Pre-inflationary homogenization of scalar field cosmologies

Physics Letters B 703 (2011) 537–542

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Physics Letters B www.elsevier.com/locate/physletb

Pre-inflationary homogenization of scalar field cosmologies Artur Alho a , Filipe C. Mena a,b,∗ a b

Centro de Matemática, Universidade do Minho, Gualtar, 4710-057 Braga, Portugal Department of Physics, Yale University, P.O. Box 208120, New Haven, CT 06520, USA

a r t i c l e

i n f o

Article history: Received 5 April 2011 Accepted 16 August 2011 Available online 22 August 2011 Editor: M. Trodden Keywords: Cosmology Early universe Inflation Stability Perturbations

a b s t r a c t We consider the evolution of covariant and gauge invariant linear density perturbations of scalar field cosmologies using a dynamical systems’ approach. We find conditions for which the perturbations decay in time, so that the spacetime approaches a homogeneous solution which inflates, for quadratic and exponential potentials. This pre-inflationary homogenization is found to be stable in the potentials’ parameter spaces. Furthermore, in each case, we determine the minimum size of the resultant homogeneous patch and show that, for quadratic potentials, the resulting inflationary solutions include those with the necessary number of e-folds. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In the inflationary scenario, an early period of accelerated expansion is usually assumed to explain the present large-scale homogeneity and spatial flatness of the universe. However, it has been shown, under fairly general assumptions such as the weak energy conditions, that inflationary models require pre-existing homogeneity over a horizon volume [1–3]. Therefore, an important question is under what conditions can a universe with initial small inhomogeneities approach a homogeneous state which develops inflation. Several past works considered this problem either by using numerical approximations or particular exact solutions to the Einstein field equations (EFEs) with scalar fields. Goldwirth and Piran [4–7] considered inhomogeneous scalar fields on Friedmann–Lemaitre–Robertson–Walker (FLRW) backgrounds in a so-called effective density approximation and using numerical schemes in spherical symmetry taking into account the backreaction in the metric. They concluded that new inflation requires homogeneity over a region of several horizon sizes and that chaotic inflation requires a sufficiently high average value of the scalar field over several horizon sizes. Other authors arrived to similar conclusions using different approaches, see e.g. [3,8]. A particularly interesting result along these lines is due to Deruelle and

*

Corresponding author at: Department of Physics, Yale University, P.O. Box 208120, New Haven, CT 06520, USA. E-mail addresses: [email protected] (A. Alho), [email protected] (F.C. Mena). 0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2011.08.044

Goldwirth [9] who considered a semi-numerical analysis, for inhomogeneous quasi-isotropic universes using a long wavelength iterative scheme, and found sufficient conditions for the onset of inflation by limiting the degree of inhomogeneity in their models. More recently, Sakai [10] used numerical approximations in a spherically symmetry model to find an example of topological inflation which allows higher degrees of inhomogeneity over horizon sizes, provided the vacuum expectation value is large enough. Exact anisotropic and inhomogeneous symmetric spacetimes have also been considered in the past to address this problem. Burd and Barrow [11] took scalar fields with exponential potentials on spatially homogeneous but anisotropic G 3 backgrounds and found that if inflation occurs, then isotropy is always reached (see also [12,13] and references therein). In turn, Ibañez et al. [14, 15] have looked at inhomogeneous G 2 exact scalar field solutions with exponential potentials and compared in each case: asymptotic isotropization, approach to inflation and the existence of inhomogeneities. They find, in particular, classes of models which do not isotropize. Also connected to this problem, we note the recent result of Bolejko and Stoeger [16] who investigated inhomogeneous dust spherical spacetimes and found non-zero measure sets of initial conditions that give rise to spontaneous homogenization of cosmological models. It would then be important to consider now non-symmetric inhomogeneous models in cosmology. This can be done using a linear perturbative analysis which includes small initial inhomogeneities. We shall address this question in the context of scalar field cosmologies which have been considered as early universe models,

