USTC-ICTS-07-07
The Measure for the Multiverse and the Probability
for Inflation
Miao Li1,2, Yi Wang2,1
1 The Interdisciplinary Center for Theoretical Study
of China (USTC), Hefei, Anhui 230027, P.R.China
2 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, P.R.China
Abstract
We investigate the measure problem in the framework of inflationary cos-
mology. The measure of the history space is constructed and applied to inflation
models. Using this measure, it is shown that the probability for the generalized
single field slow roll inflation to last for N e-folds is suppressed by a factor
exp(−3N), and the probability for the generalized n-field slow roll inflation is
suppressed by a much larger factor exp(−3nN). Some non-inflationary models
such as the cyclic model do not suffer from this difficulty.
http://arxiv.org/abs/0704.1026v1
1 Introduction
It is claimed that our vacuum is one of the 10500 possible meta-stable vacua in the
string theory landscape [1]. If this is true, then the physical parameters labeling which
vacuum we are living in can not be calculated from the first principle. Theoretically,
these parameters may only be explained by some anthropic reasoning [2], or by pure
chance.
From the cosmological point of view, in the framework of eternal inflation [3], the
vast landscape of vacua is not only a logic possibility but also the reality. If we demand
that our observable universe is not too special in the multiverse, in principle, we can
make predictions in the multiverse by calculating the probability of the corresponding
universe history.
A serious problem arises at this point. A measure of the history space is essentially
needed in order to compare different histories of the universe. But in general relativity,
it is not straightforward to construct such a measure. It is because there is no preferred
space slicing and time notation in general relativity, and singularities commonly arise
in the cosmic solutions. Even in the much simplified Friedmann-Robertson-Walker
universe, the measure problem is not easy to solve. The construction of a measure of
the history space is considered as one of the central problems in cosmology. Attempts
for this problem can be found in [4, 5].
To analyze this problem in detail, let us construct the history space of the universe
and discuss the measure. In general relativity, all the trajectories in the phase space
should lie on the hypersurface H−1(0) due to the Hamiltonian constraint H = 0. Now
we want to consider the history space, where a trajectory is represented by a single
point. So we have to identify the points in H−1(0) which can be linked by the time
evolution. Then, the history space, or the multiverse, takes the form
M = H−1(0)/R. (1)
The next step is to construct a measure on the history space. To make sense and
to be natural in physics, a measure of the history space should satisfy three conditions
[4, 6]: (i) It should be positive. (ii) It should depend only on the intrinsic dynamics
and neither on any choice of time slicing nor on the choice of dependent variables.
(iii) It should respect all the symmetries of the space of solutions.
A measure of the history space satisfying these three requirements can be con-
structed from the phase space symplectic form [4, 6, 7]. The symplectic form of the
phase space ω can be written in terms of the canonical coordinates and momenta as
dpi ∧ dq
i. (2)
where m is the number of canonical coordinates.
If we choose pm = H, then from the Hamilton’s equations, q
m = t is the time
coordinate. And the symplectic form (2) can be written as
dpi ∧ dq
i + dH ∧ dt. (3)
The Hamiltonian constraint H = 0 naturally yields a two-form transverse to the
time evolution. This is the two-form in the history space,
ωC ≡ ω|H=0 =
dpi ∧ dq
i. (4)
The measure of the history space can be constructed by raising ωC to the (m−1)th
power,
(−1)(m−1)(m−2)/2
(m− 1)!
ωm−1C . (5)
Note that ΩM is an exact form. It can be globally written as ΩM ∼ dA with
A ≡ p1dq
dpi ∧ dq
i. (6)
This measure of the history space can be applied to the inflationary cosmology in
determining the probability of inflation. At first, it was believed that the canonical
measure favors inflation [6]. But soon it is realized that both inflationary and non-
inflationary history have infinite measure [8]. So the measure problem in cosmology
remained unsolved.
Recently, Gibbons and Turok [4] suggested a solution to this measure problem.
They noticed that a universe with a very small spacial curvature at the present time
can not be distinguished from a flat one. So physically, it makes sense to cut off the
history space by identifying a universe with a very small spacial curvature with a
flat universe. As was shown in [4], the measure for some quantities, like the spacial
curvature, is cutoff dependent, and dominated by the cutoff. While the measure
for some other quantities, for example, the e-folding number of inflation, is cutoff
independent. So by applying this cutoff, the question whether a N e-folds’ inflation is
natural can be well defined, and investigated explicitly. It is shown that the history
space volume for slow roll inflation is suppressed by a factor of exp(−3N), where N
is the e-folding number.
The work [4] concentrates on a single field minimal coupled inflation model. There
is a vast variety of inflation models in addition to a single inflaton model, thus it is
interesting to ask how other models weigh in this measure. Some of the inflation mod-
els involve a modified Lagrangian density other than the minimal one, some involve
multi-fields and some modify the Einstein gravity. We want to know whether these
inflation models are also suppressed for a large e-folding number. An investigation of
these models is the main task of this paper.
