Effective Field Theory of Anisotroic Inflation Asieh Karami IPM - - PowerPoint PPT Presentation

Effective Field Theory of Anisotroic Inflation Asieh Karami IPM - School of Astronomy GC2018 February 13, 2018 In collaboration with T. Rostami and H. Firouzjahi b ased on JCAP 06(2017)p.039

Why effective field theory? ◮ There are many scenarios of inflation which are compatible with cosmological observations. So, it seems unrealistic if one can single out a particular model as the true realization of inflation in early universe. This naturally raises the question if one can classify the various inflationary scenarios either based on their main predictions or based on their theoretical constructions. Why anisotropic inflation? ◮ Vector fields and gauge fields appear in abundant in Standard Model of particle physics and in quantum field theory. It is natural to expect that they play some roles during inflation. It is conceivable that they play the role of isocurvature light fields which may also be coupled to inflaton field. This brings the interesting possibilities that light gauge fields may affect the cosmological observations by generating some observable amount of statistical anisotropies.

Outline 1 Introduction 2 Effective Action of Anisotropic Inflation 3 Free fields 4 Power Spectrum 5 Bispectrum 6 Gravitational Waves 7 Summery

Effective Field Theory There is a theory for whole energy scale - Use EFT to simplify calculations in some energy scales. ◮ Constructing EFT by considering symmetries of the model.

Anisotropic Inflation In the setup of anisotropic inflation we have the scalar field φ as the inflaton field and a U (1) gauge field A µ which is the source of electric field energy density during inflation. Because of the conformal symmetry associated with the Maxwell theory, the background electric field energy density is diluted if the gauge field is not coupled to inflaton. Therefore, in order for the background electric field energy density to survive the exponential expansion, the gauge field is coupled to the inflaton field as − f ( φ ) 2 F µν F µν / 4. The next goal is to choose the functional form of f ( φ ) such that the background electric field energy density to be a nearly constant but sub-leading fraction of the total energy density. For a given inflaton potential V ( φ ) the form of f ( φ ) can be obtained. In terms of scale factor a ( t ), it takes the time-dependent value f ( φ ) ∝ a ( t ) − 2 . At the perturbation level, this choice of f ( φ ) also yields a scale invariant power spectrum for the gauge field fluctuations.

Anisotropic Inflation The imprints of the gauge fields fluctuations in primordial curvature perturbation power spectrum P R has the form of quadrupole anisotropy which � k ) 2 � P R ( k ) = P (0) n · � 1 + g ∗ ( � , R in which P (0) is the isotropic power spectrum in the absence of gauge field, k R is the mode of interest in Fourier space and � n indicates the direction of anisotropy. In this way of parameterization, the parameter g ∗ measures the amplitude of statistical anisotropy. Observational constraints from Planck data implies | g ∗ | � 10 − 2 .

Effective Action of Anisotropic Inflation Building blocks are  δg 00 ,    X ≡ F µν F µν , Y ≡ F µν ˜ F µν ,    Z ≡ F 0 µ F 0 µ , F µν = ǫ µναβ F αβ is its dual field. F µν = ∂ µ A ν − ∂ ν A µ is the field strength and ˜ The most general action in unitary gauge in the decoupling limit is given by � � α ( t ) + B 1 ( t ) δg 00 + B 2 d 4 x √− g 2 ( t ) ( δg 00 ) 2 − M 1 ( t ) S = δX 4 4 + M 2 ( t ) δg 00 δX − M 3 ( t ) ( δg 00 ) δ ( X ) 2 − M 4 ( t ) ( δg 00 ) 2 δX 2 4 4 − N 1 ( t ) δY + N 2 ( t ) δg 00 δY − N 3 ( t ) ( δg 00 ) δ ( Y ) 2 4 2 4 − N 4 ( t ) ( δg 00 ) 2 δY − P 1 ( t ) δZ + P 2 ( t ) δg 00 δZ 4 4 2 � − P 3 ( t ) ( δg 00 ) δ ( Z ) 2 − P 4 ( t ) ( δg 00 ) 2 δZ + ... . 4 4

