1. 1 Lecture-07 Phase Transitions and Inflation Ping He ITP.CAS.CN 2006.05.31 http://power.itp.ac.cn/~hep/cosmology.htm

2. 2 7.0 Preliminary: Planck Era

3. 3 7.1 Phase Transition Spontaneous symmetry breaking (SSB): SSB can be used both in quantum field theory (particle physics) and in phase transition of statistical physics. In QFT, SSB provides a mechanism for the unification of interactions, as T decreases, new type of interactions will emerge. Strong W E SSB

4. 4 GUT transition supersymmetry transition (possible) electroweak transition quark-hadron transition

5. 5 While from the side of thermodynamics, the universe will experience a series of phase transitions, just like water from vapor to liquid to ice; In both cases, SSB is implemented by Higgs mechanism, in which there exists a scalar filed  : (1) In particle physics, it is called a Higgs Field (Boson, spin=0); (2) In statistical physics, it is called order parameter. Higgs field: a scalar field  with self-interaction, like this: symmetry for The above expression can also be used to describe 1 st order phase Transition.

6. 6 1 st order phase transition, discontinuous transition: (1) latent heat, (2) nucleation of bubbles. 2 th order phase transition, continuous transition: (1) less dramatic, no-latent heat, (2) quantified by correlation-length. F(  )  F   )  : Higgs potential ground state, or vacuum state: the state of the minimum energy. When, the vacuum expectation value 2 nd order 1 st order

7. 7 7.2 Topological Defects The type of defect produced in a symmetry-breaking phase transition depends on the symmetry and how it is broken in a complicated fashion. Topological defects are relics of phase transitions Typically, there will be the following types of topological defects: (1) magnetic monopoles; (2) cosmic strings; (3) domain walls; (4) textures. Vacuum that experiences SSB is not perfect, that is, topological defect. SSB of SU(5)  monopoles; SSB of U(1)  cosmic strings; SSB of Z 2  domain walls, etc.

8. 8 Example-1, the formation of domain wall ( 畴壁 ), by SSB of Z 2. At the boundary between different magnetism area, =0, this is just a topological defect, which is a wall- like structure, with a small thickness, so that it has energy and mass, and contributes to the cosmic matter density.

9. 9 Example-2, the formation of cosmic strings, by SSB of a U(1). A closed path encompasses a string-like structure, in which =0, with a small transverse dimension, so that it has energy and mass, and contributes to the cosmic matter density. Such a string must be either closed or infinite.

10. 10 We see that all kinds of topological defects have mass and energy, which contribute to the total cosmic mass and energy density. Any GUT in which electromagnetism (described by U(1) gauge group) is contained, with a gauge theory involving SSB of a higher symmetry, e.g., SU(5), can provide a natural explanation for the quantization of electrical charge and this implies the existence of magnetic monopoles. Monopoles are point-like defects in the Higgs filed  which appears in GUTs. See the figure, arrows indicate the 3-D orientation of  in the internal symmetry space of the theory, while the location of the arrows represents a position in ordinary space. Example-3, the formation of monopoles, by SSB of GUT gauge symmetry. Not exactly a point, but also has a small finite size, with mass and energy.

11. 11 We discuss the mass and density of monopoles. In electrostatic units, monopoles has a magnetic charge and a mass where X is the Higgs boson that mediates the GUT interaction, with mass is the energy corresponding to the GUT SSB, for typical GUTs, e.g., SU(5), we have, so that Furthermore, the size of the monopoles is very heavy

12. 12 Cosmological monopole problem At T=T GUT, GUT  SU(3)XSU(2)XU(1), if  is the characteristic dimension of the domain during symmetry breaking, the maximum number density of monopoles has the following relation. Since any single domain should be causally connected, we have where T p is the Planck temperature. It turns out that, at T GUT (~10^15GeV),

13. 13 Cosmic strings, however, assuming their existence, may be a solution rather than a problem because they may be responsible for generating primordial fluctuations which give rise to galaxies and clusters, but just a minority of cosmologists believe this. Any subsequent physical processes are very inefficient to reducing the ratio of, so the present-day number density of monopoles is equal to, or greater than the baryon density. So the density parameter of monopoles is Similar case for domain walls. So we can see that monopoles and domain walls represent a problem to cosmology, which was the essential stimulus for inflationary cosmology.

14. 14 7.3 Problems with the Standard Cosmological Model (1) The problem of horizon (homogeneity); (2) The problem of flatness; (3) The problem of the origin of structure; (4) The problem of monopole. The problem of monopole has been addressed previously.

