257. Counting states in the Bousso-Polchinski landscape

This is a guest blog post by Antonio Segui and Cesar Asenio from the University of Zaragoza. Dmitry.

This post is a summary of our recent work “A geometric-probabilistic method for counting low-lying states in the Bousso-Polchinski Landscape”. For more details and a list of references, the paper is posted on the arXiv as arXiv:0812.3247 [hep-th].

One of the recent proposals to solve the cosmological constant problem in cosmology is provided by string theory. By dimensional reduction from M-theory to 3+1 dimensions, vacua of the effective theory are classified by means of a big number of quantized fluxes leading to an enormous amount of metastable vacua, the Bousso-Polchinski (BP) Landscape. The cosmological constant problem, namely the smallness of the observed vacuum energy density in the universe, can be solved by the presence in this model of a huge number of states of very small, positive cosmological constant, together with a dynamical mechanism given by eternal inflation which allows to visit all the vacua. An anthropic selection is then advocated to explain the smallness of the observed cosmological constant.

In order to quantify this selection a counting of accessible states in the Landscape is needed; to actually realize that they indeed exist is much more difficult! Two ways of counting these low-lying states have been introduced so far:

• The simplest Bousso-Polchinski count, which computes the volume of a spherical shell of small thickness in flux space and divides it by the volume of a cell.
• The entropy count of Bousso-Yang, which computes the entropy of the occupation number of each flux assuming that they are
  independent.

We have devised a new way of counting, which is as precise as the BP count, and it seems to be generalizable enough to obtain more accurate results or estimates on the hardest numerical aspects of the BP Landscape.

The BP Landscape is a finite (but enormous ) subset of

(a lattice in flux space) where

• The number of fluxes is determined by the number of three-cycles in the compactification manifold.
• The charges are determined by the sizes of the three-cycles.
• A vacuum is characterized by the quantum numbers .
• The cosmological constant of vacuum is ()

Each vacuum lies at the center of its Voronoi cell , a translate of of volume .  Also, each defines a ball of radius . The number of states in a very small shell of width (traditionally called the “Weinberg Window’”) is then

with and .

Bousso and Polchinski used this approximation in order to guess an order of magnitude of the typical charges needed: for , but they said that this estimate is naive and rough. We therefore look for another approaches, but for the moment we are rediscovered the same formula in a very different fashion!

In order to include dynamical relaxation, Bousso and Yang use the Shannon entropy for counting.  For a subset with vacua, its uniform probability is

Then, we count with

.

We only need to estimate the Shannon entropy.  If the fluxes were independent random variables, the joint probability would split:

and the correspondent Shannon entropy would be simplified because unlike , the are much simpler to estimate by sampling a small portion of the set .  But, as long as the product of the is not constant over , its support covers a much larger region than with the symmetry of (so that symmetry of is lost).  So we can expect that

This has been verified by numerical tests.  This is the reason why we looked for another ways of counting.

Our counting proposal is based on two kinds of states one may encounter near the surface, namely:

• Boundary (or penultimate after BY)  states, which have
  1. positive cosmological constant, and
  2. at least one neighbor of negative cosmological constant.

  They are possible ends of the Brown-Teitelboim decay chain in the
  Landscape, which is the mechanism proposed for eternal inflation.

• Secant states have intersecting Voronoi cells with the surface in flux space.  They may have negative  cosmological constant.

All states in the Weinberg Window are in both categories.  Our idea is to use the secant states as an approximation to the boundary states.  Note that they are not equivalent, as can be seen in the following figure, were we show a example of the secant and boundary states of a concrete BP model.

We will count the number of states in the Weinberg Window using the following elementary formula:

where

• is the total number of secant states,
• the factor is the (first-order) approximate fraction of positive secant states,
• must be computed using the distribution of the cosmological constant as a random variable over all the secant states.

We have computed the number of secant states using a (non-constant) directional density of states given by

where , and , which allows us to compute

This formula has a nice geometrical interpretation: We divide the sphere in two halves by a hyperplane normal to each axis. This hyperplane intersects the ball inside the sphere in a ball of the same radius.  We project all the secant state Voronoi cells of volume and count how many fit the volume of the ball by dividing their volumes.  We sum for each half of the sphere (factor 2) and for each axis.

