Space Is the Place*

How to master spatial average properties of the Universe?

by Thomas Buchert, Pierre Mourier & Xavier Roy.

The question of how to define a cosmological model within General Relativity without symmetry assumptions or approximations can be approached by spatially averaging the scalar parts of Einstein’s equations. This yields general balance equations for average properties of the Universe.  One open issue that we address here is whether the form and solutions of these equations depend on the way we split spacetime into spatial sections and a global cosmological time. We also discuss whether we can at all achieve this – given the generality of possible spacetime splits.

Our CQG Letter explores the general setting with a surprisingly simple answer.

Currently most researchers in cosmology build model universes with a simplifying principle that is almost as old as General Relativity itself.  One selects solutions that are isotropic about every point, so that no properties of the model universe depend on direction. This local assumption restricts one to homogeneous geometries that define the cosmological model globally, up to the topology that is specified by initial conditions. Spacetime is foliated into hypersurfaces of constant spatial curvature, labelled by a global cosmological time-parameter. The homogeneous fluid content of these model universes is assumed to define a congruence of fundamental observers moving in time along the normal to these hypersurfaces. Einstein’s equations reduce, in this flow-orthogonal foliation, to the equations of Friedmann and Lemaître. The only gravitational degree of freedom is encoded in a time-dependent scale factor, which measures the expansion of space.


Thomas Buchert is Professor of Cosmology at the University of Lyon, working at ‘Centre de Recherche Astrophysique de Lyon’ (CRAL), (see our group)

The standard model of cosmology restricts these solutions further to the case of flat space sections. It also assumes the cosmological constant to be positive, modelling a dominating Dark Energy component that works against gravity, as an observationally justified best-fit model universe.

The dependence of the cosmological equations on the way we  separate space and time remains even in this idealised model. If we choose another foliation of spacetime, the equations of Friedmann and Lemaître will be affected: the scale factor may even become space-dependent, while the global cosmological time in these models becomes a local time. However, in this highly symmetric spacetime there is no reason to introduce a foliation that does not manifestly comply with the assumed symmetries. There is a ‘naturalness assumption’ related to the fact that a hypersurface of homogeneity is anchored to the global rest frame of the Cosmic Microwave Background radiation, a global reference system that is a priori non-existent in relativity theory (see this CQG+ article on the assumptions made).

What happens if we consider non-homogeneous geometries and no symmetry assumption or simplifying principle? Space is a relative place: we have no other choice but to consider the general setting of foliating spacetime into spatial hypersurfaces together with a congruence of the fluid flow that is, in general, tilted with respect to the normal of the spatial slices. The further operation of spatially averaging scalar variables on these slices will generally depend on the way we split spacetime, while the symmetric foliation in the standard cosmological model relies on the assumption that the homogeneous geometries describe the spatial average behaviour of a general model universe. However, we cannot expect this latter property to hold from first principles (see this CQG+ article).


Pierre Mourier is a Ph.D. student at CRAL with a main focus on this subject

An earlier answer without symmetry assumptions and approximations was given in the case of flow-orthogonal foliations, i.e. for a fluid that follows the normal congruence of the foliation, specified to non-rotating dust and later to non-rotating perfect fluids.  The first paper was an extension to General Relativity of results in Newtonian gravitation where the foliation issue plays no role. At the time, our first referee claimed that this could have been done 100 years ago. Yet it had not been done. Specifically, this paper delivered the proof, hitherto just assumed, that the average properties of an inhomogeneous Newtonian model globally comply with the assumed background model, but for a subtle reason: the general averages contain scalar invariants built from inhomogeneities, named cosmological backreaction terms, that turn out to be boundary terms in flat-space Newtonian models. Standard cosmological simulations of structure formation assume periodic boundary conditions for the inhomogeneities on some large scale (a 3-torus topology). Hence, there is no boundary with the result that the backreaction terms found have to vanish. If that were not the case, the architecture of current Newtonian simulations in cosmology would be just wrong. In the corresponding equations of General Relativity, these restrictions are gone and backreaction terms are non-vanishing, which makes these average cosmologies physically exciting. In particular, backreaction terms act like a Dark Energy component on large scales, while the same effect mimics a Dark Matter-like component on smaller scales.

In our work we consider a tilted flow for a general fluid with respect to a general foliation, without reference to any background: the true spirit of General Relativity. A first attempt to deal with such a general setting ends up with what we call an extrinsic approach: the tilted fluid-flow is averaged with respect to the volume in the hypersurfaces, and therefore `seen from outside’. Care must then be taken that the domain over which the fluid is averaged contains the same collection of fluid elements during its evolution. However, such an extrinsic description features a possibly strong foliation-dependence due to a dependence on derivatives of the normal vector, i.e. even if the tilt between the normal vector and the 4-velocity of the fluid is small (the Lorentz factor is small), derivatives can be large. This led to statements  that backreaction effects strongly depend on the foliation of spacetime.

In view of this situation, we consider in our Letter what we call a fluid-intrinsic approach, that agrees in spirit—and as it turns out also formally—with the simpler flow-orthogonal setting for non-rotational fluids. This earlier approach was intrinsic in the sense that the fluid variables were averaged within hypersurfaces generated by the fluid’s own rest frames. In the case of an irrotational dust model this means that every fluid element is in free fall, and the fluid itself foliates spacetime into flow-orthogonal hypersurfaces labelled by the fluid proper time. This is already not possible for vortical dust flows, which require a tilt between the normal to the hypersurfaces and the fluid’s 4-velocity.


Xavier Roy has been interested in this question since his Ph.D. in 2011; he is currently working in Leeds, UK, and a regular visitor at CRAL

We showed how to design a foliation that is labelled by the fluid’s proper time and still averaged on the tilted hypersurfaces. The crux of the matter is that the averaging operation has to be applied to the fluid variables, using the proper volume of the fluid for the integration. The result is intriguingly close to the fluid-orthogonal results: although the setting is general and covariant, the resulting average cosmological equations feature a global cosmological time. What is more, these general averaged cosmologies are simpler than those in the fluid-orthogonal approach of an averaged perfect fluid.

Our question of “How to master average properties of the Universe?” thus receives a simple answer that provides the final cornerstone for studying scalar averaging and backreaction in relativistic cosmology. Work on the numerical integration of the equations of General Relativity, which has just begun in the cosmology community, could benefit from this new framework.

It is always a signature of essential insight, if a more general approach yields results that are formally simpler than in a more special case.

* Sun Ra and his Arkestra 1973

Read the full article in Classical and Quantum Gravity:
Cosmological backreaction and its dependence on spacetime foliation
Thomas Buchert et al 2018 Class. Quantum Grav. 35 24LT02

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