We consider in this section the axion-like particle potential,
V = Λ4h and N an integer. We will take N = 1 so that the potential has a unique minimum at φ = πf . The decay constant has a lower bound close to the Planck mass so we are free to choose beyond that, we chose f = 7MP such that the number of e-folding is enough to generate a viable inflation.
The potential slow-roll parameters , η are,
= MP2
From (4.2.16), From (5.3.4) we can produce an expression for φ,
φ = 2f sin−1 From (4.3.7), we can calculate the tensor to scalar ratio for N = 60 before the end of inflation,
Figure 5.25: The tensor to scalar ratio vs. Q for Q < 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.
Figure 5.26: The tensor to scalar ratio vs. Q for Q > 1 using the slow-roll approximation. The dashed line indicates the upper bound given by Planck with 95% confidence level.
starts off well below the bound given by the Planck data and gets suppressed as the dissipation increases, while in figure 5.26 for high Q the tensor to scalar ratio is well below the bound and continues to do so as Q gets stronger.
The scalar spectral index (4.3.10) for N = 60 before the end of inflation can be calculated as,
ns≈ 1 − 5MP2 4(1 + Q)f2
3 − cos(φ/f ) 1 + cos(φ/f )
!
(5.3.11)
Figure 5.27: The tensor to scalar ratio vs. ns for Q < 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confi-dence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
Figure 5.28: The tensor to scalar ratio vs. ns for Q > 1 using the slow-roll approximation. The light blue region shows the range for ns with 95% confi-dence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
In figure 5.27, the tensor to scalar ratio vs. scalar spectral index data points starts from the 95% confidence region and slowly shifts towards the 68% confidence region as Q increases, while in figure 5.28 it somehow is in the border between the 68% and 95% confidence region.
Numerical Calculation
The initial conditions used in this model were ˜φ0 = 0.104 and ˜φ00 = 2.88 × 10−4 (See Appendix C).
Figure 5.29: The temperature evolution for Q = 10−5 and N ≈ 40.
Figure 5.30: The temperature evolution for Q = 30, N ≈ 3028.
Figure 5.31: The evolution of the inflaton and radiation energy density for Q = 10−5.
Figure 5.32: The evolution of the inflaton and radiation energy density for Q = 30.
From figure 5.29 and 5.30 it can be seen that the effect of the dissipation on the total duration of inflation is much more drastic than with the two models previously studied. An example where warm inflation does not have enough e-folding can be seen in figure 5.29 where the temperature T is only above H for approximately N ≈ 40. In figure 5.31, it can be seen that the radiation energy density is not comparable to the inflaton energy density at the end of inflation such that it can be the case that the behavior is close to the cold inflation, while in figure 5.32 it can be seen that the radiation energy density is comparable to the inflaton energy density.
Figure 5.33: The tensor to scalar ratio as a function of Q (Q < 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.
In figure 5.33, the tensor to scalar ratio starts off well below the bound given by the Planck data but deviated a little from the result in figure 5.25, while in figure 5.34 it can be seen that the tensor to scalar ratio is well below the bound and continues to do so as Q gets stronger.
Figure 5.34: The tensor to scalar ratio as a function of Q (Q > 1) obtained from (4.3.6). The dashed line indicates the upper bound given by Planck with 95% confidence level.
Figure 5.35: The tensor to scalar ratio as a function of the scalar spectral index ns (Q < 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
Figure 5.36: The tensor to scalar ratio as a function of the scalar spectral index ns (Q > 1) obtained from (4.3.9). The light blue region shows the range for ns with 95% confidence level and the yellow region shows the range for ns with 68% confidence level as given by Planck.
It can be seen in figure 5.35 that the tensor to scalar ratio vs. scalar spectral index data points started well inside the 68% confidence region and shifts towards 1 as the dissipation increases. This is also in contrast to the result in figure 5.27, while in figure 5.36 the tensor to scalar ratio vs. scalar spectral index data points are outside of the confidence regions given by Planck.
Chapter 6 Conclusion
We have shown that by showing explicitly the evolution of the temperature throughout inflation we could ensure that at the horizon exit the observables are within the context of warm inflation, this is in contrast to the slow-roll approach where there is no explicit process in order to determine if the warm inflation conditions are satisfied. For the numerical calculation, the result for the tensor to scalar ratio in chaotic inflation is now within the bound given by the Planck data in contrast to if it is considered in the cold inflation case, the scalar spectral index also seems to be fairly independent of the dissipation rate.
