Foundations of Physics

, Volume 48, Issue 11, pp 1617–1647 | Cite as

How Problematic is the Near-Euclidean Spatial Geometry of the Large-Scale Universe?

  • M. HolmanEmail author


Modern observations based on general relativity indicate that the spatial geometry of the expanding, large-scale Universe is very nearly Euclidean. This basic empirical fact is at the core of the so-called “flatness problem”, which is widely perceived to be a major outstanding problem of modern cosmology and as such forms one of the prime motivations behind inflationary models. An inspection of the literature and some further critical reflection however quickly reveals that the typical formulation of this putative problem is fraught with questionable arguments and misconceptions and that it is moreover imperative to distinguish between different varieties of problem. It is shown that the observational fact that the large-scale Universe is so nearly flat is ultimately no more puzzling than similar “anthropic coincidences”, such as the specific (orders of magnitude of the) values of the gravitational and electromagnetic coupling constants. In particular, there is no fine-tuning problem in connection to flatness of the kind usually argued for. The arguments regarding flatness and particle horizons typically found in cosmological discourses in fact address a mere single issue underlying the standard FLRW cosmologies, namely the extreme improbability of these models with respect to any “reasonable measure” on the “space of all spacetimes”. This issue may be expressed in different ways and a phase space formulation, due to Penrose, is presented here. A horizon problem only arises when additional assumptions—which are usually kept implicit and at any rate seem rather speculative—are made.


Cosmological flatness problem General relativity FLRW solutions Initial conditions Fine-tuning Inflation Horizon problem Second law of thermodynamics Quantum gravity 



This publication was made possible through the support of a Grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation. I thank the organizers of the Fourth International Conference on the Nature and Ontology of Spacetime, in Varna, Bulgaria, for providing an opportunity for me to present (most of) this work. Financial support from Stichting FOM (Foundation for Fundamental Research on Matter) in the Netherlands to attend the Conference is also gratefully acknowledged. Finally, I thank Andy Albrecht, Feraz Azhar, George Ellis, David Garfinkle and especially Phillip Helbig and Chris Smeenk for comments and/or discussions pertaining to the contents of this manuscript.


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Authors and Affiliations

  1. 1.Department of Physics and Astronomy, Rotman Institute of PhilosophyWestern UniversityLondonCanada

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