As we showed earlier that the power suppression can occur if one of the two following possibilities happens in the early stage of the inflation: First, the phantom equation of state (and the superinflationary expansion due to the phantomness) can induce the power suppression. Second, a positive-pressure era (with the equation-of-state parameter w > 0), such as the kinetic-energy-dominated era, at the early stage of inflation can cause the power suppression. Both scenarios are logically possible, but both ideas have their own problems. For the phantom inflation scenario, it is very difficult to construct a viable theory for the phantom matter. For the positive-pressure era, the power suppression highly depends on the choice of the vacuum state. In the de Sitter space, we have a canonical choice of the vacuum—the Bunch-Davies vacuum [45], but in the positive-pressure era, there is no such a canonical
4As regards the treatment of the two-stage inflation, our approach is closest to that of [18, 19], in which a more complicated string-motivated two-field model is considered. In [17] the two fields have no direct coupling, and certain approximations are used to obtain the analytical solutions in various regimes of the model parameters. In [11, 12, 14, 20], the single-field models are used. In [13] the system is also modeled by a single fluid.
vacuum. Moreover, if we consider an eternally inflating background (and the con-sequent Bunch-Davies vacuum), then even though the universe evolves toward a positive-pressure era, the power suppression will not be realized [30].
The existing difficulties of having a consistent explanation for the power sup-pression may imply that its origin does not lie in the semi-classical physics, but in the quantum theory of gravity. Can we explain the power suppression by quantum gravitational e↵ects? Indeed, there has been several models explaining the power suppression from quantum gravity. For example, according to the loop quantum cosmology, quantum gravitational e↵ects can induce an e↵ective phantom matter in the deep trans-Planckian regime. The phantomness thereof can explain the CMB power suppression as well as supporting the scenario of the big bounce universe [46].
In order to investigate the wave function of our universe and the power suppres-sion problem, we will rely on the Hartle-Hawking wave function, or the so-called no-boundary wave function [47]. This wave function is one of the proposals to the boundary condition of the Wheeler-DeWitt equation [48]. It is a path integral over the Euclidean compact manifolds, and can be approximated by the method of steep-est descent. Under such approximation, we can then describe the wave function as a sum of the Euclidean instantons, where each instanton should eventually be Wick-rotated into the Lorentzian signatures [49, 50] and approach real-valued functions [51, 52, 53, 54, 55]. By integrating the Lagrangian, one can estimate the probability for the history described by each instanton.
Following the work of Halliwell and Hawking [56], one can introduce perturba-tions to the background instanton solution. These perturbaperturba-tions also carry their own canonical degrees of freedom. Although in general it is very difficult to track their coupled evolution, one can consistently consider various modes separately as long as the perturbations stay in the linear regime. The probability distribution of the magnitude of each perturbation mode can then be calculated, and the
expec-tation values of these modes, or equivalently, the power spectrum, can therefore be determined.
Using the method of Laflamme [57], we can define the wave function for the Euclidean vacuum. The Euclidean vacuum gives the scale-invariant power spectrum at short-wavelength scales, hence consistent with the choice of the Bunch-Davies vacuum [45] at small scales. On the other hand, at the long-wavelength scales, the power spectrum is enhanced due to the curvature of the manifold. All these results have been known in the literature and consistent with the independent calculations from quantum field theoretical techniques [58, 59]. However, to our best knowledge, it was not emphasized that the power spectrum can be suppressed by introducing the potential term. In chapter 6, we include analytical and numerical details for the power suppression due to the potential term of the inflaton field [60].
We adopt the Planck units (c = ~ = G = 1) and the signature ( , +, +, +) throughout the thesis.
Chapter 2
Preinflationary Network of
Frustrated Topological Defects
One natural candidate that may cause the power suppression at the lowest modes of the CMB spectrum is the primordial topological defect produced during some phase transition in the preinflationary era. If the universe is populated with the topolog-ical defects before the slow-roll inflation, the expansion rate of the preinflationary universe is generally di↵erent from that of the de Sitter universe. Therefore, the curvature perturbations evolves di↵erently in the preinflationary era from the way they do in the inflationary era. If inflation sustains for just about 60 e-folds, we can then see the imprint of the transition from the preinflationary to the inflationary era on the perturbation spectrum.
