Anisotropic Infaltion --- Impact of gauge fields on inflation ---

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

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

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

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

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

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

7. What should we look at? There are two directions in studying fine structures of fluctuations! quantitative improvement-> spectral tilt qualitative improvement -> non-Gaussianity PGW Yet other possibility is the statistical anisotropy 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

9. The anomaly suggests gauge fields? gauge fields FLRW universe The key feature of gauge fields is the conformal invariance. 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, …..

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

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

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

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

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

15. The cosmic no-hair conjecture kills gauge fields! 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. Inflationary universe No gauge fields! Initial gauge fields Any preexistent gauge fields will disappear during inflation. Actually, inflation erases any initial memory other thanquantum vacuum fluctuations. Predictability is high!

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

17. Anisotropic inflation

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

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

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

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

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

24. Watanabe, Kanno, Soda, PRL, 2009 Exact Anisotropic inflation 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 For the parameter region , we found the following new solution Apparently, the expansion is anisotropic and its degree of anisotropy is given by slow roll parameter

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

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

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

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

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

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

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

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

33. Anisotropic Inflation is an attractor 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. Statistical Symmetry Breaking in the CMB

34. Phenomenology of anisotropic inflation

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

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

37. How to test the anisotropic inflation? The current observational constraint is given by WMAP constraint Pullen & Kamionkowski2007 Now, supposewe 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!

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

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

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

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

42. Summary • 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. • We have shown that anisotropic inflation withagauge kinetic function • induces the statistical symmetry breaking in the CMB. More precisely, we have given the predictions: • the statistical anisotropy in scalar and tensor fluctuations • the cross correlation between scalar and tensor • the sizable non-gaussianity • Off-diagonal angular power spectrum can be used to prove or disprove our scenario. • We have already given a first cosmological constraint on gauge kinetic functions.