This paper is organized as follows. In Section 2, we review the approach by
Gibbons and Turok [4] for gravity minimally coupled to a scalar field. It is shown
that the inflation probability can be calculated directly as a function of N . In Section
3, we discuss the measure for the scalar field with a more general Lagrangian. We
find that in this generalized case, the measure for the slow roll inflationary history is
suppressed by exactly the same factor exp(−3N). In Section 4, we consider the multi-
field inflation. It can be shown that with the assumption of slow roll for the Hubble
constant, the measure is a lot more suppressed by the exponential factor exp(−3nN),
where n is the number of inflaton fields. So it seems much more unnatural for multi-
field inflation to happen. In Section 5, we investigate the generalized Lagrangian for
multi-field inflation. We find that the generalization of the Lagrangian can not solve
the measure problem raised in Section 4. Finally, we summarize the paper in the last
section.
2 Single Field Inflation Models
In this section, we consider a single scalar field minimally coupled with gravity with
the action
−3a(N−2ȧ2 − k) +
a3N−2ϕ̇2 − a3V (ϕ)
, (7)
where N is the lapse function, and k = 0,±1 represents the spacial curvature, and
dot denotes the derivative with respect to time. For simplicity, we have set M2p ≡
1/(8πG) = 1.
By varying the action with respect to the lapse function N , we obtain the Fried-
mann equation
3H2 =
ϕ̇2 + V (ϕ)−
, (8)
where after the variation, we have set N = 1. Varying the action with respect to ϕ
leads to the scalar field equation of motion
ϕ̈+ 3Hϕ̇+ Vϕ = 0, (9)
where the subscript ϕ in V denotes derivative with respect to ϕ.
From the time derivative of (8) and using (9), we get
Ḣ = −
ϕ̇2 +
. (10)
To construct the history space, we need to slice the H = 0 hypersurface of the
phase space. A good way to do this is to choose a constant H surface H = HS as
a slicing [4], where HS is chosen low enough that it just above the end of inflation
and the universe evolves adiabatically from then on. To choose a constant H slice is
because for the flat or open universe, and non-negative potential V (ϕ), each history
trajectory crosses a constant H surface exactly once. And the reason for choosing
H low enough is that only this choice can result in a cutoff independent measure of
e-folds, and this choice is in agreement with the anthropic “top down” approach to
cosmology [9].
On a constant H surface, the measure for the history space takes the form
dpϕ ∧ dϕ, (11)
where pϕ ≡ a
3ϕ̇ denotes the canonical momentum for ϕ. It can be calculated that
dϕda 3a2
6H2S − 2V + 4ka
6H2S − 2V + 6ka
. (12)
A divergence occurs in the large a limit of (12). This is the infinity discovered in
[8]. Following [4], we set a cutoff for the spacial curvature to critical density ratio
Ωk ≡ −k/(a
2H2S) as
|Ωk| ≥ ∆Ωk, (13)
The cutoff makes sense physically because a small enough Ωk is neither geometrically
meaningful nor physically observable. As we are working on a constant HS surface,
the cutoff can be translated into the cutoff of the scale factor
a2 ≤ a2max ≡
. (14)
Recall that ΩM = dA, The measure can be reduced to a surface integral around a
constant a surface of the constant HS history space,
pϕdϕ = a
ϕ̇dϕ. (15)
To investigate the probability distribution for inflation, now concentrate on an
history space volume element A ∼ ϕ̇∆ϕ. Where we have dropped the a3max term as
it is a constant. Since the variation operation ∆ is taken on a constant H surface, it
is convenient to convert the time derivative ∂t to the derivative with respect to the
Hubble constant ∂H , using
∂t = Ḣ∂H = −
ϕ̇2∂H . (16)
Then we can take the advantage that ∂H ·∆ = ∆ · ∂H .
Note thatH do not change when we move on the history space. Then the equation
(8) leads to a constraint for the history space variation
ϕ̇∆ϕ̇ + Vϕ∆ϕ = 0, (17)
Given this constraint, the Hubble evolution for A can be calculated as
∂HA = −
A. (18)
where we have neglected the spatial curvature energy density, because it have to be
small during the last e-folds of inflation.
Note that for the e-folding number N , −H = ∂tN = Ḣ∂HN , the equation (18)
takes the form
∂HA = 3A∂HN, (19)
which can be integrated out to give
A = e3NA(HS). (20)
The above equation tells us that as we stand at the end of inflation and track back-
wards with time, a volume in the history space expands exponentially. In order not
to break the slow roll condition along the whole 60 e-folds’ inflation, The volume el-
ement A must lie in a exponentially narrow corner in the constant HS history space.
So the probability for inflation is suppressed by the exp(−3N) factor. This suppres-
sion shows that inflation is not as natural as we intuitively think. It may have not
solved the naturalness problems of the hot big bang cosmology because of its unnat-
ural nature, or there remains some unknown mechanism to produce a exponentially
sharp peak for the possibility distribution of the history space.
3 Generalized Single Field Models
In this section, we consider the action
−3aN−2ȧ2 + a3f
ϕ,N−1ϕ̇
. (21)
A good many inflation models can be described using this action. For example,
K-Inflation [10], Phantom Inflation [11], Inflation driven by the brane DBI action
[12, 13], etc.
