2.2 Inflation 

The inflation paradigm is frequently invoked to explain the flatness () of the Universe, attributing it to an era of exponential expansion at about seconds after the Big Bang. The mechanism of inflation is generally taken to be an effective cosmological constant from the dominant stress-energy of the inflation scalar field that regulates GUT symmetry breaking, particle creation, and the reheating of the Universe. In order to study whether inflation can occur for arbitrary anisotropic and inhomogeneous data, many numerical simulations have been carried out with different symmetries, matter types and perturbations. A sample of such calculations are described in the following paragraphs.

2.2.1 Plane Symmetry 

Kurki-Suonio et al. [41] extended the planar cosmology code of Centrella and Wilson [24, 25] (see § 2.5) to include a scalar field and simulate the onset of inflation in the early Universe starting with an inhomogeneous Higgs field. Their results suggest that whether inflation occurs or not can be sensitive to the shape of the potential. In particular, if the shape is flat enough, even large initial fluctuations of the Higgs field do not prevent inflation. They considered two different forms of the potential: a flat Coleman-Weinberg type which is very flat close to the false vacuum and does inflate; and a rounder `` '' type which, for their parameter combinations, does not.

2.2.2 Spherical Symmetry 

Goldwirth and Piran [33] studied the onset of inflation with inhomogeneous initial conditions for closed, spherically symmetric spacetimes containing a massive scalar field and radiation field sources (described by a massless scalar field). In all the cases they considered, the radiation field damps quickly and only an inhomogeneous massive scalar field remains to inflate the Universe. They find that small inhomogeneities tend to reduce the amount of inflation and large initial inhomogeneities can even suppress the onset of inflation. Their calculations indicate that the scalar field must have ``suitable'' initial values over a domain of several horizon lengths in order for inflation to begin.

2.2.3 Bianchi V 

Anninos et al. [5] investigated the simplest Bianchi model (type V) background that admits velocities or tilt in order to address the question of how the Universe can choose a uniform reference frame at the exit from inflation, since the deSitter metric does not have a preferred frame. They find that inflation does not isotropize the Universe in the short wavelength limit. However if inflation persists, the wave behavior eventually freezes in and all velocities go to zero at least as fast as , where is the relativistic tilt angle (a measure of velocity), and R is a typical scale associated with the radius of the Universe. Their results indicate that the velocities eventually go to zero as inflation carries all spatial variations outside the horizon, and that the answer to the posed question is that memory is retained and the Universe is never really deSitter.

2.2.4 + Gravity Waves 

In addition to the inflation field, one can consider other sources of inhomogeneity, such as gravitational waves. Although linear waves in deSitter space will decay exponentially and disappear, it is unclear what will happen if strong waves exist. Shinkai & Maeda [58] investigated the cosmic no-hair conjecture with gravitational waves and a cosmological constant in 1D plane symmetric vacuum spacetimes, setting up Gaussian pulse wave data with amplitudes and widths , where I is the invariant constructed from the 3-Riemann tensor and is the horizon scale. They also considered colliding plane waves with amplitudes and widths . They find that for any large amplitude and/or small width inhomogeneity in their parameter sets, the nonlinearity of gravity has little effect and the spacetime always evolves into a deSitter spacetime.

2.2.5 Three-D Inhomogeneous Spacetimes 

The previous paragraphs discussed results from highly symmetric spacetimes, but the possibility of inflation remains to be established for more general inhomogeneous and nonperturbative data. To this end, Kurki-Suonio et al. [42] investigated fully three-dimensional inhomogeneous spacetimes with a chaotic inflationary potential . They considered basically two types of runs: small and large scale. In the small scale run, the grid length was initially set equal to the Hubble length so the inhomogeneities are well inside the horizon and the dynamical time scale is shorter than the expansion or Hubble time. As a result, the perturbations oscillate and damp while the field evolves and the spacetime inflates. In the large scale run, the inhomogeneities are outside the horizon and they do not oscillate. They maintain their shape without damping and, because larger values of lead to faster expansion, the inhomogeneity in the expansion becomes steeper in time since the regions of large and high inflation stay correlated. Both runs have sufficient inflation to solve the flatness problem.

Physical and Relativistic Numerical Cosmology Peter Anninos http://www.livingreviews.org/lrr-1998-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de