In this chapter we consider two of the most common types of topological defects—
domain walls and cosmic strings—arising from the phase transitions at the cosmic scale. To model the transition from the preinflationary to the inflationary era, we introduce the generalized Chaplygin gas (GCG) [61, 62, 63, 64, 65]. The idea of describing the early universe by the Chaplygin gas was first suggested in Refs. [66, 67]
(see also [68, 69]) and later extended in Refs. [70, 71, 72, 73, 74, 75]. The energy density of a network of frustrated topological defects (NFTD) can be described in
a compact way, for example, as
where a is the scale factor, B1 and A1 are constants related to the energy scale of the NFTD and the de Sitter-like inflationary era, respectively, ↵1 and 1 are constants such that 1 = 1, 2 for the network of frustrated domain walls (NFDW) and the network of frustrated cosmic strings (NFCS), respectively. We assume that 1 + ↵1
is positive such that the inflationary era is preceded by a topological dominance phase. Let us be reminded in this regard that the energy density of NFDW and NFCS scales as 1/a and 1/a2, respectively [27]. It is worthy to stress that the NFTD epoch preceding the slow-roll inflationary can in principle produce inflation as well;
indeed this is the case for NFDW, but this inflation is much slower, i.e. much lazier than the slow-roll inflation. Moreover, for a NFCS dominated period the universe is increasing its size at a constant speed; i.e. with no acceleration or deceleration.
From now on whenever we refer to a preinflationary era, we will be referring to a pre-slow-roll inflationary era.
We consider a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) uni-verse filled with the matter content described by Eq. (2.1). The energy conservation gives
˙⇢ + 3H(⇢ + p) = 0, (2.2)
where a dot corresponds to a derivative with respect to the cosmic time and H stands for the Hubble rate. By inserting Eq. (2.1) into Eq. (2.2), one obtains the pressure of the matter content
p =
Note that in terms of the equation of state,
p = w⇢, (2.4)
where w is the equation-of-state parameter, the universe is in the state of w = 2/3 and w = 1/3 for = 1 (NFDW dominated) and = 2 (NFCS dominated), respectively.
In the Planck unit, the Friedmann equation reads
H2 = 2 universe began in NFTD dominated era in the past infinity, and turned into a de Sitter-like space, in which a/ 1/⌧, at later time.
2.1 Model Building and Parameters Fixing
We divide the expansion of the universe into three successive periods: the preinfla-tionary NFTD dominated era, the slow-roll inflating phase, and the standard ⇤CDM epoch. The energy density of each of these periods can be modeled as
⇢ =
Expression (2.7a) describes the energy density of the NFTD era, which was in-troduced in Eq. (2.1), followed by the de Sitter-like inflating phase. The model described by Eq. (2.7b) was previously studied within an inflationary framework in Ref. [70, 71] (see also Ref. [65]) and, under suitable constraints on A2, B2, and
↵2, can depict the transition from the de Sitter-like era to the radiation dominated
era. The energy density (2.7c) is the standard ⇤CDM model, in which ⇢r0, ⇢m0, and ⇢⇤ are the energy densities of the radiation, matter, and dark energy today, respectively. As we will show later, the parameters of the model can be constrained using observational data corresponding to the present energy density of radiation, the scalar power spectrum, and the spectral index at a given pivot scale. In addition, by requiring that the energy density is continuous at each transition, we have the conditions
A1 = A(1+↵2 1)/(1+↵2), (2.8)
B2 = ⇢r0a40 1+↵2. (2.9)
In order to obtain the scalar power spectrum, it is useful to model the matter content in the first two periods of Eq. (2.7); i.e. those described by Eqs. (2.7a) and (2.7b), through a scalar field, with the condition that the energy density and pressure of the scalar field are the same as that given by Eqs. (2.7a) and (2.7b). The energy density and pressure of the scalar field are
⇢ =
02
2 a2 + V ( ), p =
02
2 a2 V ( ), (2.10)
where the primes denote the derivatives with respect to the conformal time. The scalar field and its potential in the first period is given by
(a) = 1
where V0 = A1/(1+↵1 1). Similarly, the scalar field and its potential in the second period can be obtained by replacing ↵1 with ↵2 and setting 2 = 4 in Eq. (2.11) and
Eq. (2.12), giving
We can obtain the potential of the scalar field for the two periods as functions of the scalar field by substituting the inverse function of Eq. (2.11) into Eq. (2.12) and similarly that of Eq. (2.13) into Eq. (2.14), respectively, which leads to
V1( ) = V0
Eq. (2.16) coincides with the potential in Ref. [70, 71], as it should be. The form of Eq.(2.15) and Eq.(2.16) has been chosen such that the two periods are connected at
= 0 with the potential and its first derivative with respect to being analytically continuous at the connecting point. The result is shown in Figure 2.2.
