In the Euclidean space, we consider the scale factor solution (6.27) and a constant scalar field in the background. Neglecting the metric perturbations gn (we suppress the indices l and m in this section, since the equation of motion does not depend on them), we calculate the field perturbations fn (we ignore the tilde that denotes the c-number solution wherever no confusion arises) by numerically solving the equation of motion where we use the hat to emphasize that it is the solution in the Euclidean space.
In order to keep equation (6.61) finite, we require that both ˆfn(⌧ ) and ˆfn0(⌧ ) vanish at ⌧ = 0. More precisely, we adopt the following ansatz as the initial condition for numerical calculations:
fˆn(⌧i) = 1
2✏⌧i2, (6.62)
f˙ˆn(⌧i) = ✏⌧i, (6.63)
where ⌧i ⌧ 1 is the initial Euclidean time from which we start to integrate the dif-ferential equations, and ✏ is an arbitrary parameter. Note that since the expectation value hfn2i depends only on the ratio ˜fn/ ˙˜fn, the power spectrum is independent of the choice of ✏. In our numerical calculation, we set ⌧i = 10 4 and ✏ = 1, and evolve the Euclidean system from ⌧i to ⌧f = ⇡/2H0.
In the Lorentzian spacetime, we use the analytical solution (6.30) for the scale factor and a constant scalar field in the background to model the slow-roll inflation.
The equation of motion for the field perturbation reads d2fn
dt2 + 3H0tanh(H0t)dfn
dt +
˜
m2 +(n2 1)H02
cosh2(H0t) fn= 0. (6.64) The boundary conditions connecting the Euclidean and Lorentzian solutions are the Cauchy-Riemann conditions [51, 52, 53, 54, 55]
Re{f(ti)} = Re{ ˆf (⌧f)}, (6.65) Im{f(ti)} = Im{ ˆf (⌧f)}, (6.66) Re{ ˙f (ti)} = Im{f (⌧˙ˆ f)}, (6.67) Im{ ˙f (ti)} = Re{f (⌧˙ˆ f)}, (6.68)
where we set ti = 0 to be the initial time of integration in the Lorentzian space. We then solve the system from ti to the horizon-exit time,
texit= 1 H0
sinh 1n, (6.69)
for mode n. Note that for each mode, the expectation value (6.60), hence the power spectrum (6.55), is evaluated at its horizon-exit time.1
The Hubble parameter in the Lorentzian space is
H(t) = H0tanh(H0t). (6.70)
Therefore H0 corresponds to the Hubble constant during the exponentially growing period. To fix the value of H0, we consider the Hamiltonian constraint in Lorentzian
1The power spectrum in general does not conserve at super-horizon scales, given that we are calculating the spectrum of the inflaton field perturbations in a closed FLRW universe, adopting the wave function interpretation to deduce the expectation values of the perturbations. The relation between this formulation and the approach of traditional quantum field theory still requires further clarification, which we leave to the future works.
space,
For the case that the scalar field is massless, the constant potential, V ( ) = V0, drives the exponential growth of the scale factor a. The Hubble parameter is ap-proximately
H ⇡
r8⇡ 2
3 V0. (6.72)
We choose the normalization of the metric to be
2 = 1 V0
. (6.73)
Therefore, during the exponential growth, H ⇡ H0 ⇡p 8⇡/3.
For the case of massive scalar field, during the exponential expanding period, the Hubble parameter is approximately
H ⇡
Figure 6.1 shows the power spectrum in the massless case with H0 = p 8⇡/3.
We see that while the power spectrum is scale-invariant in the small scales, it is enhanced in the large scales. Figure 6.2 is the power spectrum for a large mass
˜
m = 1000p
0.1 with H0 = p
8⇡/3. Opposed to the massless case, we see that in this massive case the large-scale spectrum is suppressed. In Figure 6.3 we show the spectra corresponding to a range of masses, holding H0 =p
8⇡/3. We can observe the trend that, as the mass increases, the large-scale spectrum turns from being enhanced to being suppressed. We find that roughly the power is enhanced when ˜m is greater than 0.5H0, and suppressed when ˜m is less than 0.5H0.
To find out the mechanism that leads to this transition from enhancement to suppression as the mass increases, we first study the time evolution of the power
1.0 1.5 2.0 2.5 2.495
2.500 2.505 2.510 2.515 2.520
log10n log10Pn
Figure 6.1 The power spectrum obtained by numerically solving the perturbations with ˜m = 0, H0 =p
8⇡/3.
1.0 1.5 2.0 2.5
0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30
log10n log10Pn
Figure 6.2 The power spectrum obtained by numerically solving the perturbations with ˜m = 1000p
0.1, H0 =p 8⇡/3.
1.0 1.5 2.0 2.5 2.25
2.30 2.35 2.40 2.45 2.50
log10n log10Pn
Figure 6.3 The power spectrum obtained by numerically solving the perturbations with ˜m = p
0.1⇥ {0, 1, . . . , 10}, from top to bottom. All spectra are plotted with H0 =p
8⇡/3.