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see e.g. [17] and references therein. In particular, we take some of most studied inflationary solutions in FLRW spacetimes due to scalar fields with a quadratic potential (e.g. slow-roll inflation), and with an exponential potential (power-law inflation) in order to study the evolution of linear scalar perturbations. We shall use a covariant and gauge invariant perturbative approach, which, by construction, has the advantage of having clear geometrical and physical interpretations in cosmology [18–20]. We shall combine this framework together with a dynamical systems’ approach in order to prove the stability of the pre-inflationary homogenization process for quadratic and exponential potentials. The Letter is organized as follows: In Section 2, we present our perturbative framework, derive the system of perturbation equations for a general scalar field potential and calculate the system’s fixed points. Section 3, contains our main results about the stability of the pre-inflationary homogenization for scalar fields with exponential and quadratic potentials, including a phase space analysis of the respective perturbation systems. Section 4 contains a summary of our conclusions.

  12 Φ dΦ , ξ(Ψ, Φ) = − 1 + √ 6 Ψ dφ    6 Φ dΦ 6 Φ dΦ 2 ζ (Ψ, Φ) = −2 3Φ + √ 5+ √ 6 Ψ dφ 6 Ψ dφ  2 2 d Φ n + 6Φ 2 − 2 2 . dφ 2a H The use of the variable U(n) makes the analysis of the system’s stability quite transparent: If an orbit is asymptotic to an equilibrium point, the perturbation approaches a stationary state either: decaying to zero if U(n) < 0, growing if U(n) > 0 or having a constant value U(n) = 0. If the orbit is asymptotic to a periodic orbit in the cylinder, then the perturbations propagate as waves (see [22] for details). In order to study the stability of the flat background inflationary solutions, we take K = 0, in which case the fixed points P of the system (4) are given by:

dΦ

It is well known that, for linear perturbations of any FLRW background spacetime, scalar perturbation variables Δ(x, τ ) satisfy a partial differential equation (PDE) of the form

Δ + A(τ )Δ + B(τ )Δ =

D2 H2

(1)

Δ,

where A(τ ) and B (τ ) depend on the background solution, H is the Hubble function, the prime represents differentiation with respect to conformal time τ and D 2 is the Laplace–Beltrami operator. A common procedure when analyzing cosmological perturbations is to turn the PDE into an ordinary differentialequation (ODE) for Δ by using the harmonic decomposition Δ = n Δ(n) Q (n) such 2

that D 2 Q (n) = − na2 Q (n) , where n ∈ N is the wave number, a the FLRW scalar factor and Q  (n) = 0. We shall use this procedure and consider a FLRW background with a scalar field φ and potential V . Furthermore, we shall use the phase of perturbation variable (see [21,22])

U(n) :=

Δ(n)

(2)

Δ(n)

and the following expansion normalized variables (see e.g. [17])

ψ Ψ := √ , 6H

√ Φ := √

V

3H

,

K := −

3

R

6H 2

(3)

where ψ = H φ  , and 3 R is the 3-Ricci scalar. Then, we can show (see [23] for details and also [20]) that Eq. (1) coupled to the FLRW scalar field background evolution equations result in the following system for the unknown state vector ((Ψ, Φ), U ):

U(n) = −U(2n) − ξ(Ψ, Φ)U(n) − ζ (Ψ, Φ), √ dΦ   Ψ  = 2Ψ 3 − 2 + Φ 2 Ψ − 6Φ , dφ √ dΦ   Φ  = −Φ 3 + 1 + 2Ψ 2 Φ + 6Ψ dφ

6

(1 − Φ)(Φ − 4),

 1 U(±n) (P ) = −ξ(P ) ± ξ 2 (P ) − 4ζ (P ) P:

2. Covariant and gauge invariant linear density perturbations

√

dφ

=

6

(6)

2

with 2 ξ(P ) = 6ΦP − 1,



2 4 − 18ΦP − 12ΦP ζ (P ) = 24ΦP

d2 Φ dφ 2

 P

+

n2 H 2 a2

.

