Simulations of eternal inflation using FRACTINT

These simulations were produced with the open-source FractInt software for interactive plotting of two-dimensional fractals.

In all simulations, the spacetime is 2+1-dimensional. The initial square region (imitating one horizon-size region). The square is subdivided into 4 equal squares and each square is evolved independently. At each step of the simulation, the scalar field phi is modified by adding an increment delta phi (x) in each sub-square.

Initially phi=0. Thermalization occurs when phi exceeds some predefined limits, |phi| > phi*. This corresponds to "new" inflation or "topological" inflation.

The simulations differ only by the choice of the function delta phi(x).

The bluish-white color always denotes inflating regions, while other colors denote thermalized regions.

Read the poster presentation I made: in gzipped Postscript format (350 KB) or in PDF format (690 KB)

  1. delta phi is constant across each square (so neighboring squares have different values of delta phi and phi is discontinuous across any boundary).

    Boundaries between inflating (white) and thermalized (black) regions are always vertical or horizontal lines.

  2. delta phi within each square is a linear function of coordinates (x,y): delta phi = A x + B y + C, where A, B, C are random. The value of delta phi is still discontinuous across any boundary between squares.

    Boundaries between inflating and thermalized regions are always straight lines. The coloring of a thermalized region indicates the time of thermalization in that region.

  3. Four values of delta phi are chosen randomly at each corner of each square as uniform random numbers between -1 and 1. The values of delta phi inside the square are interpolated between corners using the formula delta phi = A x y + B x + Cy + D, where A,B,C,D are chosen appropriately. Since neighbor squares use each other's corners, the value of delta phi is continuous across any boundary between squares. However there are still some discontinuous boundaries for the value of phi because evolution stops at thermalization and when a region thermalizes, its value of phi freezes, i.e. delta phi in that region is taken to be zero, while neighboring regions still have nonzero delta phi, oblivious to thermalization next to them.

    On the figure, the limit value of phi is 4. Coloring of a thermalized region indicates the time of thermalization in that region. Boundaries between inflating and thermalized regions are seldom smooth lines but occasionally they are. Smooth lines within thermalized regions indicate how thermalization occurred in steps within these regions.

  4. Same algorithm as in simulation 3 with continuous delta phi. However, the step size now depends on phi: delta phi = (random step)*f(phi)+g(phi). The functions f(phi) and g(phi) are chosen to simulate larger inflation rate at top of the potential (around phi=0) and deterministic slow roll near end of inflation. This should make the regions near thermalized regions also thermalize with overwhelming probability.

    On the figure, coloring of a thermalized region indicates the time of thermalization in that region (inflating regions are white). Smooth lines within thermalized regions indicate how thermalization occurred in steps within these regions: first the centers thermalized, then the periphery. Note that although one can see these lines in thermalized regions, there are no smooth line-shaped boundaries between inflating and thermalized regions. The limiting value of phi is 4. The figure shows a neighborhood of a large thermalized region.

  5. Same simulation as above, except the four values of delta phi at each corner of each square have a continuous evolution which terminates for each corner independently when the value of phi reaches the limit value. The step delta phi at each corner is also a function of phi at that corner. The values of delta phi inside the square are interpolated between corners and now not only delta phi but also phi is continuous across any boundary between squares.


    On the top figure, the limit value of phi is 4 and the starting point is 0.3, which explains the prevalence of blue. On the bottom figure, starting point is phi=0. Coloring is of two color families: blueish represent thermalization at phi>0 and yellowish represent phi<0. There are no smooth lines between regions of different color. Occasional white points denote regions that are still inflating.

    Play an MPEG-1 animation of zooming into the picture at bottom. Small: 320x240, 2000 frames (13 MB); large: 640x480, 6000 frames (22 MB). The last movie displays zooming through slightly more than 10 orders of magnitude, until machine precision was exhausted.