暴涨宇宙论
李淼
中国科学院理论物理研究所
Cosmic Inflation
Miao Li
Institute of Theoretical Physics, Academia Sinica
Part I
Inflation
The standard cosmological model, the big bang
model, has been met with numerous successes,
including:
(1) Prediction of cosmic microwave background.
(2) Prediction of the abundance of light elements
such helium and deuterium.
(3) Of course, explanation of Hubble’s law.
……
Still, the standard big bang model does not
explain everything we observe. For example,
how the structure we see in the sky formed? Why
the universe is as old as about 14 billion years?
etc.
We need a theory of initial conditions to answer
questions that the big bang model does not
answer. Inflation was invented to partially answer
these questions.
Traditionally, three problems associated to the
initial conditions are most often quoted:
(a) The first problem is called the horizon problem.
(b) The second problem is the flatness problem.
(c) Unwanted relics.
Although to many cosmologists, the most practical
use of inflation scenario is the generation of
primordial perturbations, it is these three
“philosophical” problems that motivated Alan
Guth to invent inflation in 1981.
We now describe the three problems before
presenting the solution offered by inflation.
(a) The horizon problem.
Start with the Friedmann-Robert-Walker metric
The most distant places in early universe at time t we
can observe today is given by
For a matter-dominated universe,
if
, then
However, the particle horizon at that time, again
for a matter-dominated universe, is
The ratio of the two is
When the light last scattered, z~1000, the above ratio
is already quite small. The smaller the t, the smaller
the ratio, this is the horizon problem: why the
universe is homogeneous in a much larger scale
compared to the particle horizon?
(b) The flatness problem.
For a universe with a spatial curvature, characterized by a
number
, one of the Friedmann equations reads
where H is the Hubble “constant”
. The left hand
Side is usually denoted called the critical energy density
,
The ratio
is usually called
, thus, we have
Again, for simplicity we consider a matter-dominated
universe, the ratio of the flatness at an early time to that at
the present time is
Since the flatness is bounded at the present (in fact it is quite
close to zero) , so in at a very early time, the universe was
very flat. How does the universe choose a very flat initial
condition?
(c) The problem of relics.
In a unified theory, there are always various heavy
particles with tiny annihilation cross-section. Once
they are generated due to equilibrium in early
universe, they can “over-close” the universe, since
For a cross section
Usually, it is much greater than 1.
, we have
The solution of the inflationary universe.
We consider the simple, exponentially inflated universe.
Assume that before the hot big bang, there was such
a period:
. If the starting time is quite
early, then the particle horizon is almost constant,
The same as the Hubble horizon size
Let
be the end time of inflation , the physical
size of the particle horizon is
Suppose after inflation, the universe evolves according to
a power-law, (this is not true, but won’t effect our basic
Picture) then the physical size of the observable horizon is
The ratio of the particle horizon to the observed
horizon is
If
and
, choosing
, so
to solve the horizon problem, we need
Inflation solves the flatness in much the same way, for
example, one could assume that the observed region starts
from a maximally symmetric spatial cross section with a
non-vanishing curvature (of course more generically this
region can be more complex initially), with a
which is
not equal to one at all, we use subscript i to denote the
onset time of inflation, then
where
e-foldings.
is usually called the number of
Use the previous data, we need to achieve
If the initial
is a number of order 1, again we need
The problem of redundant relics can be easily solved too.
For a heavy thus non-relativistic particle, the energy
density scales as
, so after inflation
this can be a very tiny number. This is why we often say
that relics are inflated away during inflation era.
The Old Inflation Scenario
Alan Guth proposed the so-called old inflation model in 1981
(Alan H. Guth, Phys.Rev.D23:347-356,1981 )
There is a scalar field with a potential of the following type
In the beginning of inflation, the scalar field started from the origin
in the picture, where the potential
has a positive value.
Suppose that the kinetic energy of the scalar field can be ignored,
then according to the Friedmann equation, we have
And the reduced Planck mass is
However, this kind of inflation can not proceed forever, since
quantum tunnelling will occur spontaneously.
