Nearly scale-invariant curvature modes from entropy perturbations during graceful exit
NNearly scale-invariant curvature modes from entropy perturbations during graceful exit
Anna Ijjas a,b, ∗ , Roman Kolevatov c a Max Planck Institute for Gravitational Physics (Albert Einstein Institute), 30167 Hannover, Germany b Institute for Gravitational Physics, Leibniz University Hannover, 30167 Hannover, Germany c Department of Physics, Princeton University, Princeton, NJ 08544, USA
Abstract
In this Letter, we describe how a spectrum of entropic perturbations generated during a period of slow contraction can source anearly scale-invariant spectrum of curvature perturbations on length scales larger than the Hubble radius during the transition fromslow contraction to a classical non-singular bounce (the ‘graceful exit’ phase). The sourcing occurs naturally through higher-orderscalar field kinetic terms common to classical (non-singular) bounce mechanisms. We present a concrete example in which, by theend of the graceful exit phase, the initial entropic fluctuations have become negligible and the curvature fluctuations have a nearlyscale-invariant spectrum with an amplitude consistent with observations.
Keywords: slow contraction, entropic mechanism, graceful exit, cosmological bounce, bouncing cosmology
1. Introduction.
Observational evidence [1, 2, 3] combined with theoreticalreasoning [4, 5] strongly indicate that the gravitationally boundstructures (galaxies, galaxy clusters, etc. ) that comprise ouruniverse originate from quantum fluctuations of scalar fieldsgenerated on sub-Hubble wavelengths that evolve to induceclassical curvature perturbations with a nearly scale-invariantand gaussian spectrum on super-Hubble wavelengths. Accord-ing to the leading paradigms, the relevant quantum fluctuationsare generated during a primordial smoothing phase at energydensities su ffi ciently below the Planck density so that the cos-mological background can be described to leading order byclassical equations of motion.Two candidates for the smoothing phase are a period of ac-celerated expansion (˙ a , ¨ a >
0) and a period of slow contraction(˙ a , ¨ a < a ( t ) and the dot denotes dif-ferentiation with respect to the physical FRW time coordinate t . The key underlying idea is that, in either case, the scale fac-tor a ( t ) and the Hubble radius | H − | ≡ | a / ˙ a | evolve at di ff erentrates, | H − | ∝ a (cid:15) , (1)as determined by the equation of state (cid:15) ≡ (cid:32) + p (cid:37) (cid:33) (2)of the dominant stress-energy component with pressure p andenergy density ρ [6]. During an accelerated expansion ( (cid:15) < ∗ Corresponding author
Email address: [email protected] (Anna Ijjas)
1) phase, the Hubble radius stays nearly constant while scalarfield fluctuation wavelengths, which grow in proportion to thescale factor a , stretch at an ultra-rapid rate to become super-Hubble. In contrast, the Hubble radius during slow contraction( (cid:15) >
3) shrinks ultra-rapidly while the scale factor is nearlyconstant. For example, in a typical slow contraction phase, theHubble radius might shrink by a factor of 2 during which thescale factor decreases by only a factor of two [7]. As a result,fluctuation wavelengths that were sub-Hubble at the beginningof the phase evolve to become super-Hubble by the end.However, generating scalar field fluctuation modes withsuper-Hubble wavelengths is necessary but not su ffi cient to ex-plain cosmological observations. To explain measurements ofthe cosmic microwave background and the power spectrum ofgravitationally bound structures, the scalar field fluctuationsmust somehow source a nearly scale-invariant spectrum of co-moving curvature