The T = 1 Boundary

solution, i.e. the center manifold originating from dS₀ whose asymptotics are given by n= 1 : H ∼ −t, ϕ∼ −t, σ ∼0, ρ_pf ∼(−t)⁻², as t→ −∞ (4.55a) n= 2 : H ∼e^−23t, ϕ∼e^−3t, σ∼0, ρ_pf ∼e^{−43 t}, as t→ −∞ (4.55b) n≥3 : H ∼(−t)ⁿ⁻²ⁿ , ϕ∼(−t)ⁿ⁻²² , σ ∼0, ρ_pf ∼(−t)⁻ⁿ⁻²²ⁿ , as t→ −∞ (4.55c)

(a)The Kasner circle in the Ωϕ= 0 plane in theT = 1 boundary.

FL1

Ωφ=1( _pf=0(

Ω P₁

21

P

14

P

Σ_Φ

X

(b)Σ = 0 plane in theT = 1 bound-ary..

Figure 4.4: he invariantT = 0 boundary and the invariant subset for a monomial potential V =¹₄(λϕ)⁴.

Proof. To prove this theorem we need to use Lemma4.1alongside with generalized averaging techniques based on the methods used in [50,66,78]. We make the same approach regarding ϵ as previous seen in Chapters 2-3. So given real function f, its average over a time period associated to Ωϕis given by

⟨f⟩= 1 P(Ωϕ)

Z τ0+P⁽Ωϕ) τ0

f(τ)dτ. (4.59)

Taking the time averaging for ^dX_dτ in (4.56) and using the equation for Σϕ it gives d

dτ

XdX dτ

− dX

dτ 2

+nX²ⁿ= 0. (4.60)

Taking the time averaging for the orbit we get

*dX dτ

2+

=⟨Σ²_ϕ⟩=n⟨X²ⁿ⟩. (4.61)

Thus, using this result for a periodic orbit onT = 1 in (4.10) we get

⟨γ_ϕ⟩= 2n

n+ 1. (4.62)

We now set ϵ(τ) = 1−T(τ) and consider the system dΩϕ

dτ = 3ϵ^h(γpf−γϕ)(1−Ωϕ) + (2−γpf)Σ²_σⁱΩϕ (4.63a) dΣσ

dτ =−3

2ϵ^h(2−γ_pf)(1−Σ²σ) + (γ_pf−γ_ϕ)Ωϕ

iΣσ (4.63b)

dϵ dτ =−1

nϵ²(1−ϵ)(1 +q) (4.63c)

where (X,Σϕ,Σ+,Σ−) solves (4.13) and q+ 1 = 3

2(2Σ²ϕ+ 2Σ²++ 2Σ²−+γ_pf(1−Ωϕ−Σ²σ)). (4.64) The general idea of this averaging method is based on making the near identity transformation

Ωϕ(τ) =y(τ) +ϵ(τ)w(y, z, τ, ϵ) (4.65a) Σσ(τ) =z(τ) +ϵ(τ)g(y, z, τ, ϵ), (4.65b) and then prove that the evolution of the variables y and z are approximated, at first order, by the solution of ¯y and ¯z respectively of the averaged equation.

So starting withythe evolution equation for this new variable can be obtained using equations (4.63a) and (4.63c) together with the evolution equation for Ωϕ. This then gives

dy

dτ = 1 +ϵ∂w

∂y −1"

dΩϕ

dτ −

w+ϵ∂w

∂ϵ dϵ

dτ −ϵ∂w

∂τ −ϵ∂w

∂z dz dτ

#

= 1 +ϵ∂w

∂y −1"

3ϵ(γpf − ⟨γ_ϕ⟩)y(1−y) + (⟨γ_ϕ⟩ −γϕ)y(1−y) + (2−γpf)yz²

− 3ϵ²(γpf−γϕ) (1−2y)w+w(2−γpf)z²+g(2−γpf)y (4.66) + 3ϵ³(2−γpf)gw−(γpf −γϕ)w²+ (2−γpf)g²y−

w+ϵ∂w

∂ϵ dϵ

dτ −ϵ∂w

∂τ −ϵ∂w

∂z dz dτ

# .

Forz we made a similar thing using the equations (4.63b) and (4.63c) and we get dz

dτ = 1 +ϵ∂g

∂z −1"

dΣσ

dτ −

g+ϵ∂g

∂ϵ dϵ

dτ −ϵ∂w

∂τ −ϵ∂g

∂y dy dτ

#

= 1 +ϵ∂g

∂z −1"