It is important to note that, since the background evolution equations form an autonomous subsystem, the background fixed points are also fixed points of the system (4). Moreover, we can restrict the phase space analysis to either the state-vector (Ψ, U(n) ) or (Φ, U(n) ), since Ψ and Φ are related through the background constraint (5). 3. Pre-inflationary homogenization of scalar field cosmologies There are many examples of scalar field potentials in cosmology (see e.g. [24,25]). Among these, two of major importance are the quadratic and exponential potentials. The former are physically relevant due to the so-called slow roll regime, after which the physical process of reheating occurs [26]. In turn, exponential potentials can give rise to models which are consistent with observations of the present acceleration of the universe and their theoretical importance arises from scalar-tensor and string theories (see e.g. [27] and references therein). A dynamical systems approach to study the generality of inflation in FLRW scalar field cosmologies has been extensively used, particularly for exponential potentials. In that case, the spacetime symmetry leads to the decoupling of the Raychaudhuri equation from the evolution equations (for the normalized variables, see [17]), which can then be written as a 2-dimensional dynamical system. In the case of quadratic potentials, there is no such symmetry and the evolution equations form a higher-dimensional system. However, in that case, there are other interesting approaches to the problem (see [28–31,33]). Here, we will make use of the approach in [33], which is reviewed in Section 3.2.

(4) 3.1. Exponential potentials

subject to the background constraint equation In the case of exponential potentials one has

Ψ 2 + Φ2 + K = 1 and with

(5)

V (φ) = Λe

λφ

,

dΦ dφ

=

λ 2

Φ and

d2 Φ dφ 2

=

 2 λ 2

Φ,

(7)

A. Alho, F.C. Mena / Physics Letters B 703 (2011) 537–542

Fig. 1. Plot of p (λ) =

1 4

4

− 32 λ2 + λ4 . The non-negative region gives the allowed values

of n2 for which there are fixed points in the system (4), for an exponential potential.

539

Fig. 2. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for √ an exponential potential with λ = 2 − 1 in the long wavelength limit. The figure shows the saddle point in the region U(n) < 0.

with λ > 0 the slope (or control) parameter and Λ > 0 the cosmological constant. In the background spacetime, it is known that the fixed point in the circumference S1 is given by (see [32])

P:

√   λ 6 − λ2 , (Ψ, Φ) = − √ , √ 6

(8)

6

with the deceleration parameter qP = λ 2−2 . Thus, real solutions √ exist for 0 < λ < 6 and are inflationary (with qP < 0) if 2

0<λ<

√

2,

(9)

which corresponds to the power-law inflationary solutions studied by Halliwell in [32]. So, for the background subsystem, the only fixed point P is the one given by Eq. (8). For the perturbed system (4), we find two fixed points given by



 P , U(±n) (P ) ,

(10)

where U(±n) (P ) are given by (6) with

ξ(P ) = 5 − λ2 and ζ (P ) = 6 − λ2 +

n2 a2 H 2

Fig. 3. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for

√

an exponential potential with λ = 2 + 1 in the long wavelength limit. The figure shows the saddle point in the region U(n) > 0.

(11)

.

Thus, real solutions exist if the wave number satisfies



n2  n2crit = a2 H 2

1 4

3

λ4

2

4

− λ2 +



,

implying (see Fig. 1)

0<λ

√

2 − 1 or

√

2+1λ<

√

6.

(12)

When n = ncrit , the points U(±n) merge into a single saddle point. In √ √

particular, fixed points with λ = 2 − 1 and λ = 2 + 1 only exist in the long wavelength limit and are saddle points (see Figs. 2 and 3). For n > ncrit , the fixed points cease to exist, the orbit is periodic and the perturbations behave as waves (see Fig. 4). Furthermore, we have from (11) that ζ (P ) > 0. Thus, if (12) is satisfied, it follows that

ξ(P ) > 0

⇒

ξ(P ) < 0

⇒

U(+n) U(+n)

Fig. 4. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for an exponential potential with λ = are periodic.

