conservation of the non-linear curvature perturbation in generic single-field inflation yukawa...
TRANSCRIPT
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Conservation of the non-linear curvature perturbation
in generic single-field inflation
Yukawa Institute for Theoretical PhysicsAtsushi Naruko
In Collaboration with Misao Sasaki Based on : Class. Quantum Grav. 28 072001
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The contents of talk
1, curvature perturbation and non-Gaussianity
2, gradient expansion approach
3, conservation of non-linear curvature perturbation 3.1, G = 0 (for canonical, k-essential scalar) 3.2, G ≠ 0 (for Galileon scalar) in Einstein gravity
4, summary
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curvature perturbation and
non-Gaussianity
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Curvature perturbation
• Curvature perturbation on a slicing is regarded as Newton potential, therefore it gives the initial condition for Cosmic Microwave Background (CMB).
• Through the observations of CMB, there is a possibility that we can subtract the information of primordial universe.
• Recently, a lot of attention is paid to “non-Gaussianity” in CMB as a new window for primordial universe.
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Non-Gaussianity
• From the observation by WMAP, we know that temperature fluctuations in CMB are scale invariant and are Gaussianly distributed.
• We parameterise the deviation from Gaussian using “fNL”
• The three point function is sensitive to this type NG.
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Gradient expansion approach
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non-linear cosmological perturbations
• In order to estimate the non-Gaussianity correctly, we have to make a analysis beyond linear order.
• There are mainly two approaches to nonlinear cosmological perturbations. 1. the standard perturbative approach. 2. the gradient expansion approach • In the gradient expansion, the field equations are expanded in powers of spatial gradients, therefore, it is applicable only to perturbations on superhorizon scales. → However, the full nonlinear effects are taken into account.
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Outside Horizon
• Thanks to the talk by Yamaguchi-san, Mizuno-san and De Felice-san, we know well about the perturbation in Galileon field at horizon exit.
• To give a theoretical prediction precisely , we need to solve the evolution of curvature perturbation outside Horizon.
• It have been known that curvature perturbation is conserved on superhorizon scales in the case of under .◯◯◯ ◎◎◎
inflation
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Lyth, Malik and sasaki
• Let’s consider a perfect fluid :
and write down the energy conservation law,
• We take uniform energy density slicing
if P is the function of ρ the rhs become a function of t
curvature per. is conserved
JCAP 0505:004,2005.
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P = P (ρ) ?
• From the study by LMS, we know that if P is a function of ρ, curvature perturbation is conserved on uniform energy density slice regardless of gravity theory.
• In the case of scalar field, whether / when this condition is satisfied is not so trivial because the relation between P and ρ is complicated.
canonical :
k-essence :
Galileon : very much complicated…
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Our goal
• We investigate the evolution of curvature perturbation on superhorizon scales in rather generic single-feild inflation using gradient expansion approach.
• We show the conservation of non-linear curvature perturbation
by using scalar field equation.
• We clarify under which circumstances curvature perturbation is conserved in the case of scalar field.
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Conservation of
curvature perturbation
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metric
• We express the metric in the (3 + 1) form,
• We choose the spatial coordinate such that βi vanish and further decompose the spatial metric as
• Ψ is called as curvature perturbation because it corresponds to 3D-ricci scalar at leading order in gradient expansion.
• We define the expansion by the divergence of nμ , which is a vector orthogonal to t = const. surface.
t = const.
nμ
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scalar field equation
• We consider a rather generic scalar field.
canonical
K-essence, DBI
• After taking the variation of the above action, we obtain the scalar field equation
Galileon
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• Neglecting spatial derivatives, the equation is rewritten as
G = 0
• If the system has evolved into attractor regime,
• We choose the uniform scalar slicing ,
• The conservation of curvature perturbation is shown without specifying gravity theory.
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G ≠ 0
• The lowest order equation in gradient expansion is
• There is a second derivative of Ψ (= derivative of K), → we cannot show the conservation W/O Einstein equation.
• Once we invoke Einstein gravity, we can replace K,τ with K.
Again, in the attractor stage, we can show the conservation of curvature perturbation on the uniform scalar field slice.
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summary
• Using the gradient expansion approach, we have studied the evolution of curvature perturbation on superhorizon scales in the case of single scalar field inflation.
• In the cases of canonical and k-essential scalar field (G = 0), we have shown the conservation of curvature perturbation without specifying gravity theory using scalar field equation.
• The condition for the conservation is whether the system has evolved into a attractor regime.
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summary 2
• In the case of galileon field, the conservation has not been shown without invoking gravity theory because there appear a second derivative of Ψ in the scalar field equation.
• Once we have used the Einstein equation, we can rewrite the second derivative by first derivative and we can show the conservation of curvature perturbation in the attractor stage.