Choosing ϕ as a canonical coordinate and using the proper time, the canonical
momentum for ϕ takes the form
p = πa3, where π ≡ fϕ̇. (22)
Take variation with respect to N and ϕ, one obtains
3H2 = ϕ̇π − f, π̇ + 3Hπ − fϕ = 0, Ḣ = −
ϕ̇π. (23)
Using (23), the constraint for the variation in the history space can be written as
ϕ̇∆π = fϕ∆ϕ. (24)
And using the definition of π, we have the variation relation
∆ϕ̇ = f−1ϕ̇ϕ̇ (∆π − fϕ̇ϕ∆ϕ). (25)
where we have assumed that fϕ̇ϕ̇ 6= 0, in order that ϕ can be treated as a dynamical
degree of freedom.
Now we take the cutoff as discussed in the last section, and reduce the integration
of the history space to the boundary integration
pdϕ = a3max
πdϕ. (26)
Then it can be calculated that the variation of the volume element in the history
space A ∼ π∆ϕ evolves along the constant H surfaces as
∂HA = −
A = 3A∂HN (27)
So the conclusion is exactly the same as that of the last section. In order to get
N e-folds’ slow roll inflation, the volume element in the history space should be
exponentially fine turned.
It should be noticed that in this general case, there is the possibility that even
the history evolution is not slow rolling, accelerated expansion with a large e-folding
number can be achieved in models such as the Kflation or the phantom inflation. But
it is difficult to get a scale invariant perturbation spectrum if the slow roll condition
is not satisfied [11].
4 Multi-Field Inflation Models
Multi-field inflation models take an important part in the inflationary model build-
ing. In string theory, there can be a number of scalar fields at the inflation scale.
Phenomenally, in multi-field models, slow roll condition is less stringent and can be
satisfied in more models [14]. Moreover, there are interesting inflation models, like
the hybrid inflation model [15], which requires essentially more than one field. So it is
useful to study the measure for multi-field inflation and investigate the corresponding
probability.
The action for the multi-field inflation takes the form
−3aN−2ȧ2 +
a3N−2ϕ̇i
2 − a3V (ϕi)
, (28)
where the duplicate index i is summed over the n scalar fields.
Choosing to use the proper time, the canonical momentum for ϕi is pi ≡ a
3ϕ̇i.
And the equations of motion takes the form
3H2 =
ϕ̇2i + V, Ḣ =
ϕ̇2i , ϕ̈i + 3Hϕ̇i + Vϕi = 0. (29)
The constraint for constant HS variation is
ϕ̇i∆ϕ̇i + Vϕi∆ϕi = 0. (30)
It can be checked by direct calculation that ∂H ·∆ = ∆ · ∂H is also true operating on
ϕ̇i. In this multi-field case
A ∼ ϕ̇1 ∆ϕ1 ∧∆ϕ̇2 ∧∆ϕ2 ∧ . . . ∧∆ϕ̇n ∧∆ϕn. (31)
Using the constraint (30), each term in ∂HA is proportional to A, and ∂HA turns
out to be
∂HA = 3n
A∂HN. (32)
In a multi-field inflation model, Ḧ/(HḢ) should also be small and rolling slowly as
in the single field case. If one assumes that ϕ̈1/(Hϕ̇1) is also small and slow rolling,
then the integration can be carried out as
A = e3nNA(HS), (33)
which shows that the departure from slow-roll evolves much faster than that in the
single field case. As a result, multi-field inflation is much more unnatural then the
single-field inflation with a much smaller measure in the history space. This result
is not surprising. It is because from the first equation in (29), the Hubble constant
has contribution from the energy density of all inflation fields. While from the third
equation in (29), the Hubble constant appears as a friction in the evolution of each
single inflaton field. So in the multi-field inflation case, the friction of each single field
is contributed by all the fields, and the history space for slow roll inflation is much
more concentrated then the single field models.
As analyzed in [16], |ϕ̈1/(Hϕ̇1)| ≪ 1 may break down in some multi-field inflation
models. Now let’s see whether a fast rolling ϕ̇1 can result in something more natural.
If we want ϕ̈1/(Hϕ̇1) to cancel the exponential expansion of the history space
volume, we need
H, (34)
which amounts to demanding that
∣ϕ̇1/a
∣ increases with time. As n ≥ 2, |ϕ̇1| must
be increasing faster than a3 to make this cancellation possible. And this cancellation
need to be valid along the whole 60 e-folds of inflation. It seems impossible for ϕ̇1 to
behave like this. So even a fast rolling ϕ̇1 can not make the situation more natural.
A few words are in order here. We have picked a specific field ϕ1 out of many other
fields in studying the measure, this is just the result of integrating out pϕ1, namely,
we have allowed ϕ̇1 to vary as much as possible. We could have picked out another
field, then we would be discussing the differential measure in a different region on the
history space.
Now we see that the multi-field inflation is even more impossible than the single
field inflation. Then, if for anthropic principle or some other reasons that a 60 e-folds’
inflation has to have happened in our history, it should be single field inflation rather
than multi-field inflation, because the latter has much smaller measure.
5 Generalized Multi-Field Models
In this section, we do the generalizations one step further to consider the action
−3aN−2ȧ2 + a3f
−1ϕ̇i
, (35)
which has the features of the actions in both Section 3 and Section 4.