− 3

2ϵ(2−γ_pf)z(1−z²) + (γ_pf − ⟨γ_ϕ⟩)yz+ (⟨γ_ϕ⟩ −γ_ϕ)yz

− 3

2ϵ²(2−γ_pf)g(1−z)²−2+ (γ_pf −γ_ϕ)wz+ (γ_pf−γ_ϕ)gy (4.67)

+ 3

2ϵ³(2−γ_pf)zg²+ (2−γ_pf)g³−(γ_pf −γ_ϕ)gw−

g+ϵ∂g

∂ϵ dϵ

dτ −ϵ∂g

∂τ −ϵ∂g

∂y dy dτ

# . Setting

∂w

∂τ =f₁(y, z, τ, ϵ)− ⟨f₁(y, z, .,0)⟩= 3⟨γ_ϕ⟩y−2Σ²ϕ

y(1−y) (4.68a)

∂g

∂τ =f2(y, z, τ, ϵ)− ⟨f₂(y, z, .,0)⟩= 3 (γϕ− ⟨γ_ϕ⟩)yz (4.68b)

and expanding (4.66) and (4.67) in powers of ϵsmall enough, we get dy

dτ =ϵ⟨f₁⟩(y, z) +ϵ²h₁(y, z, w, g, τ, ϵ) +O(ϵ³) (4.69a) dz

dτ =ϵ⟨f₂⟩(y, z) +ϵ²h2(y, z, w, g, τ, ϵ) +O(ϵ³) (4.69b) where

⟨f₁⟩(y, z) =⟨f₁(y, z, .,0)⟩= 3(γ_pf− ⟨γ_ϕ⟩)y(1−y) + 3(2−γ_pf)yz² (4.70a)

⟨f₂⟩(y, z) =⟨f₂(y, z, .,0)⟩=−3

2(2−γpf)z(1−z²)−3

2(γpf− ⟨γ_ϕ⟩)yz (4.70b) h₁(.) =3(γ_pf−γ_ϕ)(1−2y)w+w(2−γ_pf)(wz²+gy)+ 1

n(1 +q)(1−ϵ)w (4.70c) 3

2

∂w

∂z

(2−γ_pf)z(1−z²) + (γ_pf− ⟨γ_ϕ⟩)yz−3∂w

∂y

(γ_pf− ⟨γ_ϕ⟩)y(1−y) + (2−γ_pf)yz² h2(.) =− 3

2

(2−γpf)g(1−z)²−2+ (γpf−γϕ)(wz−gy)+ 1

n(1 +q)(1−ϵ)w

(4.70d)

−3∂g

∂y

(γ_pf − ⟨γ_ϕ⟩)y(1−y) + (2−γ_pf)yz²+3 2

∂g

∂z

(2−γ_pf)z(1−z²) + (γ_pf − ⟨γ_ϕ⟩)yz. Notice that for large times,T ≈1,ϵ≈0, the right-hand side of (4.69a) and (4.69b) is almost

periodic meaning that ⟨γ_ϕ⟩ −γ_ϕ ≈0 which implies that w and g are bounded. So it follows from (4.65) that y and z are also bounded. We can now drop the high order terms in ϵ in (4.69) and study the truncated averaged equation, which leads to then system

d¯y

dτ = 3ϵ(γpf− ⟨γ_ϕ⟩) ¯y(1−y) + 3ϵ¯ (2−γpf) ¯yz¯² (4.71a) dz¯

dτ =−3

2ϵ(2−γ_pf) ¯z1−z¯²−3

2ϵ(γ_pf − ⟨γ_ϕ⟩) ¯yz¯ (4.71b) dϵ

dτ =−1

nϵ²(1−ϵ)(1 +q). (4.71c)

Without loss of generality we can introduce a new time variable 1

ϵ d dτ = d

dτ¯, (4.72)

that will reduce the system. The new system reads dy¯

dτ = 3 (γ_pf− ⟨γ_ϕ⟩) ¯y(1−y¯) + 3 (2−γ_pf) ¯yz¯² (4.73a) dz¯

dτ =−3

2(2−γ_pf) ¯z1−z¯²−3

2(γ_pf− ⟨γ_ϕ⟩) ¯yz¯ (4.73b) dϵ

dτ =−1

nϵ(1−ϵ)(1 +q). (4.73c)

Fixed points and stability in the case γpf ̸=⟨γ_ϕ⟩

For γ_pf ̸= ⟨γ_ϕ⟩ the dynamical system admits four fixed points. One in scalar field subset (Ωϕ= 1), two equivalent fixed points in the pure anisotropic subset (Σσ = 1) and one isolated fixed point in the pure matter subset (Ωpf = 1). The first fixed point is located on the intersection ofϵ= 0 (T = 1) with the pure scalar field subset and is

F₁ : y¯= 1, z¯= 0, ϵ= 0. (4.74) The linearisation around this fixed point yields to the eigenvalues−3(γpf−⟨γ_ϕ⟩),−³₂(2−⟨γ_ϕ⟩), and −^3⟨γ_2n^ϕ⟩ were the associated eigenvectors are the canonical basis ofR³.