Tunnelling, however, is a completely random process. The
problem with old inflation (which Guth acknowledged in his
original paper) was that some parts of the universe would
randomly tunnel to a lower energy state while others, blocked
by the potential barrier, would continue to sit at the higher
one. The fabric of spacetime would expand and these energy
states would become pre-galactic clumps of matter, but the
matter/energy density of such a universe would be much,
much less homogenous than the one we observe today.
[For his pioneering work on inflation, Alan Guth was awarded
the 2001 Benjamin Franklin Medal, and Andrei Linde, Alan
Guth, and Paul Steinhardt were all awarded the 2002 Dirac
Medal in theoretical physics. ]
To overcome this difficulty of the old inflation model, Linde,
Albrecht and Steinhardt proposed the new inflation model in
which there is no first order phase transition. The scalar slowly
rolls down its potential during inflation.
In this model, the inflaton (the scalar) has a very flat potential in
a large range, and at a given time its value is classical and there is
no thermal excitation (thus the temperature is 0).
In the end of inflation, we must generate the hot environment of
the standard big bang scenario, so the inflaton ought to decay into
relativistic particles. This is achieved by introducing a dip in the
potential. When the inflaton rolls into the dip, it starts to oscillate
and the coherent oscillation generates all sorts of particles. This is
called reheating.
The the classic inflaton satisfies equation of motion for a
spatially homogeneous field
where a dot denote derivative with respect to the co-moving
time, and prime denotes derivative with respect to the scalar.
Since during inflation, our universe expands, the Hubble
constant is positive, thus the expansion drags the inflaton against
its rolling down. For a sufficiently flat potential, we can ignore
the second derivative in the above equation, so
The Friedmann equation
One of the most important quantities is the number of e-folds
Before the end of inflation
where we used the equation of motion of inflaton and the
Friedmann equation. The above quantity is often denoted by
For the simplified equation of motion of the inflaton and the
simplified Friedmann equation to be valid, we require
,
With the help of the equations of motion, the first condition
becomes
We usually denote the quantity on the LHS by
, the first
Slow roll parameter. The first slow-roll condition is then
The second condition can be transformed into, combined with
The first slow-roll condition, the second slow-roll condition:
where
In terms of the slow-roll parameter, we have
In reality, as the inflaton rolls down its potential, due to coupling
to other particles, the motion of the inflaton brings about
generation of these particles, and this has back-reaction on the
motion of the scalar, and can be summarized in a term in the
equation of motion
where
is the decay width, for instance, for a Yakawa coupling
to a light fermion with strength g,
and
is the effective mass of the inflaton.
This damping term is operating in the short reheating period to
generate relativistic particles. For illustration purpose, let the
decay width be larger than the Hubble constant and the potential
dominated by a quadratic term, then after entering the reheating
phase,
, and
has a imaginary part inversely
proportional to . If the dip of the potential is deep enough, so
the effective mass is large, the duration of reheating period
can be very short.
One can solve the reheating equation in a more rigorous way.
Replacing the average of
motion is
by
, then the equation of
with solution
where
commence.
is the scale factor when the coherent oscillations
Thus, a good inflaton potential must be fine-tuned: it must have a
flat region for the inflaton to slowly roll down to generate enough
number of e-folds, on the other hand, it must have a deep enough
dip for inflation to quickly end to reheat the universe.
In the following, we give a few examples of often discussed
models.
(1) Power-law inflation.
The potential is
The parameter n is chosen such that the solution of the scale factor
is
For this solution to be inflation,
solved exactly too:
. The scalar field can be
This potential does not have a dip, so inflation does not end. To
end inflation, one has to add a term by hand.
The slow roll parameter s are
Let
be the field value at the ending of inflation, the number of
e-folds between
and
is
(2) Monomial potential.
With the slow roll parameters
For a reasonable
, the slow roll conditions require
That is, we are usually in a super-Planckian regime.
(3) Hybrid Inflation.
In addition to inflaton
, there is another scalar field
in
this model. The coupling between these two scalars makes
have a
dependent mass. As
starts to roll,
has a
positive mass squared, so its expectation value is zero. When
reach a critical value, the mass of
becomes vanishing and
eventually develops a negative mass squared, so its vacuum
expectation non-vanishing, and the potential of inflaton
becomes steep:
When the value of
vanishes, the most contribution to V
is from this field so inflaton rolls slowly and inflation may last
for enough time.
Primordial Perturbations
The most important role that inflaton played is not only
driving inflation, but also generating primordial curvature
perturbations. These perturbations are quantum fluctuations
of inflaton, stretched beyond the Hubble horizon then frozen
up, re-entered horizon at a later time and eventually becomes
observable, since curvatures perturbations are seeds of
structure formation, such as galaxies, clusters of galaxies. In
addition, the cosmic micro-wave background is also coupled
to curvature, thus anisotropy in CMB is due to primordial
perturbations too.
In considerations of fluctuations, one usually uses the wave-number
in the co-moving coordinates k, the physical size at a given time is
And the ratio of the Hubble scale to this perturbation scale is
This is a important quantity, when it is larger than 1, we say that the
scale is outside of the horizon, and when it is smaller than 1, we
Say that the scale entered the horizon, in particular, use the current
Hubble scale and the current scale factor, this quantity characterizes
whether we can observe this scale.
We discuss how the primordial perturbations are generated, and
only later show how these can be seeds of density perturbations.
For simplicity, we consider a single scalar case. There are two
types of perturbation, one is a combination of the scalar
curvature and the scalar field, another is tensor perturbation.
Scalar perturbation is what has been observed.
Suppose the fluctuation of the inflaton is , the curvature
perturbation (whose derivation is complicated) is
The definition of the power spectrum is
To compute
Now,
we need to compute
satisfies
since
So, ignoring the potential term,
Put things together, we have
COBE observed the spectrum at
We deduce
, and the result is
We shall see later that after quantum fluctuation crosses out the
horizon, it becomes frozen thus classical, the cross-out condition
is
It simply says that the physical size of the perturbation becomes
the same as the Hubble scale
. This relation can be
used to convert a function of k into a function of time t or vice
versa,
By the definition of number of e-folds
Let
be the scale leaving horizon when inflation ends, then
One of the quantities that CMB experiments directly measure is
the spectral index
whose definition is
Using the relation between change in k and change in time, and
Slow-roll motion of the scalar,
Using
We have
Thus, the scalar spectrum deviation from a scaling invariant
spectrum by a small quantity in slow-roll inflation.
It is also interesting to define the running of the spectral index:
where
Some Experimental Results
Beyond the slow-roll
One does not have to addict to the slow-roll approximation.
Define
with
and the conformal time
, u satisfies
One can compute
exactly:
Where
the derivative is taken with respect to t, not to the conformal
time.
We take u as a quantum field, with an action
Mode expanding u:
Canonical quantization yields
As
, a few e-folds before exiting of the mode, we
can ignore the curvature of space-time, thus the solution
Let
We have the solution
A few e-folds after exiting the horizon,
the solution asymptotes
We finally have
In the following, we enumerate a few examples
(1) Power-law inflation
so,
, a red spectrum.
(2) Natural inflation
is axion, an angle scalar. When
where
.
More generally,
b is the Euler-Mascheroni constant:
General theory of perturbations
We have by far ignored the fact that the scalar perturbation is
actually perturbation associated with the curvature perturbation.
Only in this case, structures such as the CMB anisotropy and the
large scale structure are seeded by scalar perturbation.
During inflation, density fluctuation is not a gauge invariant.
However, one can find a combination of the metric perturbation
and scalar perturbation to form an invariant.
Starting with the perturbed metric
where
is traceless. For scalar perturbation,
and
are total derivative, and only D is relevant, in particular, for
a given momentum, the scalar curvature of the spatial slice
is proportional to D. When
is present, the combination
is invariant under change of time.
