Inflationary Spacetimes Are Incomplete in Past Directions

Phys. Rev. Lett. 90, 151301 – Published 15 April 2003
Arvind Borde, Alan H. Guth, and Alexander Vilenkin


Many inflating spacetimes are likely to violate the weak energy condition, a key assumption of singularity theorems. Here we offer a simple kinematical argument, requiring no energy condition, that a cosmological model which is inflating—or just expanding sufficiently fast—must be incomplete in null and timelike past directions. Specifically, we obtain a bound on the integral of the Hubble parameter over a past-directed timelike or null geodesic. Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime.


  • Figure
  • Received 5 October 2001
  • Revised 24 January 2003
  • Published 15 April 2003

© 2003 The American Physical Society

Authors & Affiliations

Arvind Borde1,2, Alan H. Guth1,3, and Alexander Vilenkin1

  • 1Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155
  • 2Natural Sciences Division, Southampton College, New York 11968
  • 3Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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