The two equivalent fixed points located in the intersection ofϵ= 0 with the pure anisotropic subset are

F₂^± : y¯= 0, z¯=±1, ϵ= 0. (4.75) The linearisation around this fixed points yields to the eigenvalues 3(2− ⟨γ_ϕ⟩), 3(2−γ_pf) and

−_n³ whose eigenvectors are (∓2,1,0), (0,1,0) and (0,0,1).

The isolated fixed point is

F₃ : y¯= 0, z¯= 0, ϵ= 0, (4.76) the linearisation around this fixed points yields to the eigenvalues 3(γpf− ⟨γ_ϕ⟩), −³₂(2−γpf), and −^3γ_2n^pf whose eigenvectors are the canonical basis of R³.

Regarding the eigenvalues of each fixed point we need to split the analysis depending on the sign ofγ_pf− ⟨γ_pf⟩.

• Ifγ_pf >⟨γ_ϕ⟩, F1 has three negative eigenvalues since ⟨γ_ϕ⟩ ∈(0,2) so it is a hyperbolic sink. For F₂^± since γ_pf ∈ (0,2) we see that two eigenvalues are positive and the one in the ϵ-direction is negative, so F₂^± is an hyperbolic saddle, being a source in the {ϵ= 0}-subset. LastlyF₃ as two negative eigenvalues and one positive eigenvalue being a hyperbolic saddle. So in this case F1 is the future attractor, so the universe will asymptotically approach the pure scalar field solution when ¯τ →+∞.

• If γ_pf < ⟨γ_ϕ⟩, F1 has two negative eigenvalues an one positive eigenvalue so it is a hyperbolic saddle. Regarding F₂^± we have two positive eigenvalues and one negative eigenvalue, moreover in the {ϵ = 0} subset F₂^± is a hyperbolic source. Lastly, F3 has three negative eigenvalues being a hyperbolic sink. In this caseF₃is the future attractor, so the universe will asymptotically approach the the flat FL solution when ¯τ →+∞.

Convergence of the solutions in the case γpf− ⟨γ_ϕ⟩

Now we need to prove that the variables y and Ωϕ follows this evolution and the same need to be done forz and Σσ.

We start by introducing two new variables

|η(τ)|:=|y(τ)−y¯(τ)|, |ζ(τ)|:=|z(τ)−z¯(τ)| (4.77) that we need to estimate. In order to do so we define the sequences{τ_n}and {ϵ_n} such that ϵ_n=ϵ(τ_n), with n∈Nand

τn+1−τn= 1 ϵn

, τ0 = 0, ϵ0>0 (4.78)

where limτ_n= +∞ and limϵ_n= 0. So starting with the estimation of|η(τ)|we get

|η(τ)| = Z τ

τn

3ϵ(γ_pf − ⟨γ_ϕ⟩)y(1−y) + 3ϵ(2−γ_pf)yz²+ϵ²h₁(y, z, w, g, s, ϵ)ds

− Z τ

τn

(3ϵ(γpf− ⟨γ_ϕ⟩) ¯y(1−y¯) + 3ϵ(2−γpf)¯yz¯²

≤ 3ϵn

Z τ τn

|γ_pf− ⟨γ_ϕ⟩|

| {z }

|.|≤C1

|(y−y)¯ | |1−(y+ ¯y)|

| {z }

|.|≤1

ds+ 3ϵn

Z τ τn

|2−γpf|

| {z }

C2

|yz²−y¯z¯²|

| {z }

|.|≤|y−¯y|+|z²−¯z²|

ds + ϵ²_n

Z τ τn

|h₁(y, z, w, g, s, ϵ)|

| {z }

|.|M₁

ds+O(ϵ³_n)

≤ 3ϵ_n(C₁+C₂)^{Z τ}

τn

|η(s)|ds+ 3ϵ_n Z τ

τn

|z²−z¯²|

| {z }

|.|≤|z−¯z||z+¯z|≤2|z−¯z|

+ϵ²_nM₁(τ−τ_n) +O(ϵ³_n)