Using the proper time, the canonical momentum for ϕ takes the form
pi = πia
3, where πi ≡ fϕ̇i, (36)
with the equations of motion
3H2 = ϕ̇iπi − f, π̇i + 3Hπi − fϕi = 0, Ḣ = −
ϕ̇iπi. (37)
and the constraint for the history space variation
ϕ̇i∆πi = fϕi∆ϕi. (38)
We assume that the matrix fϕ̇iϕ̇j has inverse matrix. This should be true when all the
constraints in (35) are solved and ϕi only denotes the dynamical degree of freedom.
We use f ϕ̇iϕ̇j as the inverse matrix of fϕ̇iϕ̇j Then it can be shown that
∆ϕ̇i = f
ϕ̇iϕ̇j (∆πj − fϕ̇jϕk∆ϕk). (39)
In this generalized case,
A ∼ π1 ∆ϕ1 ∧∆π2 ∧∆ϕ2 ∧ . . . ∧∆πn ∧∆ϕn. (40)
using the same technique developed in Section 3 and Section 4, one finds
∂HA =
f ϕ̇1ϕ̇iπ̇i
ϕ̇1Ḣ
ϕ̇iπ̇i
ϕ̇iϕ̇k π̇k
f ϕ̇1ϕ̇jfϕ̇jϕk ϕ̇k
Ḣϕ̇1
ϕ̇iϕ̇jfϕ̇jϕkϕ̇k
A (41)
To see the implications of this equation, let us concentrate on the double field in-
flation models. It is because it seems more difficult to cancel the −3nH
term for
lager n. Lagrangian densities like f = g(ϕ1)ϕ̇
1 + h(ϕ2)ϕ̇
2 are not of special interest
here, because they can be transformed into the case discussed in Section 4 by a field
redefinition. As another example, let us consider the Lagrangian density
f = f
ϕ1, ϕ2,
(ϕ21 + ϕ
, (42)
in this case, the equation (41) takes the form
∂HA = −6A
∂HN, (43)
where
f ′ ≡
(ϕ̇21 + ϕ̇
] . (44)
So a fast rolling f ′ϕ̇21 is required to cancel the exponential expansion of the volume
of the history space.
To see the physical implications for this condition, consider the DBI inflation
model by [13]. The action of the DBI inflation model is given by
SDBI = −
1− f̃ ϕ̇2 − 1
+ V (ϕ)
where the angular motion of ϕi has been ignored, so ϕ̇
2 ≡ ϕ̇2i . Then f
′ = 1/
1− f̃ ϕ̇2 ≡
γ is just the relativistic factor defined in [13]. From the spectral index
ns − 1 = 4
, (46)
We conclude that γ should not be a fast rolling quantity along the whole history of
observable inflation. Moreover, from the equation Ḣ = −γϕ̇2/2, we see again that
γ can not be large for a long time during inflation. So the cancellation of the e−3nN
factor can not be obtained.
6 Conclusion and Discussions
In this paper, we have reviewed the measure problem in cosmology. We calculated the
measure and the probability for inflation in single and multi-field models with gener-
alized Lagrangian density. It is shown that the measure for the single field inflation
and the corresponding generalizations are suppressed by a factor of exp(−3N). While
the n-field and generalized multi-field inflation models has a measure proportional to
exp(−3nN).
This work can be understood in another way. Taking apart the discussion for the
measure and the slow roll condition, other parts of this paper can be thought of as a
proof of the attractor behavior of various kinds of generalized inflation models. On
the one hand, it is a proof that the attractor behavior is very common in inflationary
models. While on the other hand, to take the measure into consideration, we see that
it is far from obvious for an attractor to be a natural solution in cosmology. And it
is just this early time attractor combined with the requirement of slow roll that puts
inflation into a highly unnatural situation.
We did not study explicitly the inflation models with non-minimal coupling to
gravity [17]. But these models do not seem to bring large correction for the sup-
pression factor. It is because through conformal transformation, these non-minimal
coupled models are generally equivalent with the corresponding minimal coupled in-
flation models with the same number or one more inflation fields. Another reason
not to consider these models in this work is that, as the energy scale commonly drops
during inflation, near the end of inflation, the non-minimal coupling effect may not
be so important.
There are also inflation models with extra components or special spacetime prop-
erties. Examples of this kind are inflation with holographic dark energy [18, 19] or
in the non-commutative spacetime [20, 21, 22]. These models do not seem to change
the results much either. Because in the former example, the holographic dark energy
is diluted during inflation so do not seem to cause large corrections near the end
of inflation. In the latter case, although the spectrum for perturbations is greatly
modified in the non-commutative spacetime, the isotropic and homogeneous inflating
background do not change much because it belongs to a lower energy scale. So the
corrections to the probability can not be large.
As a closing remark, we noticed that some non-inflationary models do not share
the small measure problem. One example is the cyclic universe model [23]. Although
the cyclic model is controlled by gravity coupled with a scalar field, it do not have
slow roll behavior backwards in time in the cycle we live. So the key observation that
the exponentially expansion of the phase space volume breaks the slow roll condition
do not apply in the cyclic model. Nevertheless, the number of cycles in the cyclic
universe must be finite [23], so it remains to explain how all the cycles begin in the
first place.
Acknowledgments
This work was supported by grants of NSFC. We thank Bin Chen, Yi-Fu Cai, Chao-
Jun Feng, and Yushu Song for discussion.