√

2. For this value of the slope parameter the orbits

< 0, > 0. √

Therefore, when the slope parameter satisfies 0 < λ < 2 − 1, the density perturbation modes√decay (see Figs. 5 and 6). On the √ other hand, if 2 + 1 < λ < 6, there are perturbation modes

which grow (see Fig. 7). Since the values of the slope parameter, for which there exist decaying√modes, are within the values for power-law inflation (0 < λ < 2), we conclude that, in the case of an exponential potential, there is a non-zero measure set in the slope parameter space such that the process of dynamical

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A. Alho, F.C. Mena / Physics Letters B 703 (2011) 537–542

Fig. 5. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for an exponential potential with λ = 0.1 in the long wavelength limit. The figure shows the equilibrium points, one of which is the future attractor (P , U(+n) ) having U(n) < 0.

Fig. 7. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for an exponential potential with λ = 2.42 in the long wavelength limit. The figure shows the equilibrium points, one of which is the future attractor (P , U(+n) ) having U(n) > 0.





1

−

P2 :



2

1−

1−

4 9



 M2

,

1 2

 1+

1−

4 9

 M2

so that the fixed points P1 and P2 exist in the unitary circumference for 0  M  32 . Moreover, using a fixed point approximation1 it was shown in [33] that, for the background solution, P2 is the attractor on the unit circumference of constant M and that it √ is inflationary if qP2 < 0 ⇒ Ψ 2 < 13 , which implies 0 < M < 2. For M < 0.15, it was shown that the inflationary√solution has the necessary number of e-folds N > 60. For M = 2, the attractor point ceases to be inflationary, and the value of φ , at this point (which represents the end of inflation), corresponds to that of the slow-roll solution, see [33]. In the linearly perturbed case, we find that the fixed points of the system (4) are Fig. 6. Zoom of Fig. 5 around the fixed points (P , U(±n) ). This shows in more detail the local dynamics around the future attractor (P , U(+n) ).

a2 H 2

1

(13)

< . 4

Our results are summarized in Table 1. 3.2. Quadratic potentials

V (φ) =

2

dΦ

φ2,

dφ

M =√

6

dΦ 2

and

dφ 2

P1 :



−

1 2

 1+

1−

4 9

 M2 ,



1

= 0,

2

 1−

 P2 , U(±n) (P2 )

(15)

(14)

1−

4 9

M2

 ,

4

ξ(P1,2 ) = 2 ∓ 3 1 − M2 , 9

4

n2

9

H 2 a2

ζ (P1,2 ) = 3 ∓ 3 1 − M2 + 2M2 +

.

where the minus and plus signs stand for P1 and P2 , respectively. We also find that the attractor point (P2 , U(+n) (P2 )) of (4), always lies in the region U(n) < 0 of the phase-space (see Figs. 8 and 9) and it only exists if

a2

where M := m and m > 0. H For the background evolution, we recall that [33] used M as a control parameter in order to reduce the background evolution equations to a 2-dimensional system for the state vector (Ψ, Φ). In this case, the fixed points of the background (ΨP , ΦP ) ∈ [−1, 0] × [0, 1] in S1 , are given by condition (6), which reads Ψ Φ = − M 3 and gives (see also [33])





n2

In the case of quadratic potentials, one has

m2

 P1 , U(±n) (P1 ) ,

where the U(±n) (P ) are given by Eqs. (6) with

pre-inflationary homogenization exists. The minimum size of the homogeneous patch is given by the maximum allowed value for n2 , which in this case, corresponds to the maximum of p (λ) in the region where there is decay and, in turn, this is achieved for λ arbitrarily small, i.e. for

n2



 H

2

1 4

 − 3M , 2

(16)

which, in turn, implies

1

M √

12

(17)

.

Moreover, for M = √1

12

the fixed point only exist in the long

wavelength limit and it is a saddle point (see Fig. 10). For M > √1 , the orbits are periodic, see Fig. 11. Since the solutions are 12

1 This is a semi-analytic result valid for small M and is particularly useful to study the early universe inflationary dynamics.