≤ 3ϵn(C1+C2)^{Z τ}

τn

|η(s)|ds+ 6C2ϵn

Z τ τn

|ζ(s)|ds+ϵ²_nM1(τ −τn) +O(ϵ³_n). (4.79) For|ζ(τ)|=|z(τ)−z¯(τ)|, in this case we get

|ζ(τ)| = Z τ

τn

−3

2ϵ(2−γ_pf)z(1−z²)−3

2ϵ(γ_pf− ⟨γ_ϕ⟩)yz+ϵ²h2(y, z, w, g, s, ϵ)ds

− Z τ

τn

−3

2ϵ(2−γpf) ¯z(1−z¯²)−3

2ϵ(γpf − ⟨γ_ϕ⟩) ¯y¯z

≤ 3 2ϵn

Z τ τn

|2−γpf|

| {z }

|.|≤C2

|z−z| |¯ 1−zz¯−z²−z¯²|

| {z }

|.|≤2

ds+3 2ϵn

Z τ τn

|γ_pf− ⟨γ_ϕ⟩|

| {z }

|.|≤C1

|yz−y¯z|¯

| {z }

|.|≤|y−¯y|+2|z−¯z|

ds + ϵ²_n

Z τ τn

|h₂(y, z, w, g, s, ϵ)|

| {z }

|.|≤M₂

ds+O(ϵ³_n)

≤ 3ϵn(C1+C2)^{Z τ}

τn

|ζ(s)|ds+3 2ϵnC1

Z τ τn

|η(s)|ds+ϵ²_nM2(τ−τn) +O(ϵ³_n). (4.80) whereC₁,C₂,M₁, andM₂ are positive constants.

Let’s introduce a new function|ψ(τ)|=|η(τ)|+|ζ(τ)|so adding the two above equations to each other we get

|ψ(τ)| = 3ϵn(C1+C2)^{Z τ}

τn

|η(s)|ds+ 6C2ϵn

Z τ τn

|ζ(s)|ds+ϵ²_nM1(τ −τn) + 3ϵn(C1+C2)^{Z τ}

τn

|ζ(s)|ds+3 2ϵnC1

Z τ τn

|η(s)|ds+ϵ²_nM2(τ −τn) +O(ϵ³_n)

= 3ϵn(C1+C2)^{Z τ}

τn

(|η(s)|+|ζ(s)|)ds+ 6C2ϵn

Z τ τn

|ζ(s)|ds+3 2ϵnC1

Z τ τn

|η(s)|ds + ϵ²_n(M₁+M₂)(τ −τ_n) +O(ϵ³_n)

Using the fact that 6C2

Z τ τn

|ζ(s)|ds+3 2C1

Z τ τn

|η(s)|ds≤2 max{6C2,3 2C1}

Z _τ

τn

|ζ(s)|ds+^{Z τ}

τn

|η(s)|ds

we get

|η(τ)| ≤ ϵ_nC∗

Z τ τn

|ψ(s)|ds+ϵ²_nM∗(τ−τ_n) (4.81)

≤ ϵn

M∗

C∗

e^{C∗ϵn(τ−τn)}−1+O(ϵ²_n) (4.82) whereC∗= 3 (C₁+C₂) + 2 max{6C₂,³₂C₁}and M∗ =M₁+M₂ are positive constants . So for τ−τ_n∈[0,1/ϵ_n] the above inequality becomes

|ψ(τ)| ≤ϵ_nK (4.83)

where K > 0. Since |ψ(τ)|= |η(τ)|+|ζ(τ)| → 0 as ϵn → 0 it follows that |η(τ)| → 0 and

|ζ(τ)| →0 asϵ_n→0. It follows than from (4.65a) , the triangle inequality, and the fact that ϵ→0 asτ →+∞, it follows from that Ωϕhas the same limit as ¯yand, therefore converges for 0 or 1 depending on the sign ofγ_pf− ⟨γ_ϕ⟩. Regarding Σσ using (4.65b), the triangle inequality and ϵ→ 0 as τ →+∞, it follow that Σσ goes to zero since the system will converge to the point (Ωϕ,Σσ, ϵ) = (1,0,0) if γ_pf >⟨γ_ϕ⟩ and will converge to the point (Ωϕ,Σσ, ϵ) = (0,0,0) ifγ_pf <⟨γ_ϕ⟩. This completes the proof for the cases (i) and (ii) of the theorem.