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	Introduction
	Single Field Inflation Models
	Generalized Single Field Models
	Multi-Field Inflation Models
	Generalized Multi-Field Models
	Conclusion and Discussions

  We investigate the measure problem in the framework of inflationary
cosmology. The measure of the history space is constructed and applied to
inflation models. Using this measure, it is shown that the probability for the
generalized single field slow roll inflation to last for $N$ e-folds is
suppressed by a factor $\exp(-3N)$, and the probability for the generalized
$n$-field slow roll inflation is suppressed by a much larger factor
$\exp(-3nN)$. Some non-inflationary models such as the cyclic model do not
suffer from this difficulty.

Introduction
It is claimed that our vacuum is one of the 10500 possible meta-stable vacua in the
string theory landscape [1]. If this is true, then the physical parameters labeling which
vacuum we are living in can not be calculated from the first principle. Theoretically,
these parameters may only be explained by some anthropic reasoning [2], or by pure
chance.
From the cosmological point of view, in the framework of eternal inflation [3], the
vast landscape of vacua is not only a logic possibility but also the reality. If we demand
that our observable universe is not too special in the multiverse, in principle, we can
make predictions in the multiverse by calculating the probability of the corresponding
universe history.
A serious problem arises at this point. A measure of the history space is essentially
needed in order to compare different histories of the universe. But in general relativity,
it is not straightforward to construct such a measure. It is because there is no preferred
space slicing and time notation in general relativity, and singularities commonly arise
in the cosmic solutions. Even in the much simplified Friedmann-Robertson-Walker
universe, the measure problem is not easy to solve. The construction of a measure of
the history space is considered as one of the central problems in cosmology. Attempts
for this problem can be found in [4, 5].
To analyze this problem in detail, let us construct the history space of the universe
and discuss the measure. In general relativity, all the trajectories in the phase space
should lie on the hypersurface H−1(0) due to the Hamiltonian constraint H = 0. Now
we want to consider the history space, where a trajectory is represented by a single
point. So we have to identify the points in H−1(0) which can be linked by the time
evolution. Then, the history space, or the multiverse, takes the form
M = H−1(0)/R. (1)
The next step is to construct a measure on the history space. To make sense and
to be natural in physics, a measure of the history space should satisfy three conditions
[4, 6]: (i) It should be positive. (ii) It should depend only on the intrinsic dynamics
and neither on any choice of time slicing nor on the choice of dependent variables.
(iii) It should respect all the symmetries of the space of solutions.
A measure of the history space satisfying these three requirements can be con-
structed from the phase space symplectic form [4, 6, 7]. The symplectic form of the
phase space ω can be written in terms of the canonical coordinates and momenta as
dpi ∧ dq
i. (2)
where m is the number of canonical coordinates.
If we choose pm = H, then from the Hamilton’s equations, q
m = t is the time
coordinate. And the symplectic form (2) can be written as
dpi ∧ dq
i + dH ∧ dt. (3)
The Hamiltonian constraint H = 0 naturally yields a two-form transverse to the
time evolution. This is the two-form in the history space,
ωC ≡ ω|H=0 =
dpi ∧ dq
i. (4)
The measure of the history space can be constructed by raising ωC to the (m−1)th
power,
(−1)(m−1)(m−2)/2
(m− 1)!
ωm−1C . (5)
Note that ΩM is an exact form. It can be globally written as ΩM ∼ dA with
A ≡ p1dq
dpi ∧ dq
i. (6)
This measure of the history space can be applied to the inflationary cosmology in
determining the probability of inflation. At first, it was believed that the canonical
measure favors inflation [6]. But soon it is realized that both inflationary and non-
inflationary history have infinite measure [8]. So the measure problem in cosmology
remained unsolved.
Recently, Gibbons and Turok [4] suggested a solution to this measure problem.
They noticed that a universe with a very small spacial curvature at the present time
can not be distinguished from a flat one. So physically, it makes sense to cut off the
history space by identifying a universe with a very small spacial curvature with a
flat universe. As was shown in [4], the measure for some quantities, like the spacial
curvature, is cutoff dependent, and dominated by the cutoff. While the measure
for some other quantities, for example, the e-folding number of inflation, is cutoff
independent. So by applying this cutoff, the question whether a N e-folds’ inflation is
natural can be well defined, and investigated explicitly. It is shown that the history
space volume for slow roll inflation is suppressed by a factor of exp(−3N), where N
is the e-folding number.
The work [4] concentrates on a single field minimal coupled inflation model. There
is a vast variety of inflation models in addition to a single inflaton model, thus it is
interesting to ask how other models weigh in this measure. Some of the inflation mod-
els involve a modified Lagrangian density other than the minimal one, some involve
multi-fields and some modify the Einstein gravity. We want to know whether these
inflation models are also suppressed for a large e-folding number. An investigation of
these models is the main task of this paper.
