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: • 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.
We 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 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 ratiois already quite small. The smaller the t, the smallerthe 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.D23:347-356,1981 ) 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 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
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