Mimicking alternatives to inflation with interacting spectator fields
HHIP-2018-28/TH
Mimicking features in alternatives to inflation with interacting spectator fields
Guillem Dom`enech , ∗ Javier Rubio , , † and Julius Wons ‡ Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at Heidelberg,Philosophenweg 16, 69120 Heidelberg, Germany and Department of Physics and Helsinki Institute of Physics,PL 64, FI-00014 University of Helsinki, Finland
It has been argued that oscillatory features from spectator fields in the primordial power spectrumcould be a probe of alternatives to inflation. In this work, we soften this claim by showing thatthe frequency and amplitude dependence of the patterns appearing in these scenarios could bemimicked by field interactions during inflation. The degeneracy of the frequency holds for the n -point correlation functions, while the degeneracy of the amplitude is broken at the level of non-gaussianities. Keywords: Inflation; Early universe; Primordial features; Primordial standard clocks; Alternatives to infla-tion; arXiv:1811.08224
I. INTRODUCTION
The stunningly detailed maps provided by numerousCosmic Microwave Background observations [1] have con-solidated inflation [2–5] as the leading paradigm for gen-erating the primordial density fluctuations seeding struc-ture formation. In spite of its phenomenological success,some formal questions such as the initial singularity orthe trans-Planckian problem remain open for a part ofthe community. This has motivated the quest for alter-native scenarios able to generate an initial scale-invariantspectrum of density perturbations without invoking anaccelerated expansion of the Universe, but rather an al-ternative evolution of the scale factor a [6–13]. One ofthe prototypical examples is a matter contraction erataking place in the very early Universe. In this sce-nario, the scale factor a evolves according to a power-law a ∼ t p with −∞ < t < p ≈ / w eff = 2 / (3 p ) − ≈
0. Interestingly, due to a duality inthe equations of motion for the curvature perturbation[7], the spectral tilt of the primordial power spectrumgenerated by this pre hot big bang evolution [14], n s = 1 + 3 − (cid:12)(cid:12)(cid:12)(cid:12) p − (cid:12)(cid:12)(cid:12)(cid:12) ≈ , (1)turns out to coincide with that generated by an almost deSitter inflationary expansion with w eff ≈ − | p | (cid:29) ∗ [email protected] † javier.rubio@helsinki.fi ‡ [email protected] These formal aspects are not a problem for the predictionsof inflation themselves, as long as inflation is understoodas a paradigm and not as a particular model. For anextended discussion on this issue we refer the reader to https://blogs.scientificamerican.com/observations/a-cosmic-controversy/ . Several ways of distinguishing inflation from alternativeslike this have been proposed in the literature. One possi-bility is the eventual detection of primordial gravitationalwaves [13] or B-mode polarization [15]. Another optionis to look for distinctive features in the primordial spec-trum. The main idea behind this approach is that theevolution of a spectator field during a given cosmologicalera can imprint specific oscillatory signals in the densitycorrelation functions. In particular, since the clockingpattern of these probes depends on how the Universe ex-pands (or contracts), there is a chance to determine theevolution of the scale factor by carefully inspecting theamplitude and frequency of the oscillations.The evolution of spectator fields during inflation hasbeen extensively studied in the literature [16–34]. Dis-tinctive oscillatory patterns are found for instance inmulti-field inflationary models displaying sudden turnsin the inflationary trajectory [37–44] or in single-field sce-narios involving steps [45, 46] or periodic features in thepotential, as in axion monodromy [47–49].Although the idea of using primordial standard clocksas an inflationary test is certainly appealing, the fre-quency of the oscillations depends implicitly on the de-tailed structure of the theory. In particular, if the massof the spectator field gets additional contributions fromother fields, the oscillation frequency would not only de-pend on the evolution of the scale factor, but also on theadditional time-dependence inherited from interactions.To distinguish these field-dependent objects from thestandard terminology—which refers to spectator fieldswith constant masses as standard clocks —we will referto them as (classical) non-standard clocks .In view of the future prospects for detecting oscillatorysignals in the Cosmic Microwave Background [50–56], it For a review of multi-field inflationary models see for instance[35, 36] and references therein. a r X i v : . [ a s t r o - ph . C O ] J a n is important to clarify whether those associated with al-ternative scenarios could be mimicked by non-triviallyinteracting spectator fields in an inflationary setting. Inthis paper, we illustrate that this is indeed the case. Themanuscript is organized as follows. In Section II we in-troduce a general framework leading to the appearance ofclassical non-standard clocks. The associated modifica-tions of the power spectrum are computed in Section III.The degeneracy of the resulting inflationary signals withthose produced in alternatives scenarios is illustrated inSection IV. In Section V we extend our results to higher-order correlation functions. Finally, our conclusions arepresented in Section VI. II. TWO-FIELD INFLATIONARY MODEL
Let us consider a two-field inflationary model with La-grangian density
L√− g = M R − ω ( χ ) g µν ∂ µ φ∂ ν φ − V ( φ ) − f ( φ ) g µν ∂ µ χ∂ ν χ − m ( φ ) χ , (2)where M P = 1 / √ πG is the reduced Planck mass and ω ( χ ), f ( φ ), m ( φ ) and V ( φ ) are arbitrary functions oftheir corresponding arguments. This type of interactionsare generically expected to appear upon Weyl rescaling invariable gravity scenarios [57, 58] or in models involvingnon-minimal couplings of the inflaton field to gravity [35,36]. For a flat, homogeneous and isotropic Fried-mann–Lemaˆıtre–Robertson–Walker (FLRW) metric, ds = − dt + a d x , (3)the equations of motion following from Eq. (2) take theform 3 M H = 12 ω ˙ φ + V + 12 f ˙ χ + 12 m χ , (4)1 a ddt (cid:16) a ω ˙ φ (cid:17) + V ,φ + 12 m ,φ χ = 12 f ,φ ˙ χ , (5)1 a ddt (cid:0) a f ˙ χ (cid:1) + m χ = 12 ω ,χ ˙ φ , (6)with H = ˙ a/a the Hubble rate and the dots denotingderivatives with respect to the coordinate time t . Note This possibility was pointed out in Ref. [17], although withoutproviding any specific realization. Note that, although the naive application of standard effec-tive field theory arguments in Minkowski spacetime would re-strict these functions to higher-order polynomials operators sup-pressed by a given cutoff scale, this is not necessarily the casein the presence of gravity. Indeed, if the aforementioned op-erators are added in a non-minimally coupled frame, they be-come exponential-like/dilatonic functions when written in theEinstein-frame. that the inclusion of the functions f and ω changes theeffective scale factor experienced by the scalar fields, ascan be easily seen by comparing the first terms in Eqs. (5)and (6) with that of a free scalar field in a FLRW cos-mology.Within this framework, oscillatory patterns in thepower spectrum are expected in the presence of sharpturns in field space [43] or if one of the scalar fields de-velops a new minimum along its trajectory [44]. In whatfollows we will focus on the first possibility. In particu-lar, we will assume that the potential V ( φ ) renders mas-sive the field φ during the first stages of inflation whilemaking it light within the observable Cosmic MicrowaveBackground window. This choice translates into an in-flationary dynamic essentially dominated by the χ fieldat early times while driven by the φ field at late times.If sufficiently fast, the transition among these two rollingperiods leads to oscillations in the χ direction that couldpotentially impact the primordial power spectrum gen-erated by the φ field.Since the dynamics in the presence of a sharp turnis complicated and model dependent, we will focus hereon the oscillations of the χ field during the secondaryinflationary stage. To study this φ -dominated period, wewill assume the usual slow-roll conditions (cid:15) ≡ − ˙ HH (cid:28) , η ≡ ˙ (cid:15)H(cid:15) (cid:28) . (7)These conditions, together with the requirement that the χ field stays subdominant at all times, translate intorestrictions on the values and couplings of the χ field,namely δ ω (cid:28) , δ f f ˙ χ , δ m m χ (cid:28) ω ˙ φ , (8)with the quantities δ ω ≡ d ln ωdN , δ f ≡ d ln fdN , δ m ≡ d ln mdN , (9)measuring the variation of the functions ω , f and m perHubble time dN ≡ Hdt . Note that the latest requirementin Eq. (8) is not a strong restriction for an oscillating field( f ˙ χ ∼ m χ ) provided that δ f and δ m are not exces-sively large and that the function ω is not too small. Inorder to simplify the analysis, we will assume the latestquantity to be close to the canonical value ω = 1, ω ≡ ω , ∆ ω (cid:28) , (10)and restrict its potential backreaction effects on the χ -field equation of motion (6) by requiring d ∆ ωd ln χ (cid:28) m (cid:15)H χ M , (11)with (cid:15)H M P ∼ ˙ φ . We focus here on classical excitation mechanisms, postponingthe analysis of quantum ones [21, 59] to a future work.
A. Spectator field oscillations
The oscillations of the χ field in the φ -field inflation-ary background are more easily analyzed in terms of arescaled field σ ≡ f a / χ , such that the friction terms inEq. (6) are effectively removed. The resulting equationof motion becomes then that of an (undamped) harmonicoscillator, ¨ σ + m σ = 0 , (12)with time-dependent mass m ≡ m f − ¨ ff − H ˙ ff − H (cid:18) − (cid:15) (cid:19) , (13)where, in agreement with the assumption (10), we haveneglected a small ∆ ω contribution.In usual situations, the time derivatives of the function f are proportional to the instantaneous Hubble rate H and the effective mass m is dominated by the first termin Eq. (13). Assuming this to be the case, we can easilycompute the solution of Eq. (12) at the lowest order in theWKB approximation. Defining an expansion parameter µ − ≡ Hm eff (cid:28) , (14)we get χ = χ r (cid:18) aa r (cid:19) − / f r f (cid:114) m eff , r m eff sin (cid:18) Ω + θ (cid:19) + O ( µ − ) , (15)with the subindex r referring to the evaluation of the cor-responding quantity at the onset of the first oscillation, θ an (irrelevant) integration constant andΩ ≡ (cid:90) m eff dt (16)an integrated frequency.The oscillations in Eq. (15) leave an imprint in thebackground Hubble rate, which, in average and at leadingorder in the expansion parameter µ − , is only affected bythe χ -field mass m ( φ ) via the function δ m , H osc H ≈ − µ r χ r M (cid:18) δ m (cid:19) H r H (cid:18) aa r (cid:19) − sin (2Ω + θ ) . (17)This result can be easily understood by noticing that theaveraged energy density of the χ field behaves like dustirrespectively of the coupling f . Note also that, due toan accidental cancellation, the oscillations of the χ fieldfor δ m = − O ( µ − ). III. 2-POINT CORRELATION FUNCTION
The presence of the oscillating field χ modifies the evo-lution of the (gauge invariant) curvature perturbation R [60, 61]. In spite of its extensive use in the literature,we will refrain from using the standard terminology re-ferring to the background contribution (17) as “gravi-tational contribution” and to that associated with thedirect kinetic couplings between χ and φ as “direct con-tribution”. We will rather distinguish two main scenar-ios according to the behaviour of the scalar interactionsduring slow-roll.
A. Slow-roll suppressed interactions
If ∆ ω = 0 the modifications in the power spectrumare suppressed by the slow-roll conditions (8). Amongthe different contributions induced in the (gauge invari-ant) curvature perturbation R k , the most relevant one isassociated with the highest number of time derivatives,since these provide more powers of the large parameter µ .Keeping only this leading contribution, the Mukhanov-Sasaki equation for the mode functions with wavenumber k takes the form of a Mathieu equation, [60, 61] u (cid:48)(cid:48) k + (cid:18) k − z (cid:48)(cid:48) z − ∆( τ ) cos (2Ω + θ ) (cid:19) u k = 0 , (18)with the primes denoting derivatives with respect to theconformal time τ ,∆( τ ) a H ≡ − µ r χ r (cid:15)M (cid:18) δ m (cid:19) H r H (cid:18) µµ r (cid:19) (cid:18) aa r (cid:19) − , (19)and u k ≡ z R k , z ≡ aM P √ (cid:15) . (20)Although studying the resonant band structure of thisquasi-periodic equation might be interesting on its own,we will restrict ourselves to the simplest perturbativetreatment, postponing the non-perturbative analysis toa future publication. In particular, we will expand theMukhanov-Sasaki mode functions as u k = u k, + u k, + ... with u k, given by the underlying inflationary model and u k, a small perturbation satisfying u (cid:48)(cid:48) k, + (cid:18) k − z (cid:48)(cid:48) z (cid:19) u k, = ∆( τ ) cos (2Ω + θ ) u k, . (21) This classification is clearly not gauge invariant, as one can al-ways choose a constant H slicing. In this limit, the calculations using the in-in formalism and theequations of motion coincide [62].
The total curvature power spectrum is then given by P R = k π lim τ → (cid:12)(cid:12)(cid:12)(cid:12) u k, + u k, z (cid:12)(cid:12)(cid:12)(cid:12) = P R , + ∆ P R , (22)with P R , = k π lim τ → (cid:12)(cid:12)(cid:12) u k, z (cid:12)(cid:12)(cid:12) , (23)and ∆ P R = k π lim τ → Re (cid:104) u ∗ k, u k, (cid:105) z . (24)For the purposes of this paper it would be enough to ap-proximate u k, by the standard de Sitter mode functions u k, = 1 √ k (cid:18) − ikτ (cid:19) e − ikτ , (25)leaving aside a small correction of the order of slow-rollparameters. The solution of Eq. (21) can be computedusing the Green function method and yields [63] u k, ( τ ) = iu k, ( τ ) (cid:90) τ f τ dξ | u k, | ∆( ξ ) cos (2Ω + θ ) − iu ∗ k, ( τ ) (cid:90) ττ r dξ ( u k, ) ∆( ξ ) cos (2Ω + θ ) , (26)where we have assumed homogeneous boundary condi-tions at the onset of the oscillations. Note, however, thatthe exact boundary conditions used do not play a centralrole in the final result. Indeed, the integrals in this ex-pression are dominated by the time at which frequencyof the de Sitter mode function u k, coincides with thatof the oscillating field χ . In other words, the resonantbehaviour at K = Ω (cid:48) = m eff a , (27)allows to safely evaluate the integral at that time usingthe saddle-point approximation.At this point we have to specify the inflationary modelin order to analytically compute the corrections to theprimordial power spectrum. For this purpose, and inorder to facilitate the comparison with the existing lit-erature, we consider an illustrative (power-law) inflationscenario with runaway potential V ( φ ) = V e − λφ [14]. Forthis particular choice, the equation of motion of the back-ground field φ admits a slow-roll solution with a ∼ t p , φ ∼ λ ln t , p = 2 /λ . (28)For simplicity we will also consider constant rate varia-tions of the theory defining functions f and m , namely δ m , δ f = constant. This corresponds to dilatonic-likecouplings m ∼ e λδ m φ and f ∼ e λδ f φ , ubiquitously ap-pearing in non-minimally coupled theories when written in the Einstein-frame [43, 64]. Under these assumptions,the oscillatory correction to the power spectrum can bewritten as∆ P R P R , = √ π µ / r χ r (cid:15)M (cid:0) δ m (cid:1)(cid:112) δ m − δ f (cid:18) Kk r (cid:19) ν × sin (cid:34) µ r γ p p − (cid:18) Kk r (cid:19) γp + 3 π θ (cid:35) , (29)where k r ≡ a r m eff , r is the momentum associated withthe first resonant mode and we have defined an amplitudetilt ν = − γp + 2 δ m − δ f δ m − δ f , (30)with γ ≡ p δ m − p δ f δ m − δ f . (31)For m = f (or equivalently δ m = δ f or γ = 1) the effec-tive mass in Eq. (13) is (approximately) constant and ourresults reduce to those in Ref. [17]. Note also that thereexists a “singular” scenario with δ f − δ m = 1 in whichthe frequency evolves proportional to the conformal time(Ω = m eff , r τ ) and only one mode resonates. In this case,the saddle-point approximation breaks down and a next-to-leading order computation is required. We will not beinterested in this particular setting in what follows. B. Non slow-roll suppressed interactions
If ∆ ω (cid:54) = 0 the direct kinetic coupling ω ( ∂φ ) inEq. (2) can easily dominate over the other interactionterms, since, contrary to them, it is not suppressed bythe slow-roll conditions (8).Since we want the backreaction of the χ field on thebackground dynamics to stay small during the whole ob-servable window, we will assume the conditions (10) and(11) to hold and consider a simple perturbation [39]∆ ω = χ Λ , (32)with Λ a given energy scale. When this coupling is takeninto account the variable z in Eq. (20) is effectively re-placed by z ω , leading to an additional mass term pro-portional to χ (cid:48)(cid:48) in the Mukhanov-Sasaki equation (18).Taking into account Eq. (15) we obtain u (cid:48)(cid:48) k + (cid:16) k − z (cid:48)(cid:48) z − ∆ T ( τ ) (cid:17) u k = 0 , (33)with∆ T ( τ ) = ∆( τ ) cos (2Ω + θ ) + ∆ ω ( τ ) sin (Ω + θ/ , and∆ ω ( τ ) a H ≡ − µ / r χ r Λ f r f (cid:18) H r H (cid:19) / (cid:18) µµ r (cid:19) / (cid:18) aa r (cid:19) − / . (34)Performing a similar computation to that in the ∆ ω = 0,we obtain an oscillatory correction to the power spectrumdisplaying the same frequency dependence but a differ-ent amplitude tilt, namely ν = −
