# Inhomogeneous anisotropic cosmology

## Abstract

In homogeneous and isotropic Friedmann-Robertson-Walker cosmology, the topology of the universe determines its ultimate fate. If the Weak Energy Condition is satisfied, open and flat universes must expand forever, while closed cosmologies can recollapse to a Big Crunch. A similar statement holds for homogeneous but anisotropic (Bianchi) universes. Here, we prove that arbitrarily inhomogeneous and anisotropic cosmologies with “flat” (including toroidal) and “open” (including compact hyperbolic) spatial topology that are initially expanding must continue to expand forever at least in some region at a rate bounded from below by a positive number, despite the presence of arbitrarily large density fluctuations and/or the formation of black holes. Because the set of 3-manifold topologies is countable, a single integer determines the ultimate fate of the universe, and, in a specific sense, most 3-manifolds are “flat” or “open”. Our result has important implications for inflation: if there is a positive cosmological constant (or suitable inflationary potential) and initial conditions for the inflaton, cosmologies with “flat” or “open” topology must expand forever in some region at least as fast as de Sitter space, and are therefore very likely to begin inflationary expansion eventually, regardless of the scale of the inflationary energy or the spectrummore »

- Authors:

- Center for Cosmology and Particle Physics, New York University,4 Washington Place, New York, NY 10003 (United States)
- Stanford Institute for Theoretical Physics and Department of Physics, Stanford University,382 Via Pueblo Mall, Stanford, CA 94306 (United States)
- (United States)

- Publication Date:

- Sponsoring Org.:
- SCOAP3, CERN, Geneva (Switzerland)

- OSTI Identifier:
- 22572165

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2016; Journal Issue: 10; Other Information: PUBLISHER-ID: JCAP10(2016)022; OAI: oai:repo.scoap3.org:17558; cc-by Article funded by SCOAP3. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANISOTROPY; BLACK HOLES; BOUNDARY CONDITIONS; COSMOLOGICAL CONSTANT; COSMOLOGY; DE SITTER GROUP; DE SITTER SPACE; EINSTEIN FIELD EQUATIONS; ENERGY SPECTRA; GENERAL RELATIVITY THEORY; GRAVITATIONAL WAVES; INFLATIONARY UNIVERSE; MATHEMATICAL MANIFOLDS; TOPOLOGY

### Citation Formats

```
Kleban, Matthew, Senatore, Leonardo, and Kavli Institute for Particle Astrophysics and Cosmology, Stanford University and SLAC,2575 Sand Hill Road, M/S 29, Menlo Park, CA 94025.
```*Inhomogeneous anisotropic cosmology*. United States: N. p., 2016.
Web. doi:10.1088/1475-7516/2016/10/022.

```
Kleban, Matthew, Senatore, Leonardo, & Kavli Institute for Particle Astrophysics and Cosmology, Stanford University and SLAC,2575 Sand Hill Road, M/S 29, Menlo Park, CA 94025.
```*Inhomogeneous anisotropic cosmology*. United States. doi:10.1088/1475-7516/2016/10/022.

```
Kleban, Matthew, Senatore, Leonardo, and Kavli Institute for Particle Astrophysics and Cosmology, Stanford University and SLAC,2575 Sand Hill Road, M/S 29, Menlo Park, CA 94025. Wed .
"Inhomogeneous anisotropic cosmology". United States.
doi:10.1088/1475-7516/2016/10/022.
```

```
@article{osti_22572165,
```

title = {Inhomogeneous anisotropic cosmology},

author = {Kleban, Matthew and Senatore, Leonardo and Kavli Institute for Particle Astrophysics and Cosmology, Stanford University and SLAC,2575 Sand Hill Road, M/S 29, Menlo Park, CA 94025},

abstractNote = {In homogeneous and isotropic Friedmann-Robertson-Walker cosmology, the topology of the universe determines its ultimate fate. If the Weak Energy Condition is satisfied, open and flat universes must expand forever, while closed cosmologies can recollapse to a Big Crunch. A similar statement holds for homogeneous but anisotropic (Bianchi) universes. Here, we prove that arbitrarily inhomogeneous and anisotropic cosmologies with “flat” (including toroidal) and “open” (including compact hyperbolic) spatial topology that are initially expanding must continue to expand forever at least in some region at a rate bounded from below by a positive number, despite the presence of arbitrarily large density fluctuations and/or the formation of black holes. Because the set of 3-manifold topologies is countable, a single integer determines the ultimate fate of the universe, and, in a specific sense, most 3-manifolds are “flat” or “open”. Our result has important implications for inflation: if there is a positive cosmological constant (or suitable inflationary potential) and initial conditions for the inflaton, cosmologies with “flat” or “open” topology must expand forever in some region at least as fast as de Sitter space, and are therefore very likely to begin inflationary expansion eventually, regardless of the scale of the inflationary energy or the spectrum and amplitude of initial inhomogeneities and gravitational waves. Our result is also significant for numerical general relativity, which often makes use of periodic (toroidal) boundary conditions.},

doi = {10.1088/1475-7516/2016/10/022},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 10,

volume = 2016,

place = {United States},

year = {Wed Oct 12 00:00:00 EDT 2016},

month = {Wed Oct 12 00:00:00 EDT 2016}

}