Exact Inflationary Solution. Sergio del Campo

Transcription

1 Exact Inflationary Solution Sergio del Campo Instituto de Física Pontificia Universidad Católica de Valparaíso Chile I CosmoSul Rio de Janeiro, 1 al 5 de Agosto, 2011

2 Inflation as a paradigm. Models Slow-roll approximation Idea of inflation The universe underwent a quasi exponential expansion at early history: In S, a Planck size region blows up to a factor of 10 30!

3 Inflation as a paradigm. Models Slow-roll approximation Idea of inflation The universe underwent a quasi exponential expansion at early history: In S, a Planck size region blows up to a factor of 10 30! What cause inflation? Scalar field φ (Usually called the inflaton field). V (φ) Dynamic of φ (The scalar field rolls down the scalar potential)

4 Models for inflation Introduction Models Slow-roll approximation Guths original idea (1981) - Used the entropy point of view. Figure from A. Albrecht and P. Steinhardt, Phys. Rev. D, 48, 1220 (1981).

5 Models for inflation Introduction Models Slow-roll approximation Guths original idea (1981) - Used the entropy point of view. Figure from A. Albrecht and P. Steinhardt, Phys. Rev. D, 48, 1220 (1981). Now, (even in eternal inflation!) people prefer a chaotic type. No supercooling/tunneling from false vacuum. No plateau (in the scalar potential). No thermal equilibrium.

6 Inflation in the abstract Models Slow-roll approximation Inflation is an add-on to the standard Big-Bang scenario.

7 Inflation in the abstract Models Slow-roll approximation Inflation is an add-on to the standard Big-Bang scenario. Following equivalent descriptions hold: An era of accelerated expansion, ä > 0 An era of shrinking horizon ( ) ( d 1 ä ) dt ah ȧ 2 < 0 An era of a non-standard equation of state, (ρ + 3p) < 0

8 Introduction Models Slow-roll approximation During the era of inflation the field equation are: H 2 = 8π [ ] 1 2 φ 2 + V (φ) 3m 2 Pl (1) φ + 3H φ + dv (φ) dφ = 0, (2) Here φ: scalar inflaton field. V (φ): (effective) scalar potential. H: the Hubble parameter ( ȧ a, with a the scale factor). : derivatives with respect to the cosmological time, t. mpl 2 = 1/G: G the Newtons constant.

9 Models Slow-roll approximation The slow-roll approximation It is intended to solve the field Eqs. giving an explicit expression for V (φ).

10 Models Slow-roll approximation The slow-roll approximation It is intended to solve the field Eqs. giving an explicit expression for V (φ). Even in the simplest case results hard to find a solution.

11 Models Slow-roll approximation The slow-roll approximation It is intended to solve the field Eqs. giving an explicit expression for V (φ). Even in the simplest case results hard to find a solution. The slow-roll approximation it is intended. H 2 8π 3m 2 Pl V (φ) (3) dv (φ) 3H φ dφ, (4)

12 Models Slow-roll approximation The slow-roll approximation It is intended to solve the field Eqs. giving an explicit expression for V (φ). Even in the simplest case results hard to find a solution. The slow-roll approximation it is intended. H 2 8π 3m 2 Pl V (φ) (3) dv (φ) 3H φ dφ, (4) This approximation results from considering 1 2 φ 2 V (φ). It makes sure that effectively we get an accelerated expansion, since it results into ω 1.

13 Models Slow-roll approximation Successes of the inflationary model Solves a lot of problems of the Big-Bang model Gives the seed for primordial density perturbations (usually quantum fluctuations). It creates a nearly scale-free density perturbations which can grow to create structure in the universe.

14 Models Slow-roll approximation Successes of the inflationary model Solves a lot of problems of the Big-Bang model Gives the seed for primordial density perturbations (usually quantum fluctuations). It creates a nearly scale-free density perturbations which can grow to create structure in the universe. Some problems of the inflationary universe model: No embed inflation in a fundamental theory. No conclusion yet on correct functional form of inflationary scalar potential V (φ).

15 Models Slow-roll approximation Successes of the inflationary model Solves a lot of problems of the Big-Bang model Gives the seed for primordial density perturbations (usually quantum fluctuations). It creates a nearly scale-free density perturbations which can grow to create structure in the universe. Some problems of the inflationary universe model: No embed inflation in a fundamental theory. No conclusion yet on correct functional form of inflationary scalar potential V (φ). In concerning to the slow-rolling approximation, inflation fails to be applied at the end of inflation, where the kinetic energy ( 1 2 φ 2 ) becomes important.

16 Basics Introduction Advantages Amount of inflation and the H-parameters In place of giving the scalar potential V (φ) take the Hubble parameter as a function of the inflaton field φ, i.e. H = H(φ).

17 Basics Introduction Advantages Amount of inflation and the H-parameters In place of giving the scalar potential V (φ) take the Hubble parameter as a function of the inflaton field φ, i.e. H = H(φ). Set of equations φ = m2 Pl 4π 32π2 m 4 Pl dh dφ = m2 Pl 4π H (5) V (φ) = (H ) 2 12π mpl 2 H 2. (6)

18 Basics Introduction Advantages Amount of inflation and the H-parameters In place of giving the scalar potential V (φ) take the Hubble parameter as a function of the inflaton field φ, i.e. H = H(φ). Set of equations φ = m2 Pl 4π 32π2 m 4 Pl dh dφ = m2 Pl 4π H (5) V (φ) = (H ) 2 12π mpl 2 H 2. (6) This set is usually called the Hamilton-Jacobi Eqs. Eqs. (5) and (6) substitute the previous Eqs. (1) and (2). As far as H(φ) is given, the set of Eqs. (5) and (6) yields to exact solutions (No term has been dropped).

19 Also, from Eq. (5) which gives Introduction a H = 4π ah [ a(φ) = a 0 exp 4π mpl 2 Advantages Amount of inflation and the H-parameters mpl 2 φ φ 0 ] H H dφ. (7)

20 Also, from Eq. (5) which gives Introduction a H = 4π ah [ a(φ) = a 0 exp 4π mpl 2 Advantages Amount of inflation and the H-parameters mpl 2 φ φ 0 ] H H dφ. (7) Eqs.(5) and (6) allow one to generate an endless collection of exact inflationary solutions via the following procedure: Choose a form for H(φ). Use Eq. (5) to find the scalar field φ = φ(t). Use Eq. (6) to find the scalar potential V (φ). With H(φ) and φ(t) you could get H = H(t), then from here get a = a(t). H(φ) solution generating function [Carr & Lidsey, PRD (1993)]

21 Advantages Introduction Advantages Amount of inflation and the H-parameters have some advantages over the approximate slow-roll solutions. The form of the potential is readily deduced (from Eq. (6)). have direct application, where the kinetic energy of the inflaton field inevitable becomes significant. Not much work have been done with regard to this last point!

22 Advantages Amount of inflation and the H-parameters Amount of inflation and the H-parameters Inflation is commonly characterized by the number of e-folding of physical expansion that occur. ( ) af N ln, but, much better if comovil Hubble length, 1/aH is used [ ] (ah)f N ln. (ah) i Reasons ä > 0 is equivalent to d ( 1 ) dt ah < 0. The reduction of 1/aH (and not 1/a) solves the flatness and horizon problems. In evaluating perturbation it is important the relation k = ah. a i

23 The H-inflationary parameters From the field Eqs. it is possible to show where ɛ H m2 Pl 4π Also, it is found ä a = H2 (1 ɛ H ), Advantages Amount of inflation and the H-parameters ( ) H 2 H. ɛ H < 1 during inflation. ɛ H = 1 at the of inflation. where η H m2 Pl 4π dv (φ) dφ = 3 ( ) H H. mpl 2 ( 4π ɛ HH ) 3 η H,

24 ɛ H m2 Pl 4π η H m2 Pl 4π Introduction ) 2 ( H H ( H H ) In the slow-roll limit ɛ H ɛ η H η ɛ ɛ and η are the slow-roll parameters. Also, N = 2 π m Pl Advantages Amount of inflation and the H-parameters H-inflationary parameters φ φ i dφ ɛ, and N = 2 π m Pl φ (1 ɛ H ) dφ. φ i ɛ Note that N N (and N = N for ɛ H 1, the so-called extreme slow-roll approximation).

25 Spectral scalar and tensor perturbations Spectral scalar and tensor perturbations The spectral of scalar, P 1/2 R (k), and transverse-traceless tensor perturbations, P T are given by [ Lyth & Stewart, PLB (1992); Liddle & Lyth, PLB (1992).] P 1/2 R (k) = 1 ( ) H 2 = 2 ( ) H 2 2π φ mpl 2 H and P T = 16 π k=ah ( ) H 2 m 2 Pl The scalar and gravitational wave spectral indices are given by and n 1 + d ln P R d ln k n g d ln P T d ln k, respectively.

26 The brane-world The brane-world inflation (Hawkins & Lidsey, PRD 63, (2001) Rap. Comm. ) Here, the Friedman Eq. becomes ( 4π H 2 = 3λm 2 Pl ) ρ (ρ + 2λ) Introducing a new function y = y(φ), such that ρ 2λ y 2 1 y 2 (y 2 < 1 due to the weak energy condition (ρ 0)), we get 16πλ y H = 3mPl 2 1 y 2, φ = λm 2 Pl 3π y V (φ) = 2λ y 2 1 y 2 λm2 Pl 6π 1 y 2, ( y 1 y 2 )

27 The brane-world Also, [ a(φ) = exp 4π ] φ mpl 2 dφ y 3π φ y, t t 0 = λmpl 2 dφ y 2 1 φ 0 y. ( 2π ) Taking y(φ) = sech C m Pl φ, with C an arbitrary constant, gives: v(φ) = λ ( 6 C 2) ( ) Cosech 2 2π C φ 3 m Pl [ φ(t) = m ] Pl C C 2 [ 4πλ 2π Cosh 1 (t t 0) ; 4πλ C 4 m Pl 3 a(t) = 3mPl 2 This corresponds to a scaling solution. Also, ω p φ ρ φ = 1 3 ( C 2 3 ) ] 1/C 2 (t t 0) 2 1 If C 2 < 2 (C 2 > 2) = ω < 1/3 (= ω < 1/3) the universe inflates (decelerates). For small t, a(t) t 1/C 2

28 Attractor solutions δh(φ): a linear perturbation δh(φ) < exp ( 3 N N i ). could be applied to any kind of Friedmann Eq. Example H 2 = f (ρ). Note that this has been applied to a single scalar field, the inflaton field, φ. Needs to be generalized to include more fields.

29 Intermediated inflation: a(t) exp { At f } The scalar potential: V (φ) = 8A2 (β + 4) 2 [ ] φ β [6 2Aβ ( ) ] β 2 φ This is obtained from H(φ) = Af (2βA) β/4 φ β/2. Slow roll: V (φ) = 48A2 (β + 4) 2 (2Aβ)β/2 φ β