The tensor perturbation is given by
There are 6 components in this perturbation, due to the
traceless condition, 5 remain. For a given momentum, we
Further impose 3 on-shell conditions, finally only 2
components are left. Explicitly
with
These components have an action
where
The equation of motion is
As before, we find the power spectrum of tensor modes
where
For power-law inflation
For natural inflation
More generally,
where
Density Perturbation
Density can be viewed as caused by the primordial
Perturbation through Einstein equations. In other words, the
primordial perturbation is seeds for density perturbation. One
compute linear perturbation when the amplitude
is small.
As the fluctuation evolves, the amplitude becomes larger and
larger and eventually enters the nonlinear region. Here we are
concerned only with linear perturbation.
Define the scalar potential that appear in the Poisson equation
Let
be the unperturbed density, the perturbation of the
Scalar potential is defined by
then
or
We have seen that cosmic perturbations were generated during
Inflation and exited horizon. After inflation ends, the expansion
ff universe decelerate, thus
decreases. For a radiation
dominated universe,
so
. Eventually,
For some k,
becomes smaller than 1, thus the
Perturbation scale enters horizon.
We already have a formula relating the density perturbation to
scalar potential. The question is, how this potential evolves as
It enters the horizon?
The quantity
is conserved, and for
a universe dominated by a component with equation of state
parameter w,
so
. For radiation
thus
for a total density contrast.
,
For adiabatic perturbation, which is what we are mostly interested
in,
are all equal for all physical quantities. Let x denote
a species and
since,
its density contrast,
, then
.
The above is a theory for linear perturbation. Study of nonlinear
perturbation requires numerical simulation.
Anisotropy of CMB
Photons in CMB we observe have rarely scattered since a
Time between the epoch of decoupling and the epoch of
reionization. So we can say that all CMB photons we see
today originate the surface called the last scattering surface.
The radius of this last scattering surface as measured today
is the size of the particle horizon:
It turns out that the anisotropy of CMB, to the first order
approximation, can be described by fluctuation of temperature
of black-body radiation, namely
where
is the unit vector pointing to the direction of the
observation. Perform the expansion
where
are multi-poles.
The temperature fluctuation is related to primordial perturbation
by a transfer function, since various effects such as the SachsWolfe effect.
Let
where
can be computed using
.
Since
we have
Due to rotational invariance, the temperature multi-pole
Is related to curvature perturbation through
is the transfer function to be computed.
The correlation of multi-pole is called the angular power
spectrum
Using the transfer relation we find
Define
then
Polarization.
Choosing x and y as Cartesian coordinates perpendicular to the
direction of observation, the polarization of CMB radiation is
determined by
and
. One of the Stokes parameter is
in contrast to the total intensity
The temperature fluctuation is related to intensity fluctuation
While the polarization “fluctuation” is defined by
This is similar to
we have
Define
. Let the transfer function be
,
Then
Acoustic peaks.
For scales smaller than
, a range of physical effects
take place before the last scattering, in particular, peaks in the
angular power spectrum show up due to the standing-wave
oscillation of the baryon density, these are called acoustic peaks.
The angular power spectra must be computed numerically, using
computer program such as CMBFAST.
In the following we show some pictures taken from the WMAP
experiment.
Total power spectrum
Cross power spectrum
Other experiments prior WMAP
The improvement of WMAP over COBE
Part II Non-commutative Inflation
The non-commutative inflation scenario was invented by Ho
and Brandenberger:
R Brandenberger , P-M Ho, Phys.Rev.D66:023517,2002,
hep-th/0203119.
Their initial motivation is to replace inflation itself by effects of
non-commutative space-time, this goal is not achieved.
Later, it turned out that, due to observation of Huang and Li, that
this scenario can be used to understand some unusual feature
of the WMAP results.
Namely, the large running of the spectral index can be naturally
explained as an effect of the non-commutative inflation.
On the other hand, anomalously suppressed low multi-poles can
not be explained.
In any case, study of non-commutative is a first step towards
unraveling string effects in early universe.
Space-time Noncommutativity in String
Theory
Space-time noncommutativity is a universal feature in string
theory.
The first observation of this effect was made by Yoneya in
the end of eighties.
T Yoneya, Mod.Phys.Lett.A4:1587, 1989.
