Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton

Transcription

1 Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton Lorenzo Sorbo UMass Amherst Padova, 17/02/2011 LS,

2 Plan of the talk Inflation, radiative stability and pseudo-nambu-goldstone Bosons Parity violation in the CMB from pseudoscalar inflaton Constraints and discussion

3 5 INTERESTING FACTS ABOUT THE UNIVERSE - it is old and very large - in first approximation it is homogeneous and isotropic - it is approximately flat - structure grew out of small, scale invariant perturbations - spectrum of primordial perturbations was gaussian

4 All these facts can be explained by INFLATION := period of accelerated expansion in the very early Universe a=scale factor of the Universe. Obeys H 2 = 8πG 3 ρ ρ 3MP 2 H. a a during inflation require H constant (not so easy, since ρ dilutes away for ordinary matter...)

5 V(φ) How to get some slowly diluting matter? very early Universe filled by scalar field φ, potential V(φ)>0 φ to induce acceleration, V(φ) must be flat V (φ) <<V (φ)/mp to have long enough inflation, V(φ) must stay flat for long enough V (φ) <<V (φ)/mp 2

6 Simple way of realizing V (φ) <<V (φ)/mp, V (φ) <<V (φ)/mp 2 : monomial potential, with φ large enough Most famous example: quadratic potential (chaotic inflation) Linde 1983 V(φ)=m 2 φ 2 /2 Amplitude of perturbations produced during inflation m ~ GeV

7 ...but, in general, quantum loops will contribute to V and V (and V etc...)

8 Radiative corrections can disrupt the inflationary potential in two ways 1- affect the functional form of V(φ) 2- affect value of the parameters that appear in V(φ) Chaotic inflation example 1- adds terms φ n, n=4, 6, push m to larger values (e.g. MP - cf EW hierarchy pbm) How to make sure that radiative effects are under control?

9 A very well-known system that contains controllably small quantities is the Standard Model: small quantities are protected against radiative effects by symmetries If a model has a symmetry, quantum effects cannot violate it (unless the symmetry is anomalous...) If the symmetry is broken, quantum effects cannot make the breaking much larger (ie the breaking parameter is controllably small)

10 A field φ has a shift symmetry if the theory that describes it is invariant under the transformation φ φ + c (c=arbitrary constant) If this symmetry is exact, the only possible potential for φ is V(φ)=constant (i.e. a cosmological constant) an exact shift symmetry is an overkill......but we can break the symmetry a bit and generate a potential

11 An (important) example If φ is a phase, then shift symmetry global U(1) Theory with a spontaneously broken global U(1) Decompose where δh is massive and φ is a massless Goldstone boson (pseudoscalar) The global U(1) is broken e.g. by gravitational instantons A potential is generated: +... ( S = instanton action, MP n ) Pseudo-Nambu-Goldstone boson PNGb

12 ...using a pngb as an inflaton... Natural inflation Freese et al 1990 V(φ)=µ 4 [ cos(φ/f)+1] 2µ 4 V(φ) 0 π φ/f

13 Because of its radiative stability, A pngb gives an extremely well motivated model of inflation from the point of view of effective field theory

14 What about data? f>3.5 MP from Savage et al, 2006

15 Stringy models of natural inflation? YES, in principle (string theory contains a plethora of pngbs) However Banks, Dine, Fox and Gorbatov 03 String Theory appears to require f<mp n-instanton actions contribute e - (n M P/f) cos(n φ/f) to pngb potential first f/mp harmonics in V(φ) matter

16 Ways out? Kim, Nilles and Peloso Use two pngbs Blanco-Pillado et al Use pngbs and moduli - Use many pngbs Dimopoulos et al 2005

17 Ways out? Siverstein and Westphal, Use monodromy Kaloper and LS, Mixing with 4-form L=

18 I hope I convinced you... There are many well motivated models of pngb inflation The pngb is a pseudoscalar: macroscopic parity violation in the Early Universe Is it possible to observe the effect of such parity violation?

19 Imprinting parity violation on the CMB If inflaton is a pseudoscalar (in particular a pngb), it interacts with the electromagnetic field via L φff = φ f αβγδ F αβ F γδ (f=constant with dimensions of a mass) in Coulomb gauge A0=0, A=0, decompose into helicity modes A(x,t)= λ=± (does not need to be the same f that appears in V(φ) ) d 3 k (2 π) 3/2 a λ k A k λ(t) e λ (k)e ikx + a λ k A λ k (t) e λ (k)e ikx

20 Transferring parity violation to the gauge modes The mode functions Aλ k (t) are sourced by φ0(t), and obey Ä λ + H A λ + k 2 φ a 2 + λ f k a A λ =0 for λ=-, the mass term is negative for ~1 Hubble time: Exponential amplification of left handed modes only! parity violation is transferred to the electromagnetic field A L exp π 2 φ fh

21 Primordial gravitational waves Let us now focus on the tensor components of the metric g µν (x, t) dx µ dx ν = dt 2 + a 2 (t) (δ ij + h ij (x, t)) dx i dx j δ ij h ij = i h ij =0 ij i the tensor mode has two components (=helicity ±2) so we can decompose it, in momentum space, into left handed and right handed modes h ij (k, t)=h L (k, t) L ij(k)+h R (k, t) R ij(k)

22 Transferring parity violation to the gravitational waves The energy of the electromagnetic field sources gravitational waves: (note: this is an operator equation) ḧ λ +3ȧ a ḣλ + k2 a 2 h λ = 2 M 2 P Π ij λ T ij EM Tλ Projector on helicity-λ components Spatial components of gauge field stress-energy tensor since the RHS is known (computed in previous slides), can obtain hλ formally with retarded propagator

