The accelerated expansion of the universe during the primordial inflationary era con-verts the initial quantum fluctuations in the universe into macroscopic cosmological perturbations, which leads to the inhomogeneity we observe nowadays in the CMB [78, 79]. Following the standard approach, we will use gauge invariant quantities that involve the metric perturbations and the scalar field fluctuations [80]. For con-venience, we will choose the comoving curvature perturbation, R, which in addition is conserved on large scales [81, 82].
We expect that a NFTD in the very early universe and just before the infla-tionary era could give the appropriate corrections to the quadrupole modes of the CMB data as observed nowadays [77]. We will next quantify the quantum cosmolog-ical perturbations during that period and obtain the power spectrum of the scalar perturbations.
The scalar perturbations can be described by introducing the variable (see, for example, Ref. [83])
u = zR, (2.26)
where z ⌘ a ˙H. The variable u can be decomposed into Fourier modes, uk, which fulfill the field equation [83]
d2uk
The modes uk can be mapped to the spectrum of the comoving curvature perturba-tions which reads [83]
2⇡2
k3 PR(k) = |uk|2
z2 . (2.28)
Given that we are dealing with adiabatic perturbations, the comoving curvature perturbations remain constant on large scales and consequently we can equate the power spectrum at the horizon exit with the power spectrum of the primordial scalar perturbations at the horizon reentry as observed on the CMB. Therefore, for a given
10 !10 100 10 14 10 26 10 38 10 50 10 !15
10 !5 10 5 10 15 10 25 10 35 10 45
z '' !z a '' !a
Figure 2.4 The green dashed curve corresponds to z00/z and the black curve corre-sponds to a00/a. As can be seen that the approximation z00/z ⇡ a00/a holds during the NFDW dominated and the power-law inflationary eras.
mode k, the spectrum is evaluated at the horizon exit; i.e. when k = acrossH, where across stands for the value of the scale factor when the mode exists the horizon.
We next obtain the evolution of the mode function uk(⌧ ) for each comoving wave number k in order to obtain the curvature perturbation spectrum. We will tackle this issue numerically rather than using the standard results for slow-roll inflation [78, 84, 83], because those conditions are not fulfilled at very early time when the NFTD is dominant. It is easier to solve Eq. (2.27) numerically by splitting it into two first order di↵erential equations,
8>
><
>>
:
X0 = Y
Y0 = k2 zz00 X,
(2.29)
where we have set X = uk.
In addition, we need to impose a set of boundary conditions at the time when the wavelength of a given mode k is much smaller than the Hubble radius; that is,
10 !10 100 10 14 10 26 10 38 10 50
Figure 2.5 The green dashed curve corresponds to z00/z and the black curve corre-sponds to a00/a. As can be noticed that the approximation z00/z ⇡ a00/a holds during the NFCS dominated and the power-law inflationary eras.
k aH. Those boundary conditions will depend on the specific NFTD scenario and also on the given scale of the mode, as we will explain shortly. It is also worthy to stress two things: (i) the approximation z00/z ⇡ a00/a holds during the NFTD dominance and during the power-law inflationary era (See Figure 2.4 and Figure 2.5), and (ii) the analytical solutions for the power-law inflation [85] can be used as boundary conditions for the modes with comoving wave numbers larger than roughly 10 3 Mpc 1 as explained next. We first recall that the power-law solutions for X and Y are [84] where l is the exponent characterizing the power-law expansion in terms of the conformal time ⌧ ; i.e. a / ⌧l. For our model, l = ((1 + 2)/(2(1 + 3)) 1 ) 1.