15. 15 7.3.1 The cosmological horizon problem (Same as homogeneity problem) Big-bang singularity, with limited light speed  existence of particle horizon comoving particle horizon: Hubble radius (length): comoving Hubble radius (length): In the above, we use So,

16. 16 and the size of particle horizon at last-scattering surface: In the above calculation, we have used The angular-diameter distance so at the last scattering surface The angle subtended by particle horizon at the last-scattering surface: P Q Since no causal connection between P and Q, how T P =T Q ?

17. 17 7.3.2 The flatness problem From Friedmann equation, we have (see Lecture-02): so, we have Since,, so at the Planck epoch,, that is So, the present-day non-flatness evolves from the discrepancy of at the initial (Planck) time. A fine-tuning problem!

18. 18 7.3.3 The problem of structure formation At the Planck time, any physical scale should be, so That is, the scale of the largest structure at the present-day should not be larger than 100km! t t0t0 scale t pl l pl

19. 19 Inflation: an extremely fast expansion at the birth of the Universe, was driven by the vacuum energy of some material fields. Modern cosmology predicts that a short period of inflation occurred at the extremely early epoch, which can immediately overcome or explain the above problems in the standard cosmology: 7.4 Inflationary Cosmology

20. 20 7.4.1 inflation driven by vacuum energy Friedmann equation, with flat geometry: For the very early universe: Before the phase transition, that is, the vacuum energy can be neglected, so that the dynamics is totally controlled by radiation. The temperature decreases as the expansion of the universe, when, phase transition will not promptly begin, and the universe is in metastable state, which is overcooled. F(  )   : Higgs potential 1 st order

21. 21 The expansion of the universe rapidly reduced the density of radiation (   T^4), while leaving the vacuum energy unchanged, so that the energy density is vacuum-dominated. The dynamic equation is Here, T c is a constant, so that H is also a constant, and hence For GUT, T c =10^15GeV, so that This is inflation. If inflation lasts for merely, then a will increase times Definition of e-foldings: For the above example, the e-foldings N=100

22. 22 7.4.2 evolution of scale and temperature Log (physical scale) Log (comoving scale)  Inflation  physical Hubble length comoving Hubble length physical scale comoving scale Log (time) (1) Horizon is nearly unchanged during inflation and otherwise increasing. (2) A physical scale is initially inside the horizon, but crosses outside some time before the end of inflation, reentering long after inflation is over. |  scale  | |  amp  | Comoving wave number

23. 23 T(t) a(t) T(t) a(t) Standard cosmology Inflationary cosmology inflation reheating (1) Because of the inflation, the temperature dramatically decreased. Its evolution is adiabatic, so aT~ const, that is, ; (2) When inflation stopped, the latent heat of the vacuum energy was released; (3) The universe was re-heated again to the temperature of about T c ; (4)Reheating is non-adiabatic, hence, large amount of entropy was generated during reheating. (5) After the end of reheating, the universe restored to the adiabatic expansion.

24. 24 7.4.3 inflationary solution to homogeneity problem (horizon problem) Log (physical scale)  Inflation  physical Hubble length physical scale Log (time) lss For example, the super-horizon scale at the last-scattering surface (lss) was causally connected before inflation, thus in this way, the homogeneity problem is overcome. For this, the e-foldings should be larger than N>60. (details omitted)

25. 25 7.4.4 inflationary solution to flatness problem From Eq-7.20 At the inflationary stage, density is dominated by vacuum energy, which is constant. That is, Assuming, the e-foldings N=100, that is, a increases 43 orders of magnitude. From Eq-7.30, we can see that decreases for 86 orders of magnitude. After the cancellation of 60 orders of magnitude of pre-inflationary discrepancy, there are still net 26 orders of magnitude of decrease left.

26. 26 It not just a solution to the flatness problem, it is also a prediction of flatness, from Eq-7.30, we see that where i, and f indicate the start and end of inflation, since even remarkably deviates from 1, then after inflation, The prediction is model-dependent, there is also open-inflationary model.

27. 27 7.4.5 inflationary solution to structure formation problem Since a increases ~43 orders during inflation, so From Eq-7.22, we know that The largest structure at the present-day universe is galaxy cluster, which has typically the size of about ~ Mpc. Thus, structure formation problem is overcome! The current idea about the formation of structure is: Quantum fluctuations on microscopic scales during the inflationary epoch can, by virtue of the enormous expansion, lead to fluctuations on very large scales today.

28. 28 7.4.6 inflationary solution to monopole problem Monopoles are generated during the phase transition by SSB of GUT. If inflation took place after (or during) the phase transition, then since scale factor increases by 43 orders, so the density of monopoles is diluted by ~130 orders of magnitude. That is, from Eq-7.10 The same is true for cosmic strings and all other topological defects. However, if there were other phase transitions after the epoch of inflation, defects could have been formed again.

29. 29 7.5 Dynamics of Scalar Fields The previous example is based on Guth’s “old inflation” model (1981), which was settled upon 1 st order phase transition, and is now abandoned, because that, being a 1 st order transition, it occurs by a process of bubble nucleation. So that: (1) They are too small to be identified with our observable universe, and (2) They are carried apart by the expanding phase too quickly for them to coalesce and produce a large bubble which is identified with our universe. So that (3) The end state of this model would therefore be a highly chaotic universe, quite the opposite of what is intended. 7.5.1 Guth’s “Old Inflation” model

30. 30 7.5.2 “New Inflation” model (Linde 1982; Albrecht & Steinhardt 1982) To obtain inflation, we need material with the unusual property of a negative pressure. Such a material is a scalar, describing spin=0 particles. The scalar field responsible for inflation is often called the inflaton. The Lagrangian density of the scalar field  is Gives a contribution to the energy-momentum tensor of the form For a homogeneous state, the spatial gradient terms vanishes, and becomes the type of the perfect fluid:

31. 31 The equation of motions of  is: F   )  : Higgs potential 2 nd order To produce a long enough period of inflation and a rapid reheating after inflation, the potential V(  ) should be like the following of the right one. Such a potential may be implemented by super-symmetry theories.

32. 32 This is a 2 nd order phase transition. As the temperature of the Universe lowers, the state should slowly roll down to the minima of the potential. That is, the potential should satisfy the slow-roll conditions: With slow-roll approximation, Eq-7.39 becomes After the slow-rolling phase the field  falls rapidly into the minimum at  0 and undergoes oscillations, and a rapid liberation of energy which was trapped in the false vacuum, that is re-heating. The oscillations are damped by the creation of particles coupled to  field.

33. 33 Problems with this “New Inflation” model, is that it suffers severe fine-tuning problems. (1)The potential must be very flat near the origin to produce enough inflation and to avoid excessive fluctuations due to the quantum field. (2) The field  is assumed to be in thermal equilibrium with the other matter fields before the onset of inflation, which requires that  be coupled fairly strongly to the other fields. But the coupling constant would induce corrections to the potential which would violate the previous constraint. The ending of the inflation is It is unlikely to achieve thermal equilibrium in a self-consistent way that inflation can start under this slow-roll conditions.

34. 34 7.5.3 “Chaotic Inflation” model (Linde 1983) It is an improvement of the “New Inflation” model, which is also based upon a scalar field, but the potential takes some simplest form, e.g., Here m is the mass of the filed. Slow-roll conditions can be satisfied if Chaotic inflation model assumes that at some initial time, perhaps just after the Planck time, the  field varied from place to place in an arbitrary manner. If any region satisfies the above conditions it will inflate and eventually encompass our observable universe. The results are locally flat and homogeneous, but on scales larger than the horizon the universe is highly curved and inhomogeneous. In this model, no need for GUT or super-symmetry, and no requirement for any phase transition. The field  at the Planck time is completely decoupled from all other physics.

35. 35 7.5.4 Other inflationary models (1) “Stochastic Inflation” model (Linde 1994), also called eternal inflation. The basic idea is the same as the chaotic one in that the universe is globally extremely inhomogeneous. The stochastic inflation model takes into account quantum fluctuations during the evolution of . In this case, the universe at any time will contain regions which are just entering into an inflationary phase. The picture of this universe is a continuous branching process in which new mini-universes expand to produce locally smooth Hubble patches within a highly chaotic background universe. (2) “Open inflation” model (Coles and Ellis 1994). Before WMAP results, The universe may have the possibility that it is open, so that open inflation model is constructed, in which after this kind of inflation, the universe is homogeneous but is curved by invoking a kind of quantum tunneling from a meta-stable false vacuum state immediately followed by a 2 nd order of phase transition of inflation. The tunneling creates a bubble inside which the space-time resembles an open universe. After WMAP results, less interesting now. (3) Many …

36. 36 References E.W. Kolb & M.S. Turner, The Early Universe, Addison-Wesley Publishing Company, 1993 P. Coles & F. Lucchin, Cosmology, 2 nd edtion, John Wiley & Sons, 2002 A.R. Liddle & D.H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, 2000 L. Bergstrom & A. Goobar, Cosmology and Particle Astrophysics, Springer, 2004

37. 37 俞允强，热大爆炸宇宙学，北京大学出版 社， 2001 范祖辉， Course Notes on Physical Cosmology, See this site.this site