The equations of motion for this Lagrangian are given by: ¨ ϕ(6X − 1) + 3H(2X − 1) ˙ϕ − ϕ2(3X2− X) + 2 σ(σ2+ 1)(2X − 1) ˙ϕ ˙σ = 0 (3.59) and ¨ σ + 3H ˙σ + m2σ − 8σ Cϕ2(σ2+ 1)2(X 2 − X) = 0 (3.60) To get k-inflation with the first field we have to claim 1 1/σ2and H ≈q 4
Cϕ2(−X + 3X2).
Inserting these two conditions into (3.59), we obtain the same equation of motion as in section 3.3.1: ¨ ϕ = 6 C|ϕ| √ 3X2− X(2X − 1) ˙ϕ + 2 ϕ(X − 3X 2) 6X − 1 , (3.61)
with the fixed points given by (3.48). During this stage of inflation the second field σ decreases slowly, while ϕ increases. I.e. the k-inflation-term vanishes. The normal ”Klein- Gordon equation” is left and from now on the evolution is determined by the second field σ. σ oscillates and as we have chosen a quadratic potential, the averaged equation of motion of the oscillating field σ is zero (compare section 2.1 or [22])
The number of e-foldings is given by
N ≡ ln a(ta(t)end)
!
=
Z tend
tin
H(t)dt, (3.62)
where tend is given through the time, when the condition 1 >> f (σ) is violated.
Numerical Calculations
We investigate the properties of the full model (3.58) numerically. The parameters and initial conditions are set by: m = 0.001, γ = 1/20, ϕ(0) = 1.5, σ(0) = 3.63, ˙ϕ(0) = 1,
˙σ(0) = −0.5. I.e. a ∼ t40/3.
Figure 3.9 shows that w = γ − 1 ≈ −0.95 and thus power-law inflation takes place. After a sufficiently large number of e-foldings, N ≈ 77, the graceful exit occurs and the k-inflation-term vanishes. As we have chosen a quadratic potential, < w >→ 0 (see figure 3.10).
We vary again the initial conditions and parameters in the program and it follows that these can lie in a wide range, but the above mentioned initial conditions have to be fulfilled.
3.4
Slow-roll Hybrid-k-Inflation
In the section before we considered power-law hybrid-k-inflation, where a ∼ t3(w+1)2 . Now
we want to come back to the exponential inflation with a ∼ exp (√ε t). Analogously to the slow-roll k-inflationary model, mentioned in section 2.3 and [6], we want to derive a model where slow-roll inflation is made by the first field ϕ (called k-essence). The graceful exit is brought by the interaction with a second field σ, i.e. we add the kinetic and potential term (Y − V (σ)) and build in a further function K(ϕ, σ) = s(σ)k(ϕ).
Lagrangian and Initial Conditions We consider the following Lagrangian:
p = s(σ)k(ϕ)X + X2+ Y − V (σ). (3.63) s is a function of the second field and initially should be negative due to the desired inflationary behaviour. k = k(ϕ) is a positive function of the first field and has to be introduced to have a non-zero squared speed of sound for the k-inflationary phase. Possible functions for k(ϕ) are given in [6] and [5].The idea is that if s(σ) changes its sign, the k- inflation-term s(σ)k(ϕ)X + X2 becomes positive and the exit occurs. The energy density
can be calculated, using (3.6)
ε = s(σ)k(ϕ)X + 3X2+ Y + V (σ). (3.64) To get an analogous model to the one of section 2.3, we set the initial conditions as follows:
• s(σ) ≈ −1
• (Y + V (σ)) (s(σ)k(ϕ)X + 3X2), i.e. m H with H in ≈
q
−k(ϕ)X + 3X2.
Equations of Motion and Analysis
The equations of motion are given by (3.11) and (3.12).In first order ∂k/∂ϕ and ∂s/∂σ are small.
First of all we consider the k-inflationary stage. Therefore we can set s(σ) = −1 and thus we can do the same analysis as in section 2.3 with K(ϕ) = −k(ϕ) < 0. The zeroth-order slow-roll solution corresponds to:
Xsr= 1 2k(ϕsr), (3.65) ˙ ϕsr = σ q k(ϕsr), (3.66) εsr = 1 4k 2(ϕ sr), (3.67) Hsr = 1 2k(ϕsr). (3.68)
The number of e-foldings for the k-inflationary phase is given by
N = Z tend tin H(t)dt = Z σend σin H ˙σdσ, (3.69) with σend = σ∗. σ∗is the critical value where the initial condition for k-inflation (s(σ) ≈ −1)
3.4 Slow-roll Hybrid-k-Inflation 47
The squared speed of sound during the k-inflationary stage is given by (2.45)
c2s = p,X ε,X ≈
−k + 2X
−k + 6X. (3.70)
A possible function k(ϕ), which satisfies the stability requirement (2.46) and the slow roll condition (2.41) δX Xsr ' 2 3 ∂ ∂ϕ 1 √ k ! 1 (3.71) is k(ϕ) = 1/ϕ.