23 The amplitude of the helicity-λ gravitational waves If Gk(t,t ) is retarded propagator for operator d 2 /dt 2 +3 H d/dt+k 2 /a 2, then h λ (k, t)= 2 M 2 P h λ (k, t)h λ (q, t) = 4 M 4 P where dt G k (t, t ) T λ (k, t) and from this we obtain the amplitude dt G k (t, t ) T λ (k, t ) T λ (q, t ) dt G q (t, t )T λ (k, t )T λ (q, t ) is quartic in the gauge field A and can be computed in terms of the functions Aλ k (t)

24 ...note that to the special solution of the inhomogeneous equation for h λ one should add the general solution of the homogeneous equation ḧ λ +3ȧ a ḣλ + k2 a 2 h λ = 0 2 MP 2 Π ij λ T ij EM parity-invariant uncorrelated to component sourced by φ exists in standard inflation models

25 Parity violating gravitational waves AL and AR have different amplitudes <TLTL> <TRTR> Denoting h λ (x, t) h λ (y, t) = P R (k) = P L (k) = H2 π 2 M 2 P H2 π 2 M 2 P d 3 k P λ (k) (2 π) 3 k H2 M 2 P H2 MP 2 e 4πξ ξ 6 e 4πξ ξ 6 e ik(x y) standard parity-invariant part parity-violation! ξ φ 2 fh 1

26 How can primordial gravitational waves be detected? The CMB radiation is polarized! Polarization is measured by two quantities The gradient (curl-free) component of polarization are denoted as E-mode The curl (divergence-free) component is denoted as B-mode From Baumann et al,

27 How can primordial gravitational waves be detected? The CMB radiation is polarized! E-modes are generated by currents of plasma associated to gradients of density in the CMB B-modes not related to any preferred direction in CMB, so must be associated to primordial tensor modes Not yet observed. Planck (2013?) will improve constraints by O(3). Other experiments likely to improve by O(20)

28 How can primordial parity violation be detected? With T (=temperature fluctuations), E and B can measure many two point functions: <TT>, <TE>, <EE> Already observed <BB> Wanted! <TB>, <EB> Should vanish in parity invariant CMB T and E are parity-even B is parity-odd Lue, Wang and Kamionkowski 98

29 Detection prospects related to observability of nonzero <EB> and/or <TB> Depend on two parameters Saito Ichicki Taruya 07, Contaldi Maguejio Smolin 08, Gluscevic Kamionkowski 10 r = P R + P L P T tensor-to-scalar ratio χ =2 P R P L P R + P L chirality of primordial perturbations From Gluscevic Kamionkowski 10

30 For our system χ = e4 πξ H 2 ξ 6 MP e 4 πξ H 2 ξ 6 MP 2. ξ φ 2 fh = 2 M P f O Exponential dependence on the coupling 1/f Δχ is either very small or very close to unity In principle parity violation detectable for significant part of parameter space. But...

31 Constraints from nongaussianities The produced electromagnetic modes backreact on the inflaton, contributing to its three-point function Barnaby Peloso 10 NONGAUSSIANITIES f equil NL H 6 3 M 6 P e 6 πξ ξ 9.

32 Constraints from nongaussianities (2) WMAP constrains fnl equil <266 ξ<2.6 Δχ<<1 Parity violation will not be detectable in the simplest model without violating constraints from nongaussianities

33 Ways out i) A CURVATON Most of the primordial perturbation is due to a second field with nearly-gaussian perturbations. ii) MANY GAUGE FIELDS Contributions to fnl add incoherently. With ~10 3 gauge fields f NL safely small constraint from nongaussianities is evaded

34 Ways out E.g. parameter space for curvaton making 90% of primordial perturbations HM P Parity violation!"<!" min too small Too many r>0.24 tensor modes Detectable parity violation Ξ

35 Discussion (I) Planck will improve bound on tensor modes and might even detect them! It will be important to look for nonvanishing <EB> and <TB> Here we have presented a scenario that can give rise to those correlators

36 Discussion (II) Nonvanishing <EB> and <TB> could also be produced by some late-universe effect (e.g. pseudoscalar quintessence) Gluscevic and Kamionkowski 2010 have however shown that it is possible to distinguish a primordial <EB> and <TB> from a late one

37 Discussion (III) Another coupling discussed in the past that might lead to nonvanishing <EB> and <TB> is δl = φ f αβγδ R αβ µν R µν γδ...however in order to have Δχ large enough one needs to study the theory in a strongly coupled regime Lyth Quinbay Rondriguez 05 Satoh 10

38 Discussion (IV) Cook, LS in progress P R (k) = P L (k) = H2 π 2 M 2 P H2 π 2 M 2 P H2 M 2 P H2 MP 2 e 4πξ ξ 6 e 4πξ ξ 6 ξ increases during inflation GWs produced towards the end of inflation (i.e. at smaller scales) have larger amplitude ξ φ 2 fh 1 might be detected by advanced LIGO!

39 Discussion (V) Standard relationship between amplitude of gravitational waves and H does not apply! HM P r>0.24 r= !"<!" min Ξ

40 Conclusions Models of pseudoscalar inflation very well motivated We have shown they naturally lead to a chiral spectrum of gravitational waves In simplest model, strong constraints from nongaussianities However, candidate explanations if nonvanishing <EB> and <TB> will be observed