Musings

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, $\phi=\phi_0(t)$, the fluctuations, $\delta \phi(\vec{x},t)$, about this solution, transform inhomogeneously under temporal diffeomorphisms, $$t \to t + \xi^0(\vec{x},t),\qquad \delta\phi \to \delta\phi + \dot{\phi}_0(t)\xi^0$$ If we wish, we can use this to fix to “unitary gauge” where $\delta\phi \equiv 0$, 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

• $(g^{0 0}+1)$
• variations of the extrinsic curvature $$\delta K_{\mu\nu} = K_{\mu\nu} - a^2 H h_{\mu\nu}$$ of the constant-$\phi$ slices
• variations of the Riemann tensor $$\delta R_{\mu\nu\rho\sigma} = R_{\mu\nu\rho\sigma} -2 (H+k) h_{\mu [\rho} h_{\sigma]\nu} -(\dot{H}+H^2) a^2 h_{\mu\sigma}\tensor{\delta}{^0_\nu}\tensor{\delta}{^0_\rho} -\text{perm}$$
• covariant derivatives, $\nabla_\mu$
• arbitrary functions of time

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

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

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

Vanilla single-field slow-roll inflation — a scalar field with a minimal kinetic term and slow roll potential $V(\phi)$ — satisfies $${\dot{\phi}_0(t)}^2 = -2 M_{\text{pl}}^2 \dot{H},\qquad V(\phi_0(t)) = M_{\text{pl}}^2 (3H^2 +\dot{H})$$ 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 $\mathcal{L} = f(X,\phi)$, $X = g^{\mu\nu}\partial_\mu \phi \partial\nu\phi$. Expanding about a solution, $\phi_0(t)$, this yields an action of the form (1), where $$M_n(t)^4 = {\dot{\phi}_0}^{2n}\left.\tfrac{\partial^n f}{{\partial X}^n}\right\vert_{\phi=\phi_0(t)}$$

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, $$\pi(x) \to \tilde{\pi}(\tilde{x}(x)) = \pi(x) -\xi^0(x)$$ We convariantize each of the terms in (1) with respect to this action. For instance

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 $\pi$ $$\tfrac{1}{2} g^{0 0}\dot{\pi}_c^2, \qquad \text{where}\quad \pi_c = M_{\text{pl}}{\left(2 \dot{H}\right)}^{1/2} \, \pi$$ and the leading mixing term with the tensor modes $${\dot{H}}^{1/2} \partial_i \pi_c h^i,\qquad \text{where}\quad h^i = \sqrt{2} M_{\text{pl}} g^{0 i}$$ In the slow roll approximation, $\epsilon = \tfrac{M_{\text{pl}}^2}{2} {\left(\frac{V'}{V}\right)}^2 \simeq - \frac{\dot{H}}{H^2}\ll 1$, there’s a parametrically large range of energies

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

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

Typically, for extracting predictions from inflation, we’re interested in computing various quantities at Horizon-crossing, $E\sim H$, which lies well within this range. Here and below, we’re neglecting the additional $\pi$-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 $\pi$ and, in some models, it may even dominate over (2). The quadratic part of the action (4) is $$(- M_{\text{pl}}^2 \dot{H} +2 M_2^4) \dot{\pi}^2 + M_{\text{pl}}^2 \dot{H} \tfrac{{\partial_i\pi}^2}{a^2}$$ The canonically-normalized $\pi$ field is $$\pi_c = {\left(2(- M_{\text{pl}}^2 \dot{H} +2 M_2^4)\right)}^{1/2}\pi$$ and there’s a nontrivial speed of sound $$c_s^2 = \frac{- M_{\text{pl}}^2 \dot{H}}{- M_{\text{pl}}^2 \dot{H} +2 M_2^4}$$

In (3), I estimated the cutoff for this effective theory as $\Lambda^2 \sim 4\pi \epsilon M_{\text{pl}}^2$. When the speed of sound departs significantly from $1$, Cheung et al claim a lower energy cutoff (since the effective theory is non-relativistic, we need to distinguish this from the momentum cutoff) $$\Lambda^2 \sim 4\pi \epsilon^{1/2} M_{\text{pl}} H \frac{c_s^{5/2}}{{(1-c_s^2)}^{1/2}}$$ At the very least, we should demand $H\ll \Lambda$, 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, $n_s$, 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 $M_2^4$ term dominates over (2). In this limit (of small $c_s$), the extrinsic curvature terms in the third line of (1) become important and lead to an $$\int d^4 x \sqrt{-g} \left[ - \frac{\overline{M}_2^2 +\overline{M}_3^2}{2}\frac{{(\partial^2_i \pi)}^2}{a^4} \right]$$ term in the quadratic part of the action, i.e. to a nonrelativistic dispersion relation for $\pi$.