fluctuations of the metric with the amplitudeof ∼ − .In general, scalar field fluctuations source two types of met-ric fluctuations: adiabatic fluctuations on constant mean curva-ture hypersurfaces; and entropic fluctuations on hypersurfacesof constant energy density [8, 9]. Notably, scalar fields in back-grounds undergoing accelerated expansion can generate bothtypes. If it can be arranged that the metric fluctuations arepurely adiabatic and of the correct small amplitude, they canpotentially account for the observed temperature fluctuationsof the cosmic microwave background. However, acceleratedexpansion also inevitably stretches rare, large-amplitude scalarfield fluctuations that source large-amplitude adiabatic fluctu-ations of the metric. These large metric fluctuations triggerthe well-known quantum runaway problem, an e ff ect that spoilsthe spectrum and destroys homogeneity and isotropy altogether[10, 11, 12]. Slow contraction, on the other hand, can onlyamplify entropic modes. Adiabatic modes (as well as gravita- Preprint submitted to Elsevier February 9, 2021 a r X i v : . [ g r- q c ] F e b ional waves [13]) experience a growing anti-friction due to therapidly decreasing Hubble radius which leads to their decay.This eliminates the quantum runaway problem, an importantand distinctive advantage of the slow contraction scenario. Todate, we do not know of any other smoothing mechanism thatcould do the same.It is well-known that, during smoothing slow contraction, anon-linear sigma type kinetic interaction between two scalarfields naturally leads to a nearly scale-invariant and gaussianspectrum of super-Hubble relative field fluctuations – purelyentropic modes – which are quantum generated long before themodes leave the Hubble radius [14, 15, 16, 17]. In this Letter,we demonstrate how these entropic modes can source curvatureperturbations on super-Hubble scales during the ‘graceful exit’phase, i.e. , the transition from slow contraction to the bouncestage. We show that the sourcing is due to a common feature ofclassical (non-singular) bounce models in which higher-orderkinetic terms associated with the scalar matter fields becomeimportant during graceful exit [18, 19]. For concreteness, wepresent an example in which the only significant fluctuations atthe beginning of the graceful exit phase are entropic but, by theend of the phase, the entropic fluctuations have become negligi-ble and the curvature fluctuations have a nearly scale-invariantspectrum with an amplitude consistent with observations.
2. Cosmological model
In the scenario that we shall consider the cosmological evo-lution is sourced by two kinetically-coupled scalar fields φ and χ both of which are minimally coupled to Einstein gravity. Thecorresponding Lagrangian density is defined as L = R − ( ∂ µ φ ) − Σ ( φ )( ∂ µ χ ) + Σ ( φ )( ∂ µ χ ) − V ( φ, χ ) , (3)where R is the Ricci scalar; Σ ( φ ) is the quadratic kinetic cou-pling function; Σ ( φ ) is the quartic kinetic coupling functionand V ( φ, χ ) is the scalar potential depending on both φ and χ fields. The potential is steep and negative along the φ directionwhile nearly constant along the χ direction. Throughout, weuse reduced Planck units.Our interest here is analyzing what happens after a periodof slow contraction has already homogenized and isotropizedspacetime well-described by an FRW metric. Varying the ac-tion given through Eq. (3) with respect to the scalar fields andevaluating for the FRW background yields the evolution equa-tions for φ and χ :¨ φ + H ˙ φ + V ,φ = (cid:16) Σ ,φ + Σ ,φ ˙ χ (cid:17) ˙ χ , (4a) (cid:32) + Σ ˙ χ Σ + Σ ˙ χ (cid:33) ¨ χ + (cid:32) H + Σ ,φ + Σ ,φ ˙ χ Σ + Σ ˙ χ ˙ φ (cid:33) ˙ χ = (4b) = − V ,χ Σ + Σ ˙ χ . Note that the symmetries of the FRW space-time geometry leadto spatially homogeneous background field distributions, i.e. , φ = φ ( t ), χ = χ ( t ). Variation of Eq. (3) with respect to the metric yields thestress-energy tensor T µν . On an FRW background, scalar fields(collectively) act as perfect fluids and can be associated with anenergy density ρ and pressure p being given by the temporaland spatial components of T µν : ρ = − T = ˙ φ + (cid:16) Σ + Σ ˙ χ (cid:17) ˙ χ + V , (5a) p = T ii = ˙ φ + (cid:16) Σ + Σ ˙ χ (cid:17) ˙ χ − V . (5b)With Eq. (2), we can define the e ff ective equation of state asso-ciated with the ‘fluid’ as follows: (cid:15) = − Σ ˙ χ H − VH . (6)Finally, the Friedmann constraint and evolution equation takethe form: 3 H = ρ = ˙ φ + (cid:16) Σ + Σ ˙ χ (cid:17) ˙ χ + V , (7a) − H = ρ + p = ˙ φ + (cid:16) Σ + Σ ˙ χ (cid:17) ˙ χ . (7b)
3. Entropy Modes from Slow Contraction
As an example, we consider a scalar field potential that, dur-ing the slow contraction phase, is negative and steeply gradedalong the φ direction: V ( φ, χ ) ≈ − V e φ/ M . (8)Here V > M is the characteristic massscale associated with φ . At low energies, especially during thesmoothing slow contraction phase, higher-order kinetic termsas well as the χ field’s potential energy density are negligible,such that the Einstein-scalar system reduces to the simple set ofevolution and constraint equations:¨ φ + H ˙ φ − V M e φ/ M ≈ , (9a)¨ χ + (cid:32) H + Σ ,φ Σ ˙ φ (cid:33) ˙ χ ≈ , (9b)3 H ≈ ˙ φ + Σ ˙ χ − V e φ/ M . (9c)For Σ = e φ/ m , where m (cid:46) M , it is straightforward to show(see Ref. [17]) that the unique attractor scaling solution of theEinstein-scalar system of equations (9) is: φ ≈ − M × ln ( − At ) , a ≈ ( − t ) (cid:15) , (cid:15) ≈ (cid:18) M Pl M (cid:19) , (10a)˙ χ ≈ , (10b)where A = M − (cid:15) √ V / ( (cid:15) −
3) and (cid:15) is the equation of state asdefined in Eq. (2). The FRW time coordinate t runs from largenegative to small negative values and we normalized a such that a = | H − | ≈ a (cid:15) shrinking exponentiallyfaster than the scale factor a . For example, for M / M Pl ∼ . (cid:15) ∼
50 such that | H − | shrinks by a factor of 2 while a de-creases by a factor of 2, the case described in the Introduction.2 key to the background dynamics during the smoothingphase is the non-linear σ -type kinetic interaction Σ ( φ )( ∂ µ χ ) between the φ and χ fields. As can be seen from Eq. (9b), thiscontribution changes the Hubble anti-friction term ( ∝ H ) intoa friction term3 H + ˙ φ m ≈ − t ) − (cid:32) MM Pl (cid:33) + Mm (cid:29) , (11)if M Pl / M > m / M Pl ). As a result, the two scalar fields ex-hibit very di ff erent dynamics. The χ field is being continuouslydamped by the friction in Eq. (9b) until it eventually ‘freezes’at some constant value χ . At the same time, the φ field, whichonly experiences anti-friction according Eq. (9a), is being blue-shifted due to the pure Hubble anti-friction and hence keepsrolling down its negative potential energy curve, rapidly becom-ing the dominant stress-energy component, which robustly ho-mogenizes and isotropizes the cosmological background [20].While the χ -field does not contribute to the backgroundsmoothing, it plays an important role at perturbative order:quantum fluctuations in the φ field, which experience the sameHubble anti-friction as the background, blue-shift. The oppo-site is true for the χ field. Due to the modified damping term asgiven in Eq. (11), quantum fluctuations in the χ field ‘see’ a de-Sitter-like background and red-shift. Consequently, by meansanalogous to the case of inflation, they can lead to a nearlyscale-invariant spectrum of χ -fluctuations with super-Hubblewavelengths.This