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January 4, 2008

Effective Field Theory of Inflation

One of the grand organizing principles of my corner of Physics is that of effective field theory. Rather than analysing a whole bunch of different models, and hoping to discern a pattern, effective field theory gives us a way of parametrizing our ignorance about physics (at high energies, in the early universe …) that we don’t know, and give a model-independent analysis of its leading effects.

In a beautiful recent paper, Cheung et al give an effective field theory analysis of generic models of single-field inflation.

We’re interested in a fairly generic theory of a scalar field coupled to gravity, which happens to have an inflating FRW solution. Having such a scalar field provides a natural clock, “breaking” the symmetry under temporal diffeomorphisms. Spatial diffeomorphisms, however, remain unbroken, and we can use this as a tool to organize an effective Lagrangian.

If we expand about a homogeneous solution, ϕ=ϕ 0(t)\phi=\phi_0(t), the fluctuations, δϕ(x,t)\delta \phi(\vec{x},t), about this solution, transform inhomogeneously under temporal diffeomorphisms, tt+ξ 0(x,t),δϕδϕ+ϕ˙ 0(t)ξ 0 t \to t + \xi^0(\vec{x},t),\qquad \delta\phi \to \delta\phi + \dot{\phi}_0(t)\xi^0 If we wish, we can use this to fix to “unitary gauge” where δϕ0\delta\phi \equiv 0, and all of the fluctuations are in the metric. In the spirit of effective field theory, we should, in unitary gauge, write down a Lagrangian (for the fluctuations of the metric) containing all terms invariant under spatial, but not necessarily temporal diffeomorphisms. That is, we should allow terms involving

  • (g 00+1)(g^{0 0}+1)
  • variations of the extrinsic curvature δK μν=K μνa 2Hh μν \delta K_{\mu\nu} = K_{\mu\nu} - a^2 H h_{\mu\nu} of the constant-ϕ\phi slices
  • variations of the Riemann tensor δR μνρσ=R μνρσ2(H+k)h μ[ρh σ]ν(H˙+H 2)a 2h μσδ 0 ν δ 0 ρ perm \delta R_{\mu\nu\rho\sigma} = R_{\mu\nu\rho\sigma} -2 (H+k) h_{\mu [\rho} h_{\sigma]\nu} -(\dot{H}+H^2) a^2 h_{\mu\sigma}\tensor{\delta}{^0_\nu}\tensor{\delta}{^0_\rho} -\text{perm}
  • covariant derivatives, μ\nabla_\mu
  • arbitrary functions of time

but still covariant under spatial diffeomorphisms. (Here, h μν=g μν+n μn νh_{\mu\nu}= g_{\mu\nu}+ n_\mu n_\nu is the induced metric on the constant ϕ\phi surfaces and n μn^\mu is the unit normal.)

Demanding that we’re expanding about a solution fixes the terms linear in the fluctuations about FRW. S=d 4xg[12M pl 2R+M pl 2(H˙ka 2)g 00M pl 2(3H 2+H˙+2ka 2)+] S = \int d^4 x \sqrt{-g} \left[\tfrac{1}{2}M_{\text{pl}}^2 R + M_{\text{pl}}^2\left(\dot{H}- \tfrac{k}{a^2}\right)g^{0 0} - M_{\text{pl}}^2\left(3H^2 +\dot{H} +2\tfrac{k}{a^2}\right)+\dots\right] where “\dots” indicate terms quadratic and higher in the fluctuations.

Choosing the spatially-flat (k=0k=0) case, one can systematically expand the “\dots” as

(1)S= d 4xg[12M pl 2R+M pl 2H˙g 00M pl 2(3H 2+H˙) +12!M 2(t) 2(g 00+1) 2+13!M 3(t) 3(g 00+1) 3+ 12M¯ 1(t) 3(g 00+1)δK μ μ 12M¯ 2(t) 2(δK μ μ ) 212M¯ 3(t) 2δK μ ν K ν μ +]\begin{aligned} S =& \int d^4 x \sqrt{-g} \left[\tfrac{1}{2}M_{\text{pl}}^2 R + M_{\text{pl}}^2 \dot{H} g^{0 0} - M_{\text{pl}}^2\left(3H^2 +\dot{H}\right)\right.\\ & + \tfrac{1}{2!} M_2(t)^2 (g^{0 0}+1)^2 + \tfrac{1}{3!} M_3(t)^3 (g^{0 0}+1)^3+\dots\\ & \left.- \tfrac{1}{2} \overline{M}_1(t)^3 (g^{0 0}+1)\delta \tensor{K}{^\mu_\mu} - \tfrac{1}{2} \overline{M}_2(t)^2 {(\delta \tensor{K}{^\mu_\mu})}^2 - \tfrac{1}{2} \overline{M}_3(t)^2 \delta \tensor{K}{^\mu_\nu}\tensor{K}{^\nu_\mu}+\dots \right] \end{aligned}