Initially the scalar field σ is chosen to be large and during the k-inflationary stage it rolls down very slowly, as H m. As a possible function for s(σ) one can chose s(σ) = − tanh (σ − σ∗). Then the change of sign occurs when σ crosses the critical value
σ∗.
Like in the power-law model 3.3.1, there are two ways of exiting inflation:
In the first case σ crosses the critical value σ∗ when the Hubble parameter H is still
larger than the mass m of the field σ. Then the k-inflation term becomes negligible in comparison to the potential V (σ). The last stages of inflation are driven by V (σ) and the exit occurs like in normal slow-roll-inflation. The time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied.
In the other case the Hubble parameter H is smaller than m of the field σ. The latter begins to oscillate and then crosses the critical value σ∗. Then we do not have a
potential-driven stage.
After the change of sign of the function s(σ), the energy density of the second field σ dominates the energy density of the first field. X towards zero and thus
lim
X→0wϕ ' limX→0
skX + X2
skX + 3X2 = +1,
i.e. εϕ ∼ a−6. As this energy density decreases much faster than the energy density of
radiation (εrad ∼ a−4) and the energy density of dust (εdust ∼ a−3), the first field ϕ vanishes
very quickly.
From now on the evolution is determined by the second field σ and the equation of motion for a potential V (σ) = 12m2σ2 is given by:
¨
σ + 3H ˙σ + m2σ = 0. (3.72) In [22] the approximate solution for this equation is given. For the case that the Hubble parameter H becomes smaller than the mass m, the WKB solution gives
σ ∼ m−1/2a−3/2sinZ mdt, (3.73)
i.e. σ starts to oscillate.
As we haven chosen a potential V ∼ σ2, we obtain with (2.16) an averaged equation of
and thus the energy density falls as a−3. In the case of a quartic potential we would obtain
< w >→ 1
3. I.e. the oscillating field σ mimics an ultra-relativistic fluid with a decreasing
energy density a−4.
Numerical considerations with V (σ) = 12m2σ2
We want to consider the model (3.63) numerically with V (σ) = 1 2m
2σ2, k(ϕ) = 1/ϕ
and s(σ) = − tanh (σ − σ∗). First of all we choose the following parameters and initial
conditions: σ∗ = 4, m = 0.0003, ϕ(0) = 100, σ(0) = 9.15, ˙ϕ(0) = 1, ˙σ(0) = −0.001. With
this parameters and initial conditions we obtain the figures 3.12 till 3.14.
Figure 3.13 shows that w ≈ −1 and thus quasi-exponential inflation takes place. During this k-inflationary stage the squared speed of sound is ≈ 0.005. After a large number of e-foldings, N > 65, the function s(σ) changes its sign and the energy density of the second field σ dominates. The k-inflationary stage ends and standard slow-roll inflation, driven by the potential V (σ), takes place. Then c2
s = 1, compare figure 3.14. After the graceful
exit at N > 75, σ starts oscillating and < w >→ 0, as we have chosen V ∼ σ2. For a
quartic potential we would get < w >→ 13.
We change the initial conditions and take σ∗ = 8.5, m = 0.0003, ϕ(0) = 100, σ(0) =
10.5, ˙ϕ(0) = 1, ˙σ(0) = −0.001. Then the function s(σ) changes its sign at N ≈ 20 and we have a longer potential-driven inflationary stage. The squared speed of sound for these initial conditions is illustrated in figure 3.11.
Like in the power-law model we see, that the time of the transit from k-inflation to the potential-driven inflation, can be varied by changing the initial value of σ and the parameter σ∗. This feature of the model may be important for the formation of structure,
because during these two inflationary stages the cosmological perturbations have different sound speeds and therefore, different amplitudes.
10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 PSfrag replacements N c2s
Figure 3.11: Illustration of the squared speed of sound c2
s for the initial condition σ(0) =
10.5 and the parameter σ∗ = 8.5. During the k-inflationary phase c2
s ≈ 0. At N ≈ 20 the
transit to the potential-driven inflationary phase occurs and from now on c2 s = 1.
3.4 Slow-roll Hybrid-k-Inflation 49 10 20 30 40 50 60 70 80 -1 -0.5 0.5 1 PSfrag replacements s N
Figure 3.12: Illustration of the function s(σ) = − tanh (σ − 4) for (3.63). When s → 0 the energy density of the k-essence field ϕ becomes negligible in comparison to the one of the second field.
10 20 30 40 50 60 70 80 -1 -0.5 0.5 1 PSfrag replacements w N
Figure 3.13: Illustration of the equation of state w = w(N ) for (3.63) with V = 1 2m
2σ2.
Till N = 65 we have k-inflation, driven by the first field ϕ. The last e-foldings are made by the inflation with the potential V (σ). The exit occurs when we reach N ≈ 77.
10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 PSfrag replacements c2s N
Figure 3.14: Illustration of the squared speed of sound c2
s. During the k-inflationary phase
c2
Chapter 4
Conclusions
The inclusion of non-canonical kinetic terms in the Lagrangian of a scalar field can have quite non-trivial and unexpected cosmological effects. Non-linear terms of this type are expected to appear in any effective field theory and do indeed arise in most models unifying gravity with other particle forces, including supergravity and superstring models.
