1. Anisotropic Infaltion --- Impact of gauge fields on inflation --- Jiro Soda Kyoto University ExDiP 2012, Hokkaido, 11 August, 2012

2. 2 Standard isotropic inflation Isotropic homogeneous universe Friedman eq. Action K-G eq. inflation

3. 3 Origin of fluctuations Curvature perturbations Tensor perturbations The relationyields The tensor to the scalar ratio Action for GW initial inflation end 2 polarizations scale invariant spectrum

4. 4 From COBE to WMAP! CMB angular power spectrum COBE gravitational red shift The predictions have been proved by cosmological observations. We now need to look at a percent level fine structure of primordial fluctuations! WMAP provided more precise data!

5. E and B Polarizations Primordial gravitational waves via B-modes Primordial gravitational waves

6. 6 : prefered direction Groeneboomn & Eriksen (2008) The preferred direction may be produced by gauge fields. Eriksen et al. 2004 Hansen et al. 2009 Statistical Anisotropy?

7. 7 quantitative improvement -> spectral tilt qualitative improvement -> non-Gaussianity PGW There are two directions in studying fine structures of fluctuations! Yet other possibility is the statistical anisotropy What should we look at? If there exists coherent gauge fields during inflation, the expansion of the universe must be anisotropic. Thus, we may have statistical anisotropy in the primordial fluctuations.

8. Gauge fields and cosmic no-hair 8

9. The anomaly suggests gauge fields? 9 gauge fields FLRW universe In terms of a conformal time, we can explicitly see the conformal invariance cancelled out Thus, gauge fields are decoupled from the cosmic expansion and hence no interesting effect can be expected. In particular, gauge fields are never generated during inflation. However, ….. The key feature of gauge fields is the conformal invariance.

10. Gauge fields in supergravity 10 While, the role of gauge kinetic function in inflation has been overlooked. Cosmological roles of Kahler potential and super potential in inflation has been well discussed so far. Supergravity action The non-trivial gauge coupling breaks the conformal invariance because the scalar field has no conformal invariance. : Kahler potential : superpotential : gauge kinetic function Thus, there is a chance for gauge fields to make inflation anisotropic. Why, no one investigated this possibility?

11. Black hole no-hair theorem 11 Black hole has no hair other than M,J,Q gravitational collapse Israel 1967,Carter 1970, Hawking 1972 any initial configurations event horizon By analogy, we also expect the no-hair theorem for cosmological event horizons. Let me start with analogy between black holes and cosmology.

12. Cosmic no-hair conjecture 12 deSitter Gibbons & Hawking 1977, Hawking & Moss 1982 cosmic expansion Inhomogeneous and anisotropic universe no-hair ？

13. Cosmic No-hair Theorem Dominant energy condition: Strong energy condition: Assumption: Statement: The universe of Bianchi Type I ~ VIII will be isotropized and evolves toward de Sitter space-time, provided there is a positive cosmological constant. Wald (1983) : energy density & pressure other than cosmological constant ex) a positive cosmological constant Type IX needs a caveat.

14. Sketch of the proof 14 Ricci tensor (0,0) Einstein equation (0,0) Bianchi Type I ~ VIII : Strong Energy Condition Dominant Energy Condition in time scale : No Shear = Isotropized Spatially Flat No matter

15. The cosmic no-hair conjecture kills gauge fields! 15 No gauge fields! Inflationary universe Any preexistent gauge fields will disappear during inflation. Actually, inflation erases any initial memory other than quantum vacuum fluctuations. Predictability is high! Since the potential energy of a scalar field can mimic a cosmological constant, we can expect the cosmic no-hair can be applicable to inflating universe. Initial gauge fields

16. Can we evade the cosmic no-hair conjecture? All of these models breaking the energy condition have ghost instabilities. ・ Vector inflation with vector potential Ford (1989) ・ Lorentz violation Ackerman et al. (2007) ・ A nonminimal coupling of vector to scalar curvature Golovnev et al. (2008), Kanno et al. (2008) Himmetoglu et al. (2009) ・・・ Fine tuning of the potential is necessary. ・・・ Vector field is spacelike but is necessary. ・・・ More than 3 vectors or inflaton is required One may expect that violation of energy conditions makes inflation anisotropic.

17. Anisotropic inflation 17

18. Gauge fields in inflationary background 18 de Sitter background abelian gauge fields It is possible to take the gauge gauge symmetry f depends on time. canonical commutation relation

19. Do gauge fields survive? 19 Thus, it is easy to obtain power spectrums blue red The blue spectrum means no gauge fields remain during inflation. The red spectrum means gauge fields survive during inflation. Canonical commutation relation leads to commutation relations and the normalization vacuum

20. Mode functions on super-horizon scales 20 sub-horizon super-horizon matching at the horizon crossing Take a parametrization Since we knowwe get Length time Sub-horizon Super-horizon

21. Gauge fields survive! 21 red spectrum For a large parameter region, we have a red spectrum, which means that there exists coherent long wavelength gauge fields! Finally, at the end of inflation, we obtain the power spectrum of gauge fields on super-horizon scales

22. We should overcome prejudice! 22 According to the cosmic no-hair conjecture, the inflation should be isotropic and no gauge fields survive during inflation. However, we have shown that gauge fields can survive during inflation. It implies that the cosmic no-hair conjecture does not necessarily hold in inflation. Hence, there may exist anisotropic inflation.

