We have seen that inflation is a powerful and predictive theory of the physics of the very early universe. Unexplained properties of a standard FRW cosmology, namely the flatness and homogeneity of the universe, are natural outcomes of an inflationary expansion. Furthermore, inflation provides a mechanism for generating the tiny primordial density fluctuations that seed the later formation of structure in the universe. Inflation makes definite predictions, which can be tested by precision observation of fluctuations in the CMB, a program that is already well underway.
In this section, we will move beyond looking at inflation as a subject of experimental test and discuss some intriguing new ideas that indicate that inflation might be useful as a tool to illuminate physics at extremely high energies, possibly up to the point where effects from quantum gravity become relevant. This idea is based on a simple observation about scales in the universe. As we discussed in Sec. 2, quantum field theory extended to infinitely high energy scales gives nonsensical (i.e., divergent) results. We therefore expect the theory to break down at high energy, or equivalently at very short lengths. We can estimate the length scale at which quantum mechanical effects from gravity become important by simple dimensional analysis. We define the Planck length _{Pl} by an appropriate combination of fundamental constants as
(133) |
For processes probing length scales shorter than _{Pl}, such as quantum modes with wavelengths < _{Pl}, we expect some sort of new physics to be important. There are a number of ideas for what that new physics might be, for example string theory or noncommutative geometry or discrete spacetime, but physics at the Planck scale is currently not well understood. It is unlikely that particle accelerators will provide insight into such high energy scales, since quantum modes with wavelengths less than _{Pl} will be characterized by energies of order 10^{19} GeV or so, and current particle accelerators operate at energies around 10^{3} GeV. ^{(12)} However, we note an interesting fact, namely that the ratio between the current horizon size of the universe and the Planck length is about
(134) |
or, on a log scale,
(135) |
This is a big number, but we recall our earlier discussion of the flatness and horizon problems and note that inflation, in order to adequately explain the flatness and homogeneity of the universe, requires the scale factor to increase by at least a factor of e^{55}. Typical models of inflation predict much more expansion, e^{1000} or more. We remember that the wavelength of quantum modes created during the inflationary expansion, such as those responsible for density and gravitational-wave fluctuations, have wavelengths which redshift proportional to the scale factor, so that so that the wavelength _{i} of a mode at early times can be given in terms of its wavelength _{0} today by
(136) |
This means that if inflation lasts for more than about N ~ 140 e-folds, fluctuations of order the size of the universe today were smaller than the Planck length during inflation! This suggests the possibility that Plank-scale physics might have been important for the generation of quantum modes in inflation. The effects of such physics might be imprinted in the pattern of cosmological fluctuations we see in the CMB and large-scale structure today. In what follows, we will look at the generation of quantum fluctuations in inflation in detail, and estimate how large the effect of quantum gravity might be on the primordial power spectrum.
In Sec. 2.4 we saw that the state space for a quantum field theory was a set of states (n(k_{1}), ..., n(k_{i})) representing the number of particles with momenta k_{1},...,k_{i}. The creation and annihilation operators ^{†}_{k} and _{k} act on these states by adding or subtracting a particle from the state:
(137) |
The ground state, or vacuum state of the space is just the zero particle state,
(138) |
Note in particular that the vacuum state |0> is not equivalent to zero. The vacuum is not nothing:
(139) |
To construct a quantum field, we look at the familiar classical wave equation for a scalar field,
(140) |
To solve this equation, we decompose into Fourier modes u_{k},
(141) |
where the mode functions u_{k}(t) satisfy the ordinary differential equations
(142) |
This is a classical wave equation with a classical solution, and the Fourier coefficients a_{k} are just complex numbers. The solution for the mode function is
(143) |
where _{k} satisfies the dispersion relation
(144) |
To turn this into a quantum field, we identify the Fourier coefficients with creation and annihilation operators
(145) |
and enforce the commutation relations
(146) |
This is the standard quantization of a scalar field in Minkowski space, which should be familiar. But what probably isn't familiar is that this solution has an interesting symmetry. Suppose we define a new mode function u_{k} which is a rotation of the solution (143):
(147) |
This is also a perfectly valid solution to the original wave equation (140), since it is just a superposition of the Fourier modes. But we can then re-write the quantum field in terms of our original Fourier modes and new operators _{k} and ^{†}_{k} and the original Fourier modes e^{ik . x} as:
(148) |
where the new operators _{k} are given in terms of the old operators _{k} by
(149) |
This is completely equivalent to our original solution (141) as long as the new operators satisfy the same commutation relation as the original operators,
(150) |
This can be shown to place a condition on the coefficients A and B,
(151) |
Otherwise, we are free to choose A and B as we please.
This is just a standard property of linear differential equations: any linear combination of solutions is itself a solution. But what does it mean physically? In one case, we have an annihilation operator _{k} which gives zero when acting on a particular state which we call the vacuum state:
(152) |
Similarly, our rotated operator _{k} gives zero when acting on some state
(153) |
The point is that the two "vacuum" states are not the same
(154) |
From this point of view, we can define any state we wish to be the "vacuum" and build a completely consistent quantum field theory based on this assumption. From another equally valid point of view this state will contain particles. How do we tell which is the physical vacuum state? To define the real vacuum, we have to consider the spacetime the field is living in. For example, in regular special relativistic quantum field theory, the "true" vacuum is the zero-particle state as seen by an inertial observer. Another more formal way to state this is that we require the vacuum to be Lorentz symmetric. This fixes our choice of vacuum |0> and defines unambiguously our set of creation and annihilation operators and ^{†}. A consequence of this is that an accelerated observer in the Minkowski vacuum will think that the space is full of particles, a phenomenon known as the Unruh effect [43]. The zero-particle state for an accelerated observer is different than for an inertial observer.