32 + 12 γp + δ m − δ f δ m − δ f . (35)This scenario is particularly interesting since for m = f (or equivalently δ m = 2 δ f ) the signal mimics that of amodel with power-law index ¯ p ≡ p/γ . This correspon-dence is expected since the equation of motion for the χ field effectively reproduces in this case that of a FLRWuniverse with scale factor ¯ a = f a [65, 66]. Note indeedthat the frequency of the oscillations in the curvature per-turbation is determined by the ratio of two time scales:the effective mass m eff and the expansion rate H . For aconstant mass, the change in µ = m eff /H is solely dueto the Hubble rate H and therefore the frequency probesdirectly the expansion rate of the Universe. When themass gets a time dependence we see a combination ofboth. The time dependence of the mass can be, however,absorbed into a new redefined Hubble rate ¯ H , namely µ = m eff , r ¯ H where ¯ H ≡ H m r m ff r . (36)When m = f the quantity ¯ H coincides with the expan-sion rate of a universe with scale factor ¯ a = f a . That isthe reason why a time dependent spectator field massis able to mimic the frequency of the oscillating fea-tures above. Note that this intuitive argument is notlinked to matter domination. In particular, an exponen-tial/dilatonic choice of the theory defining functions f ( φ )and m ( φ ) is able to mimic any oscillatory feature gen-erated by standard spectator fields in cosmologies withpower-law scale factor evolutions, as, for instance, ekpy-rotic scenarios with 0 < p (cid:28) IV. MIMICKING ALTERNATIVE SCENARIOS
The typical behaviours of the oscillatory signals gen-erated by a free spectator field during a matter contrac-tion era and during an inflationary stage are illustratedin Fig. 1. Let us discuss how this picture is modified in Strictly speaking, the frequency is halved with respect to theprevious case since the corrections depend now on χ instead of χ . K/k r ∆ P R / P R , Matter Contraction Inflation
FIG. 1. Illustration of the generic standard clock signals (i.e.constant m , f and w ) appearing in matter contraction (blueline) and inflationary scenarios (orange line). We have fixedthe value of k r and, thus, the signal of the matter contractionevolves towards smaller K values since the frequency of theoscillations ( K = m eff a ) decreases in a contracting phase. k/k r p = 2 / , δ m = δ f = 0 p = 57 , δ m = − . , δ f = 0 K/k r ∆ P R / P R , p = 2 / , δ m = δ f = 0 p = 57 , δ m = − . , δ f = 0 FIG. 2. Comparison of the corrections to the power spec-trum associated with a standard clock signal during mattercontraction (blue line) and a non-standard one during infla-tion and with the same frequency (orange line). The cases∆ ω = 0 and ∆ ω (cid:54) = 0 are displayed in the upper and lowerpanel, respectively. For the matter contraction era we choose µ r = 25 and p = 2 /
3, while for inflation we take µ r = 50, p = 57 and δ m = − .
96. For an easier comparison, we leftthe amplitude in arbitrary units and fixed to be the same at K = k r . the presence of non-trivial interactions. To this end, notethat by taking δ m − δ f = 1 − ¯ p/p ¯ p − . (37)one can always adjust the scale dependence of the fre-quency to that of a universe with a power-law evolution a ∼ t ¯ p . For a fixed wavenumber k r , the frequencyrescales as well, mimicking that of a field with effective p = 2 / , δ m = δ f = 0 p = 57 , δ m = − . , δ f = − . K/k r ∆ P R / P R , p = 2 / , δ m = δ f = 0 p = 57 , δ f = − .