0.5 1.0 1.5 2.0 0.5
1.0 1.5 2.0 2.5 3.0 3.5
log10n log10Pn
Figure 6.4 The time evolution of power spectrum in the case of ˜m = 0, H0 =p 8⇡/3.
The darker curves correspond to the spectra at later times. The lightest curve is the initial Lorentzian spectrum at time ti. For each n mode, the power is evaluated up to its horizon crossing time.
spectrum in the Lorentzian space. The time evolution of spectrum in the massless case is given in Figure 6.4. For massive case, the time evolution of the spectra for the cases of ˜m = 0.5H0, H0, and 2H0 is given in Figures 6.5 to 6.7.
Through the spectrum evolution, we find that the power enhancement or sup-pression are reflected in the initial spectra in the Lorentzian space. At the small scales, before the horizon exit the slopes of the spectra are close to that of the spec-trum of the Bunch-Davis vacuum. At the horizon crossing, the small-scale spectra are nearly scale-invariant. At the large scales, we see that at the horizon crossing the spectra is enhanced or suppressed determined by the mass of the scalar field as we showed before. Moreover, we note that even before the horizon crossing, al-ready in the initial spectra in the Lorentzian space there are corresponding power enhancement or suppression relative to the small-scale Bunch-Davis vacuum. The
0.5 1.0 1.5 2.0 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
log10n log10Pn
Figure 6.5 The time evolution of power spectrum in the case of ˜m = 0.5H0, H0 = p8⇡/3. The darker curves correspond to the spectra at later times. The lightest
curve is the initial Lorentzian spectrum at time ti. For each n mode, the power is evaluated up to its horizon crossing time.
0.5 1.0 1.5 2.0 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
log10n log10Pn
Figure 6.6 The time evolution of power spectrum in the case of ˜m = H0, H0 = p8⇡/3. The darker curves correspond to the spectra at later times. The lightest
curve is the initial Lorentzian spectrum at time ti. For each n mode, the power is evaluated up to its horizon crossing time.
0.5 1.0 1.5 2.0 0
1 2 3
log10n log10Pn
Figure 6.7 The time evolution of power spectrum in the case of ˜m = 2H0, H0 = p8⇡/3. The darker curves correspond to the spectra at later times. The lightest curve is the initial Lorentzian spectrum at time ti. For each n mode, the power is evaluated up to its horizon crossing time.
origin of the power enhancement or suppression therefore lies on the Lorentzian initial condition, or, equivalently, on the Euclidean final spectrum.
To find out the e↵ect of mass on the Euclidean final spectrum, we note that the Euclidean equation of motion (6.61) can also be analytically solved, yielding the solution
fˆn(⌧ ) = AP⌫n[cos(H0⌧ )]
sin(H0⌧ ) , (6.75)
where A is an overall coefficient that has no e↵ect on the final Euclidean spectrum,
⌫ = 1 +p
9 4 ˜m2/H02
2 , (6.76)
and we have picked the solution that is consistent with the no-boundary initial condition. When ˜m2/H02 > 9/4, ⌫ and ˆfn(⌧ ) become complex. The power spectrum at the beginning of the Lorentzian time can be evaluated using the Euclidean solution at ⌧ = ⇡/2H0 through the boundary conditions. When evaluating the ratio ˜fn/ ˙˜fn with complex ˜fn, we interpret it as the amplitude | ˜fn/ ˙˜fn|. We then have the initial power spectrum in the Lorentzian space as
P (n) = 3n3H02 8⇡
P⌫n(0)
P⌫n0(0) . (6.77)
In the large-mass limit, we can intuitively understand the power suppression of the initial power spectrum induced by the mass term in the following way. In such a limit, the solution to the equation of motion (6.61) roughly consists of an expo-nentially growing mode, exp( ˜m⌧ ), and an exponentially decaying mode, exp( ˜m⌧ ).
Hence, the amplitude| ˜fn/ ˙˜fn| is roughly of the order of 1/ ˜m, which is suppressed by
˜
m = m/p
V0. Note that the large-mass limit actually lies beyond the linear regime of perturbations, and the purpose of considering it is only to provide an intuitive understanding. As shown in Figure 6.3, the long-wavelength spectrum is already suppressed as ˜m2/H02 is as small as roughly 0.1p
6/p
8⇡/3 ⇡ 0.43. Therefore, it only requires a moderate mass to induce the e↵ect of suppression.
Chapter 7
Conclusions and Discussions
In this thesis we explore the initial conditions to the inflationary universe using the long-wavelength suppression of the CMB spectrum as an important observational clue. We start by considering the toy model to the preinflationary era that consists of the NFTD of two kinds: (i) a NFDW that induces an accelerating rate smaller than that of the standard slow-roll inflation, and (ii) a NFCS that describes a universe expanding in a constant rate. We model such a matter content for the very early universe as a kind of the generalized Chaplygin gas described by Eq. (2.18) (see also Eq. (2.1)), which gives a smooth transition from a NFTD dominated era to a de Sitter-like phase or a power-law inflationary era. We constrain our model using the WMAP7 data for the power spectrum of the scalar perturbations, Ps = 2.45⇥ 10 9, and the spectral index, ns= 0.963, at the pivot scale, k0 = 0.002 Mpc 1 [77]. After fixing the parameters of the model and imposing the approximate initial vacuum states for the perturbations, we obtain the curvature power spectra for the cases of a NFDW and a NFCS, as shown in Figures 2.6 and 2.7. The most important feature of the spectra is the drop of the power for the long-wavelength modes with k 0.002 Mpc 1. Nevertheless, through a more detailed and systematic analysis presented later, the initial vacua are found to have di↵erent structures from the approximated ones used here. In particular, we will see that for the case of the preinflationary
NFDW dominated era, the large-scale spectrum is actually enhanced, rather than suppressed.
