Numerical GR cosmology papers of interest
Dust fluid models:
- Early, very simple (proof-of-principle) setup using Einstein Toolkit: Bentivegna & Bruni 2016: Effects of Nonlinear Inhomogeneity on the Cosmic Expansion with Numerical Relativity.
- Giblin, Mertens & Starkman 2016: Departures from the Friedmann-Lemaitre-Robertston-Walker Cosmological Model in an Inhomogeneous Universe: A Numerical Examination (the code is mainly described in this earlier paper: Integration of inhomogeneous cosmological spacetimes in the BSSN formalism).
- Follow-up by the same authors and the same code, with ray-tracing (also 2016): Observable Deviations from Homogeneity in an Inhomogeneous Universe.
- Macpherson, Price & Lasky 2019 (using Einstein Toolkit): Einstein's Universe: Cosmological structure formation in numerical relativity (the code is described in more detail here: Inhomogeneous cosmology with numerical relativity); see also Macpherson, Lasky & Price 2018 on the Hubble constant (but without ray-tracing): The Trouble with Hubble: Local versus Global Expansion Rates in Inhomogeneous Cosmological Simulations with Numerical Relativity. NB : In these papers, realistic initial conditions are set up using linear perturbation theory at an initial time corresponding to z = 1000, and then evolved in full GR up to z=0. There are however substantial local violations of the Hamiltonian constraint at late times.
- Daverio, Dirian & Mitsou 2017 (mostly a code description paper): A numerical relativity scheme for cosmological simulations.
- Clesse, Roisin & Füzfa 2017: Mimicking Dark Energy with the backreactions of gigaparsec inhomogeneities.
- East, Wojtak & Abel 2018: Comparing fully general relativistic and Newtonian calculations of structure formation.
Attempts at N-body GR models (note that there are mostly follow-up works by the same groups as above):
Note that the East/Wojtak/Abel/Pretorius group uses substantially different resolution techniques to solve the Einstein equations than all other groups (they do not rely on the BSSN scheme) but the N-body part is mostly similar to that of the other two groups above.
For completeness, there are also "partially GR" simulations which aim at including GR effects without fully solving the Einstein equations, typically within weak-field approximations. This is in particular the case of:
NB: While interesting for other uses, I do not think such approximate frameworks can actually help fully determining the cosmological consequences of inhomogeneous dynamics.
There is also a code comparison paper (under review) with 4 of the above codes (two solving the full Einstein equations and two using various weak-field approximations): Adamek, Barrera-Hinojosa, Bruni, Li, Macpherson & Mertens 2020, Numerical solutions to Einstein's equations in a shearing-dust Universe: a code comparison
. This comparison considers a very specific setup with an initial metric dominated by a large vector mode (in terms of standard perturbation theory, although the setup is not perturbative) corresponding to gravitomagnetic effects, on top of an (initially) Einstein-de Sitter expansion.
Requirements for a numerical investigation of structure formation effects on cosmological dynamics and observables
Here is what would be my personal "wish list" of conditions a simulation should fulfill to make sure the most important effects are not missed:
- Solving the full Einstein equations (except possibly for initial conditions setup, where some approximations could be done).
- Evolve perturbations covering an appropriate (large) range of spatial scales, from around last-scattering epoch (z ≈ 1000) to present time.
- Simultaneously resolve smal scales and probe the dynamics on very large scales. An adaptative mesh refinement is likely to be required for this.
- Deal with the shell-crossings and virialisation where structures form, which requires going beyond the dust fluid approximation: phase space distribution (Einstein-Vlasov), N-body, or at least refined fluid models.
- An implentation of ray-tracing would be useful to look for physical observables, beyond the spatial averaging procedure and corresponding "backreaction" terms (which have to depend on the spatial slicing at least to some moderate extent). This may be replaced or complemented by an overall past-light-cone finder, along with light-cone averaging procedures.
- Ideally, finding some way to get rid of the periodic boundary conditions that may constrain the dynamics. At least trying some other simple boundary conditions for comparison. This may "simply" be solved by considering a large enough simulation box to encompass the whole past light-cone of the observer. Boundaries would then have no causal relation to the observer and one could take any convenient boundary conditions, e.g. periodic ones.
More theoretical work is needed to investigate the importance of the periodic boundary conditions (point 6); this concern is motivated by the Newtonian result that "kinematical backreaction" is strictly zero on a boundary-free (e.g. periodic) domain, even though this does not directly apply in GR.
The Macpherson et al.
group is the only one so far with published results matching points 1 and 2, but the current code cannot deal with any of the other points (although AMR is in principle possible within the Einstein Toolkit, and I think point 5 is underway). There are also some concerns regarding the constraints violation.
Other groups have implemented some of the other points, either with weak field approximations or without actually running it within a full cosmological setup. This is the case of the Giblin, Mertens et al.
group in particular: their code has the potential to satisfy all conditions except (maybe) point 6, but they have no published result yet with a full cosmological setup ran for a long time (point 2).
Papers of interest about Einstein-Vlasov, Maxwell-Vlasov, Klein-Gordon formalisms mimicking Vlasov-Poisson, N-body simulations, and ray-tracing
Einstein-Vlasov in spherical symmetry (with discussion of the numerical problem in section IV), Akbarian & Choptuik 2014: Critical collapse in the spherically symmetric Einstein-Vlasov model
Numerical N-body (particle-in-cell) solver for Maxwell-Vlasov, Kormann & Sonnendrücker 2019: Energy-conserving time propagation for a geometric particle-in-cell Vlasov--Maxwell solver
Klein-Gordon formalism to mimic structure formation according to Vlasov-Poisson: Uhlemann, Rampf, Gosenca, Hahn 2019: Semiclassical path to cosmic large-scale structure
; Uhlemann & Kopp 2016: Beyond single-stream with the Schrödinger method
; Uhlemann 2018: Finding closure: approximating Vlasov-Poisson using finitely generated cumulants
Test of the accuracy of (Newtonian) N-body simulations, Sylos Labini 2013: A toy model to test the accuracy of cosmological N-body simulations
Test of the accuracy of computing observables from approximate ray-tracing in Newtonian N-body simulations, comparing this to exact ray-tracing in exact GR (LTB and Szekeres) models, Koksbang & Hannestad 2015: Studying the precision of ray tracing techniques with Szekeres models
. (This arXiv version includes a correction —also published separately as an erratum— with respect to the original published paper.)
Also: a paper of interest about (the lack of a) homogeneity scale / homogeneity at large scales, Park, Hyun, Noh, Hwang 2017: The cosmological principle is not in the sky
Some review papers on Inhomogeneous Cosmology and "Backreaction"
Coley & Ellis 2020: Theoretical cosmology
Bolejko & Korzyński 2017: Inhomogeneous cosmology and backreaction: Current status and future prospects
Buchert & Räsänen 2012: Backreaction in Late-Time Cosmology
Buchert 2011: Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects
Buchert 2008: Dark Energy from structure: a status report
Wiltshire 2011: What is dust?—Physical foundations of the averaging problem in cosmology
Clarkson, Ellis, Larena, Umeh 2011: Does the growth of structure affect our dynamical models of the universe? The averaging, backreaction and fitting problems in cosmology
Ellis 2011: Inhomogeneity effects in cosmology
Kolb 2011: Backreaction of inhomogeneities can mimic dark energy
Räsänen 2011: Backreaction: directions of progress