Eternal inflation and localization on the landscape
Eternal inflation and localization on the landscape
D. Podolsky1∗ and K. Enqvist1,2
1 Helsinki Institute of Physics, P.O. Box 64 (Gustaf Hällströmin katu 2), FIN-00014, University of Helsinki, Finland and
2 Department of Physical Sciences, P.O. Box 64, FIN-00014, University of Helsinki, Finland
(Dated: November 4, 2018)
We model the essential features of eternal inflation on the landscape of a dense discretuum of
vacua by the potential V (φ) = V0 + δV (φ), where |δV (φ)| ≪ V0 is random. We find that the
diffusion of the distribution function ρ(φ, t) of the inflaton expectation value in different Hubble
patches may be suppressed due to the effect analogous to the Anderson localization in disordered
quantum systems. At t → ∞ only the localized part of the distribution function ρ(φ, t) survives
which leads to dynamical selection principle on the landscape. The probability to measure any but
a small value of the cosmological constant in a given Hubble patch on the landscape is exponentially
suppressed at t → ∞.
PACS numbers: 98.80.Bp,98.80.Cq,98.80.Qc
String theory is believed to imply a wide landscape [1]
of both metastable vacua with a positive cosmological
constant and true vacua with a vanishing or a negative
cosmological constant; the latter are called anti-de Sitter
or AdS vacua, where space-time collapses into a singular-
ity. In regions with positive cosmological constant, or in
de Sitter (dS) vacua, the universe inflates, and because
of the possibility of tunneling between different de Sitter
vacua inflation is eternal.
The problem of calculating statistical distributions
of the landscape vacua is very complicated [2] and is
even considered to be NP-hard [3] (the total number
of vacua on the landscape is estimated to be of order
10100 ÷ 101000). Our aim is to consider how eternal in-
flation proceeds on the landscape by using the mere fact
that the number of vacua within the landscape is ex-
tremely large, so that their distribution can have signif-
icant disorder. The dynamics of eternal inflation is then
described by the Fokker-Planck equations in the disor-
dered effective potential.1 In that case, the landscape
dynamics may have some interesting parallels in solid
state physics, as we will discuss in the present paper.
Eternal inflation on the landscape can be modeled as
follows [5, 6]. Let us numerate vacua on the landscape
by the discrete index i and define Pi(t) as the probability
to measure a given (positive) value of the cosmological
constant Λi in a given Hubble patch. If the rates of
tunneling between the metastable minima i and j on the
landscape are given by the time independent matrix Γij ,
then the probabilities Pi satisfy the system of “vacuum
dynamics” equations [7]
Ṗi =
j 6=i
(ΓjiPj − ΓijPi)− ΓisPi. (1)
The last term in this equation corresponds to tunneling
∗On leave from Landau Institute for Theoretical Physics, 119940,
Moscow, Russia.
1 An approach somewhat similar to ours was also presented in [4].
between the metastable de Sitter vacuum i and a true
vacuum with a negative cosmological constant (an AdS
vacuum), i.e. tunneling into a collapsing AdS space-time
[8]. The collapse time tcol ∼ MP /V
is much shorter
than the characteristic time trec ∼ exp
M4P /VdS
for tun-
neling back into a de Sitter metastable vacuum, so that
the AdS true vacua effectively play the role of sinks for
the probability current (1) describing eternal inflation on
the landscape [5].
In what follows we will assume that the effect of the
AdS sinks is relatively small; otherwise the landscape will
be divided into almost unconnected “islands” of vacua
[6], preventing the population of the whole landscape by
eternal inflation.
In the limit of weak tunneling only the vacua closest
to each other are important. It is convenient to classify
parts (islands) of the landscape according to the typical
number of adjacent vacua within each part. Technically,
the landscape of vacua of the string theory can be repre-
sented as a graph with 10100÷101000 nodes and a number
of connections between them of the same order. By an
island on the landscape, we mean a subgraph relatively
weakly connected to the major ”tree”. The dimension-
ality of the island can then be defined as the Hausdorff
dimension NH of the corresponding subgraph [17]. For
example, if there are only two adjacent vacua for any
vacuum in a given island, then NH = 1 for this island
and we denote it as quasi-one-dimensional; a domain of
vacua with NH = 2 is quasi-two-dimensional, and so on.