becomes particularly clear if we follow the evolution ofthe canonically normalized perturbation variable v χ = a √ Σ δχ ,where δχ is the linearized field variable associated with χ .For each Fourier mode with wavenumber k , the correspondingMukhanov-Sasaki equation takes the simple form: v (cid:48)(cid:48) χ + (cid:32) k − z (cid:48)(cid:48) z (cid:33) v χ = , (12)where z ≡ √ Σ a , and prime denotes di ff erentiation w.r.t. theconformal time coordinate τ defined through d τ = a − dt . Eval-uating for the scaling solution as given in Eq. (10a), we find thevariable z as a function of τ : z ∝ ( − τ ) − (cid:15) − (cid:16) Mm (cid:15) − (cid:17) , (13)and, hence, z (cid:48)(cid:48) / z ∝ /τ turning Eq. (12) into a Bessel equa-tion. At the onset of slow contraction, the energy density ∼ H is small and space-time is leading-order classical, suchthat it is natural to assume Bunch-Davies boundary conditions( v χ = e − ik τ / √ k for τ → −∞ ). The corresponding solution tothe Bessel equation (12) then takes the well-known form: v χ = (cid:114) π − τ ) H (1) ν ( − k τ ) , (14)where H (1) ν is a Hankel function of the first kind, and ν = + τ z (cid:48)(cid:48) z = + Mm (cid:15) − (cid:15) − . (15) On large scales ( − k τ (cid:28) v χ ∝ ( − τ ) − ν − · k − ν . (16)such that the spectral tilt of the χ perturbations is given by n s − = − ν = − Mm (cid:15) − (cid:15) − . (17)Note that strictly equal mass scales ( M = m ) lead to an exactlyscale-invariant spectrum ( n s − = M is slightly greater than the scale m associated with the kineticinteraction ( e.g., M = . m ), the spectrum is slightly red inagreement with microwave background observations ( n s − (cid:39)− . χ with entropyperturbations because, in field space, the χ field defines a direc-tion perpendicular to the adiabatic background trajectory; see, e.g. , Ref. [21]. However, this geometric interpretation has lim-ited applicability: it is only valid in cases where all scalar matterfields have canonical kinetic energy density; see Ref. [22]. Amore general and precise statement is that δχ sources (macro-scopic) entropy modes, S ≡ H (cid:32) δ p ˙ p − δρ ˙ ρ (cid:33) ≡ H δ p nad ˙ p , (18)provided it generates a non-zero pressure contribution on hy-persurfaces of constant density, i.e. , δ p nad ≡ δ p − ˙ p ˙ ρ δρ (cid:44) . (19)As we will show next, this occurs naturally during gracefulexit from slow contraction to the onset of the bounce stage.Furthermore, we demonstrate that the same entropy modessource super-Hubble curvature modes consistent with cosmicmicrowave background observations.
4. Sourcing Curvature Modes during Graceful Exit
The smoothing slow contraction phase comes to an end whenthe scalar field kinetic energy increases relative to the potentialenergy such that (cid:15) →
3. The phase that connects to the bouncestage is called ‘graceful exit.’ In scenarios where the cosmolog-ical bounce occurs at high yet sub-Planckian energies, this in-termediate stage is dominated by the kinetic energy of the fields.In particular, this is precisely where one expects higher-orderkinetic terms to start playing a role; see, e.g.
Refs. [18, 19, 23].As we will see, this naturally leads to the sourcing of super-Hubble curvature modes by the fluctuations in χ generated dur-ing the smoothing phase.In Ref. [22], we have shown that, on large scales ( k (cid:28) a | H | ),the conservation of stress-energy leads to a simple relation de-scribing the evolution of curvature fluctuations R as a functionof the entropy modes:˙ R ≈ − H ˙ p ˙ ρ S = H δ p nad ρ + p . (20)3n spatially-flat gauge, the co-moving curvature perturbationcan be expressed as a function of the perturbed scalars as fol-lows: R ≡ H ˙ φδφ + (cid:112) Σ + Σ ˙ χ ˙ χδχ ˙ φ + (cid:0) Σ + Σ ˙ χ (cid:1) ˙ χ , (21)and, for our model as described in Eq. (3), the non-adiabaticpressure contribution is given by δ p nad ≈ c S (cid:104) − Σ ˙ χ ˙ φ × ˙ F (22) + (cid:16)(cid:0) Σ + Σ ˙ χ (cid:1)(cid:0) V ,φ + Σ ,φ ˙ χ (cid:1) ˙ χ − V ,χ ˙ φ − Σ ˙ χ ¨ φ (cid:17) × F (cid:105) . Here