96 = δ m / FIG. 3. Comparison of the corrections to the power spec-trum associated with a standard clock signal during mattercontraction (blue line) and a non-standard one with the samefrequency and amplitude during inflation (orange line). Inthe top panel we compare the signals coming from a ∆ ω = 0scenario in matter contraction and from a ∆ ω (cid:54) = 0 scenarioin inflation. We chose µ r = 25 and p = 2 / µ r = 50, p = 57, δ m = − .
98 and δ f = − .
017 for the inflationary case. In the lower panel, wecompare the signal coming from two scenarios with ∆ ω (cid:54) = 0.We choose µ r = 25 and p = 2 / µ r = 50, p = 57 and δ f = − .
96 = δ m / K = k r . mass ¯ µ r ≡ µ r (cid:12)(cid:12)(cid:12)(cid:12) p ¯ p ¯ p − p − (cid:12)(cid:12)(cid:12)(cid:12) . (38)The simultaneous choice of δ m and δ f allows therefore tofix the frequency and tilt of the oscillations in order toimitate alternative signals. However, if ∆ ω = 0 the scaledependence of the amplitude breaks the degeneracy sinceboth ν and ¯ p = p/γ depend only on the combination δ m − δ f (cf. Fig. 2). On the other hand, if the termproportional to ∆ ω dominates, the dependence of tilton δ m and δ f changes and we can choose δ m = 2 δ f suchthat the amplitude of the oscillations mimics also that auniverse with power-law index ¯ p , i.e. ν = − / / (2¯ p )(cf. Fig 3).Incidentally, we have found a degeneracy between thegravitational signal from a matter contraction and a di-rect signal from inflation with p ≈ δ m ≈ − .
98 and δ f ≈ − . δ f > δ m the effective mass of the field χ decreases withtime, such that it might eventually dominate the back-ground evolution. We assume that such regime is neverreached. K/k r ∆ P R / P R , p = 57 , δ m = − δ f = 5 p = 57 , δ m = δ f = 0 FIG. 4. Comparison of particular signal coming from a non-standard clock with ∆ ω (cid:54) = 0 (blue line) and a standard clockone (orange line). For both signals we choose µ r = 25 and p = 57. For the non-standard clock we take δ m = − δ f = 5. Before ending this section, we would like to point out aparticular scenario, only present for non-standard clockslike the ones under consideration. That would be anoscillatory signal with γ = 1 (i.e. with a frequency thatcould be interpreted as that of an standard clock duringinflation) but with an increasing amplitude. This willhappen whenever δ f < (1 − p ) / p (cf. Fig. (4)). V. HIGHER-ORDER CORRELATIONS
The above discussion illustrates how the combinationof sudden turns and time-dependent masses can eas-ily produce varying-frequency oscillatory patterns in thepower spectrum mimicking the scale dependence of al-ternative inflationary scenarios. Given this result, it isimportant to check if higher-order correlation functionscould allow to break the degeneracy among these scenar-ios.The overall scale dependence of the three-point corre-lation function can be studied by looking at the equilat-eral configuration. As before, we consider two cases ac-cording to the behaviour of the scalar interactions duringslow-roll. For ∆ ω = 0, the leading contribution to thebispectrum reads [67–69] L ,I ∝ a M (cid:15) ˙ η R ˙ R , (39)with (cid:15) and η the first and second slow-roll parameters.Using the in-in formalism, we find that, up to a shapefunction, the associated non-gaussianity is given by f osc NL ≈ √ πµ / r χ r (cid:15)M (cid:0) δ m (cid:1)(cid:112) δ m − δ f (cid:18) Kk r (cid:19) ν × sin (cid:34) µ r γ p p − (cid:18) Kk r (cid:19) γp + 3 π θ (cid:35) , (40) Note that if m = f the effective mass is constant and our resultscoincide with those in Refs. [17] and [21]. with K ≡ k + k + k and ν = − γp + 3 δ m − δ f δ m − δ f . (41)On the other hand, if ∆ ω (cid:54) = 0 the leading contributionto the three-point correlation function is rather given by[40] L ,I ∝ a M χ Λ (cid:15) R ˙ R . (42)In this case the scale dependence of the amplitude is mod-ified to ν = −