In the quasi-one-dimensional case (neglecting the AdS
sinks) the system (1) reduces to
Ṗi = −Γi,i+1Pi+Γi+1,iPi+1−Γi,i−1Pi+Γi−1,iPi−1. (2)
While in general Γij 6= Γji, we will take 〈Γij〉 = 〈Γji〉
on the average.2 Furthermore, suppose that the initial
2 This condition is never satisfied for the Bousso-Polchinski land-
scape [9], where the adjacent vacua are those with closest values
http://arxiv.org/abs/0704.0144v3
condition for Eq. (2) is
Pi(0) = 1, Pj 6=i(0) = 0. (3)
so that the initial state is well localized. Naively, one may
expect that the distribution function Pi(t) would start
to spread out according to the usual diffusion law and
the system of vacua would exponentially quickly reach a
“thermal” equilibrium distribution of probabilities for a
given Hubble patch to be in a given dS vacuum. However,
there exists a well known theorem [10] from the theory of
diffusion on random lattices stating that the distribution
function Pi remains localized near the initial distribution
peak for a very long time, with its characteristic width
behaving as
〈i2(t)〉 ∼ log
t . (4)
This is a surprising result when applied to eternal infla-
tion where the general lore (see for example [11]) is that
the initial conditions for eternal inflation will be forgot-
ten almost immediately after its beginning. Instead, in
what follows we will argue that the memory about the
initial conditions may survive during a very long time on
the quasi-one-dimensional islands of the landscape.
We will model the landscape by a continuous inflaton
potential
V (φ) = V0 + δV (φ), (5)
where V0 is constant, and δV (φ) is a random contribution
such that |δV (φ)| ≪ V0, and φ is the inflaton or the or-
der parameter describing the transitions. As in stochas-
tic inflation [16], in different causally connected regions
fluctuations have a randomly distributed amplitude and
observers living in different Hubble patches see differ-
ent expectation values of the inflaton. When stochastic
fluctuations of the inflaton are large enough, the expec-
tation value of the inflaton in a given Hubble patch is
determined by the Langevin equation [16]
φ̇ = −
+ f(t), (6)
where the stochastic force f(φ, t) is Gaussian with corre-
lation properties
〈f(t)f(t′)〉 =
δ(t− t′). (7)
From (6) one can derive the Fokker-Planck equation,
which controls the evolution of the probability distribu-
tion ρ(φ, t) describing how the values of φ are distributed
of the effective cosmological constant. However, the spectrum
of states on Bousso-Polchinski landscape is not disordered, so
that the analysis based on averaging over disorder is not appli-
cable. Disorder appears in more realistic multithroat models of
the string theory landscape.
among different Hubble patches in the multiverse. One
finds [16]
∂ρ(φ, t)
. (8)
The general solution to Eq. (8) is given by
ρ = e
4π2δV (φ)
cnψn(φ)e
0 (t−t0)
4π2 , (9)
where ψn and En are respectively the eigenfunctions and
the eigenvalues of the effective Hamiltonian
Ĥ = −
+W (φ). (10)
W (φ) =
is a functional of the scalar field potential V (φ). It is
often denoted as the superpotential due to its “super-
symmetric” form: the Hamiltonian (10) can be rewrit-
ten as Ĥ = Q̂†Q̂, where Q̂ = −∂/∂φ + v′(φ) with
v(φ) = 4π2δV (φ)/(3H40 ).
The eigenfunctions of the Hamiltonian (10) satisfy the
Schrödinger equation
+ (En −W (φ))ψn = 0, (12)
and its solutions have the following well known features
[16]:
1. The eigenvalues of the Hamiltonian (10) are all pos-
itive definite.
2. The contributions from eigenfunctions of excited
states ψn>0(φ) to the solution Eq. (9) become ex-
ponentially quickly damped with time. However,
if one is interested in what happens at time scales
∆t . 1/En, the first n eigenfunctions should be
taken into account. In particular, if the spectrum
of the Hamiltonian (10) is very dense, as in the case
of the string theory landscape, knowing the ground
state alone is not enough for complete understand-
ing dynamics of eternal inflation.