Next we tackle the issue of analyzing the potentials (2.15) and (2.16). First of all, we consider that the scalar field potential (2.15) has a unique minimum at = 0 to maximize the amount of inflation during the first period. Notice that unless this condition is imposed, the scalar field might roll down the potential till it reaches the minimum of V1( ) and then would have to climb up to reach the local maximum
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Κ !1"Α
1" Β
1Φ 2 1.00
1.01 1.02 1.03 1.04
V
1!Φ"#V
0Figure 2.1 The scalar field potential for the NFDW era ( 1 = 1) for di↵erent values of ↵1. The solid curve corresponds to ↵1 = 1 (has unique minimum at = 0) and the dashed curve corresponds to ↵1 = 7 (has minimum at > 0). The situation is similar for the case of NFCS ( 1 = 2), which we omit here.
located at = 0, as shown in Figure 2.1. Imposing that the potential (2.15) has a unique minimum reached at = 0 implies a condition on the parameters ↵1 and 1
that
↵1 < 6 1 1
. (2.17)
Therefore, bearing in mind that (i) 1 = 1, 2 for NFDW and NFCS, respectively, and (ii) 0 < 1 + ↵1 so that the phase of FNTD precedes the inflationary phase, we conclude that 1 < ↵1 < 5 for NFDW and 1 < ↵1 < 2 for NFCS. We show the shape of the potential V1( ) for di↵erent cases when the condition (2.17) is fulfilled and violated in Figure 2.1.
In addition, we can constrain our model, potentials (2.15) and (2.16), using the methodology in Ref. [70, 71]. More precisely, we can use the WMAP7 observation of the power spectrum of the comoving curvature perturbation, Ps = 2.45⇥ 10 9, and the spectral index, ns = 0.963, at the pivot scale k0 = 0.002 Mpc 1 to fix the parameters in our model [77]. We can as well impose a bound on the number of
e-folds, Nc, since a given mode exits the horizon until the end of inflation as done in Ref. [70, 71]. This gives the best-fit values for ↵2, V0, and, therefore, A2. Notice that once V0is fixed, the parameter A1is fixed for a given ↵1 as well, since V0 = A1/(1+↵1 1). The parameter B1 in Eq. (2.7a) fixes the energy density of the NFTD, which strongly a↵ects the lowest modes that exited the horizon around the onset of infla-tion, and causes a significant drop on the lowest modes of the primordial spectrum of the curvature perturbation. Although we expect that the NFTD would a↵ect the lowest modes, we must make sure that the curvature power spectrum Ps and the spectral index ns at the pivot scale k0 are consistent with the observations. There-fore, we choose the value of B1 such that Ps and ns match the observed values at k0, and that the amplitude of Ps drops at the scales whose comoving wave numbers are smaller than k0. Roughly speaking, the parameter B1 controls the horizontal shift of the curvature power spectrum.
However, following this procedure, it turns out that we can not find a set of values for the parameter B1 that satisfies the constraint on the spectral index. In fact, in this model B1 turns out to be always smaller than 0.9. This drawback originates from the second period described by Eq. (2.7b), which corresponds to the transition from the slow-roll inflationary era to the radiation dominated period.
More precisely, the model described by Eq. (2.7) does not give enough e-folds during the slow-roll inflationary era. We show, as an example, in Figure 2.2 how the scalar field rolls too quickly and the radiation dominated phase is reached too early in the case corresponding to a NFDW.
We therefore suggest an alternative model described by
⇢ =
where 1 = 1, 2 discriminates the NFDW and the NFCS as in the former model,
!4 !2 0 2 4 Φ 0.5
1.0 1.5
V Φ !V 0
V
1!V
0V
2!V
0Figure 2.2 The scalar field potential for 1 = 1 (preinflationary NFDW). The up-ward convex curve is V1 (cf. Eq. (2.15)) and the downward cacave curve is V2
(cf. Eq. (2.16)). We can choose 0 for V1 and 0 for V2 to describe the potential of the scalar field . In this model the scalar field rolls down too quickly in the slow-roll inflationary era and the radiation dominated phase is reached too early.
and 2, 3, B1, A2, B2 are constants we will explain below. The major di↵erence of this model from out previous one is that here we model the slow-roll inflation by a power-law expansion. We choose this model because it generates an almost flat curvature spectrum for modes larger than the pivot scale k0 = 0.002 Mpc 1 and gives enough e-folds during the power-law inflationary period. In addition, the model introduces in a natural way that a NFTD precedes the power-law inflation as (1 + 2)/(1 + 3) < 1 (please see also the conditions (2.19), (2.20) and (2.21)).