23. 23 In this case, it is well known that there exists an isotropic power law inflation Gauge fields and backreaction In this background, one can consider generation of gauge fields gauge kinetic function power-law inflation our universe There appear coherent gauge fields in each Hubble volume. Thus, we need to consider backreaction of gauge fields.

24. Exact Anisotropic inflation 24 For the parameter region, we found the following new solution Kanno, Watanabe, Soda, JCAP, 2010 For homogeneous background, the time component can be eliminated by gauge transformation. Let the direction of the vector to be x – axis. Then, the metric should be Bianchi Type-I Watanabe, Kanno, Soda, PRL, 2009 Apparently, the expansion is anisotropic and its degree of anisotropy is given by slow roll parameter

25. The phase space structure 25 Isotropic inflation Anisotropic inflation After a transient isotropic inflationary phase, the universe enter into an anisotropic inflationary phase. Kanno, Watanabe, Soda, JCAP, 2010 The result universally holds for other set of potential and gauge kinetic functions. Quantum fluctuations generate seeds of coherent vector fields. anisotropy scalar vector

26. More general cases Hamiltonian Constraint Scale factor Anisotropy Scalar field const. of integration

27. Behavior of the vector is determined by the coupling Conventional slow-roll equations (The 1st Inflationary Phase) Hamiltonian Constraint Scalar field To go beyond the critical case, we generalize the function by introducing a parameter Vector grows Vector remains const. Vector is negligible Now we can determine the functional form of f Critical Case The vector field should grow in the 1 st inflationary phase. Can we expect that the vector field would keep growing forever?

28. Attractor mechanism Hamiltonian Constraint Scalar field Define the ratio of the energy density Typically, inflation takes place at The opposite force to the mass term Irrespective of initial conditions, we have The growth should be saturated around

29. Inflaton dynamics in the attractor phase Hamiltonian Constraint Scalar field const. of integration We find becomes constant during the second inflationary phase. The modified slow-roll equations: The second inflationary phase E.O.M. for Φ : (2nd inflationary phase) (1st inflationary phase) Remember Energy Density Solvable

30. Phase flow: Inflaton Numerically solution at Scalar field (2nd inflationary phase) (1st inflationary phase)

31. The degree of Anisotropy Anisotropy The degree of anisotropy is determined by The slow-roll parameter is given by Attractor point We find that the degree of anisotropy is written by the slow-roll parameter. Attractor point Attractor Point :Hamiltonian Constraint Ratio Compare : A universal relation

32. Evolutions of the degree of anisotropy 32 Numerically solution at Initially negligible grows fast becomes constant disappears increase

33. Anisotropic Inflation is an attractor 33 Statistical Symmetry Breaking in the CMB It is true that exponential expansion erases any initial memory. In this sense, we have still the predictability. However, the gauge kinetic function generates a slight anisotropy in spacetime.

34. 34 Phenomenology of anisotropic inflation

35. What can we expect for CMB observables? 35 vector-tensor vector-scalar In the isotropic inflation, scalar, vector, tensor perturbations are decoupled. The power spectrum is isotropic However, in anisotropic inflation, we have the following couplings length t Preferred direction

36. 36 Predictions of anisotropic inflation statistical anisotropy in curvature perturbations cross correlation between curvature perturbations and primordial GWs statistical anisotropy in primordial GWs TB correlation in CMB Thus, we found the following nature of primodordial fluctuations in anisotropic inflation. These results give consistency relations between observables. Watanabe, Kanno, Soda, PTP, 2010 Dulaney, Gresham, PRD, 2010 Gumrukcuoglu,, Himmetoglu., Peloso PRD, 2010 preferred direction

37. 37 WMAP constraint Pullen & Kamionkowski 2007 Now, suppose we detected Then we could expect statistical anisotropy in GWs cross correlation between curvature perturbations and GWs If these predictions are proved, it must be an evidence of anisotropic inflation! How to test the anisotropic inflation? The current observational constraint is given by

38. How does the anisotropy appear in the CMB spectrum? 38 The off-diagonal part of the angular power spectrum tells us if the gauge kinetic function plays a role in inflation. For isotropic spectrum,, we have Angular power spectrum of X and Y reads For anisotropic spectrum, there are off-diagonal components. For example,

39. We should look for the following signals in PLANCK data! 39 When we assume the tensor to the scalar ratio and scalar anisotropy The off-diagonal spectrum becomes The anisotropic inflation can be tested through the CMB observation! Watanabe, Kanno, Soda, MNRAS Letters, 2011

40. Non-gaussianity in isotropic inflation For c=1, we have the scale invariant spectrum. deSitter Barnaby, Namba, Peloso 2012 anisotropy

41. Non-gaussianity in anisotropic inflation Scale invariantIs an attractor!

42. Summary 42  We have shown that anisotropic inflation with a gauge kinetic function induces the statistical symmetry breaking in the CMB.  Off-diagonal angular power spectrum can be used to prove or disprove our scenario. More precisely, we have given the predictions:  We have already given a first cosmological constraint on gauge kinetic functions. the statistical anisotropy in scalar and tensor fluctuations the cross correlation between scalar and tensor the sizable non-gaussianity  Anisotropic inflation can be realized in the context of supergravity.  As a by-product, we found a counter example to the cosmic no-hair conjecture.