We now recall that the potential V (φ) is a random func-
tion of the inflaton field and has extremely large number
of minima. This allows us to draw several conclusions
about the form of the eigenfunctions ψn(φ) using the for-
mal analogy between Eq. (12) and the time-independent
Schrödinger equation describing the motion of carriers
in disordered quantum systems such as semiconductors
with impurities. The physical quantities in disordered
systems can be calculated by averaging over the random
potential of the impurities.3
A famous consequence of the random potential gener-
ated by impurities in crystalline materials is the strong
suppression of the conductivity, known as Anderson lo-
calization [12, 13]. This effect is essential in dimensions
lower than 3 and completely defines the kinetics of carri-
ers in one-dimensional systems. There, impurities create
a random potential for Bloch waves with the correlation
properties
〈u(r)u(r′)〉 =
δ(r − r′), 〈u(r)〉 = 0, (13)
where τ is the mean free path for electrons and ν is the
density of states per one spin degree of freedom of the
electron gas at the Fermi surface. As a consequence, in
the one-dimensional case all eigenstates of the electron
hamiltonian become localized with
ψn(r) ∼ exp
|r − rn|
at t → ∞, where rn are the positions of localization
centers, and the localization length L is of the order of
the mean free path lτ = 〈v〉τ . As a result, the probability
density ρ(R, t) to find electron at the point R at time t
asymptotically approaches the limit ρ(R) ∼ exp(−R/L)
for R ≫ L, or ρ(R) ∼ Const for R ≪ L at t → ∞.
The one-dimensional Anderson localization takes place
for an arbitrarily weak disorder and arbitrary correlation
properties of the random potential u(r) [13].
Also, in a two-dimensional case all the electron eigen-
states in a random potential remain localized. However,
the localization length grows exponentially with energy,
the rate of growth being related to the strength of the dis-
order. In three-dimensional case, the localization prop-
erties of eigenstates are defined by the Ioffe-Regel-Mott
criterion: if the corresponding eigenvalue of the Hamil-
tonian of electrons En satisfies the condition En < Eg
where Eg is so called mobility edge, then the eigenstate
is localized. The mobility edge Eg is a function of the
strength of the disorder. In higher dimensional cases the
situation is unknown.
Let us now return to the discussion of eternal infla-
tion described by the Fokker-Planck equation (8). Since
the localization is the property of the eigenfunctions of
the time-independent hamiltonian (10), it is also a nat-
ural consequence of the effective randomness of the po-
tential of the string theory landscape.4 The diffusion
3 Observe that the typical number of these impurities varies be-
tween 1012 to 1017 per cm3 while the number of vacua on the
string theory landscape is 10100 ÷ 101000.
4 The Anderson localization on the landscape of string theory was
discussed before in [14] in the context of the Wheeler-deWitt
equation in the minisuperspace. The possibility to have the An-
derson localization on the landscape was also mentioned in [15].
of the probability distribution (4) is suppressed due to
the localization of the eigenfunctions ψn(φ) contributing
to the overall solution (9). This counteracts the general
wisdom that eternal inflation rapidly washes out any in-
formation of the initial conditions. Indeed,in the quasi-
one-dimensional case all the wave functions ψn(φ) are
localized, i.e., for a particular realization of disorder they
behave as
ψn(φ) ∼ exp
|φ− φn|
. (15)
where φn define the ”localization centers” as in the Eq.
(14), and L is the localization length which is of the same
order of magnitude as the “mean free path” related to the
strength of the disorder in the superpotential W (φ).