Fixed points and stability in the case γpf =⟨γ_ϕ⟩

Now we analyse the caseγ_pf =⟨γ_ϕ⟩. In this case our averaged system (4.73) reads d¯y

dτ = 3(2−γpf)¯y¯z² (4.84a)

dz¯ dτ =−3

2(2−γ_pf)¯z(1−z¯²) (4.84b) dϵ

dτ =−1

nϵ(1−ϵ)(1 +q). (4.84c)

The system admits a line of fixed points in the intersection of the scalar field subset with ϵ= 1 that is given by

L_y : y¯=y₀, z¯= 0, ϵ= 0 (4.85) the linearisation around this line of fixed points yields to the eigenvalues 0,−_n+1³ , and −_n+1³ whose eigenvectors are the canonical basis ofR³. In this case we have two negative eigenvalues (sinceγ_pf ∈(0,2)) and a zero eigenvalue corresponding to a center manifold. To solve this we will introduce adapted variables

˜

y= ¯y−y¯0, z˜= ¯z, ˜ϵ=ϵ (4.86) which leads to the adapted system

dy˜

dτ =F(˜y,z, ϵ¯ ), dz¯ dτ =−3

2(2−γ_pf)¯z+G(˜y,z, ϵ¯ ), dϵ

dτ =−3γ_pf

2n ϵ+N(˜y,z, ϵ¯ ) (4.87) where F,G and N are high order functions. With this new adapted variables we relocated the fixed points on the lineL_y to the origin. The 1-dimensional center manifold can be locally represented by the graph h :E^c→ E^s, i.e., (¯z, ϵ) = (h1(˜y), h2(˜y)) satisfying the fixed point, h(0) = 0 and the tangency, ^dh(0)_d˜y = 0 conditions which solves the following equations

F(˜y, h1(˜y), h2(˜y))h^′₁(˜y) =−3

2(2−γpf)h1(˜y) +G(˜y, h1(˜y), h2(˜y)) (4.88a) F(˜y, h₂(˜y), h₂(˜y))h^′₂(˜y) =−3γ_pf

2n h₂(˜y) +N(˜y, h₁(˜y), h₂(˜y)) (4.88b) In general is not possible to find the exact solution for hi and Taylor expansion are used, however in this particular case the can find an exact solution for h₁ andh₂ so we get

h₁(˜y) =±^q1 +e^2Cz(˜y+y₀), h₂(˜y) = 1 1 +_1+C ^{y+y˜ 0}

ϵe^{2Cz( ˜y+y0)}

γpf 2n(2−γpf )

(4.89)

Therefore, it follows that the center manifold reads dy˜

dτ = 3y(1 +e^2Cz(y+y₀))(2−γ_pf) (4.90) which shows explicitly that each point on the lineLy are center saddles.

The system also admits two more fixed points in the intersection of the subset{z= 1} and {ϵ= 0}given by

F₂^± : y= 0, z=±1, ϵ= 0. (4.91)

The linearisation around this fixed points yields to the eigenvalues _n+1³ , _n+1³ , and−_n³ were the associated eigenvectors are the canonical basis of R³. In this case we have two positive eigenvalues and a negative eigenvalue, so we are in a presence of a saddle. Moreover in the {ϵ= 0}subset we see thatF₂^± are sources.

Therefore the line of fixed is normally hyperbolic and each point on the line is the ω-limit point of a unique interior orbit. So there exists an orbit of the dynamical system (4.84) with ϵ >0 initially that converges to (y₀,0,0) for each y₀ asτ →+∞.

Convergence of solutions in the case γ_pf =⟨γ_ϕ⟩

Just as in the proof of the cases (i) and (ii), we can again estimate|η(τ)|and|ζ(τ)|however the estimation is very similar to the ones introduced in (4.79) and (4.80) since the only difference is thatγ_pf− ⟨γ_ϕ⟩= 0. Asϵ_n→0 then|η(τ)| →0 and so yand ¯yhave the same limit and the same can be said regarding z and ¯z. Finally, from equation (4.65a), the triangle inequality, and the fact that ϵ → 0 as τ → +∞, it follows from that Ωϕ has the same limit as ¯y and therefore will converge to a point between (0,1). In the case of Σσ using (4.65b), the triangle inequality andϵ→0 asτ →+∞, it follow that Σσ goes to zero. This concludes the proof of the case (iii).

The global profile for our solutions in the (X,Σϕ, T)-plane can be seen in Fig. 4.5. While the qualitative solutions for the (Σ+,Σ−, T)-plane can be found in Fig.4.6

(a)Global picture forγpf= 1. (b)Global picture forγpf=⁴₃. (c)Global picture forγpf=³₂.

Figure 4.5: Qualitative solutions for the scalar field potential V(ϕ) = ¹₄(λϕ)⁴ in the (X,Σϕ, T)-plane for various matter of state.

Figure 4.6: Qualitative solutions for the scalar field potential V(ϕ) = ¹₄(λϕ)⁴ in the (Σ+,Σ−, T)-plane forγpf = ⁴₃.

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