F ≡ (cid:32) ˙ φ ˙ χ ˙ φ + Σ ˙ χ + Σ ˙ χ (cid:33) (cid:32) δχ ˙ χ − δφ ˙ φ (cid:33) (23)describes the relative field fluctuations; and the formal quantity c S ≡ ˙ φ + Σ ˙ χ + Σ ˙ χ ˙ φ + Σ ˙ χ + Σ ˙ χ (24)denotes the propagation speed of the adiabatic mode. The de-tailed dynamics of R and S can be determined by integratingthe closed system of Eqs. (A.3-A.4) under spatially-flat gaugeconditions, as detailed in the Appendix.During slow contraction, δ p nad ≈
0, as can be seen whenevaluating Eq. (22) for the scaling attractor solution for which˙ χ ≈
0. During graceful exit, on the other hand, ˙ χ is non-zeroand Σ is non-negligible. In contrast to the smoothing phase,the relative field fluctuations F lead to a non-zero non-adiabaticpressure which, in turn, sources super-Hubble co-moving cur-vature modes.Since the curvature perturbations R are sourced by S onsuper-Hubble scales, as indicated in Eq. (20), R automaticallyinherits the nearly scale-invariant form of the entropic spec-trum.A particular example illustrating the sourcing of R ( t ) by S ( t )is presented in Fig. 1. In this example, the kinetic couplingfunctions in the action, Eq. (3), have a simple exponential form: Σ = e φ/ m , Σ = e φ/ m , with m = .
67 and m = − .
5. We havealso taken the dependence of the potential on φ in this transi-tion phase after slow contraction to be negligible and on χ tobe small: V ( φ, χ ) = V χ , with V = × − such that V ( φ, χ )is small compared to the total kinetic energy density through-out the bounce phase (from the end of slow contraction at time t = t i = − × to the bounce itself at t = t f = − , expressedin reduced Planck time units), as expected when approaching abounce. The background initial conditions are the following: φ i = − .
5, ˙ φ i = . · − , χ i =
0, ˙ χ i = − . · − . Thesebackground conditions were chosen such that the energy den-sity in φ dominates over the energy density in χ at the end of theslow contraction phase, as expected in bouncing scenarios: thatis Ω φ / Ω χ | t = t i (cid:29)
1. Conversely, the initial ratio of the curvatureperturbation is set to be negligible compared to the entropic per-turbation (( R / S ) ≈ − (cid:28) |S(t)||R(t)| t -5 -6 -6 -7 t -8 -7 -5 -30 000 -20 000 -10 000-30 000 -20 000 -10 00010 -6 Figure 1: A plot of the magnitude of the curvature perturbation on co-movinghypersurfaces, |R ( t ) | (top), and the entropy perturbation, |S ( t ) | (bottom), as afunction of time t (expressed in reduced Planck units) for the example discussedin the text. The evolution of R ( t ) and S ( t ) for a super-Hubble radiusmode with k / a | H | (cid:28) R by the entropic perturbation causes R (cid:28) − to grow to an amplitude consistent with observations(see Fig. 2), R ≈ − . The total curvature perturbation powerspectrum amplitude is then (cid:68) R ( x ) (cid:69) = (cid:90) d k (2 π ) (cid:32) R k ν (cid:33) = (cid:90) dkk (cid:32) R π (cid:33) k − ν = (cid:90) dkk ∆ R ( k ) . (25)Taking the R obtained from the numerical integration to corre-spond to k ∗ = . M pc − , we obtain ∆ R ( k ) | t = t f = . · − with n s (cid:39) . , (26)in accord with current observations [1]. Over the same period ofevolution, Fig. 2 shows that ( R / S ) grows to be greater than 10 such that the fractional contribution of isocurvature perturba-tions to the total power spectrum β iso becomes negligible, alsoin accord with current observations.