The physics of space-time uncertainty in string theory is very
simple. Let us start with the Heisenberg uncertainty relation
in the natural units. This relation is the basic reason why a
QFT is normally UV divergent, since as we probe very short
distances, more and more energy modes set in. In string theory,
however, energy is also deposited on extensive objects such as
strings, letting string expands with energy
Combining the two relations, we have
The above relation can be obtained by a direct string state
analysis. For a string in a state of oscillating mode
This state can be achieved by scattering the string with a
state carrying momentum p. The spatial extension of this
state is
. Averaging over n, we have
Consider a parallelogram on the string world-sheet, with
Physical size A in one direction, and B in another direction.
The amplitude is proportional to
where
and a and b are world-sheet lengths. Let
thus
and
The space-time uncertainty relation can also be obtained by
analyzing the scattering amplitude of 2 strings.
where
is the scattering angle:
The space-time uncertainty relation also generalizes to
non-perturbative string theory. For instance, as we increase
string coupling constant in type IIA string theory, the light
degrees of freedom are no-longer strings, but D0-branes.
The interaction between these objects can be analyzed by
using matrix mechanics, which can be derived as the low
energy theory of open strings stretched between D0-branes,
thus, it is of no surprise that the space-time uncertainty still
applies to this case.
Although we have enough evidence to believe that space-time
uncertainty is a generic feature in string theory, we have no
Mathematical formalism to realize it explicitly in string theory
yet.
The Heisenberg uncertainty relation
can be regarded
as a consequence of the mathematical statement
In string theory, we can not simply write down
This relation is neither covariant, nor it is meaningful when
applied to any degrees of freedom.
Thus, we must keep in mind that any model utilizing a simpleminded commutators can not be the real story. Nevertheless,
such a model may capture the some physical ingredient of string
cosmology.
The Brandenberger-Ho non-commutative inflation model is just
such a model.
One way is realize the commutation relation
use the star-product:
is to
For instance,
When apply space-time noncommutativity to cosmology, it is
not convenient to use the usual co-moving time in the FRW
metric. BH introduced another time
in which the metric
reads
This is because the space-time uncertainty is valid for
physical space and time, and can be translated into
One may apply this idea to a 1+1 dimensional universe to gain
some experience. In such a universe, in addition to time t,
there is only one spatial dimension x. The Einstein action in
terms of FRW metric is modified into
In terms of Fourier modes
(The meaning of
will be explained later.)
The action now reads
where
Define
Then,
with
The action written in this form is identical to the one without
spacetime noncommutativity.
We are ready to generalize the above construction to a d+1
space-time. In order to preserve translational invariance and
rotational invariance, we should not work with Cartesian
coordinates, since commutators of these coordinates with
time break these symmetries. Instead, we work directly with
the radial coordinate in the momentum space and generalize
the above action directly:
where
The scalar satisfies an equation which is identical to the old one
except that z is replaced by
and
replaced by
.
Now let us go back to explain the meaning of the upper
momentum cut-off
.
The effective wave-length of the mode k is
Similarly, the effective uncertainty in the physical time is the
same, due to the space-time uncertainty, we must have
, this leads to the cut-off
.
We note that since the wave-length cannot be taken as
uncertainty, this upper bound must be taken with a grain of salt.
In addition to the UV cut-off, there is also an IR cut-off, we
shall mention it shortly.
We have written down the action for the inflaton fluctuation.
The scalar perturbation is a combination of this fluctuation
and the spatial curvature fluctuatio, and shall satisfy the same
equation. In short, we have for
Thus,
The cross-out horizon condition is
Since
also depends on k, it is possible that the above
condition, unlike in the commutative case, can never be met,
namely, we always have
we say that very long wave-lengths are generated
outside horizon.
Now, one solve the wave equation and quantize u, as usual,
the power spectrum formula is
where the conformal time as a function of k is determined
by the crossing-out horizon condition.
Compared with the commutative inflation, two ingredients
set in and modify the power spectrum. First, the definition
Of the conformal time is changed, as in
, second,
changed too, thus the
crossing-out horizon condition modified.
