Mimicking alternatives to inflation with interacting spectator fields

It has been argued that oscillatory features from spectator fields in the primordial power spectrum could be a probe of alternatives to inflation. In this work, we soften this claim by showing that the frequency and amplitude dependence of the patterns appearing in these scenarios could be mimicked 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.


I. INTRODUCTION
The stunningly detailed maps provided by numerous Cosmic Microwave Background observations [1] have consolidated inflation [2][3][4][5] as the leading paradigm for generating the primordial density fluctuations seeding structure formation. In spite of its phenomenological success, some formal questions such as the initial singularity or the trans-Planckian problem remain open for a part of the community. 1 This has motivated the quest for alternative scenarios able to generate an initial scale-invariant spectrum of density perturbations without invoking an accelerated expansion of the Universe, but rather an alternative evolution of the scale factor a [6][7][8][9][10][11][12][13]. One of the prototypical examples is a matter contraction era taking place in the very early Universe. In this scenario, the scale factor a evolves according to a powerlaw a ∼ t p with −∞ < t < 0 and p ≈ 2/3 related to the effective equation-of-state parameter at that epoch, w eff = 2/(3p) − 1 ≈ 0. Interestingly, due to a duality in the equations of motion for the curvature perturbation [7], the spectral tilt of the primordial power spectrum generated by this pre hot big bang evolution [14], turns out to coincide with that generated by an almost de Sitter inflationary expansion with w eff ≈ −1 and |p| 1. * domenech@thphys.uni-heidelberg.de † javier.rubio@helsinki.fi ‡ wons@thphys.uni-heidelberg.de 1 These formal aspects are not a problem for the predictions of inflation themselves, as long as inflation is understood as a paradigm and not as a particular model. For an extended discussion on this issue we refer the reader to https://blogs.scientificamerican.com/observations/ a-cosmic-controversy/.
Several ways of distinguishing inflation from alternatives like this have been proposed in the literature. One possibility is the eventual detection of primordial gravitational waves [13] or B-mode polarization [15]. Another option is to look for distinctive features in the primordial spectrum. The main idea behind this approach is that the evolution of a spectator field during a given cosmological era can imprint specific oscillatory signals in the density correlation functions. In particular, since the clocking pattern of these probes depends on how the Universe expands (or contracts), there is a chance to determine the evolution of the scale factor by carefully inspecting the amplitude and frequency of the oscillations.
Although the idea of using primordial standard clocks as an inflationary test is certainly appealing, the frequency of the oscillations depends implicitly on the detailed structure of the theory. In particular, if the mass of the spectator field gets additional contributions from other fields, the oscillation frequency would not only depend on the evolution of the scale factor, but also on the additional time-dependence inherited from interactions. To distinguish these field-dependent objects from the standard terminology-which refers to spectator fields with constant masses as standard clocks-we will refer to them as (classical) non-standard clocks.
In view of the future prospects for detecting oscillatory signals in the Cosmic Microwave Background [50][51][52][53][54][55][56], it is important to clarify whether those associated with alternative scenarios could be mimicked 3 by non-trivially interacting spectator fields in an inflationary setting. In this paper, we illustrate that this is indeed the case. The manuscript is organized as follows. In Section II we introduce a general framework leading to the appearance of classical non-standard clocks. The associated modifications of the power spectrum are computed in Section III. The degeneracy of the resulting inflationary signals with those produced in alternatives scenarios is illustrated in Section IV. In Section V we extend our results to higherorder correlation functions. Finally, our conclusions are presented in Section VI.

II. TWO-FIELD INFLATIONARY MODEL
Let us consider a two-field inflationary model with Lagrangian density where M P = 1/ √ 8πG is the reduced Planck mass and ω(χ), f (φ), m(φ) and V (φ) are arbitrary functions of their corresponding arguments. This type of interactions are generically expected to appear upon Weyl rescaling in variable gravity scenarios [57,58] or in models involving non-minimal couplings of the inflaton field to gravity [35,36]. 4 For a flat, homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) metric, that the inclusion of the functions f and ω changes the effective scale factor experienced by the scalar fields, as can be easily seen by comparing the first terms in Eqs. (5) and (6) with that of a free scalar field in a FLRW cosmology. Within this framework, oscillatory patterns in the power spectrum are expected in the presence of sharp turns in field space [43] or if one of the scalar fields develops a new minimum along its trajectory [44]. 5 In what follows we will focus on the first possibility. In particular, we will assume that the potential V (φ) renders massive the field φ during the first stages of inflation while making it light within the observable Cosmic Microwave Background window. This choice translates into an inflationary dynamic essentially dominated by the χ field at early times while driven by the φ field at late times. If sufficiently fast, the transition among these two rolling periods leads to oscillations in the χ direction that could potentially impact the primordial power spectrum generated by the φ field.
Since the dynamics in the presence of a sharp turn is complicated and model dependent, we will focus here on the oscillations of the χ field during the secondary inflationary stage. To study this φ-dominated period, we will assume the usual slow-roll conditions These conditions, together with the requirement that the χ field stays subdominant at all times, translate into restrictions on the values and couplings of the χ field, namely with the quantities measuring the variation of the functions ω, f and m per Hubble time dN ≡ Hdt. Note that the latest requirement in Eq. (8) is not a strong restriction for an oscillating field (f 2χ2 ∼ m 2 χ 2 ) provided that δ f and δ m are not excessively large and that the function ω 2 is not too small. In order to simplify the analysis, we will assume the latest quantity to be close to the canonical value ω 2 = 1, and restrict its potential backreaction effects on the χfield equation of motion (6) by requiring with H 2 M 2 P ∼φ 2 .

A. Spectator field oscillations
The oscillations of the χ field in the φ-field inflationary background are more easily analyzed in terms of a rescaled field σ ≡ f a 3/2 χ, such that the friction terms in Eq. (6) are effectively removed. The resulting equation of motion becomes then that of an (undamped) harmonic oscillator,σ with time-dependent mass where, in agreement with the assumption (10), we have neglected a small ∆ω 2 contribution. In usual situations, the time derivatives of the function f are proportional to the instantaneous Hubble rate H and the effective mass m 2 eff is dominated by the first term in Eq. (13). Assuming this to be the case, we can easily compute the solution of Eq. (12) at the lowest order in the WKB approximation. Defining an expansion parameter we get with the subindex r referring to the evaluation of the corresponding quantity at the onset of the first oscillation, θ an (irrelevant) integration constant and an integrated frequency. The oscillations in Eq. (15) leave an imprint in the background Hubble rate, which, in average and at leading order in the expansion parameter µ −1 , is only affected by the χ-field mass m(φ) via the function δ m , This result can be easily understood by noticing that the averaged energy density of the χ field behaves like dust irrespectively of the coupling f . Note also that, due to an accidental cancellation, the oscillations of the χ field for δ m = −3 enter the background evolution at order O(µ −2 ).

III. 2-POINT CORRELATION FUNCTION
The presence of the oscillating field χ modifies the evolution 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 referring to the background contribution (17) as "gravitational contribution" and to that associated with the direct kinetic couplings between χ and φ as "direct contribution". 6 We will rather distinguish two main scenarios according to the behaviour of the scalar interactions during slow-roll.

A. Slow-roll suppressed interactions
If ