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 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, 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 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 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 Scenario Alan Guth proposed the so-called old inflation model in 1981 (Alan H. Guth, Phys.Rev.D23: ,1981 )Alan H. Guth 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 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 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.

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 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 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 and the conformal time, u satisfies with

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:

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. General theory of perturbations

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,. 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 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 Sachs- Wolfe 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. Let the transfer function be, we have Define

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 BrandenbergerR Brandenberger, P-M Ho, Phys.Rev.D66:023517,2002,P-M Ho hep-th/ 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.T Yoneya

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 and thus

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 simple- minded 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 is to use the star-product:

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 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/ JCAP 0311:001,2003,astro-ph/ ; astro-ph/

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 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 is the modified Conformal time.

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 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 generalized to the second order, done in the work: H. S. Kim, G. S. Lee and Y. S. Myung, hep-th/ ; hep-th/ 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: ,1998, astro-ph/ Supernova Cosmology Project (S. Perlmutter et al.), Astrophys.J.517: ,1999, astro-ph/

Taken from Riess et al.