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

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 Inflation

Still, the standard big bang model does notexplain 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 notanswer. 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, againfor a matter-dominated universe, is

The ratio of the two is

When the light last scattered, z~1000, the above ratiois 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 , we have

Usually, it is much greater than 1.

The solution of the inflationary universe.We consider the simple, exponentially inflated universe.Assume that before the hot big bang, there was sucha period: . If the starting time is quiteearly, 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 anon-vanishing curvature (of course more generically this region can be more complex initially), with a which isnot 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.

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 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

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 ina 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 oscillateand 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 conditionbecomes

We usually denote the quantity on the LHS by , the firstSlow 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 couplingto 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 potentialdominated 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, sothe 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 by , then the equation of motion is

with solution

where is the scale factor when the coherent oscillations commence.

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 factoris

For this solution to be inflation, . The scalar field can besolved 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.

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 inthis 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. Whenreach 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 Vis from this field so inflaton rolls slowly and inflation may lastfor 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 becomesobservable, 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-numberin 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, weSay that the scale entered the horizon, in particular, use the currentHubble 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 we need to compute since

Now, satisfies

So, ignoring the potential term,

Put things together, we have

COBE observed the spectrum at , and the result is

We deduce

We shall see later that after quantum fluctuation crosses out thehorizon, it becomes frozen thus classical, the cross-out conditionis

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, andSlow-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-rollOne does not have to addict to the slow-roll approximation.

Define and the conformal time , u satisfies

with

One can compute exactly:

Where

the derivative is taken with respect to t, not to the conformaltime.

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:

We have by far ignored the fact that the scalar perturbation isactually perturbation associated with the curvature perturbation.Only in this case, structures such as the CMB anisotropy and thelarge 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 perturbations

Starting with the perturbed metric

where is traceless. For scalar perturbation, andare total derivative, and only D is relevant, in particular, fora given momentum, the scalar curvature of the spatial sliceis 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, weFurther 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, theprimordial perturbation is seeds for density perturbation. Onecompute linear perturbation when the amplitude is small.

As the fluctuation evolves, the amplitude becomes larger andlarger and eventually enters the nonlinear region. Here we areconcerned 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 duringInflation 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 toscalar potential. The question is, how this potential evolves asIt enters the horizon?

The quantity is conserved, and fora universe dominated by a component with equation of state

parameter w, . For radiation ,

so 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 its density contrast, , then

since, .

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 temperatureof 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 perturbationby a transfer function, since various effects such as the Sachs-Wolfe effect.

Let

where

can be computed using .

Since

we have

Due to rotational invariance, the temperature multi-poleIs 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 thedirection 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 . Let the transfer function be , we have

Define

Then

Acoustic peaks.

For scales smaller than , a range of physical effectstake place before the last scattering, in particular, peaks in theangular power spectrum show up due to the standing-waveoscillation 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 Hoand 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, thatthis scenario can be used to understand some unusual featureof the WMAP results.

Namely, the large running of the spectral index can be naturallyexplained as an effect of the non-commutative inflation.

On the other hand, anomalously suppressed low multi-poles cannot 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 stringtheory.

The first observation of this effect was made by Yoneya inthe end of eighties.

T Yoneya, Mod.Phys.Lett.A4:1587, 1989.

The physics of space-time uncertainty in string theory is verysimple. 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 shortdistances, more and more energy modes set in. In string theory,however, energy is also deposited on extensive objects such asstrings, letting string expands with energy

Combining the two relations, we have

The above relation can be obtained by a direct string stateanalysis. 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 and

thus

The space-time uncertainty relation can also be obtained byanalyzing 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 increasestring coupling constant in type IIA string theory, the lightdegrees 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-timeuncertainty is a generic feature in string theory, we have no Mathematical formalism to realize it explicitly in string theoryyet.

The Heisenberg uncertainty relation can be regardedas 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 simple-minded commutators can not be the real story. Nevertheless,such a model may capture the some physical ingredient of stringcosmology.

The Brandenberger-Ho non-commutative inflation model is justsuch a model.

One way is realize the commutation relation is to

use the star-product:

For instance,

When apply space-time noncommutativity to cosmology, it isnot convenient to use the usual co-moving time in the FRWmetric. 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 gainsome 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 withoutspacetime 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 withthe 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 thesame, 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 determinedby the crossing-out horizon condition. Compared with the commutative inflation, two ingredientsset in and modify the power spectrum. First, the definitionOf 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 constructa model which may have observable effects to be discovered infuture. However, they didn’t foresee that WMAP may alreadybe an experiment with possible signature of some effects.

In 2003, I and my student Qing Guo Huang learned that WMAPFirst 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 werealized that there are just too many parameters in this model.

I was aware of BH’s work, and was clear of the fact that thereis only one parameter, the string scale, in addition to whateverparameters in the inflaton potential. Thus, a simple non-commutative 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 , time as in

is

so

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 thisassumption, 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 dependsonly 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 possiblesince 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 importantIn 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 assumethe 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 is the modifiedConformal time.

In discussing the running of the spectral index, we need anotherparameter

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 and for a fixed s.

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 was later generalizedto the second order, done in the work:

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 experimentaldata.

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.