The higgs-like inflation and axion inflation are both an example of small-field models such that its tensor to scalar ratio are well inside the bound and are not surprising since they do not have the same problem in the cold inflation case as in chaotic inflation. The scalar spectral index results indicate that there seems to be more consistency for weak dissipation since for the axion inflation, the scalar spectral index is out of the region for the strong dissipation case, while for the higgs inflation either case are within the region. In general, the slow-roll approach and the numerical approach gives a different result, however, for the Chaotic inflation it just so happened that the scalar spectral index derived from slow-roll is approximately independent of the dissipation which in contrast to the Higgs and Axion case where it is dependent on the dissipation.
Appendix A
Scalar Field Theory
For a single scalar field, the Lagrangian density L is given by,
L = −1
2∂µφ∂µφ − V (φ). (A.0.1) where φ is the scalar field, the first term in the Lagrangian density is the kinetic term and V (φ) is the scalar field potential.
In curved spacetime, the action takes the form,
S = Z
d4x√
−gL, (A.0.2)
where g is the determinant of the Friedmann-Robertson-Walker metric gµν. By varying the action, we get
φ + 3H ˙¨ φ − a−2∇2φ + V,φ = 0. (A.0.3) For a homogeneous scalar field, we have
φ + 3H ˙¨ φ + V,φ = 0. (A.0.4) This equation is called the Klein-Gordon equation.
The Klein-Gordon equation is similar to the harmonic oscillator equation,
¨
x + β ˙x + kx = 0 (A.0.5)
such that the friction term in the Klein Gordon equation in accordance to the friction β in the harmonic oscillator equation is 3H.
Assuming that the scalar field φ is large initially and suppose we take a po-tential of the form V = 12m2φ2, then from the Friedmann equation,
H2 ≈ V
3MP2, (A.0.6)
we have,
H ≈ mφ
√6MP. (A.0.7)
This means that if the scalar field φ is large then H is also large which implies that the friction is large so that the scalar field φ moves very slowly down the potential in such a way that it approximately does not move from its original position, therefore H is approximately constant.
H = ˙a
a ≈ constant (A.0.8)
a ≈ eHt (A.0.9)
Appendix B
Perturbations during Inflation
In this appendix we follow closely the treatment by Baumann [5]. We consider the metric tensor of the perturbed FRW universe gµν,
gµν = ¯gµν+ δgµν, (B.0.1) where ¯gµν is the background metric and δgµν is the assumed small pertur-bations. The line element for the perturbed FRW universe in conformal time is given by,
ds2 = a2(τ )[−(1 + 2A)dτ2− 2Bidηdxi+ ((1 − 2D)δij + 2Eij)dxidxj]. (B.0.2)
B.1 Scalar, vector, tensor decomposition
The homogeneous, isotropic, and spatially flat background possesses sym-metry such that these symmetries allow a decomposition of the metric into scalar, vector, and tensor components. In other words, the metric tensor is symmetric so that instead of having sixteen degrees of freedom, it has only ten degrees of freedom. A and D transform as scalars, Bi transforms as a 3-vector, and Eij as a 3-tensor.
From vector calculus, a vector field can be decomposed into two parts such that,
B = ~~ BS+ ~BV, (B.1.1)
with ∇ × ~BS = 0 (ijk∂iBj = 0) and ∇ · ~BV = 0 (δij∂jBiV = 0); so that we can write ~BS = −∇B for some scalar B. In component notation, we have
Bi = ∂iB + BiV. (B.1.2)
We see that the first term of (35) is a scalar and the second term is a vector.
BiV has one constraint so it has two independent components.
Also, any rank-2 symmetric tensor can be decomposed such that,
Eij = EijS + EijV + EijT (B.1.3) where EijV satisfies the constraint as in the second term of (34). EijT satisfies the four constraints EiiT = 0 (traceless), δijEijT = 0 (transverse/divergenceless) so that it has only two independent components. Also, EijS and EijV can be written as,
EijS = (∂i∂j− 1
3δij∇2)E (B.1.4)
EijV = 1
2(∂jEi+ ∂iEj) (B.1.5) The ten degrees of freedom of the metric tensor have now been decomposed into a 4+4+2 degrees of freedom where the scalar part consists of A, B, D, and E; the vector part consists of BiV and Ei; the tensor part consists of EijT.
At first order perturbation, it can be seen that the scalar, vector, and tensor part do not couple to each other and evolve independently so that we can study them separately. We are interested in studying the scalar and tensor perturbations which are responsible for the density fluctuations and
gravitational waves. Vector perturbations are not important in our context since it couples to the rotational velocity perturbations in the cosmic fluid which tend to decay as the universe expands.