This paper is organized as follows. In Section 2, we review the approach by
Gibbons and Turok [4] for gravity minimally coupled to a scalar field. It is shown
that the inflation probability can be calculated directly as a function of N . In Section
3, we discuss the measure for the scalar field with a more general Lagrangian. We
find that in this generalized case, the measure for the slow roll inflationary history is
suppressed by exactly the same factor exp(−3N). In Section 4, we consider the multi-
field inflation. It can be shown that with the assumption of slow roll for the Hubble
constant, the measure is a lot more suppressed by the exponential factor exp(−3nN),
where n is the number of inflaton fields. So it seems much more unnatural for multi-
field inflation to happen. In Section 5, we investigate the generalized Lagrangian for
multi-field inflation. We find that the generalization of the Lagrangian can not solve
the measure problem raised in Section 4. Finally, we summarize the paper in the last
section.
2 Single Field Inflation Models
In this section, we consider a single scalar field minimally coupled with gravity with
the action
−3a(N−2ȧ2 − k) +
a3N−2ϕ̇2 − a3V (ϕ)
, (7)
where N is the lapse function, and k = 0,±1 represents the spacial curvature, and
dot denotes the derivative with respect to time. For simplicity, we have set M2p ≡
1/(8πG) = 1.
By varying the action with respect to the lapse function N , we obtain the Fried-
mann equation
3H2 =
ϕ̇2 + V (ϕ)−
, (8)
where after the variation, we have set N = 1. Varying the action with respect to ϕ
leads to the scalar field equation of motion
ϕ̈+ 3Hϕ̇+ Vϕ = 0, (9)
where the subscript ϕ in V denotes derivative with respect to ϕ.
From the time derivative of (8) and using (9), we get
Ḣ = −
ϕ̇2 +
. (10)
To construct the history space, we need to slice the H = 0 hypersurface of the
phase space. A good way to do this is to choose a constant H surface H = HS as
a slicing [4], where HS is chosen low enough that it just above the end of inflation
and the universe evolves adiabatically from then on. To choose a constant H slice is
because for the flat or open universe, and non-negative potential V (ϕ), each history
trajectory crosses a constant H surface exactly once. And the reason for choosing
H low enough is that only this choice can result in a cutoff independent measure of
e-folds, and this choice is in agreement with the anthropic “top down” approach to
cosmology [9].
On a constant H surface, the measure for the history space takes the form
dpϕ ∧ dϕ, (11)
where pϕ ≡ a
3ϕ̇ denotes the canonical momentum for ϕ. It can be calculated that
dϕda 3a2
6H2S − 2V + 4ka
6H2S − 2V + 6ka
. (12)
A divergence occurs in the large a limit of (12). This is the infinity discovered in
[8]. Following [4], we set a cutoff for the spacial curvature to critical density ratio
Ωk ≡ −k/(a
2H2S) as
|Ωk| ≥ ∆Ωk, (13)
The cutoff makes sense physically because a small enough Ωk is neither geometrically
meaningful nor physically observable. As we are working on a constant HS surface,
the cutoff can be translated into the cutoff of the scale factor
a2 ≤ a2max ≡
. (14)
Recall that ΩM = dA, The measure can be reduced to a surface integral around a
constant a surface of the constant HS history space,
pϕdϕ = a
ϕ̇dϕ. (15)
To investigate the probability distribution for inflation, now concentrate on an
history space volume element A ∼ ϕ̇∆ϕ. Where we have dropped the a3max term as
it is a constant. Since the variation operation ∆ is taken on a constant H surface, it
is convenient to convert the time derivative ∂t to the derivative with respect to the
Hubble constant ∂H , using
∂t = Ḣ∂H = −
ϕ̇2∂H . (16)
Then we can take the advantage that ∂H ·∆ = ∆ · ∂H .
Note thatH do not change when we move on the history space. Then the equation
(8) leads to a constraint for the history space variation
ϕ̇∆ϕ̇ + Vϕ∆ϕ = 0, (17)
Given this constraint, the Hubble evolution for A can be calculated as
∂HA = −
A. (18)
where we have neglected the spatial curvature energy density, because it have to be
small during the last e-folds of inflation.
Note that for the e-folding number N , −H = ∂tN = Ḣ∂HN , the equation (18)
takes the form
∂HA = 3A∂HN, (19)
which can be integrated out to give
A = e3NA(HS). (20)
The above equation tells us that as we stand at the end of inflation and track back-
wards with time, a volume in the history space expands exponentially. In order not
to break the slow roll condition along the whole 60 e-folds’ inflation, The volume el-
ement A must lie in a exponentially narrow corner in the constant HS history space.
So the probability for inflation is suppressed by the exp(−3N) factor. This suppres-
sion shows that inflation is not as natural as we intuitively think. It may have not
solved the naturalness problems of the hot big bang cosmology because of its unnat-
ural nature, or there remains some unknown mechanism to produce a exponentially
sharp peak for the possibility distribution of the history space.
3 Generalized Single Field Models
In this section, we consider the action
−3aN−2ȧ2 + a3f
ϕ,N−1ϕ̇
. (21)
A good many inflation models can be described using this action. For example,
K-Inflation [10], Phantom Inflation [11], Inflation driven by the brane DBI action
[12, 13], etc.
Choosing ϕ as a canonical coordinate and using the proper time, the canonical
momentum for ϕ takes the form
p = πa3, where π ≡ fϕ̇. (22)
Take variation with respect to N and ϕ, one obtains
3H2 = ϕ̇π − f, π̇ + 3Hπ − fϕ = 0, Ḣ = −
ϕ̇π. (23)
Using (23), the constraint for the variation in the history space can be written as
ϕ̇∆π = fϕ∆ϕ. (24)
And using the definition of π, we have the variation relation
∆ϕ̇ = f−1ϕ̇ϕ̇ (∆π − fϕ̇ϕ∆ϕ). (25)
where we have assumed that fϕ̇ϕ̇ 6= 0, in order that ϕ can be treated as a dynamical
degree of freedom.
