.pdf

vachaspati_lecture3.mp4

48 Gravitational Wave Stairway from Topological Defects

We investigate how gravitational wave (GW) observations can reveal the structure of hidden sectors beyond the Standard Model. We make use of GW's from Local/Global string to analyze cases with sequential decoupling of new degrees of freedom which can produce a distinctive “stairway” pattern in the GW spectrum. Focusing on two benchmarks—a dark QCD-like theory and a model with a Kaluza-Klein tower—we try to study how such features could distinguish different hidden sectors. We assess the possibility in a simplified study of the SNR in LISA, in which a more rigorous statistical study and potential full reconstruction are for the future

Speaker: Nicklas Ramberg (SISSA Trieste)

11:00 Coffee break

49 Non-topological solitons and the effects of an external field on them

After reviewing the basic properties of non-topological solitons like Q-balls and oscillons, I will discuss the effects of an external field on them.

Speaker: Masahide Yamaguchi

nordita25_yamaguchi.pdf

12:30 Lunch break

50 Terasite simulations of Axion string networks

In the post-inflationary scenario, if PQ symmetry breaking occurs, a network of global cosmic strings are expected to form. At the time of the QCD phase transition, the previously scaling network of such defects will collapse by forming domain walls attached to strings. Given that all of the energy density of the string network is left behind into axion waves, in order to improve current estimates of axion dark matter density, estimates of the scaling density of axion strings are required. In this talk, I will review recent work on the dynamics of axion strings which uses extreme scale computing resources for simulating 16384ˆ3 lattices. I will address and comment on the recent claims of logarithmic violations of scaling.

Speaker: José Correia (University of Helsinki)

correia.mp4

Stockholm_July.pdf

Thursday 14 August

51 Cosmic string loop fragmentation

Now that GW observatories at all frequencies have become able to detect/constrain the existence of cosmic strings, it is not acceptable that the community has not yet reached a consensus on the cosmic string abundance. In this talk, we will start by an in-depth review of two popular models and highlight their differences and limits. I will try to convince you that something is missing, and that fragmentation can fill in this gap. Then, I will present some new and unpublished results from Nambu-Goto simulations motivating a new model of cosmic string loop fragmentation.

Speaker: Pierre Auclair (APC)

auclair.mp4

11:00 Coffee break

52 Gravitational wave and particle emission from cosmic string loops

Cosmic strings are one-dimensional topological defects predicted to form in the early universe after a phase transition by a variety of extensions of the Standard Model, ranging from axion models to grand-unified theories. After forming, strings are expected to decay via the emission of particles and gravitational waves (GWs), leading to the formation of a GW background (GWB) that may be detected by ongoing or future GW experiments. Traditionally, predictions of the GWB produced by cosmic strings are based on the Nambu-Goto approximation, in which the internal structure of the strings and their decay into particles are neglected. However, a correct interpretation of a positive GW signal would require of accurate predictions that go beyond the Nambu-Goto approximation and take into account the field-theory nature of the strings. As a first step to accurately characterize the GWB from a network of cosmic strings incorporating field-theory effects, we are focusing on understanding the evolution and decay-routes of closed strings, called loops, which in the NG approximation are expected to dominate the productions of GWs. Using field-theory lattice simulations, we have successfully characterized the emission of particles and GWs from loops, reaching a separation of scales of four orders of magnitude between the loop length and its core radius. In this talk, I will present the results from this study, showing evidence that, for the types of loops expected to appear in the early universe, the emission of particles always dominates over the production of GWs.

Speaker: Jorge Baeza-Ballesteros (DESY Zeuthen)

baeza-ballesteros.mp4

12:30 Lunch break

Friday 15 August

Panel discussion week 3

Conveners: Amelia Drew (ICTP, Trieste), Daniel Figueroa

12:00 Lunch break