5. Conclusion
Scalar field perturbations of quantum origin can source adia-batic or entropic fluctuations on super-Hubble scales. Entropic4
R/S) tt -2 -4 -2 -4 -6 β iso -30 000 -20 000 -10 000-30 000 -20 000 -10 000 Figure 2: A plot of [ R ( t ) / S ( t )] (top) for the example discussed in the text. Atthe end of the slow contraction phase and entering the bounce phase, |R| (cid:28) S isnegligible; but, by the time the bounce would occur ( t ≈ −
100 in this example),the sourcing of the curvature perturbation by the entropic perturbation leads to |R| (cid:29) S . Consequently, the fractional contribution of the entropy modes tothe total power spectrum, | β iso ( t ) | ≡ S / ( R + S ), is nearly one entering thebounce phase but negligibly small by the time the bounce occurs, consistentwith current observations. fluctuations on super-Hubble scales can source curvature modeson super-Hubble scales as a consequence of stress-energy con-servation. If our universe has undergone a phase of slow con-traction that connects to the current expanding phase througha cosmological bounce, adiabatic and gravitational wave fluc-tuations from the smoothing phase decay and therefore cannotcontribute to the observed fluctuation spectra of the cosmic mi-crowave background. Rather, we would expect that the tem-perature anisotropies stem from super-Hubble entropy modesgenerated during slow contraction that sourced curvature modesbefore the onset of decelerated expansion.In this paper, we described a scenario for how this mech-anism might naturally occur during graceful exit when slowcontraction ended but the bounce has not yet occurred. Fur-thermore, we presented an explicit example that generates aspectrum of primordial perturbations that agrees with currentcosmological observations.The key ingredients of this new mechanism are:- a non-linear σ -type kinetic interaction between two scalarfields that acts as a friction term (typical of de Sitter-likeexpansion) on one of the fields which it ‘freezes,’ leading to a nearly scale-invariant spectrum of this field’s quantumfluctuations;- a higher-order quartic kinetic term that typically comes todominate at the end of slow contraction and at the onset ofthe classical (non-singular) bounce stage. This term natu-rally leads to a non-adiabatic pressure contribution, sourc-ing super-Hubble curvature modes before the bounce oc-curs.This novel kinetic sourcing mechanism opens up several newavenues for future research. For example, it will be interestingto see if di ff erent graceful exit and bounce mechanisms leavedi ff erent detectable imprints on the spectrum when the curva-ture modes are being sourced by entropy modes during gracefulexit. Acknowledgements
We thank Paul J. Steinhardt for helpfulcomments and discussions. The work of A.I. is supported bythe Lise Meitner Excellence Program of the Max Planck Soci-ety and by the Simons Foundation grant number 663083. R.K.thanks the Max Planck Institute for Gravitational Physics (Han-nover) for hospitality, where parts of this work were completed.
