Multi-Field Inflation in Cosmology

1. Multi-Field Inflation in Cosmology By Iftikhar Ahmad GUCAS, College of Physical Sciences, Beijing China

2. Plan of the Talk • What is cosmology ? • Big Bang Cosmology. • Problem with Big bang Cosmology. • What is Inflation ? • Cosmic Microwave Background • History of Inflation. • Types of Inflation Models. • Perturbation spectrum of Multi-field inflation with Small-field potential. • Conclusion.

3. What is Cosmology? The Study of the Universe: its structure, origin, evolution, and destiny.

4. What is Big Bang Cosmology? • The Big Bang Model is a broadly accepted theory for the origin and evolution of our universe . • All matter started at the one point with a big bang . • The Big Bang theory predicts that the early universe was a very hot place. • A hot, dense expanding universe, should be predominantly hydrogen, helium. • Universe is ~75% hydrogen, ~25% helium by mass.

5. Problems with the Big Bang • The horizon problem • The flatness problem • The Horizon Problem: • The horizon problem tells us the large scale homogeneity and isotropy of the universe must be a part of initial conditions but Hot Big Bang theory is unable to explain it.

6. Flatness problem • The flatness problem is simple that during radiation or matter domination aH is decreasing function of time. • In ‘standard’ cosmology: =1 means universe is flat If 1,  moves quickly away from 1 after big bang Today universe is close to flat Inflation t Matter dominated Radiation dominated

7. What is Inflation In Cosmology? • The rapid expansion in the first 10-35 s of the Universe. • Inflation is simply an epoch during which the scale factor of the universe is accelerating

8. Inflation solved the following problems • The homogeneity on large scales. • The Horizon Problem. • The flatness Problem. • Alen Guth(1981) proposed an idea which could resolve the horizon problem

9. Cosmic Microwave Background • It all began in 1964 when pigeons were accused of roosting and “messing” in a new microwave dish atBell Labs in New Jersey, when Arno Penzias and Robert Wilson, then at Bell Labs, noticed a small discrepancy in their microwave instruments that indicated an excess of radiation coming in from space. Not content to ignore it, they soon made one of the profound discoveries of the 20th century:

10. CMBR • CMBR photons emanate from a cosmic photosphere like the surface of the Sun except that we are inside it looking out • The cosmic photosphere has a temperature which characterizes the radiation that is emitted • Photons in CMBR come from surface of last scattering where they stop interacting with matter and travel freely through space • It has cooled since it was formed by more than 1000 to 2.73 degrees K

11. History of Inflation • In 1981 Alan Guth proposed model (now it is called old inflation ) which based on theory of supercooling. • In 1982, Andrie Linde , Andreas Albrecht and Paul Steinhardt proposed a model (Which is now called new inflation model ) in which (φ) is roll slowly, field φ is call the inflaton. • Linde(1983) proposed one of the most popular inflationary model is chaotic inflation • After this there are so many inflation model has been proposed.

12. Enter Horizon Exit Horizon Provide seeds of CMB Large scale Structure of Universe Horizon Matter dominated Radiation dominated Inflation t Evolution Of Universe

13. Motion of scalar field Slow roll Oscillating Inflation Reheating The new inflationary potential is shown. The potential curvature is very flat in order to permit the field to slow roll down the hill to yield enough e-folds of inflation during that time. Inflation begins at some φi and ends at φf when the field begins to evolve rapidly to its stable symmetry–breaking state φ = v, around which the field oscillates until reheating.

14. Types of Inflation Models • Large-Field Models. • Small-Field Models. • Hybrid Models.

15. Single-Field model with Small-Field potential Consider a potential of type Then one finds this The scalar spectral index n Reference (L. Alabidi and D.H. Lyth, JCAP05 (2006) 016)

16. Perturbation spectrum of multi-fieldinflation with small-field potential • The multi-field inflation model relaxes the difficulties suffered by single field inflation models, and thus may be regarded as an attractive implementation of inflation. • Small-field models are typically characterized by V ′′(φ) < 0. • We take a potential for multi-fields • Where i and µi are the parameters describing the height and tilt of potential.

17. The Scalar Spectrum of Spectral Index for p > 2 To find number of e-folds using Eq. (1) (1) After putting the vales and simplifications one gets (2)

18. under approximation conditions (3) Using this condition in Eq.(3) we get (4)

19. Sasaki -Stewart formulism (5) Using this formula (5) we can find the spectral index For sake of simplicity of our result we make some substitution like (6)

20. (7) With condition Using this condition then first two terms on right side of (7) must vanish so, (8)

21. Now putting the values of A1, A2, A4 and A5 in above one can gets Finally we get (9) When i = j, R(wi) = 0 then it implies that

22. In this case the scalar spectrum multi-field will be the same as that of its corresponding single (Reference Alabidi Land Lyth D H, JCAP05(2006)016) then R(wi) is always positive so spectrum is more redder than its corresponding single field. When then R(wi) is always negative therefore the spectrum is less red than its corresponding single field. However, for more general cases, it seems that dependent of parameters of fields and initial conditions, there is not a definite conclusion. In the case of two scalar fields for i = 1, 2; j = 1, 2

23. The Scalar Spectrum for p = 2 • Equation (8) is only valid for p>2,sowe need to separately calculate the case Using Eq(9) to find number of e-folds (10) The spectral index (11) (12) From equation of motion we get (13)

24. With the help of Eq.(13) one gets (15) Further if we have same µ i, we will obtain (14)

25. Conclusion • For p>2 we found that the spectrum may be redder or bluer than of its corresponding single field. • The result is dependent of the value of fields and their effective masses with this value at the horzion-crossing. • Our result is different from that of multi-field inflation with power law potential, in which the definite conclusion that the spectrum is redder or bluer than of its corresponding single field may be obtained. • When the effective masses of all fields are equal, the spectrum will be the same with that of its corresponding single field.

26. Conclusion • By studying the spectrum behavior for p = 2 , it is noted that the spectral index lies between that of the single field with largest µ k and that of the single field with smallest µ k. • In this case we observed that spectrum may be redder or bluer then of its corresponding single field φk. • then the definite conclusion that the spectrum is more redder or bluer than of its corresponding single field φk may not obtained. • But when we fixed μk = μj = μ0, then the spectrum will be the same with that of its single field φk. • We have assume that isocurvature perturbation may be neglected.

27. Classic Ending