32 + 12 γp + 2 δ m − δ f δ m − δ f . (43)We see therefore that even in the case m = f there isa correction to the scale dependence of the amplitude ofthe non-gaussianities as compared to that in an scenariowith power-law index ¯ p . This implies that, at least in thesimple case we studied, the degeneracy would be brokenby an eventual measurement of the scale dependence ofnon-gaussinities.Independently of their origin, the generated non-gaussianities display the same frequency dependence asthe power spectrum. This result is due to the existenceof a single oscillation frequency and can be easily ex-tended to higher-order correlation functions. We con-clude therefore that, if the existence of oscillatory fea-tures in the power spectrum would be eventually estab-lished, one would need the full information of the ampli-tudes and the frequency in order to probe the evolutionof the scale factor. VI. CONCLUSIONS AND DISCUSSION
The frequency and momentum dependence of poten-tial oscillatory features in the primordial density fluc-tuations has been advocated as a way to distinguish theinflationary paradigm from alternatives scenarios such asbounces or matter contraction eras [16–26]. These fea-tures are typically imprinted by the oscillations of a spec-tator field, provided that this is excited at some point inthe primordial Universe.In this work, we have shown that if the mass of a spec-tator field receives an explicit time dependence duringinflation—due for instance to non-trivial interaction withother fields—the oscillatory signal in the power spectrumand other higher-order correlation functions can easilymimick the one appearing in alternatives scenarios toinflation, both in its frequency and momentum depen-dence. To illustrate this result we considered two par-ticular examples, one with no dominant kinetic mixingamong the spectator field and the inflaton kinetic termand one involving a relevant kinetic coupling χ ( ∂φ ) . Us-ing a WKB approximation we showed that the oscilla- tions in the power spectrum take the generic form∆ P R P R ∝ (cid:18) Kk r (cid:19) ν sin (cid:34) C (cid:18) Kk r (cid:19) γp + θ (cid:35) (44)with C and θ constants, p the power-law index of the scalefactor and γ a parameter related to the time-dependentspectator field mass and its kinetic term normalization.Interestingly, only the combination p/γ is probed by thefrequency, but not just the power-law index p . The scaledependence of the amplitude in Eq. (44) is given by ν = − γp + 2 δ m − δ f δ m − δ f if ∆ ω = 0 , − + γp + δ m − δ f δ m − δ f if ∆ ω (cid:54) = 0 , (45)with δ m and δ f free parameters of the model related tothe change rate of the mass and the kinetic coefficient ofthe χ field, respectively. In the former case (∆ ω = 0),one can choose the interaction among fields in such away that the frequency matches that of an alternativematter contraction scenario, but the scale dependence ofthe amplitude breaks the degeneracy. In the second case(∆ ω (cid:54) = 0), the amplitude of the power spectrum can alsofollow that of the alternative matter contraction scenariofor a particular choice of the interactions. Regardingthe frequency, this result holds even when one consid-ers higher order correlations functions, since the singleoscillation frequency of the spectator field gets imprintedin the same way in all of them. However, the degener-acy is broken by the scale dependence of the amplitudeof higher-order correlations functions. We illustrated thisresult by explicitly computing the bispectrum in the equi-lateral configuration, which takes the schematic form f NL ∝ (cid:18) Kk r (cid:19) ν sin (cid:34) C (cid:18) Kk r (cid:19) γp + θ (cid:35) , (46)with ν = − γp + 3 δ m − δ f δ m − δ f if ∆ ω = 0 , − + γp + δ m − δ f δ m − δ f if ∆ ω (cid:54) = 0 . (47)The mimicking mechanism proposed in this paper doesnot require special fine-tunings or large number of pa-rameters. Although we focused for concreteness on mat-ter contraction scenarios leading to an almost scale in-variant primordial power spectrum, our results can beeasily extended to other alternative scenarios by prop-erly choosing the form of the interactions. Our findingssoften the claim that oscillatory features from spectatorsfields during inflation can probe inflation and its alterna-tives without a full knowledge of the n -point correlationfunctions. To do so, one would need the complete infor-mation of the frequency and amplitudes of, at least, the2- and 3-point correlation functions. ACKNOWLEDGMENTS
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