The noncommutative inflation model was constructed by Ho
and Brandenberger in 2002, their purpose is simply to construct
a model which may have observable effects to be discovered in
future. However, they didn’t foresee that WMAP may already
be an experiment with possible signature of some effects.
In 2003, I and my student Qing Guo Huang learned that WMAP
First year results indicate a possible running power spectral
index, soon after WMAP papers appearing in February, we
started thinking about explaining the running of power index.
We wanted to examine the brane-world cosmology, then we
realized that there are just too many parameters in this model.
I was aware of BH’s work, and was clear of the fact that there
is only one parameter, the string scale, in addition to whatever
parameters in the inflaton potential. Thus, a simple noncommutative inflaton model must have enough predictive
power.
The work I did with Huang was summarized in papers:
Q. G Huang and M. Li, JHEP 0306:014, 2003, hep-th/0304203
JCAP 0311:001,2003,astro-ph/0308458;
astro-ph/0311378.
Let us apply this model to the power-law inflation. We start
with
is
so
, time
as in
To simplify the picture, we define a scale l,
and
. We are ready to solve the crossing-out
condition. Let
and the crossing-out time with
:
The problem cannot be solved exactly. In order to solve it
approximately, we assume that
where
is the uncertainty in time for mode k. With this
assumption, we expand
and
Thus, the approximate crossing-out time, to the first order, is
And the power spectrum
By the definition of the spectral index
We find
and by the definition of its running
where x denotes
Now, we want to fit the following data of WMAP
Using the result at
, we fix two parameters
which in turned are used to “predict”
We have seen that the shape of the power spectrum depends
only on the two parameters already determined. There are
actually three parameters in our model: in addition to n and l,
there is the string scale
. The two scales determine
,
corresponding to a macroscopic scale. This is made possible
since n is large.
In order to determine all the parameters, we need to use the
normalization of the power spectrum. The full formula is
Since
we fix
The IR modes created outside the horizon may be important
In explaining the suppressed low multipoles, we shall not
discuss them here.
Finally, we present a fit to the angular power spectrum
More generally, we can study noncommutative inflation by
assuming, just like the slow-roll conditions, another small
parameter related to string scale. Indeed, in order not to spoil
too much the standard slow-roll inflation results, we need to
assume that the Hubble scale is larger than the string scale:
Unlike the other two slow-roll parameters, this parameter
depends on k explicitly, the reason is that we need to assume
the perturbation mode to have small enough energy. Expansion
in
is a double expansion in small string scale and low energy.
Together with the usual definition of slow-roll parameters
we have
and
where
is the usual conformal time, and
Conformal time.
is the modified
In discussing the running of the spectral index, we need another
parameter
We now solve the equation
to obtain
for
.
The power spectrum
to compute the spectral index, we the relation between time
and k, obtained using the crossing-out horizon condition
and
The final result is
To see whether a noncommutative inflation model can fit
the data, we plot a curve in the plane of
fixed s.
and
for a
Let us apply this analysis to the model with potential
In terms of the number of e-folds
we have
For
, we take N=50, p=2 and find
so it is impossible to explain the observed running, this is a
general result for commutative inflation.
For noncommutative inflation, take
as an
example, we plot the fitting in the plane of two parameters
p and
.
The blue line corresponds to
and
The first order calculation in terms of
to the second order, done in the work:
was later generalized
H. S. Kim, G. S. Lee and Y. S. Myung, hep-th/0402018;
hep-th/0402198.
The result is sufficiently complicated, we cite only one formula
However, it is not attempted to compare with experimental
data.
Part III Dark Energy
The discovery of dark energy can be viewed as the most exciting
progress made in the last two decade in cosmology.
The first ground-breaking results appeared in:
Supernova Search Team (Adam G. Riess et al.)
Astron.J.116:1009-1038,1998, astro-ph/9805201.
Supernova Cosmology Project (S. Perlmutter et al.),
Astrophys.J.517:565-586,1999, astro-ph/9812133.
Taken from Riess et al.
Taken from Riess et al.