For those modes whose k 10 3Mpc 1, we can start the numerical integration of Eq. (2.29) during the power-law inflationary era where the condition k aH is still satisfied. However, for scales roughly smaller than 10 3 Mpc 1, we split the boundary conditions imposed on the NFDW and the NFCS separately.2 Despite the general solutions (2.30) and (2.31) are still valid for the NFTD, the exponent describing the power-law expansion, l, depends on the specific characters of the NFTD:
• During NFDW dominant era, ⇢ ⇠ B1/a 1 with 1 = 1, which also implies a power-law expansion because a(⌧ ) / ⌧1/( 1/2 1). Thus we can use the solutions (2.30) and (2.31) as boundary conditions with l = 1/( 1/2 1).
• During the NFCS dominant era, ⇢ ⇠ B1/a 1 with 1 = 2. Note that for this value of 1 the exponent l in Eqs. (2.30) and (2.31) is not well-defined, so we need to fix the initial condition in this case in a di↵erent way. It can be shown from Eq.(2.5) that a00/a is a constant if ⇢ / 1/a2. Introducing a new variable k˜⌘p
k2 a00/a, the wave equation (2.27) becomes d2uk
d⌧2 + ˜k2uk = 0, (2.32)
which has two properly normalized linearly independent solutions,
uk(⌧, k) = e i˜k⌧
p2˜k
, ei˜k⌧
p2˜k
. (2.33)
We then choose the solution with exponent i˜k⌧ because it reduces to the Minkowski initial condition [78].
2We will show in the next chapter that, through a more detailed and systematic analysis, the initial vacua have di↵erent structures from the approximation made here. As a consequence, the power spectra obtained from the initial vacua adopted in the next chapter also di↵er from those in this chapter. 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.
Finally, it is easier to use the scale factor as the independent variable in the numerical integration of Eq. (2.29) instead of the conformal time. The relation between the conformal time ⌧ and the scale factor a is
⌧ =
The resulting curvature power spectra are shown in Figure 2.6 and Figure 2.7, corresponding to the NFDW and the NFCS, respectively. Let us recall that all the parameters used here have been fixed by imposing the observational constraints in the way stated in Sec. 2.1. For modes k such that 10 3Mpc 1 < k < 105Mpc 1, we obtain a constant slope for the spectrum of the curvature perturbation. This is a simple consequence of the intermediate phase given in Eq.(2.18b) previously analyzed in Ref. [72, 73], and implies a power-law inflation. Our new and important result is the drop of PR for the modes whose k 10 3Mpc 1, which is helpful in explaining the quadrupole anomaly through an alternative way from those used in Refs. [6, 86, 87, 8, 88]. Such a decrease of PR is a consequence of the NFTD era just before the inflation.
We calculate the CMB temperature anisotropy spectrum by the numerical pack-age CMBFAST [89, 90, 91] with minor modifications to the form of specifying the primordial power spectrum. In the modified version of CMBFAST, the primordial power spectrum is fed into the code as an interpolating function instead of a func-tional form. This adjustment is made so that rather than accepting only the nearly-scale-invariant spectrum as its initial condition, CMBFAST is now compatible with any general shape of initial spectrum. This feature is essential to our case since the primordial spectra obtained from our scenarios severely deviate from the scale-invariant form in large scales whose wave numbers are smaller than 10 3Mpc 1.
0 5 10
log ! k
Mpc !1 "
!9.4
!9.2
!9.0
!8.8
!8.6 log !P R "
Figure 2.6 This plot corresponds to the curvature perturbation spectrum for the inspired modified GCG model with 1 = 1 (see Eq. (2.18a)), which describes a NFDW dominated era followed by a power-law inflationary period. We choose
3 = 1.05. The vertical dashed line corresponds to the pivot scale k = 0.002 Mpc 1.
Figure 2.8 shows the CMB spectra generated by the NFTD scenarios along with that generated by the standard inflationary model assuming a power-law initial spectrum. The data of WMAP 7-year observation [77] are also marked in the plot.
