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Inflation

Inflation. Università Milano Bicocca,2005. Margutti Raffaella. 1.Introduction. The standard cosmology is a successful framework for interpreting observations. In spite of this fact there were certain questions which remained unsolved until 1980s.

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Inflation

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  1. Inflation Università Milano Bicocca,2005 Margutti Raffaella

  2. 1.Introduction The standard cosmology is a successful framework for interpreting observations. In spite of this fact there were certain questions which remained unsolved until 1980s. For many years it was assumed that any solution of these problems would have to await a theory of quantum gravity. The great success of cosmology in 1980s was the realization that an explanation of some of these puzzles might involve physics atlower energies: “only” 1015 Gev, vs 1019Gev of quantum gravity. THE CONCEPT OF INFLATION WAS BORN.

  3. What follows is an outline of the main features of inflation in his “classical” form; The reader will find more than one model of inflation in scientific literature; here we will refer to the standard inflation which involves a first order cosmological phase transition.

  4. 2.Classical problems of standard isotropic cosmology 2.1 The horizon problem From CBR observations we know that: On angular scales >> 1°. Sandard cosmology contains a particle horizon of radius: In the radiation dominated era,when a(t) ~t 1/2. (We will use natural units, c=1). In the matter dominated era (a(t) ~t 2/3) : R po(t0)=3t0~6000 Mpc h-1 At t= tls (last scattering) Rpo(tls)=3tls Because of the expansion of the universe the universe at last scattering is now:

  5. Subtending an angle of about 1° • The microwave sky shows us homogeneity and isotropy on angular scales >>1° • Why do we live in a nearly homogeneous universe even though some parts of the universe are not (or not yet ) causally connected???

  6. 2.2 The flatness problem From the first Friedmann equation: We have (see appendix 1): At the Plank epoch: Remembering that

  7. We have: To get Ω0 1 today requires a FINE TUNNING of Ω in the past. At the Plank epoch which is the natural initial time, this requires a deviation of only 1 part in 1061 !!! However , if Ω =1 from the beginning  Ω =1 forever But a mechanism is still required to set up such an initial state

  8. 3.The idea of inflation

  9. In the figure we have • ACCELARATED EXPANSION This is the most general features of what become known as the INFLATIONARY UNIVERSE. Equation of state (from the second Friedmann equation): To solve the horizon problem and allow causal contact over the whole of the region observed at last scattering requires a universe that expands more than linearly (yellow in the previous figure) We want The general concept of inflation rests on being able to achieve a negative-pressure equation of state.  This can be realized in a natural way using quantum field theory.

  10. 4.Basic concepts of quantum field theory 4.1 The Lagrangian density Real scalar field Potential of the real scalar field , usually in the form where m is the mass of the field in natural units • The restriction to scalar field is not simply for reasons of simplicity but because is expected in many theory of unification that additional scalar field such as the Higgs field will exist. • The scalar field is in general complex. We will use a realone only for simplicity.

  11. 4.2 Energy momentum tensor and equation of state • The Lagrangian density written above is obviously invariant under space-time translations of the origin of the reference system. • The existence of a global symmetry leads directly to a CONSERVATION LAW, according to the Noethern’s theorem.(See appendix 2 for details). The conserved energy-momentum tensor is From this we read off the energy density and pressure, since: With the conventions that

  12. If we add the requirement of homogeneity of the scalar field: If The equation of state is : This is of the type we need in order to solve the horizon problem!(p< -1/3 ρ).

  13. 4.3 Dynamics of the field From the Euler –Lagrange equation of motion: We now derive the equation of motion for the scalar field. In order to be correct in general relativity the lagrangian density L needs do take the form of an invariant scalar times the jacobian In a Friedmann-Walker-Robertson model: The Euler-Lagrange equation than becomes: From which it’s not difficult to obtain: With the requirements of homogeneity of the field:

  14. 5.Cosmological implications 5.1 Evolution of the energy density If: • The universe is dominated by the scalar field Φ with Lagrangian and p= -ρ , that’s to say • The scalar field is not coupled with anything From the relation Adding the equation of state for the field (p= -ρ) and solving we have: and since with From the first of the Friedmann equation:

  15. 5.2 Exponential expansion From the first Friedmann equation: • More then linear expansion: • this is what we need in order to solve the horizon problem