Let us now discuss how eternal inflation proceeds on
islands where the typical number of adjacent vacua is
larger than two. In the quasi-two-dimensional case the
network of vacua within a given island is described by a
composite index ~i = (i, j). The distribution function ρ
for finding a given value of the cosmological constant in a
given Hubble patch is a two-dimensional matrix. Again,
all the eigenstates of the corresponding tunneling hamil-
tonian Ĥ are localized. However, since the localization
length grows exponentially with energy, the distribution
function effectively spreads out almost linearly with
〈~i2(t)〉 ∼ t
1 + c1
logα t
+ · · ·
, (16)
where α > 0 are constants depending on the correlation
properties of the disorder on the landscape [18]. The
low energy eigenstates (namely, the states with E < Eg
where Eg is the mobility edge) are localized with a rela-
tively small localization length.
In the quasi-higher-dimensional cases the distribution
function spreads out according to the linear diffusion law
at intermediate times. Again, there exists a mobility edge
Eg such that the eigenstates of the tunneling Hamilto-
nian with energies E < Eg are localized. These low
energy eigenstates define the asymptotics of the distri-
bution function ρ at
t≫ E−1g . (17)
The value of the mobility edge Eg strongly depends on
the dimensionality of the island and the strength of the
disorder, and the higher is the dimensionality, the lower
is the mobility edge [17].
Localization of the low energy eigenstates in two- and
higher-dimensional cases introduces an effective dynam-
ical selection principle for different vacua on the land-
scape (5): in the asymptotic future, not all of them will
be populated, but only those near the localization centers
φn, and the probability to populate other minima will be
suppressed exponentially according to the Eq. (15).
It is interesting to note that in condensed matter sys-
tems the localization centers are typically located near
the points where the effective potential has its deepest
minima [13]. In the case of eternal inflation, it means
that the probability to measure any but very low value of
the cosmological constant in a given Hubble patch will be
exponentially suppressed in the asymptotic future [17].
Finally, we discuss the effect of sinks on the dynamics
of tunneling between the vacua. On the string theory
landscape, dS metastable vacua are typically realized by
uplifting stable AdS vacua (as, for example, in the well
known KKLT model [19]). The probability to tunnel
from the uplifted dS state i back into the AdS vacuum
is related to the value of gravitino mass m3/2 in the dS
state [8] and given by
tAdS ∼ Γ
is ∼ exp
Const.M2P
3/2,i
. (18)
The gravitino mass after uplifting [20] has the order of
magnitude m3/2,i ∼ |VAdS,i|
1/2/MP . Since at long time
scales VAdS,i can also be regarded as a random quantity,
our analysis of the general solution of “vacuum dynam-
ics” equations (1) does not have to be modified in any
essential way [17].
In addition to AdS sinks, Hubble patches where eternal
inflation has ended (stochastic fluctuations of the infla-
ton expectation value became smaller than the effect of
classical force) also effectively play a role of sinks for the
probability current described by the Eq. (8). In par-
ticular, the Hubble patch we live in is one of such sinks.
Related to the effect of sinks, there exists a time scale tend
for eternal inflation on the landscape (5) such that the
unitarity of the evolution of the probability distribution
ρ breaks down at t ≫ tend [17]. Our discussion remains
valid if t ≪ tend. It is unclear whether the probability
distribution ρ has achieved the late time asymptotics in
the corner of the landscape we live in.
In summary, we have argued that eternal inflation
on the landscape may lead to a strong localization of
the inflaton distribution function among different Hub-
ble patches. This is a consequence of the high density
of the vacua, which effectively implies a random poten-
tial for the order parameter responsible for inflation. We
found that the inflaton motion is analogous to the mo-
tion of carriers in disordered quantum systems, and there
exists an analogue of the Anderson localization for eter-
nal inflation on the landscape. Physically, this means
that not all the vacua on the landscape are populated by
eternal inflation in the asymptotic future, but only those
near the localization centers of the inflaton effective po-
tential. They are located near the deepest minima of the
potential, which implies that the probability to measure
any but very low value of the cosmological constant in
a given Hubble patch is exponentially suppressed at late
times.
Acknowledgements
The authors belong to the Marie Curie Research Train-
ing Network HPRN-CT-2006-035863. D.P. is thankful to
I. Burmistrov, N. Jokela, J. Majumder, M. Skvortsov,
K. Turitsyn ad especially to A.A. Starobinsky for the
discussions. K.E. is supported partly by the Ehrnrooth
foundation and the Academy of Finland grant 114419.
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