The first period in Eq. (2.18a) describes the matter content of the universe during a period that transits from a NFTD dominated phase to a power-law inflationary
era. The parameters B1 and A2 are associated with the energy scale of the NFTD and that of the power-law inflation, respectively. The second period with the energy density (2.18b) was previously studied within another inflationary framework in Ref. [72, 73]. It connects smoothly a power-law inflating phase with a radiation dominated universe, and the constraints on the parameters 2 and 3 are,1
1 + 2 < 0, (2.19)
1 + 3 < 0, (2.20)
2(1 + 3) < 1 + 2. (2.21)
These constraints imply that (i) there is a power-law inflating phase, (ii) the in-flationary era precedes the radiation dominated period, and (iii) the null energy condition is always fulfilled so that there is no superinflationary phase. Finally, the energy density described by (2.18c) corresponds to the ⇤CDM model as that described by Eq. (2.7c).
Although there seems to be many free parameters, they can be fixed down to only one by the following procedure: (i) Fix B2 by the current amount of radiation for a given 3. (ii) Constrain the power-law expansion quantified by A(1+2 3)/(1+ 2) by the WMAP7 data of the curvature power spectrum Ps and the spectral index ns. Since there are three parameters (A2, 2 and 3) to be constrained by only two conditions (Ps and ns), we are left with the only one free parameter, which we choose to be 3. (iii) Fix B1 such that Ps and ns remain the correct values at the pivot scale k0, and Ps drops only at comoving wave numbers smaller than k0.
Again, it is suitable to introduce a scalar field that mimics the matter content described in Eqs. (2.18a) and (2.18b); i.e. we describe the dynamics of the model through a scalar field with a potential whose energy density and pressure can be obtained from Eq. (2.10). During the NFTD period (cf. Eq. (2.18a)), the mapping
1The notation is di↵erent from the one used in the work [72, 73]. The parameters and ↵ in Ref. [72, 73] are denoted as 2 and 3here, respectively.
between the scalar field and the perfect fluid of our model leads to the scalar field potential during this period. Similarly, we map the perfect fluid with the energy density (cf. Eq. (2.18b)) to the scalar field with a new potential V2( ) [72, 73] obtained in Ref. [72, 73]. Such a potential, with an appropriate initial condition, drives a power-law inflation and mimics a radiation dominated universe afterwards.
Unlike the previous model described by Eq. (2.11)-(2.12) and Eq. (2.13)-(2.14), here it is not feasible to find analytically the inverse functions of Eq. (2.22) and Eq. (2.24), so we cannot obtain the analytical forms of the potential as functions of . We thus connect the scalar field potential numerically. V1(a) and V2(a) are connected at the intersection of the first two periods (Eq. (2.18a) and Eq. (2.18b)),
Figure 2.3 This plot shows the rescaled potentials given in Eqs. (2.23) and (2.25) versus the scalar field , where V0 = A1/(1+2 3)(A2/B2) (1+ 2)/(q(1+ 3)). The blue curve corresponds to the NFCS case, and the red curve corresponds to the NFDW case. The energy scale of inflation, V0, in our model is about 1015 GeV for both NFDW and NFCS.
where the second term of Eq. (2.18a) dominates over its first term, and the first term of Eq. (2.18b) dominates over its second term so that the potential and its first derivatives with respect to are approximately continuous at the intersection of the first two periods. It is worthy to notice that an integration constant appears when we integrate Eq. (2.10) after mapping it to the energy density and pressure of a given perfect fluid. Therefore, we can always choose the constant properly such that the scalar field is continuous at the connecting point. As a result, we can use the scale factor a as a parametric parameter to plot V1( ) and V2( ), which are shown in Figure 2.3 as an example. The scalar field starts with a negative value and rolls down the potential as the universe inflates until it reaches the radiation dominated era.