Vanilla single-field slow-roll inflation — a scalar field with a minimal kinetic term and slow roll potential V(ϕ)V(\phi) — satisfies ϕ˙ 0(t) 2=2M pl 2H˙,V(ϕ 0(t))=M pl 2(3H 2+H˙) {\dot{\phi}_0(t)}^2 = -2 M_{\text{pl}}^2 \dot{H},\qquad V(\phi_0(t)) = M_{\text{pl}}^2 (3H^2 +\dot{H}) which, at the classical level, gives the first line of (1), with all the subsequent terms set to zero. But these will surely be generated by quantum corrections, and are present, even at tree level, in more exotic models.

DBI inflation is one of a class of models where the Lagrangian for the scalar field takes the form =f(X,ϕ)\mathcal{L} = f(X,\phi), X=g μν μϕνϕX = g^{\mu\nu}\partial_\mu \phi \partial\nu\phi. Expanding about a solution, ϕ 0(t)\phi_0(t), this yields an action of the form (1), where M n(t) 4=ϕ˙ 0 2n nfX n| ϕ=ϕ 0(t) M_n(t)^4 = {\dot{\phi}_0}^{2n}\left.\tfrac{\partial^n f}{{\partial X}^n}\right\vert_{\phi=\phi_0(t)}

And so forth. Any single field inflation model can be cast in the form of (1), for some choice of coefficients of the higher terms in the effective Lagrangian.

The unitary gauge Lagrangian, (1), describes the physics of two tensor modes and a scalar mode. As in spontaneously-broken gauge theories, there is — at sufficiently high energies — an approximate decoupling of the scalar mode from the tensor modes and an Equivalence Theorem that relates the self-interactions of the former to those of the Goldstone mode.

To get at the latter, one reintroduces the Goldstone mode via the Stückelberg trick. Under a temporal diffeomorphism, π(x)π˜(x˜(x))=π(x)ξ 0(x) \pi(x) \to \tilde{\pi}(\tilde{x}(x)) = \pi(x) -\xi^0(x) We convariantize each of the terms in (1) with respect to this action. For instance

(2)M pl 2H˙g 00M pl 2H˙(t+π(x))[g 00(1+π˙) 2+2(1+π˙) iπg 0i+g ij iπ jπ]M_{\text{pl}}^2 \dot{H} g^{0 0} \to M_{\text{pl}}^2 \dot{H}(t+\pi(x))\left[g^{0 0} (1+\dot{\pi})^2 + 2 (1+\dot{\pi})\partial_i \pi g^{0 i} + g^{i j} \partial_i \pi \partial_j \pi\right]

The result is, at first blush, a complete mess, and would be of no help, were we not interested in a slow-roll background.

(2) contains the kinetic term for π\pi 12g 00π˙ c 2,whereπ c=M pl(2H˙) 1/2π \tfrac{1}{2} g^{0 0}\dot{\pi}_c^2, \qquad \text{where}\quad \pi_c = M_{\text{pl}}{\left(2 \dot{H}\right)}^{1/2} \, \pi and the leading mixing term with the tensor modes H˙ 1/2 iπ ch i,whereh i=2M plg 0i {\dot{H}}^{1/2} \partial_i \pi_c h^i,\qquad \text{where}\quad h^i = \sqrt{2} M_{\text{pl}} g^{0 i} In the slow roll approximation, ϵ=M pl 22(VV) 2H˙H 21\epsilon = \tfrac{M_{\text{pl}}^2}{2} {\left(\frac{V'}{V}\right)}^2 \simeq - \frac{\dot{H}}{H^2}\ll 1, there’s a parametrically large range of energies

(3)ϵH 2E 24πϵM pl 2\epsilon H^2 \ll E^2 \ll 4\pi\epsilon M_{\text{pl}}^2

where the effective Lagrangian is valid and the mixing of π\pi with the tensor modes can be neglected.