  16. 5.3 Necessity of Cosmological Phase Transition • The discussion so far indicates a possible solution of the problems of standard cosmology, but has a critical, missing ingredient. • In the period of inflation the dynamics of the universe is dominated by the scalar field Φ, which has as equation of state. There remains the difficulty of returning to a “normal” equation of state: • THE UNIVERSE IS REQUIRED TO UNDERGO A COSMOLOGICAL PHASE TRANSITION

  17. 5.4 Necessity of Reheating • The exponential expansion produces a universe that is essentially devoid of normal matter and radiation; • Because of this the temperature of the universe becomes <<T, if T was the temperature at the beginning. • We know that at the end of the inflation the temperature has to be high enough in order to allow the violationof the barion number and nucleosynthesis. • A phase transition to a state of 0 vacuum energy, if istantaneous, would transfer the energy of the field to matter and radiation as latent heat.  THE UNIVERSE WOULD THEREFORE BE REHEATED

  18. 6.The potential of the scalar field and the SRD approximation • In order to solve the equation of motion of Φ we have to specify a particular form of the potential. • Different forms of V(Φ) have been explored during the years and each of them produces a different type of expansion of the universe. Requirements on V(Φ) : 1.In order to have negative perssure: From this system we derive a(t)

  19. 2.THE SRD (SLOW-ROLLING-DOWN) APPROXIMATION: The solution of the equation of motion become tractable if we make the socalled SRD approximation: From the equation of motion we have: • The condition than becomes a condition on Φ: (using the first of Friedmann equations)

  20. In the SRD approximation:

  21. (From Friedmann equation). • We will use a potential of the form: In the figure we can see the temperature dependent potential of the form written above, illustrated at various temperatures: At T>T1 only false vacuum is available; At T<T2, once the barrier is small enough, quantum tunneling can take place and free the scalar field to move: we have a first order transition to the vacuum state. It’s important to remark that the energy density difference between the two vacuum states is

  22. 7.The Inflation solution of standard cosmology problems 7.1 The horizon problem In order to solve the horizon problem we need the horizon of the inflationary epoch to be now bigger than ours: Horizon during inflation: Our horizon (matter dominated expansion) Growth of inflationary horizon from the end of inflation up to now Expansion of the horizon during inflation If ti<<te

  23. If the comoving entropy is conserved, then: a3T3=const (This is non true when p=p(T,Θ) , that’s to say: when pressure is not only function of the temperature.This is what happens for example during phase transition at a temperature different from the critical one) Remembering that in natural units: • From SRD (1 st Fried.equat.) • If we are dealing with a quantum field at temperature μ, then en energy density is expected in the form of vacuum energy. Where μ 10 15-16 Gev (From GUT theories

  24. We define: Te= Temperature at the end of inflation Its value is strongly dependent on reheating A phase transition to astate of zero vacuum energy , if instantaneous, would transfer the energy To normal matter and radiation (case of perfect reheating) the universe would therefore be reheated. In approximation of “perfect” reheating: It will be proved below that this is also exactly the number needed to solve the flatness problem

  25. 7.2 The flatness problem As we have already seen, from the first of Friedmann equations we have (see appendix 1 for details): We take: t*=ti and t=te Remembering that ρ is nearly constant during inflation, we have: Exponential expansion:

  26. We deduce: because of the factor We would like to have an estimate of the parameter Ω(t) at the present epoch Ω(t0)  Ω0 again the relation with tt0 te

  27. If we have perfect reheating:

  28. 7.3 Number of e-foldings: criteria for inflation As we have already seen, successful inflationin any model requires more than 60 e-foldings of the expansion.The implications of this fact are easily calculated using the SRD equation: Using the first of Friedmann equations:

  29. N > if V’< A model in which the potential is sufficiently flat (V’<<) that slow-rolling down can begin will probably achieve the critical 60 e-foldings. • The criterion for successful inflation is thus that the initial value of the field exceeds the Plank scale (mp)

  30. 8.Ending of inflation • The relative importance of time derivatives of Φincreases as Φrolls down the potential and V approaches zero. The inflationary phase will cease! • The field will oscillate about the bottom of the potential, with oscillation becoming damped because of the friction term.