We then systematically investigate the power spectrum in the general background geometry. We study the curvature perturbations of a scalar field in the FLRW universe parameterized by the equation of state parameter w, and find that the large-scale spectrum at the end of inflation reflects the super-horizon spectrum of the initial state. We show that if the universe begins in the superinflation era (w <
1) or that with positive pressure (w > 0), the large-scale curvature perturbation spectrum is suppressed due to the blue-tilted super-horizon spectrum in the initial era. We first find the scaling relation of the super-horizon spectrum for a scalar field in the FLRW background with constant equation of state. At the large scales, the spectrum is blue-tilted for positive-pressure (w > 0) or superinflation (w < 1) era, and red-tilted for the era with 1 < w < 0, except the singular case with w = 1/3. In the slow-roll (w' 1) and zero-pressure (w = 0) background, the super-horizon spectrum is scale-invariant. We also point out that the conclusions are drawn from assuming the mode function approaches the Minkowski limit at small scales.
Although being natural in the accelerating universe, this assumption becomes a posteriori in the decelerating universe since the sub-horizon modes are evolved from the super-horizon modes, which are initially across causally disconnected regions before entering the horizon.
We develop the method of spectrum evolution to analyze how the expansion of the background geometry transforms the initial spectrum in the preinflationary era into the super-horizon spectrum in the inflationary era. By analyzing three scenar-ios: a single slow-roll era, a slow-roll era preceded by a kinetic era, and two successive slow-roll eras connected by a kinetic era, we show the following two facts. First, the large-scale power suppression in the model with a single preinflation kinetic era stems from the blue-tilted super-horizon spectrum of the initial kinetic era. Second,
the additional slow-roll era preceding the kinetic era changes the super-horizon ini-tial spectrum, so the large-scale power is enhanced, rather than suppressed. These results show that the large-scale spectrum depends sensitively on the initial vacuum.
In the universe beginning with the positive-pressure era, as we pointed out earlier, the super-horizon modes are initially across causally disconnected regions, and the well-motivated assumption on the initial state is still lacking. Some investigations about the e↵ect of the di↵erent initial vacuum on the spectrum have been carried out in the literature [31, 32, 33, 34]. We also explore the case that is free from the acausal issue: a superinflation era preceding the slow-roll era, and show that the large-scale spectrum is suppressed due to the initially blue-tilted spectrum in the superinflation era.
To show that the pattern of evolution we obtain through the single-field analysis also applies to the two-field systems, we calculate the curvature perturbation and CMB spectra of a two-stage inflation model from the given two-field potential. We show that the large-scale power is enhanced due to the initial spectrum set in the first accelerating era, and the e↵ect of the intermediate decelerating era on the spectrum is connecting the two plateaus generated in the two accelerating eras, which agrees with the picture obtained through the ad hoc single-field analysis.
In view of the shortcomings of the positive-pressure era and the superinflation era as consistent initial conditions to the universe, we explore other possibilities pro-vided by the quantum gravitational theory. Particularly, we investigate the power spectrum of perturbations with the no-boundary wave function [47] as the initial con-dition of the inflationary universe. We have relied on very conservative approaches, including the canonical quantization [48], the Euclidean path integral approach and the steepest descent approximation [47], and the use of instantons at the background as well as perturbation levels [56], which are consistent with traditional techniques of quantum field theory in several regimes [99]. We find that the inflationary universe is
approximately scale-invariant at the short-wavelength scales, while the power spec-trum of the pure de Sitter space is enhanced at the long-wavelength scales. We also find that the power spectrum can be either enhanced or suppressed due to the detailed choice of the potential; for example, the mass term of the inflaton field. In particular, as long as the inflaton is moderately massive, the long-wavelength spec-trum is suppressed. This opens a possibility that the power suppression is indeed a hint to that our universe starts from an instanton with a massive inflaton field that approximates the Hartle-Hawking wave function.
This line of exploration definitely needs more work. It will be interesting to see more detailed calculations for other realistic inflationary scenarios; e.g., the Starobinsky-type inflation models [100]. Also, we investigated for compact and ho-mogeneous instantons, but there are other instantons that also explain the origin of our universe; e.g., the Coleman-De Luccia instantons [101] or the Euclidean worm-holes [102, 103, 104]. Another brave question is: what is the relation between the big bounce model of the loop quantum cosmology [46] and the Hartle-Hawking wave function [47]? Both approaches explain the power suppression, but it is yet unclear which one is more suitable as the model of the beginning of our universe. We leave these interesting issues as future research topics.
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