A. Alho, F.C. Mena / Physics Letters B 703 (2011) 537–542

541

Table 1 Summary of the results of Sections 3.1 and 3.2 about the behavior of density perturbations in flat FLRW scalar field backgrounds with exponential and quadratic potentials. In this table, we use the notation ΨP2 = −

1 2



1−



1−

4 M2 9



.

Potentials

Future attractor

Parameter space

V (φ) = Λe λφ

Point Ψ = − √λ , U(+n) < 0

0<λ<

√

6

Point Ψ = − √λ , U(+n) > 0

√

6

2−1<λ<

6

V (φ) =

m 2

φ2

2 − 1, n2 < n2crit

2+1<λ<

Periodic orbit Ψ = − √λ 2

√

Point Ψ = ΨP2 , U(+n) < 0 Periodic orbit Ψ = ΨP2

√ √

M>

Inflationary solutions

Pert. decays

0<λ<

6, n2 < n2crit

Pert. grows

2+1

Pert. is a wave

0 < M < √1 , n2 < n2crit √1 12

Physical meaning

12

Pert. decays

√

0
2

√

2

Pert. is a wave

Fig. 8. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for a quadratic potential with M = 0.2 in the long wavelength limit. The figure shows the equilibrium points (P1 , U(±n) ) and (P2 , U(±n) ). The future attractor (P2 , U(+n) ) has U(n) < 0.

Fig. 10. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for a quadratic potential with M = √1 in the long wavelength limit. The figure shows

Fig. 9. Zoom of Fig. 8 around the fixed points (P2 , U(±n) ). This shows in more detail

Fig. 11. Density perturbations described by orbits in the phase plane (Ψ, U(n) ) for a quadratic potential with M = 0.3. For this value of the control parameter the orbits are periodic.

the local dynamics around the future attractor (P2 , U(+n) ).

12

the two saddle points.

√

inflationary for 0 < M < 2, the inequality (17) implies the existence of an open subset, 0 < M < √1 , in parameter space such

the necessary number of e-folds. Our results are summarized in Table 1.

that there is pre-inflationary homogenization. Furthermore, the minimum size of the homogeneous patch corresponds to arbitrarily small values of M, i.e. to

4. Conclusion

12

n2 a2 H 2

1

(18)

< . 4

This in agreement with the numerical results of Goldwirth and Piran [5,6] (see also a discussion of results in [3]), and gives analytical constraints on the size of the inhomogeneities on the onset of inflation. Finally, we note that the condition M < √1 includes 12

M < 0.15, which corresponds to inflationary solutions that have

We have considered the problem of pre-inflationary homogenization of scalar field cosmologies using a dynamical systems approach in covariant and gauge-invariant linear perturbation theory. We have established conditions under which the linearly perturbed inhomogeneous cosmologies homogenize having inflationary solutions as global attractors, for quadratic and exponential potentials. We have shown that the pre-inflationary homogeniza-

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A. Alho, F.C. Mena / Physics Letters B 703 (2011) 537–542

tion occurs for open sets in the control parameter spaces and, therefore, that this process is stable in these settings. For all cases considered here, the models lead to a homogeneous patch whose minimum size corresponds to arbitrarily small values of the control parameters. For quadratic potentials, these results confirm previous numerical studies [5,6] and, furthermore, include inflationary solutions which have the necessary number of e-folds [33].

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Acknowledgements

[18] [19] [20] [21] [22]

F.C.M. thanks Vincent Moncrief and hospitality from the Department of Physics, Yale University, where this work has been completed. The authors were supported by projects PTDC/MAT/108921/ 2008 and CERN/FP/116377/2010, by CMAT, Univ. Minho, through FCT plurianual funding and by FCT grants SFRH/BD/48658/2008 and SFRH/BSAB/967/2010. References [1] [2] [3] [4] [5] [6] [7]

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