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暴涨宇宙论
李淼
中国科学院理论物理研究所
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Cosmic Inflation
Miao Li
Institute of Theoretical Physics, Academia Sinica
null 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.
……Part I InflationnullStill, 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. nullTraditionally, three problems associated to the
initial conditions are most often quoted:
The first problem is called the horizon problem.
The second problem is the flatness problem.
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.
nullWe now describe the three problems before
presenting the solution offered by inflation.
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 bynullFor 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?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?null(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
nullAgain, 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?null
null(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 , we have
Usually, it is much greater than 1.
nullThe 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 nullSuppose 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 neednullInflation 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 is usually called the number of
e-foldings.
nullUse the previous data, we need to achieve
If the initial is a number of order 1, again we need
nullThe 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 ScenarioThe Old Inflation ScenarioAlan 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
nullIn 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.
nullTunnelling, 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. null[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.nullIn 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.
nullThe 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
nullThe 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
nullFor 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
nullThe 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
nullIn 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.
nullThis 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.
nullOne can solve the reheating equation in a more rigorous way.
Replacing the average of by , then the equation of
motion is
with solution
where is the scale factor when the coherent oscillations commence.nullThus, 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
nullThe parameter n is chosen such that the solution of the scale factor
is
For this solution to be inflation, . The scalar field can be
solved exactly too:
This potential does not have a dip, so inflation does not end. To end inflation, one has to add a term by hand.nullThe slow roll parameter s are
Let be the field value at the ending of inflation, the number of
e-folds between and is
null(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.null(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:
nullWhen the value of vanishes, the most contribution to V
is from this field so inflaton rolls slowly and inflation may last
for enough time. nullPrimordial 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.nullIn 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.
nullWe 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
nullThe definition of the power spectrum is
To compute we need to compute since
Now, satisfies
nullSo, ignoring the potential term,
Put things together, we have
COBE observed the spectrum at , and the result is
We deduce
nullWe 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,
nullBy 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
nullUsing the relation between change in k and change in time, and
Slow-roll motion of the scalar,
Using
We have
nullThus, 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 ResultsSome Experimental ResultsnullnullBeyond the slow-rollBeyond the slow-rollOne does not have to addict to the slow-roll approximation.
Define and the conformal time , u satisfies
with
nullOne can compute exactly:
Where
the derivative is taken with respect to t, not to the conformal
time.
nullWe take u as a quantum field, with an action
Mode expanding u:
Canonical quantization yields
nullAs , 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
nullA few e-folds after exiting the horizon,
the solution asymptotes
We finally have
nullIn the following, we enumerate a few examples
Power-law inflation
so, , a red spectrum.
null(2) Natural inflation
is axion, an angle scalar. When
where .nullMore generally,
b is the Euler-Mascheroni constant:
nullWe 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.
General theory of perturbationsnullStarting 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.nullThe 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
nullThese components have an action
where
The equation of motion is
nullAs before, we find the power spectrum of tensor modes
where
For power-law inflation
nullFor natural inflation
More generally,
where Density PerturbationDensity PerturbationDensity 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. nullDefine 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
nullWe 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? nullThe quantity is conserved, and for
a universe dominated by a component with equation of state
parameter w, . For radiation ,
so thus
for a total density contrast.nullFor adiabatic perturbation, which is what we are mostly interested
in, are all equal for all physical quantities. Let x denote
a species and its density contrast, , then
since, .
The above is a theory for linear perturbation. Study of nonlinear perturbation requires numerical simulation.Anisotropy of CMBAnisotropy of CMBPhotons 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:
nullIt 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.
nullThe temperature fluctuation is related to primordial perturbation
by a transfer function, since various effects such as the Sachs-
Wolfe effect.
Let
where
can be computed using .
nullSince
we have
Due to rotational invariance, the temperature multi-pole
Is related to curvature perturbation through
is the transfer function to be computed.
nullThe correlation of multi-pole is called the angular power
spectrum
Using the transfer relation we find
Define
then
nullPolarization.
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 param
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