  31. If the equation of motion remains the one written above (absence of coupling), then: • We will have a stationary field that continues to inflate without end, if V(Φ=0)>0. • We will have a stationary field with 0 energy density. • BUT • If we introducein the equation the couplings of the scalar field to matter field: • this thing will cause the rapid oscillatory phase to produce particles, leading to reheating

  32. 8.1 Absence of coupling From the relation: It’s not difficult to derive: And in presence of the scalar field and radiation: Remembering that: Equation of motion  0

  33. This extra term is often added empirically to represent the effect of particle creation; • The effect of this term is to remove energy from the motion of Φ and damping it in the form of a radiation background; • Φundergoes oscillations of declining amplitude after the end of inflation and Γonly changes the rate of damping. • For more detailed models of reheating see Linde (1989) and Kofman , Linde & Starobinsky (1997). 8.2 Adding a term of coupling: It’s the same thing as varying the equation of motion of the scalar field We have in this way: and also (harmonic oscillations) = Temperature of reheating G=degree of freedom Energy density for relativistic particles in the case of perfect reheating Because of the factor even in the case of perfect reheating is < of the initial one

  34. A plot of the exact solution for the scalar field in a model with a potential. The top panel shows how the absolute value of the field falls smoothly with time during the inflationary phase, and then starts to oscillate when inflation ends. The bottom panel shows the evolution of the scale factor a(t). We see the initial exponential behavior flattening as the vacuum ceases to dominate The two models shown have different starting conditions: the former (upper lines in each panel) gives about 380 e-foldings of inflation; the latter only 150. (From Peacock,1999).

  35. 9. Relic fluctuations from Inflation 9.1 Fluctuation spectrum • During inflation there is a true event horizon, of proper size 1/H • This fact suggest that there will be thermal fluctuations present, in analogy with black holes for which the Hawking temperature is: • The analogy is close but imperfect, and the characteristic temperature here is: • The inflationary prediction is of a horizon scale amplitude fluctuation The main effect of these fluctuations is to make different parts of the universe have fields that are perturbed by an amount δΦ with:

  36. We are dealing with various copies of the same rolling behavior Φ(t) but viewed at different times, with: • The universe will then finish inflation at different times, leading to a spread in energy density. • The horizon scale amplitude is given by the different amounts that the universe have expanded following the end of inflation: (Indetermination on the scalar field, from quantum theory of fields. See Peacock, 1999 for details) This plot shows how fluctuations in the scalar field transform themselves into density fluctuations at the end of inflation. Inflation finishes at times separated by in time for the two different points, inducing a density fluctuation

  37. 9.2 Inflation coupling From the SRD equation, we know that the number of e-foldings of inflation is: • If Since N ≈ 60 and the observed value of fluctuations (Really weak coupling!!!)

  38. If From the first of Friedmann equations: And since is needed for inflation, From CBR observations This constraints appear to suggest a defect in inflation, in that we should be able to use the theory to explain why , rather than using observations to constrain the theory

  39. 9.3 Gravity Waves Inflationary models predict a background of gravitational waves of expected rms amplitude: It’s not easy to show from a mathematical point of view how such a prediction arises. Here is enough to say that everything comes from the fact that in linear theory any quantum field is expanded into a sum of oscillators with the usual creation and annihilation operators.  The fluctuations of the scalar field are transmuted into density fluctuations, but gravity waves will survive to the present day.

  40. 10. Conclusion To summarize, inflation: • Is able to give a satisfactory explanation to the horizon and flatness problem; • Is able to predict a scale invariant spectrum, but problems arises with the amplitude of the fluctuations predicted (or alternatively with the coupling constant λ ); • Is strongly linked with quantum field theory.

  41. 11.References • Kofman, Linde, Starobinsky,1997:hep-ph/9704452 • Linde,1989:Inflation and quantum cosmology, Academic Press. • Lucchin,1990:Introduzione alla cosmologia, Zanichelli. • Peacock,1999:Cosmological physics, Cambridge University Press. • Ramond:Quantum field theory. • Weinberg,1972:Gravitation and cosmology, John Wiley and sons.

  42. Appendix 1 At t=t0  Substituting this result in the first equation: And remembering that It’s not difficult to get the following equation:

  43. Appendix 2 Given a lagrangian density L for the field and the transformations: A Def: If L is invariant for “A”: And This is the Noethern’s theorem

  44. A special case: INVARIANCE with respect to SPACE-TIME TRANSLATIONS We have: No variations of the filed Def If we take a Lagrangian density CONSERVED

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