Now we take the cutoff as discussed in the last section, and reduce the integration
of the history space to the boundary integration
pdϕ = a3max
πdϕ. (26)
Then it can be calculated that the variation of the volume element in the history
space A ∼ π∆ϕ evolves along the constant H surfaces as
∂HA = −
A = 3A∂HN (27)
So the conclusion is exactly the same as that of the last section. In order to get
N e-folds’ slow roll inflation, the volume element in the history space should be
exponentially fine turned.
It should be noticed that in this general case, there is the possibility that even
the history evolution is not slow rolling, accelerated expansion with a large e-folding
number can be achieved in models such as the Kflation or the phantom inflation. But
it is difficult to get a scale invariant perturbation spectrum if the slow roll condition
is not satisfied [11].
4 Multi-Field Inflation Models
Multi-field inflation models take an important part in the inflationary model build-
ing. In string theory, there can be a number of scalar fields at the inflation scale.
Phenomenally, in multi-field models, slow roll condition is less stringent and can be
satisfied in more models [14]. Moreover, there are interesting inflation models, like
the hybrid inflation model [15], which requires essentially more than one field. So it is
useful to study the measure for multi-field inflation and investigate the corresponding
probability.
The action for the multi-field inflation takes the form
−3aN−2ȧ2 +
a3N−2ϕ̇i
2 − a3V (ϕi)
, (28)
where the duplicate index i is summed over the n scalar fields.
Choosing to use the proper time, the canonical momentum for ϕi is pi ≡ a
3ϕ̇i.
And the equations of motion takes the form
3H2 =
ϕ̇2i + V, Ḣ =
ϕ̇2i , ϕ̈i + 3Hϕ̇i + Vϕi = 0. (29)
The constraint for constant HS variation is
ϕ̇i∆ϕ̇i + Vϕi∆ϕi = 0. (30)
It can be checked by direct calculation that ∂H ·∆ = ∆ · ∂H is also true operating on
ϕ̇i. In this multi-field case
A ∼ ϕ̇1 ∆ϕ1 ∧∆ϕ̇2 ∧∆ϕ2 ∧ . . . ∧∆ϕ̇n ∧∆ϕn. (31)
Using the constraint (30), each term in ∂HA is proportional to A, and ∂HA turns
out to be
∂HA = 3n
A∂HN. (32)
In a multi-field inflation model, Ḧ/(HḢ) should also be small and rolling slowly as
in the single field case. If one assumes that ϕ̈1/(Hϕ̇1) is also small and slow rolling,
then the integration can be carried out as
A = e3nNA(HS), (33)
which shows that the departure from slow-roll evolves much faster than that in the
single field case. As a result, multi-field inflation is much more unnatural then the
single-field inflation with a much smaller measure in the history space. This result
is not surprising. It is because from the first equation in (29), the Hubble constant
has contribution from the energy density of all inflation fields. While from the third
equation in (29), the Hubble constant appears as a friction in the evolution of each
single inflaton field. So in the multi-field inflation case, the friction of each single field
is contributed by all the fields, and the history space for slow roll inflation is much
more concentrated then the single field models.
As analyzed in [16], |ϕ̈1/(Hϕ̇1)| ≪ 1 may break down in some multi-field inflation
models. Now let’s see whether a fast rolling ϕ̇1 can result in something more natural.
If we want ϕ̈1/(Hϕ̇1) to cancel the exponential expansion of the history space
volume, we need
H, (34)
which amounts to demanding that
∣ϕ̇1/a
∣ increases with time. As n ≥ 2, |ϕ̇1| must
be increasing faster than a3 to make this cancellation possible. And this cancellation
need to be valid along the whole 60 e-folds of inflation. It seems impossible for ϕ̇1 to
behave like this. So even a fast rolling ϕ̇1 can not make the situation more natural.
A few words are in order here. We have picked a specific field ϕ1 out of many other
fields in studying the measure, this is just the result of integrating out pϕ1, namely,
we have allowed ϕ̇1 to vary as much as possible. We could have picked out another
field, then we would be discussing the differential measure in a different region on the
history space.
Now we see that the multi-field inflation is even more impossible than the single
field inflation. Then, if for anthropic principle or some other reasons that a 60 e-folds’
inflation has to have happened in our history, it should be single field inflation rather
than multi-field inflation, because the latter has much smaller measure.
5 Generalized Multi-Field Models
In this section, we do the generalizations one step further to consider the action
−3aN−2ȧ2 + a3f
−1ϕ̇i
, (35)
which has the features of the actions in both Section 3 and Section 4.
Using the proper time, the canonical momentum for ϕ takes the form
pi = πia
3, where πi ≡ fϕ̇i, (36)
with the equations of motion
3H2 = ϕ̇iπi − f, π̇i + 3Hπi − fϕi = 0, Ḣ = −
ϕ̇iπi. (37)
and the constraint for the history space variation
ϕ̇i∆πi = fϕi∆ϕi. (38)
We assume that the matrix fϕ̇iϕ̇j has inverse matrix. This should be true when all the
constraints in (35) are solved and ϕi only denotes the dynamical degree of freedom.