It can be seen that the preinflation NFTD era alleviates the quadrupole anomaly of the CMB. Note that the NFCS has a stronger e↵ect on reducing the amplitude of the lower modes than the NFDW does. This is a result of the initial slope and the turn-around point of the curvature perturbation spectrum induced by NFDW and NFCS as shown in Figure 2.6 and 2.7. Also note that regarding the lower modes of the CMB, it is irrelevant that the potentials and the first derivatives of the scalar field with respect to the cosmic time are not rigorously continuous at the connecting point. The reason is that major contributions to the lower modes of CMB came from the scalar perturbations whose comoving wave numbers are about 10 5 to 10 3Mpc 1. These modes had already exited the Hubble radius during the
0 5 10
log ! k
Mpc !1 "
!9.4
!9.2
!9.0
!8.8
!8.6 log !P R "
Figure 2.7 This plot corresponds to the curvature perturbation spectrum for the modified GCG model with 1 = 2 (see Eq. (2.18a)), which describes a NFCS domi-nated era followed by a power-law inflationary period. We choose 3 = 1.05. The vertical dashed line corresponds to the pivot scale k = 0.002 Mpc 1.
first period so that the power spectrum in this regime is obtained without the need to integrate across the connecting point.
10 100 500 1000 0
1000 2000 3000 4000 5000 6000
10 100 500 1000
0 1000 2000 3000 4000 5000 6000
Multipole Moment!l"
l!l!1"ClTT#!2Π"$ΜK2%
Figure 2.8 The CMB temperature anisotropy spectrum. The dots with error bars are the WMAP 7-year data. The solid line is the prediction of standard inflation with a power-law spectrum. The dashed and dotted lines are the spectra of the scenarios of NFCS and NFDW, respectively.
Chapter 3
Power Spectrum in the Universe with Constant Equation of State
In the previous chapter, we studied two scenarios of the preinflationary era that may potentially account for the power suppression of the long-wavelength CMB spectrum. There are, however, a few things we want to improve. First of all, in the previous treatment, we introduced the generalized Chaplygin gas (GCG) to model the preinflationary and inflationary (as well as the following ⇤CDM) eras. The GCG model is actually a complicated model in the sense that there are many model parameters. Moreover, not any type of such models can fit a given set of observation results. As we have already seen in the previous chapter, a careful model selection is needed to ensure that we can fit the model with the observations. Second of all, the initial conditions in the preinflationary era rely on the solutions (2.30) and (2.31) that approximate the background evolution by the power-law expansion. It would be preferable if we can find an exact solution in the general background.1 Last of all, although we studied two possibilities, there have been many other models proposed to explain the power suppression. What are the common characters that are essential for a model to be viable?
1Note that in the case of NFCS, the initial conditions were fixed in a di↵erent way.
We conduct a series of investigations to address the above issues in this and the following chapters. In this chapter, we first find the single-field model that gives the background evolution with a given equation-of-state parameter, w, and the corresponding general solutions to the curvature perturbations. We then discuss the common assumption on the small-scale behavior of the solution and its causal character. In the end we find the power spectrum in the background with given w and its power-law relation with respect to the wavenumber k.
3.1 Perturbations with Constant Equation of State
We consider a scalar field in an expanding Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe with the action
S = Z
d4xp
gP (X, ), (3.1)
where the kinetic term
X = 1
2@µ @µ . (3.2)
The energy-momentum tensor can be put into the form of the perfect fluid,
Tµ⌫ = P gµ⌫ + (⇢ + P )uµu⌫, (3.3)
by identifying P as the pressure, and the energy density and the velocity as
⇢ =2X@P
@X P, (3.4)
uµ = @µ
p2X. (3.5)
The background evolution with constant equation of motion, 1 < w 1, can be modeled by the Lagrangian,
P = X V ( ), (3.6)
where, in the homogeneous and isotropic background, X = ˙2/2 0, and assuming the potential V 0.2 The dots denote the time derivatives. If one only requires
⇢ = X + V to be positive and allows V to be negative, then one can have w > 1 when X < V < 0.3
In the conformal Newtonian gauge, the perturbed FLRW metric is
ds2 = (1 + 2 )dt2+ a2(t)(1 + 2 )dx2, (3.7) where a is the scale factor, and and are the metric perturbations. The equa-tion of moequa-tion of the curvature perturbaequa-tion, R, is given by the Mukhanov-Sasaki equation in the Fourier space [94, 95],
R00+ 2A0
k is the wavenumber, H = ˙a/a is the Hubble parameter, is the field perturbation.
