Testing gravity theories using tensor Jan Novk perturbations Jan - PowerPoint PPT Presentation

Testing gravity theories using tensor perturba- tions Testing gravity theories using tensor Jan Novák perturbations Jan Novák Technical university in Liberec, Czech republic 7.9.2017, Praha 1 / 15

Graviton Testing gravity Central role in quantization of the gravitational field is played by theories using tensor the graviton, which is a massless particle of spin 2. perturba- tions Jan Novák Spin 2 , massless 2 / 15

Propagation equation Testing gravity ds 2 = − dt 2 + a 2 ( t )( δ ij + h ij ) dx i dx j , theories (1) using tensor perturba- tions h k + a 2 h k + 3 ˙ a Jan Novák ¨ ˙ k 2 h k = 16 π G Π T k , (2) a Harmonic oscillator 3 / 15

Relativistic theories of gravity other than GR Testing gravity theories using tensor perturba- tions Relativistic theories of gravity other than GR can Jan Novák change the damping rate of the graviational waves in the second term, or they can modify the speed of propagation of the gravitational waves in the third term or add some source term on the RHS. 1 1 At second order in perturbations anisotropic stress always appears even when the matter consists completely of dust. It means that there will be a RHS term at second order in perturbations. 4 / 15

Modified propagation equation Testing gravity theories using tensor perturba- tions Jan Novák k + 2 g ′ k + k 2 h k = F k ( η ) , h ′′ g h ′ (3) where we included all terms on the RHS in one function F k ( η ) . 5 / 15

Gravitational perturbation inside horizon Testing gravity theories using tensor k + 2 g ′ perturba- k + k 2 h k = 0 h ′′ g h ′ (4) tions Jan Novák When we use the substitution W = gh k , we obtain this equation W ′′ + ( k 2 − a ′′ a ) W = 0 . (5) We obtain the solution √ 16 π G � h k ( t ) → exp ( − ik d η ) (6) ( 2 π ) 3 / 2 √ 2 kg in early times of inflation. 6 / 15

Horizon Testing gravity theories using tensor perturba- tions Jan Novák 7 / 15

Gravitational perturbation outside horizon Testing gravity theories Under this assumption, we have a = − 1 H η , where H is using tensor perturba- tions constant expansion rate during inflation. And the above equation (4) becomes Jan Novák k − 2 ( 1 + ˜ ν 0 ) k + k 2 h k = 0 . h ′′ h ′ (7) τ 3 2 +˜ ν 0 y , the above If we let x = − k η and h k = x equation becomes x 2 d 2 y dx + [ x 2 − ( 3 dx 2 + x dy ν 0 ) 2 ] y = 0 . 2 + ˜ (8) 8 / 15

Inflation Testing gravity theories using tensor perturba- tions Jan Novák 9 / 15

Tensor mode spectrum Testing gravity theories using tensor perturba- tions Jan Novák Bessel equation t 2 ¨ y + [ t 2 − ν 2 ] y = 0 y + t ˙ (9) From this we obtain the tensor mode spectrum ∆ t = G ( 2 H ) 2 ( 1 +˜ ν 0 ) [Γ( 3 ν 0 )] 2 2 + ˜ . (10) π 3 k 3 + 2 ˜ ν 0 10 / 15

Inflation consistency equation Testing gravity theories using tensor perturba- ν 0 we can identify Since we get the dependence k − 3 − 2 ˜ tions the tensor spectral index as Jan Novák n T = − 2 ˜ ν 0 . (11) It vanishes in GR. Let’s look at slow roll inflation. We obtain the result n T = − 2 ˜ ν 0 − 2 ǫ, (12) where ǫ = − ˙ H H 2 . 11 / 15

Inflation Testing gravity theories using tensor perturba- tions Jan Novák 12 / 15

Modified inflation consistency equation Testing gravity n T = − 2 ˜ ν 0 − r / 8 , (13) theories using tensor perturba- ˜ ν 0 affects the CMB B-mode power spectrum tions Jan Novák 13 / 15

Bimetric theories of gravity Testing gravity theories using tensor perturba- tions Jan Novák Anisotropic stress is a sign of a modification of gravity! Source term on the RHS: a 2 Γ γ ij , Γ = Γ( a b , a i ) (14) 14 / 15

Testing gravity theories using tensor perturba- tions Jan Novák Pictures were taken from www.scienceblogs.com, Inflation in brane world gravity (A.Banersee) and TASI lectures on Inflation (D.Baumann). 15 / 15