In this way, we can demand that the volume of a single -quadrant of the intersection ball be large when compared with a “mean projection volume” of the cell

We obtain the condition

In this way, we can let grow while the product is constant, which resembles the so-called t’Hooft coupling in the planar limit of gauge theory.

We need a model to compute the probability for to be in the positive side of the Weinberg Window.  To achieve this, we assume two basic features of the secant states in a BP Landscape:

• The Voronoi cells of the secant states are small enough to replace the sphere which intersects the cell by its tangent hyperplane.
• The orientations of the hyperplane intersecting the cell are random if one picks a secant state at random.

In this way, we propose to study the set of secant states by choosing a probability measure on the secant hyperplane set.  This measure will give us the probability we are looking for.

First we parametrize the “hyperplane space”:  If is the set of hyperplanes in , our hyperplane space is

To specify a hyperplane we need:

• A normal vector of unit norm .
• A point , e.g., the closest point to the center of .

Note that (), so that only and must be retained:

The maximum value of is direction dependent:

The previous figure shows the construction of the hyperplane space in the case.  The blue contour comprises all points which represent a different hyperplane intersecting the cell.  The corner is used to construct the first quadrant part of the contour, and .  The green points have been generated using on the third quadrant.

Now we need a probability measure in the hyperplane space. The simplest choice is the uniform probability on :

where

• is the characteristic function on .
• is a normalization constant:

  

We may ask: Why the uniform probability measure?  Because:

• It is simple.
• It maximizes the Shannon entropy in our compact
  space.
• It reproduces the correct directional density
  of states:

  

• Numerically we find that the actual (geometric) distribution is less random than a (simulated) uniform sample, but covers hyperplane space more efficiently (as qMC sequences do), allowing their description by a uniform measure.

We have called the hyperplane space together with the uniform probability measure on it “Random Hyperplane Model”.

In order to compute the desired probability we need the marginal distribution of the parameter, which will be called :

It turns out that and are related:

so that the distribution of is

Integrating in the parameter we obtain

where the domain of integration is

is the whole sphere when and has connected components when .

Values of can be computed using Monte Carlo integration.

The properties of the function include:

• It has a Fermi-like profile,
• It is exactly constant for low :

  

• It is exactly zero for .
• It represents a mean value around which actual discrete distributions oscillate (all these are known to be staggered in earlier works on statistics on the BP Landscape by Schwartz-Perlov and Vilenkin).
• We have tested numerically that this behavior is robust against the Brown-Teitelboim decay chain even on high dimensional BP Landscapes.

Finally, allows the computation of the last piece of our formula:

so that we find

This is the BP count again!

This raises the question: Is our model too crude or the BP count works better than expected?  In order to ask this question we have performed a numerical test which shows the unexpected precision of the BP formula.  This test is shown in the following figure:

The stair-like line represents the actual count of the number of states in a shell of variable width.  The plain red line is the result predicted for the BP count.  We find a remarkable agreement.  The validity condition is satisfied for this example:
.

Two small improvements are possible:

1. Replacing the fraction in by the ratio

which is the fraction of volumes of the shells comprised between , and , .  We choose as an effective width of the shell of positive secant states.

2. Adding the factor to the RHM measure (now ) to take into account the curvature of the sphere.  This replaces the function by

We now summarize the conclusions achieved in this work:

• We have re-derived the BP count in a very different way.
• Our method provides a validity condition which is easy to verify in advance for each different BP Landscape.
• It also contains a distribution for the values of which is constant near .
• Numerical tests seem to confirm the accuracy of the approach, and thereby the accuracy of the BP count.
• Further corrections can take into account the asymmetry between the positive and negative secant states.

As part of our future work, we would like to know if we can adapt this method for the boundary rather than secant states, which are the relevant states for this problem.  Also, we have doubts on the origin of the apparent robustness of the Fermi-like profile against dynamical selection: It is characteristic of the geometry of the BP model or it is a feature of the BT decay chain?