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Testing gravity theories using tensor perturba- tions Jan Novák

Testing gravity theories using tensor perturbations

Jan Novák

Technical university in Liberec, Czech republic

7.9.2017, Praha

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Testing gravity theories using tensor perturba- tions Jan Novák

Graviton

Central role in quantization of the gravitational field is played by the graviton, which is a massless particle of spin 2.

Spin 2, massless

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Testing gravity theories using tensor perturba- tions Jan Novák

Propagation equation

ds2 = −dt2 + a2(t)(δij + hij)dxidxj, (1) ¨ hk + 3 ˙ a a ˙ hk + a2 k2 hk = 16πGΠT

k ,

(2)

Harmonic oscillator

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Testing gravity theories using tensor perturba- tions Jan Novák

Relativistic theories of gravity other than GR

Relativistic theories of gravity other than GR can change the damping rate of the graviational waves in the second term,

• r they can modify the speed of propagation of the

gravitational waves in the third term

• r add some source term on the RHS.1

1At 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.

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Testing gravity theories using tensor perturba- tions Jan Novák

Modified propagation equation

h′′

k + 2g′

g h′

k + k2hk = Fk(η),

(3) where we included all terms on the RHS in one function Fk(η).

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Testing gravity theories using tensor perturba- tions Jan Novák

Gravitational perturbation inside horizon

h′′

k + 2g′

g h′

k + k2hk = 0

(4) When we use the substitution W = ghk, we obtain this equation W ′′ + (k2 − a′′ a )W = 0. (5) We obtain the solution hk(t) → √ 16πG (2π)3/2√ 2kg exp(−ik

• dη)

(6) in early times of inflation.

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Testing gravity theories using tensor perturba- tions Jan Novák

Horizon

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Testing gravity theories using tensor perturba- tions Jan Novák

Gravitational perturbation outside horizon

Under this assumption, we have a = − 1

Hη, where H is

constant expansion rate during inflation. And the above equation (4) becomes h′′

k − 2(1 + ˜

ν0) τ h′

k + k2hk = 0.

(7) If we let x = −kη and hk = x

3 2 +˜

ν0y, the above

equation becomes x2 d2y dx2 + x dy dx + [x2 − (3 2 + ˜ ν0)2]y = 0. (8)

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Testing gravity theories using tensor perturba- tions Jan Novák

Inflation

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Testing gravity theories using tensor perturba- tions Jan Novák

Tensor mode spectrum

Bessel equation t2¨ y + t ˙ y + [t2 − ν2]y = 0 (9) From this we obtain the tensor mode spectrum ∆t = G(2H)2(1+˜

ν0)[Γ( 3 2 + ˜

ν0)]2 π3k3+2˜

ν0

. (10)

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Testing gravity theories using tensor perturba- tions Jan Novák

Inflation consistency equation

Since we get the dependence k−3−2˜

ν0 we can identify

the tensor spectral index as nT = −2˜ ν0. (11) It vanishes in GR. Let’s look at slow roll inflation. We

• btain the result

nT = −2˜ ν0 − 2ǫ, (12) where ǫ = − ˙

H H2 .

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Testing gravity theories using tensor perturba- tions Jan Novák

Inflation

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Testing gravity theories using tensor perturba- tions Jan Novák

Modified inflation consistency equation

nT = −2˜ ν0 − r/8, (13) ˜ ν0 affects the CMB B-mode power spectrum

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Testing gravity theories using tensor perturba- tions Jan Novák

Bimetric theories of gravity

Anisotropic stress is a sign of a modification of gravity! Source term on the RHS: a2Γγij, Γ = Γ(a b, ai) (14)

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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).

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