Effective Action of Anisotropic Inflation Background Equations The background Friedmann equations are given by � � � � 1 1 H 2 = ˙ H + H 2 B 1 ( t ) + α ( t ) , = − 2 B 1 ( t ) − α ( t ) . 3 M 2 3 M 2 Pl Pl Solving for α and B 1 , in the small anisotropy limit we have α ( t ) ≃ 3 M 2 P H 2 , B 1 ( t ) ≃ − ǫH 2 M 2 P . The fraction of gauge field energy density to total energy is 2 ) a − 2 ˙ 1 2 ( M 1 − P 1 A 2 x R = . 3 M 2 P H 2 The background Maxwell equation is �� � � M 1 − P 1 a ( t ) ˙ ∂ t A x = 0 . 2 From above equations we can conclude that M 1 − P 1 A x ( t ) ∝ a 3 ( t ) , 2 ∝ a − 4 ( t ) .

The Second Order Action One can restore the inflaton fluctuations by performing the transformation x 0 → x 0 ′ = x 0 + π , π plays the role of Goldstone boson and encodes the fluctuations of inflaton in an arbitrary coordinate system. g 00 transforms as ∂x ′ 0 ∂x ′ 0 g ′ 00 ( x ′ ) ∂x ν g µν ( x ) = ∂x µ g 00 − 2 ˙ πg 00 + ∂ i π∂ j πg ij + ˙ π 2 g 00 , = and δg 00 → 2 ˙ π + a − 2 ( π ,i ) 2 − ˙ π 2 .

The Second Order Action By going to Coulomb-radiation gauge, the full second order action is given by � � � π ) 2 + a − 2 ( π ,i ) 2 � d 4 x √− g π 2 − 1 4 M 1 δX (2) + M 2 ˙ πδX (1) + S = B 1 − ( ˙ + B 2 ˙ P 1 πδZ (1) � − 1 4 P 1 δZ (2) + P 2 ˙ πδZ (1) + ˙ , where δX (1) = − 4 δZ (1) = 2 a − 2 ˙ a 2 ˙ A x δ ˙ A x δ ˙ A x , A x , and � � δX (2) = 2 y + 1 x,y + 1 y,x − 2 − δ ˙ x − δ ˙ A 2 A 2 a 2 δA 2 a 2 δA 2 a 2 δA x,y δA y,x , a 2 δZ (2) = 1 a 2 [ δ ˙ x + δ ˙ A 2 A 2 y ] .

The Free Fields Scalar Field The free action of π is given by � �� � � d 4 x √− g ( − B 1 ) 1 − B 2 π 2 − a − 2 ( π ,i ) 2 S ( π ) = ˙ . 2 B 1 Note that B 1 ∝ ˙ H < 0 so the kinetic energy has the correct sign. The free wave function of π with the Minkowski initial conditions deep inside the horizon is H (1 + ikc s τ ) e − ikc s τ . π ( k ) = 2 k 3 / 2 � c s | B 1 |

The Free Fields Vector Field The free field action for δA i fluctuations is given by � ǫ ijk δA i,j � 2 � � � � 2 − M 1 a − 2 � d 4 x √− g 1 ( M 1 − P 1 S ( δA ) δ ˙ = 2 ) A i . 2 2 a 2 For simplicity we define M 1 ≡ M 1 − P 1 A x ( t ) ≡ Aa 3 , ˜ ˙ 2 ≡ M 1 a − 4 , in which M 1 and A are constants. With these definitions 2 R = M 1 A P H 2 . 6 M 2

