Multifield Inflation After Planck: the Case for Nonminimal Couplings
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View metadata, citation and similar papers at core.ac.uk brought to you by CORE week endingprovided by DSpace@MIT PRL 112, 011302 (2014) PHYSICAL REVIEW LETTERS 10 JANUARY 2014 Multifield Inflation after Planck: The Case for Nonminimal Couplings † David I. Kaiser* and Evangelos I. Sfakianakis Department of Physics and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 11 April 2013; published 7 January 2014) Multifield models of inflation with nonminimal couplings are in excellent agreement with the recent results from Planck. Across a broad range of couplings and initial conditions, such models evolve along an effectively single-field attractor solution and predict values of the primordial spectral index and its run- ning, the tensor-to-scalar ratio, and non-Gaussianities squarely in the observationally most-favored region. Such models can also amplify isocurvature perturbations, which could account for the low power recently observed in the cosmic microwave background power spectrum at low multipoles. Future measurements of primordial isocurvature perturbations could distinguish between the currently viable possibilities. DOI: 10.1103/PhysRevLett.112.011302 PACS numbers: 98.80.Cq Early-Universe inflation remains the leading framework We consider this class of models to be well motivated for for understanding a variety of features of our observable several reasons. Realistic models of particle physics include Universe [1,2]. Most impressive has been the prediction multiple scalar fields at high energies. In any such model, of primordial quantum fluctuations that could seed large- nonminimal couplings are required for self-consistency, scale structure. Recent measurements of the spectral tilt since they arise as renormalization counterterms when of primordial (scalar) perturbations ns find a decisive quantizing scalar fields in curved spacetime [7]. More- departure from a scale-invariant spectrum [3,4].The over, the nonminimal coupling constants generically rise Planck Collaboration’svalue,ns ¼ 0.9603 Æ 0.0073,dif- with energy under renormalization-group flow with no fers from ns ¼ 1 by more than 5σ. At the same time, UV fixed point [8], and hence one expects jξj ≫ 1 at infla- observations with Planck constrain the ratio of tensor- tionary energy scales. In such models inflation occurs for to-scalar perturbations to r<0.11 (95% C.L.), and are field values and energy densities well below the Planck consistent with the absence of primordial non- scale (see [9–11] and references therein). Higgs inflation ∼ 0 Gaussianities, fNL [4,5]. [11] is an elegant example: in renormalizable gauges The Planck team also observes less power in the angular (appropriate for high energies) the Goldstone modes remain power spectrum of temperature anisotropies in the cosmic in the spectrum, yielding a multifield model [10,12,13]. microwave background (CMB) radiation at low multipoles, We demonstrate here for the first time that models of this l ∼ 20–40, compared to best-fit ΛCDM cosmology: a broad class exhibit an attractor behavior: over a wide range 2.5–3σ departure on large angular scales, θ > 5° [6]. of couplings and fields’ initial conditions, the fields evolve Many physical processes might ultimately account for along an effectively single-field trajectory for most of infla- the deviation, but a primordial source seems likely given tion. Although attractor behavior is common for single- the long length scales affected. One plausible possibility field models of inflation [14], the dynamics of multifield is that the discrepancy arises from the amplification of iso- models generally show strong sensitivity to couplings curvature modes during inflation [4]. and initial conditions (see, e.g., [15] and references In this Letter we demonstrate that simple, well-motivated therein). This is not true for the class of multifield models multifield models with nonminimal couplings match the examined here, thanks to the shape of the effective potential latest observations particularly well, with no fine-tuning. in the Einstein frame. The multifield attractor behavior This class of models (i) generically includes potentials that demonstrated here means that, for most regions of phase are concave rather than convex at large field values, space and parameter space, this general class of models (ii) generically predicts values of r and ns in the most- yields values of ns, r, the running of the spectral index α favored region of the recent observations, (iii) generically ¼ dns=d ln k, and fNL in excellent agreement with