We use f ϕ̇iϕ̇j as the inverse matrix of fϕ̇iϕ̇j Then it can be shown that
∆ϕ̇i = f
ϕ̇iϕ̇j (∆πj − fϕ̇jϕk∆ϕk). (39)
In this generalized case,
A ∼ π1 ∆ϕ1 ∧∆π2 ∧∆ϕ2 ∧ . . . ∧∆πn ∧∆ϕn. (40)
using the same technique developed in Section 3 and Section 4, one finds
∂HA =
f ϕ̇1ϕ̇iπ̇i
ϕ̇1Ḣ
ϕ̇iπ̇i
ϕ̇iϕ̇k π̇k
f ϕ̇1ϕ̇jfϕ̇jϕk ϕ̇k
Ḣϕ̇1
ϕ̇iϕ̇jfϕ̇jϕkϕ̇k
A (41)
To see the implications of this equation, let us concentrate on the double field in-
flation models. It is because it seems more difficult to cancel the −3nH
term for
lager n. Lagrangian densities like f = g(ϕ1)ϕ̇
1 + h(ϕ2)ϕ̇
2 are not of special interest
here, because they can be transformed into the case discussed in Section 4 by a field
redefinition. As another example, let us consider the Lagrangian density
f = f
ϕ1, ϕ2,
(ϕ21 + ϕ
, (42)
in this case, the equation (41) takes the form
∂HA = −6A
∂HN, (43)
where
f ′ ≡
(ϕ̇21 + ϕ̇
] . (44)
So a fast rolling f ′ϕ̇21 is required to cancel the exponential expansion of the volume
of the history space.
To see the physical implications for this condition, consider the DBI inflation
model by [13]. The action of the DBI inflation model is given by
SDBI = −
1− f̃ ϕ̇2 − 1
+ V (ϕ)
where the angular motion of ϕi has been ignored, so ϕ̇
2 ≡ ϕ̇2i . Then f
′ = 1/
1− f̃ ϕ̇2 ≡
γ is just the relativistic factor defined in [13]. From the spectral index
ns − 1 = 4
, (46)
We conclude that γ should not be a fast rolling quantity along the whole history of
observable inflation. Moreover, from the equation Ḣ = −γϕ̇2/2, we see again that
γ can not be large for a long time during inflation. So the cancellation of the e−3nN
factor can not be obtained.
6 Conclusion and Discussions
In this paper, we have reviewed the measure problem in cosmology. We calculated the
measure and the probability for inflation in single and multi-field models with gener-
alized Lagrangian density. It is shown that the measure for the single field inflation
and the corresponding generalizations are suppressed by a factor of exp(−3N). While
the n-field and generalized multi-field inflation models has a measure proportional to
exp(−3nN).
This work can be understood in another way. Taking apart the discussion for the
measure and the slow roll condition, other parts of this paper can be thought of as a
proof of the attractor behavior of various kinds of generalized inflation models. On
the one hand, it is a proof that the attractor behavior is very common in inflationary
models. While on the other hand, to take the measure into consideration, we see that
it is far from obvious for an attractor to be a natural solution in cosmology. And it
is just this early time attractor combined with the requirement of slow roll that puts
inflation into a highly unnatural situation.
We did not study explicitly the inflation models with non-minimal coupling to
gravity [17]. But these models do not seem to bring large correction for the sup-
pression factor. It is because through conformal transformation, these non-minimal
coupled models are generally equivalent with the corresponding minimal coupled in-
flation models with the same number or one more inflation fields. Another reason
not to consider these models in this work is that, as the energy scale commonly drops
during inflation, near the end of inflation, the non-minimal coupling effect may not
be so important.
There are also inflation models with extra components or special spacetime prop-
erties. Examples of this kind are inflation with holographic dark energy [18, 19] or
in the non-commutative spacetime [20, 21, 22]. These models do not seem to change
the results much either. Because in the former example, the holographic dark energy
is diluted during inflation so do not seem to cause large corrections near the end
of inflation. In the latter case, although the spectrum for perturbations is greatly
modified in the non-commutative spacetime, the isotropic and homogeneous inflating
background do not change much because it belongs to a lower energy scale. So the
corrections to the probability can not be large.
As a closing remark, we noticed that some non-inflationary models do not share
the small measure problem. One example is the cyclic universe model [23]. Although
the cyclic model is controlled by gravity coupled with a scalar field, it do not have
slow roll behavior backwards in time in the cycle we live. So the key observation that
the exponentially expansion of the phase space volume breaks the slow roll condition
do not apply in the cyclic model. Nevertheless, the number of cycles in the cyclic
universe must be finite [23], so it remains to explain how all the cycles begin in the
first place.
Acknowledgments
This work was supported by grants of NSFC. We thank Bin Chen, Yi-Fu Cai, Chao-
Jun Feng, and Yushu Song for discussion.
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	Introduction
	Single Field Inflation Models
	Generalized Single Field Models
	Multi-Field Inflation Models
	Generalized Multi-Field Models
	Conclusion and Discussions