Using two scalar fields σ,ϕ, we have shown that inflation can be even driven by the mix- term of both kinetic-energy terms XY . For this case of two kinetically coupled massless ghost-like scalar fields we have shown, that de-Sitter inflation is a late time asymptotic. This solution corresponds to an attractive fixed point, which is of the type of a stable star. This toy model might be a candidate for a dynamical dark energy. Another interesting feature of this model is a dynamical crossing of the so-called “ phantom divide” w = −1. This crossing is not possible for general one-scalar-field models of dark energy [29]. However the observations of dark energy even slightly prefer this transition through w = −1 [1]. In the case when the Lagrangian depends not only on the field derivatives, but also on the fields, the exit from inflation can be implemented.
We have introduced a further model called hybrid-k-inflation, in which the k-essence field ϕ drives inflation, while the second field σ brings the exit due to an interaction with ϕ. This hybrid-k-inflationary model can produce a power-law accelerating expansion, where a ∼ t2/(3(w+1)), or a quasi-exponentially inflationary stage (slow-roll model), where
a ∼ exp (Ht).
We presented two classes of power-law models. In the first case the pressure changed its sign as the field σ rolled down and crossed a critical value σ∗. The last stages of inflation
were driven by the potential of the second field σ and the exit from inflation occurred in the same way as in standard slow-roll-inflation. The time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied by choosing different initial conditions and parameters. Furthermore we realized a graceful exit in this model as follows: When the Hubble parameter H became less than the mass m of the field σ the latter began to oscillate, crossed a critical value σ∗ and then changed the sign of the pressure of the
k-essence. In this case no potential-driven inflationary stage occurred.
In the second class of power-law models the graceful exit was realized in a different way. When the Hubble parameter H became less than the mass m of the field σ the latter began
to oscillate near vacuum and suppressed due to a specific interaction the contribution of the k-essence ϕ to the Friedmann equations. In this model we only had k-essence and no potential-driven inflation.
For the slow-roll hybrid-k-inflationary model we have shown that after a sufficiently long stage of inflation the second field σ intervened and the graceful exit from inflation can be implemented in the same way as in the power-law model. In this model the time-duration of the k-inflationary stage and the potential-driven inflationary stage can be varied as well. As in the power-law case, this model allows also to have no potential-driven stage.
We have shown that the cosmological dynamics governed by two non-trivially coupled scalar fields is very rich. The models of this type can lead to two observationally dif- ferent inflationary stages, occurring one after another. The transition to the Friedmann decelerating universe can occur with different energy scales, which can be independent on the generated spectrum of fluctuations. Thus the presence of an additional degree(s) of freedom makes it difficult to distinguish the various inflationary models by observations.
Appendix A
Power-law Inflation
Analogously to power-law inflation in [6] the formulas for a more general Lagrangian
p = F (ϕ)(−X + αX2) (A.1) will be derived. α is a positive constant (normally taken = 1) and 0 < γ < 23. Note that γ = w + 1 = (p/ε) + 1. I.e. by selection of a certain γ the power of t is set, since the scale factor evolves as
a ∼ t3γ2 . (A.2)
The energy density can be calculated with (3.6) and gives p = F (ϕ)(−X +3αX2). Inserting
p and ε into ε + p = γε one gets the fixed point
X = 2 − γ
α(4 − 3γ). (A.3)
Expressing ε in terms of p and substituting (A.1) with X given by (A.3) into
˙εtot = −3H(εtot+ ptot),
one gets the differential equation for F (ϕ):
∂F ∂ϕ !2 f (ϕ)−3 = 9 4C (A.4) with C = 4 − 3γ (γ − 1)2 2 − γ 4 − 3γ(γ − 1 + 1 α) − γ !2 . (A.5)
Note, that if α = 1 we obtain the same result as in [6]: C = 4−3γ4γ2 . The differential equation has the following solutions:
F1(ϕ) =
4
F2(ϕ) =
4
(√Cϕ + ϕ∗)2,
with ϕ∗ =constant.
Thus the Lagrangian for power-law k-inflation is:
p = 4
Cϕ2(−X + αX
2), (A.6)
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Danksagung
Zun¨achst m¨ochte ich meinem Betreuer Prof. Dr. Viatcheslav Mukhanov danken, der mir diese Arbeit erm¨oglicht und mit Anregungen meine Arbeit gef¨ordert hat. Weiterhin gilt mein Dank dem Zweitkorrektor Prof. Dr. Dieter L¨ust. Ein großer Dank geht an Alexander Vikman, der immer zu Diskussionen bereit war und mir mit Rat und Tat zur Seite stand. Desweiteren m¨ochte ich mich bei Dr. Serge Winitzki, meinen Zimmerkollegen, Kommili- tonen, Freunden und Eltern f¨ur die stete Hilfsbereitschaft und Unterst¨utzung bedanken.
Erkl¨arung
Ich versichere, diese Arbeit selbst¨andig angefertigt und dazu nur die im Literaturverzeich- nis angegebenen Quellen benutzt zu haben.
M¨unchen, den 31.01.2006