Numerical Simulations of Early Universe Sources of Gravitational Waves

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

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)
• Kari Rummukainen (University of Helsinki)
• 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 (University of Cambridge)

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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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)

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)

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)

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)

11:00 Coffee break

7 Lectures on Inflation. Part II

Speaker: Matteo Fasiello

12:30 Lunch break

8 Stochastic Inflation in (Numerical) General Relativity

Speaker: Gerasimos Rigopoulos

Wednesday 30 July

9 Backreaction and cosmic butterflies: what simulations can tell us about inflation

Speaker: Angelo Caravano

10 Primordial gravitational waves from from fully relativistic inflation simulations

Speaker: Paul Shellard

11:00 Coffee break

11 Lectures on Inflation. Part III

Speaker: Matteo Fasiello

12:30 Lunch break

12 GW background signals from Inflation: lattice calculation

Speaker: Joanes Lizarraga

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)

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

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)

12:30 Lunch break

16 TBA

Speaker: Emanuela Dimastrogiovanni

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

11:00 Coffee break

18 Numerical Simulations and Primordial Blak Holes

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

12:30 Lunch break

Panel discussion week 1

Convener: Daniel Figueroa

Monday 4 August

19 TBA

Speaker: Marek Lewicki

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

Speaker: Nicklas Ramberg (SISSA Trieste)

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

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)

Tuesday 5 August

23 The bubble wall velocity in first order phase transitions

Speaker: Jorinde van de Vis

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)

11:00 Coffee break

25 TBA

Speaker: Ryusuke Jinno (DESY)

12:30 Lunch break

26 Evolution of primordial magnetic fields

Speaker: Jennifer Schober (University of Bonn)

Wednesday 6 August

27 Gas from first order phase transitions

Speaker: Mark Hindmarsh (University of Sussex)

28 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)

11:00 Coffee break

29 Phase transitions on the lattice

Speaker: Kari Rummukainen (University of Oulu)

12:30 Lunch break

30 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)

Thursday 7 August

31 Higgsless simulations: Gravitation waves from sound waves

Speaker: Henrique Rubira

32 Gravitational waves from decaying sources: strong phase transitions

Speaker: Isak Stomberg (DESY)

11:00 Coffee break

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

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

12:30 Lunch break

34 TBA

Speaker: Kenneth Marschall

Friday 8 August

35 Gravitational Waves from First-Order Phase Transitions

Speaker: Antonino Midiri

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

Speaker: Madeline Salome

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)

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)

39 Generating gravitational wave spectra from equations of state for phase transitions in the early universe

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)

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

Speaker: Tanmay Vachaspati (Arizona State University)

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)

11:00 Coffee break

42 TBA

Speaker: Lara Sousa

12:30 Lunch break

43 TBA

Speaker: Jose Juan Blanco-Pillado

Tuesday 12 August

44 Lectures on topological defects. Part II

Speaker: Tanmay Vachaspati (Arizona State University)

45 Simulating primordial black hole formation from domain wall collapse

Speaker: Matthew Elley

11:00 Coffee break

46 TBA

Speaker: Malte Buschmann

12:30 Lunch break

47 Gravitational waves from axions

If the Peccei-Quinn symmetry associated to an axion has ever been restored after inflation, 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 on the post-inflationary scenario 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 describe straightforward ways in which the Peccei-Quinn symmetry can be restored after inflation for such decay constants. We also comment on the recent possible NANOgrav signal in light of our results.

Speaker: Marco Gorghetto (DESY Hamburg)

Wednesday 13 August

48 Lectures on topological defects. Part III

Speaker: Tanmay Vachaspati (Arizona State University)

49 Gravitational Wave Stairway from Topological Defects.

Speaker: Nicklas Ramberg (SISSA Trieste)

11:00 Coffee break

50 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

12:30 Lunch break

51 TBA

Speaker: José Correia (University of Helsinki)

Thursday 14 August

52 Cosmic string loop fragmentation

Speaker: Pierre Auclair (APC)

11:00 Coffee break

53 Gravitational wave and particle emission from cosmic string loops

Speaker: Jorge Baeza-Ballesteros (University of Valencia/IFIC)

12:30 Lunch break

Friday 15 August

Panel discussion week 3

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

12:30 Lunch break