In this regime, the physics of the scalar mode is encoded in

(4)S π=d 4xg[12M pl 2RM pl 2H˙(π˙ 2( iπ) 2a 2)+2M 2 4(π˙ 2+π˙ 3π˙( iπ) 2a 2)43M 3 4π˙ 3+]S_\pi = \int d^4 x \sqrt{-g} \left[ \tfrac{1}{2} M_{\text{pl}}^2 R - M_{\text{pl}}^2 \dot{H} \left(\dot{\pi}^2 - \tfrac{{(\partial_i \pi)}^2}{a^2}\right) + 2 M_2^4 \left(\dot{\pi}^2 +\dot{\pi}^3 -\dot{\pi} \tfrac{{(\partial_i \pi)}^2}{a^2}\right) - \tfrac{4}{3} M_3^4 \dot{\pi}^3 + \dots \right]

Typically, for extracting predictions from inflation, we’re interested in computing various quantities at Horizon-crossing, EHE\sim H, which lies well within this range. Here and below, we’re neglecting the additional π\pi-dependence, coming from Taylor-expanding the time-dependence of the coefficients, which is legitimate, if the time-dependence of the coefficients is slow (suppressed by a slow-roll parameter) compared to the Hubble time.

Actually, as you can see from (4), I lied slightly above. The first term on the second line of (1) also contributes to the kinetic term of π\pi and, in some models, it may even dominate over (2). The quadratic part of the action (4) is (M pl 2H˙+2M 2 4)π˙ 2+M pl 2H˙ iπ 2a 2 (- M_{\text{pl}}^2 \dot{H} +2 M_2^4) \dot{\pi}^2 + M_{\text{pl}}^2 \dot{H} \tfrac{{\partial_i\pi}^2}{a^2} The canonically-normalized π\pi field is π c=(2(M pl 2H˙+2M 2 4)) 1/2π \pi_c = {\left(2(- M_{\text{pl}}^2 \dot{H} +2 M_2^4)\right)}^{1/2}\pi and there’s a nontrivial speed of sound c s 2=M pl 2H˙M pl 2H˙+2M 2 4 c_s^2 = \frac{- M_{\text{pl}}^2 \dot{H}}{- M_{\text{pl}}^2 \dot{H} +2 M_2^4}

In (3), I estimated the cutoff for this effective theory as Λ 24πϵM pl 2\Lambda^2 \sim 4\pi \epsilon M_{\text{pl}}^2. When the speed of sound departs significantly from 11, Cheung et al claim a lower energy cutoff (since the effective theory is non-relativistic, we need to distinguish this from the momentum cutoff) Λ 24πϵ 1/2M plHc s 5/2(1c s 2) 1/2 \Lambda^2 \sim 4\pi \epsilon^{1/2} M_{\text{pl}} H \frac{c_s^{5/2}}{{(1-c_s^2)}^{1/2}} At the very least, we should demand HΛH\ll \Lambda, which sets a lower limit on the speed of sound in the context of this effective Lagrangian.

There are a great many applications of (4). One can calculate the tilt, n sn_s, or the strength of the non-Gaussianities, etc, in a fairly unified and model-independent fashion. And it gives a nice way of thinking about the particular features of various different models of inflation.

Particularly nice is their analysis of models (like the ghost condensate) in which the M 2 4M_2^4 term dominates over (2). In this limit (of small c sc_s), the extrinsic curvature terms in the third line of (1) become important and lead to an d 4xg[M¯ 2 2+M¯ 3 22( i 2π) 2a 4] \int d^4 x \sqrt{-g} \left[ - \frac{\overline{M}_2^2 +\overline{M}_3^2}{2}\frac{{(\partial^2_i \pi)}^2}{a^4} \right] term in the quadratic part of the action, i.e. to a nonrelativistic dispersion relation for π\pi.

Posted by distler at January 4, 2008 3:06 AM

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3 Comments & 1 Trackback

Re: Effective Field Theory of Inflation

Thats indeed a very nice and carefull paper. Thanks for the link.

Posted by: Robert on January 9, 2008 5:00 AM | Permalink | Reply to this

Re: Effective Field Theory of Inflation

Dear Jacques,

I’m just a grad student here.

This paper reminds me of a question I’ve been wondering about. In Minkowski space QFT, we classify terms in the lagrangian by their mass dimensions to figure if they are “relevant,” “marginal,” or “irrevelant” in the low energy limit. Does this classification still hold in curved spacetimes? And if so, is dimensional analysis still a valid way to carry out this classification? Is there a place that discusses these and related issues at a pedagogical and basic level?

Thank you.

Posted by: wandering.the.cosmos on January 12, 2008 10:17 PM | Permalink | Reply to this

Re: Effective Field Theory of Inflation

In a non-Lorentz-invariant effective theory, the power counting can be rather different from what you would find in a Lorentz-invariant theory. The last paragraph of this blog post (and the corresponding section of the paper) provides one example of this. Another would be Polchinski’s discussion of the effective field theory of superconductivity.

If you have something specific in mind, we could discuss that further…

Posted by: Jacques Distler on January 12, 2008 10:33 PM | Permalink | PGP Sig | Reply to this
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