Appendix A. Evolving the linearized scalars δφ and δχ In this Appendix, we derive the evolution equations for theperturbed scalars δφ and δχ that we used to numerically com-pute the example presented in Figs. 1 and 2 above.Scalar variables of the linearly perturbed line element for aspatially-flat Friedmann-Robertson-Walker (FRW) space-timeare given by ds = − (1 + α ) dt + a ( t ) ∂ i β dt d x i (A.1) + a ( t ) (cid:104) (1 − ψ ) δ i j + ∂ i ∂ j E (cid:105) dx i dx j , where α and β are the linearized lapse and (scalar) shift pertur-bations, respectively, and − ψδ i j + ∂ i ∂ j E is the scalar part of thelinearized spatial metric.With φ = φ ( t ) + δφ ( t , x ), χ = χ ( t ) + δχ ( t , x ) denoting smallinhomogeneities in the scalar fields around the homogeneousbackground, the linearized action (3) takes the form: L = a (cid:32) − ψ + k a ψ − k a σ sh (cid:16) ˙ ψ + H α (cid:17) − (cid:16) H ˙ ψ + k a ψ (cid:17) α + (cid:16) ˙ φδφ + (cid:0) Σ + Σ ˙ χ (cid:1) ˙ χδχ (cid:17)(cid:16) ψ + k a σ sh (cid:17) − (cid:16) H − ˙ φ − (cid:0) Σ + Σ ˙ χ (cid:1) ˙ χ (cid:17) α − (cid:16) ˙ φδ ˙ φ + (cid:0) Σ ,φ ˙ χ + Σ ,φ ˙ χ + V ,φ (cid:1) δφ (cid:17) α − (cid:16)(cid:0) Σ + Σ ˙ χ (cid:1) ˙ χδ ˙ χ + V ,χ δχ (cid:17) α + δ ˙ φ − k a δφ + (cid:16) Σ ,φφ ˙ χ + Σ ,φφ ˙ χ − V ,φφ (cid:17) δφ + (cid:16) Σ + Σ ˙ χ (cid:17) δ ˙ χ − (cid:16) Σ + Σ ˙ χ (cid:17) k a δχ − V ,χχ δχ + (cid:16) Σ ,φ ˙ χ + Σ ,φ ˙ χ (cid:17) δφδ ˙ χ − V ,φχ δφδχ (cid:33) , (A.2)where σ sh ≡ a (cid:16) a ˙ E − β (cid:17) is the scalar part of the linearized shear.5ariation of Eq. (A.2) with respect to α and β leads to thelinearized Hamiltonian and momentum constraints: − k a (cid:16) ψ + H σ sh (cid:17) = (cid:16) H − ˙ φ − Σ ˙ χ − Σ ˙ χ (cid:17) α (A.3a) + H ˙ ψ + ˙ φδ ˙ φ + (cid:16) Σ + Σ ˙ χ (cid:17) ˙ χδ ˙ χ + (cid:16) Σ ,φ ˙ χ + Σ ,φ ˙ χ + V ,φ (cid:17) δφ + V ,χ δχ , H α + ˙ ψ = (cid:16) ˙ φδφ + (cid:16) Σ + Σ ˙ χ (cid:17) ˙ χδχ (cid:17) . (A.3b)Varying Eq. (A.2) with respect to δφ and δχ yields the evolu-tion equations for the perturbed scalar fields: δ ¨ φ + H δ ˙ φ + (cid:16) k a + V ,φφ − Σ ,φφ ˙ χ − Σ ,φφ ˙ χ (cid:17) δφ (A.4a) − (cid:16) ¨ φ + H ˙ φ − Σ ,φ ˙ χ − Σ ,φ ˙ χ − V ,φ (cid:17) α − ˙ φ (cid:16) ˙ α + ψ + k a σ sh (cid:17) − (cid:16) Σ ,φ + Σ ,φ ˙ χ (cid:17) ˙ χδ ˙ χ + V ,φχ δχ = (cid:16) Σ + Σ ˙ χ (cid:17)(cid:16) δ ¨ χ + H δ ˙ χ (cid:17) + (cid:16) Σ + Σ ˙ χ (cid:17) . δ ˙ χ (A.4b) + (cid:16) Σ + Σ ˙ χ (cid:17) k a δχ + V ,χχ δχ − (cid:16) Σ + Σ ˙ χ (cid:17)(cid:16) ¨ χ + H ˙ χ (cid:17) α − (cid:16) Σ + Σ ˙ χ (cid:17) . ˙ χα + V ,χ α − (cid:16) Σ ˙ χ + Σ ˙ χ (cid:17) ˙ α − (cid:16) Σ ˙ χ + Σ ˙ χ (cid:17) (cid:16) ψ + k a σ sh (cid:17) + (cid:16) Σ ,φ + Σ ,φ ˙ χ (cid:17) ˙ χδ ˙ φ + V ,φχ δφ + (cid:16) Σ ,φ + Σ ,φ ˙ χ (cid:17)(cid:16) ¨ χ + H (cid:17) δφ + (cid:16) Σ ,φ + Σ ,φ ˙ χ (cid:17) . ˙ χδφ = . In spatially-flat gauge ( ψ, E ≡ δφ and δχ which we used above to numerically compute the gauge-invariant quantities S and R defined in Eqs. (18) and (21), re-spectively. References [1] E. Komatsu et al. (WMAP), Astrophys. J. Suppl. , 330 (2009), .[2] J. L. Sievers et al. (Atacama Cosmology Telescope), JCAP , 060(2013), .[3] N. Aghanim et al. (Planck), Astron. Astrophys. , A6 (2020), .[4] J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys.Rev.
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