The primes denote the derivative with respect to the conformal time ⌘, defined by
2With (3.6), the evolution of constant w > 1 and given initial values i and ˙ican be realized by the ad hoc potential
V ( ) = ˙2
i(1 w)
2(1 + w) exphp
24⇡(1 + w) ( i)i .
3A negative potential with w > 1 may be invoked, for example, in the cyclic universe scenario [92]. It requires special care to treat the perturbations in such models [93]. If we assume the universe is not cyclic, and starts from a big bang followed by a decelerating era (which includes the case w > 1), then, as we point out in this work, the initial adiabatic vacuum is acausal.
In the cyclic universe scenario, the density perturbations in the post-bounce expanding phase is proposed to be seeded by the perturbations generated in the pre-bounce contracting phase.
However, the treatment of perturbations in the contracting phase and across the bounce is still under investigation. See, for example, [93] for a review.
dt = ad⌘. Introducing the new variable u = AR, we can get rid of the first-derivative term, turning (3.8) into
u00+
If the evolution of the universe is described by a constant w > 1, there is a simple relation A00/A = a00/a since
A =
r3(1 + w)
8⇡ a. (3.12)
The scale factor evolves as
a(⌘) = ai(1 + ↵⇠)1/↵, (3.13)
Note that w = 1/3 is a singular case.4 Corresponding ↵ and µ for some reference values of w are listed in Table 3.1.
4The super-horizon spectrum is asymptotically divergent for w = 1/3 (or µ = 0). To un-derstand why, first note that in this case the universe does not accelerate nor decelerate (¨a = 0), so the comoving scale of Hubble horizon is constant in time. If w is slightly smaller than 1/3, the universe accelerates but slowly. It takes a long time for the horizon to shrink a little. At the meantime the amplitudes of the fluctuations inside the horizon keep decaying, therefore the amplitude of the power spectrum changes much within a small range of k.
Table 3.1. Corresponding values of ↵ and µ for some reference equation-of-state parameter w. The parameter ↵ = (1 + 3w)/2 is related to the scale factor by (3.13),
and µ =|3(1 w)/2(1 + 3w)| describes the general solution of the perturbation through (3.16). Note that for the accelerating universe with w < 1/3, one has
↵ < 0, while for the decelerating universe with w > 1/3, one has ↵ > 0.
w 1 1 2
If we try to model the slow-roll evolution by assigning w = 1, we will end up with A = 0 and cannot proceed in the way we did in the previous paragraph. The way around that is to use the attractor solution of the slow-roll era. By writing the density and pressure in terms of field, we have
A =
0
H, (3.18)
assuming 0 < 0 without loss of generality. In the attractor regime, H as well as
˙ = 0/a are approximately constant, so we can write
A = ˙i
Ha, (3.19)
which is proportional to a as it is in (3.12). Also it can be verified by solving the Friedmann equation with constant H that (3.13) reproduces the scale factor in the slow-roll case, so the equation of motion (3.15) still holds. We will refer to the slow-roll limit as w' 1 in this paper.
For the superinflationary universe with w < 1, we model it by the Lagrangian,
P = X V ( ), (3.20)
with the sign of the kinetic term reversed. With the requirement ⇢ = X + V > 0, one generally has V > X > 0, and the scale factor still evolves as (3.13). By substituting the original definition of A with
A = ap
⇢ P
H , (3.21)
it turns out that the equation of motion of the curvature perturbation can still be written in the form of the Mukhanov-Sasaki equation (3.8), and the rest of the analysis follows.