So far we have been working within the context of field theory in special relativity. What about in an expanding universe? The generalization to a curved spacetime is straightforward, if a bit mysterious. We will replace the metric for special relativity with a Robertson-Walker metric,
(155) |
Note that we have written the Robertson-Walker metric in terms of conformal time d = dt/a. This is convenient for doing field theory, because the new spacetime is just a Minkowski space with a time-dependent conformal factor out front. In fact we define the physical vacuum in a similar way to how we did it for special relativity: the vacuum is the zero-particle state as seen by a geodesic observer, that is, one in free-fall in the expanding space. This is referred to as the Bunch-Davies vacuum.
Now we write down the wave equation for a free field, the equivalent of Eq. (140) in a Robertson-Walker space. This is the usual equation with a new term that comes from the expansion of the universe:
(156) |
Note that the time derivatives are with respect to the conformal time , not the coordinate time t. As in the Minkowski case, we Fourier expand the field, but with an extra factor of a() in the integral:
(157) |
Here k is a comoving wavenumber (or, equivalently, momentum), which stays constant as the mode redshifts with the expansion a, so that
(158) |
Writing things this way, in terms of conformal time and comoving wavenumber, makes the equation of motion for the mode u_{k}() very similar to the mode equation (142) in Minkowski space:
(159) |
where a prime denotes a derivative with respect to conformal time. All of the effect of the expansion is in the a"/a term. (Be careful not to confuse the scale factor a() with the creation/annihilation operators _{k} and ^{†}_{k}!)
This equation is easy to solve. First consider the short wavelength limit, that is large wavenumber k. For k^{2} >> a"/a, the mode equation is just what we had for Minkowski space
(160) |
except that we are now working with comoving momentum and conformal time, so the space is only quasi-Minkowski. The general solution for the mode is
(161) |
Here is where the definition of the vacuum comes in. Selecting the Bunch-Davies vacuum is equivalent to setting A = 1 and B = 0, so that the annihilation operator is multiplied by e^{-ik} and not some linear combination of positive and negative frequencies. This is the exact analog of Eq. (147). So the mode function corresponding to the zero-particle state for an observer in free fall is
(162) |
What about the long wavelength limit, k^{2} << a"/a? The mode equation becomes trivial:
(163) |
with solution
(164) |
The mode is said to be frozen at long wavelengths, since the oscillatory behavior is damped. This is precisely the origin of the density and gravity-wave fluctuations in inflation. Modes at short wavelengths are rapidly redshifted by the inflationary expansion so that the wavelength of the mode is larger than the horizon size, Eq. (164). We can plot the mode as a function of its physical wavelength = k/a divided by the horizon size d_{H} = H^{-1} (Fig. 19), and find that at long wavelengths, the mode freezes out to a nonzero value.
The power spectrum of fluctuations is just given by the two-point correlation function of the field,
(165) |
This means that we have produced classical perturbations at long wavelength from quantum fluctuations at short wavelength.
What does any of this have to do with quantum gravity? Remember that we have seen that for an inflationary period that lasts longer than 140 e-folds or so, the fluctuations we see with wavelengths comparable to the horizon size today started out with wavelengths shorter than the Planck length _{Pl} ~ 10^{-35} cm during inflation. For a mode with a wavelength that short, do we really know how to select the "vacuum" state, which we have assumed is given by Eq. (162)? Not necessarily. We do know that once the mode redshifts to a wavelength greater than _{Pl}, it must be of the form (161), but we know longer know for certain how to select the values of the constants A(k) and B(k). What we have done is mapped the effect of quantum gravity onto a boundary condition for the mode function u_{k}. In principle, A(k) and B(k) could be anything! If we allow A and B to remain arbitrary, it is simple to calculate the change in the two-point correlation function at long wavelength,
(166) |
where the subscript B-D indicates the value for the case of the "standard" Bunch-Davies vacuum, which corresponds to the choice A = 1, B = 0. So the power spectrum of gravity-wave and density fluctuations is sensitive to how we choose the vacuum state at distances shorter than the Planck scale, and is in principle sensitive to quantum gravity.
While in principle A(k) and B(k) are arbitrary, a great deal of recent work has been done implementing this idea within the context of reasonable toy models of the physics of short distances. There is some disagreement in the literature with regard to how big the parameter B can reasonably be. As one might expect on dimensional grounds, the size of the rotation is determined by the dimensionless ratio of the Planck length to the horizon size, so it is expected to be small
(167) |
Here we have introduced a power p on the ratio, which varies depending on which model of short-distance physics you choose. Several groups have shown an effect linear in the ratio, p = 1. Fig. 20 shows the modulation of the power spectrum calculated in the context of one simple model [44].
Figure 20. Modulation of the power spectrum of primordial fluctuations for a rotation B ~ H/m_{Pl}. |
Others have argued that this is too optimistic, and that a more realistic estimate is p = 2 [45] or even smaller [46]. The difference is important: if p = 1, the modulation of the power spectrum can be as large as a percent or so, a potentially observable value [47]. Take p = 2 and the modulation drops to a hundredth of a percent, far too small to see. Nonetheless, it is almost certainly worth looking for!
^{12} This might not be so in "braneworld" scenarios where the energy scale of quantum gravity can be much lower [42]. Back.