The Free Fields Vector Field Decomposing the gauge field fluctuations in terms of its polarization base ǫ s i ( k ) in Fourier space � δA ( s ) ( k, t ) ǫ s δA i = i ( k ) . s The polarization vector can have either the linear polarization form with s = 1 , 2 or the circular (helicity) polarization form with s = ± . Now imposing the Minkowski initial condition for the gauge field fluctuations deep inside the horizon we obtain 1 2 c v M 1 Hτ 3 (1 + ikc v τ ) e − ikc v τ , δA ( s ) = k 3 / 2 � i in which c v represents the speed of gauge field fluctuations v = M 1 c 2 , ˜ M 1

Power Spectrum The leading interactions involving π and δA i fluctuations are given by � � � S ( πδA ) = dτd 3 x 4 HA M 1 πδA ′ x + Aa − 1 M 2 π ′ δA ′ , x M 2 ≡ M 2 a − 4 with M 2 being a constant. where M 2 − P 2 / 2 ≡ ˜ For simplicity we choose the wave number as k = k (cos θ, sin θ, 0) , where θ is the angle between the wave number and the preferred direction ˆ n , i.e. cos θ = � k · � n . � � 2 2 = 8 H 2 c 5 δP total s M 1 A 1 + 3 M 2 N 2 sin 2 θ . P (0) B 1 c v M 1 π Comparing the above expression with the amplitude of quadrupole anisotropy g ∗ yields � � 2 g ∗ = − 48 Rc 5 s N 2 1 + 3 M 2 , ǫ c v M 1

Bispectrum The leading cubic interactions in conformal time are � � a − 4 HM 1 πδA ′ 2 + a − 5 M 2 π ′ δA ′ 2 � S (3) = dτd 3 x 2 , The leading anisotropic contributions to bispectrum is given by � � ijk = (2 π ) 3 δ 3 ( k 1 + k 2 + k 3 ) B ijk ( k 1 , k 2 , k 3 ) , π ( k 1 ) π ( k 2 ) π ( k 3 ) where � � 3 2 36 c 6 s H 5 R 1 + 27 M + 45 M + 15 M 2 2 2 B tot ( k 1 , k 2 , k 3 ) = N k 1 N k 2 N k 3 ǫ 3 c 2 v M 4 3 2 M 1 M M P 1 1 � C ( k 2 , k 3 ) � × + 2c . p . , k 3 2 k 3 3 and k 2 ) 2 − ( � k 3 ) 2 + ( � n . � n . � n . � n . � k 3 ) ( � k 2 . � C ( k 2 , k 3 ) ≡ 1 − ( � k 2 ) ( � k 3 ) .

Gravitational Waves In the limit of small anisotropy the tensor perturbations in metric are ds 2 = a ( τ ) 2 � − dτ 2 + ( δ ij + h ij ) dx i dx j � . The quantum operators � h ij ( k , τ ) are decomposed in terms of the annihilation and creation operators as usual � h s ( k , τ ) e ( s ) � � � h s ( k , τ ) = h s ( k, τ ) a s ( k ) + h ∗ s ( k, τ ) a † h ij ( k , τ ) = ij ( k ) , s ( − k ) , s =+ , × where e ( s ) ij ( k ) is polarization base in Fourier space and the tensor excitations has the standard profile � � 2 iHτ 1 − i e − ikτ , √ h s ( k, τ ) = kτ M P 2 k

Gravitational Waves Power Spectrum the interaction Hamiltonians involving the mixing of tensor perturbations with the scalar and gauge field fluctuations have the following form H int = H πh + + H π ′ h + + H δA x h + + H δA V h × in which √ √ 2 H sin 2 θa 4 πh + , 2 M 2 sin 2 θa 3 π ′ h + H πh + = 2 2 M 1 A H π ′ h + = 2 2 A and H h + δA x = − M 1 A H h × δA V = iM 1 A 2 δA ′ √ √ x h + , sin θδA V h × . 2 The total anisotropy in tensor power spectrum is given by � � 2 � � � � RH 2 P c v k 3 N 2 sin 2 θ . c 2 − c 2 δ hh tot = − 3 c s v − 3 v + 1 M 2