recent ∼ 0 predicts fNL except for exponentially fine-tuned initial observations. The well-known empirical success of single- field values, (iv) generically predicts ample entropy pro- field models with nonminimal couplings [11,16] is thus duction at the end of inflation, with an effective equation preserved for more realistic models involving multiple ∼ 0 1 3 of state weff ½ ; = , and (v) generically includes isocur- fields. Whereas the attractor behavior creates a large obser- vature perturbations as well as adiabatic perturbations, vational degeneracy in the r versus ns plane, the isocurva- which might account for the low power in the CMB power ture spectra from these models depend sensitively upon spectrum at low multipoles. couplings and initial conditions. Future measurements of 0031-9007=14=112(1)=011302(5) 011302-1 © 2014 American Physical Society week ending PRL 112, 011302 (2014) PHYSICAL REVIEW LETTERS 10 JANUARY 2014 primordial isocurvature spectra could therefore distinguish among models in this class. In the Jordan frame, the fields’ nonminimal couplings remain explicit in the action Z pffiffiffiffiffiffi 1 4 −~ ϕI ~ − δ ~μν∂ ϕI∂ ϕJ − ~ ϕI SJ ¼ d x g fð ÞR 2 IJg μ ν Vð Þ ; (1) FIG. 1 (color online). Potential in the Einstein frame, VðϕIÞ. Left: λχ ¼ 0.75λϕ, g ¼ λϕ, ξχ ¼ 1.2ξϕ. Right: λχ ¼ g ¼ λϕ, where quantities in the Jordan frame are marked by ξ ξ ξ ≫ 1 0 λ 1 a tilde. Performing the usual conformal transforma- ϕ ¼ χ. In both cases, I and < I, g< . ~ → 2 −2 ϕI ~ tion, gμνðxÞ gμνðxÞ¼ MPl fð ðxÞÞgμνðxÞ, where ≡ 8π −1=2 2 43 1018 MPl ð GÞ ¼ . × GeV is the reduced surements of the Higgs boson mass near the electroweak Planck mass, we may write the action in the Einstein frame symmetry-breaking scale require λϕ ≃ 0.13. Under renorm- as [9] alization-group flow, λϕ will fall to the range 0 < λϕ < 0.01 Z λ 0 01 ξ ≥ ffiffiffiffiffiffi 2 1 at the inflationary energy scale; ϕ ¼ . requires ϕ 4 p MPl μν I J I 780 S ¼ d x −g R − G g ∂μϕ ∂νϕ − Vðϕ Þ : to satisfy the constraint on HðtÃÞ=MPl, which in turn E 2 2 IJ 1 2 requires ξϕ ∼ Oð10 –10 Þ at low energies [17]. For our (2) general class of models, we therefore consider couplings at the inflationary energy scale of order λI;g∼ The potential in the Einstein frame VðϕIÞ is stretched −2 3 Oð10 Þ and ξI ∼ Oð10 Þ [18]. by the conformal factor compared to the Jordan-frame Expanding the scalar fields to first order, potential: ϕIðxμÞ¼φIðtÞþδϕIðxμÞ, we find [9,10] 4 I MPl ~ I 2 _ 2 Vðϕ Þ¼ Vðϕ Þ: (3) M 2 3f 4f2ðϕIÞ σ_ 2 ¼ G φ_ Iφ_ J ¼ Pl ϕ_ þ χ_2 þ : (5) IJ 2f f The nonminimal couplings induce a curved field-space manifold in the Einstein frame with metric We also expand the spacetime metric to first order around a G ϕK 2 2 δ 3 IJð Þ¼½MPl=ð fÞ½ IJ þ f;If;J=f, where f;I ¼ spatially flat Friedmann-Robertson-Walker metric. Then ∂f=∂ϕI [9]. We adopt the form for fðϕIÞ required for the background dynamics are given by [9] renormalization [7], 1 1 1 X 2 σ_ 2 _ − σ_ 2 1 H ¼ 2 þ V ; H ¼ 2 ; f ϕI M2 ξ ϕI 2 : 3M 2 2M ð Þ¼2 Pl þ Ið Þ (4) Pl Pl I I I IK Dtφ_ þ 3Hφ_ þ G V;K ¼ 0; (6) Here we consider two-field models, I, J ¼ ϕ, χ. D D I ≡ As emphasized in [9–11], the conformal stretching of the where t is the (covariant) directional derivative, tA φ_ JD I _ I ΓI Jφ_ K Einstein-frame potential, Eq. (3), generically leads to con- JA ¼ A þ JKA [9,19]. The gauge-invariant cave potentials at large field values, even for Jordan-frame Mukhanov-Sasaki variables for the linearized perturbations potentials that are convex. In particular, for a Jordan-frame QI obey an equation of motion with a mass-squared matrix ~ I 4 potential of the simple form Vðϕ Þ¼ðλϕ=4Þϕ þ given by [9,19] 2 2 4 ðg=2Þϕ χ þðλχ=4Þχ , Eqs. (3) and (4) yield a potential MI ≡ GIJ D D − RI φ_ Lφ_ M in the Einstein frame that is nearly flat for large field values, J ð J KVÞ LMJ ; (7) ϕI → λ 4 4ξ2 Vð Þ JMPl=ð JÞ (no sum on J), as the Jth component ϕI of becomes arbitrarily large. This basic feature leads to where RI is the Riemann tensor for the field-space “ ” LMJ extra-slow roll evolution of the fields during inflation. If manifold. λ ξ ϕI the couplings J and J are not equal to each other, Vð Þ To analyze inflationary dynamics, we use a multifield develops ridges separated by valleys [9]. Inflation occurs in formalism (see [2,20] for reviews) made covariant with the valleys as well as along the ridges, since both are respect to the nontrivial field-space curvature (see [9,19] ≠ 0 regions of false vacuum with V . See Fig. 1. and references therein). We define adiabatic and isocurva- Constraints on r constrain the energy scale of inflation, ture directions in the curved field space via the unit vectors 3 7 10−5 HðtÃÞ=MPl < .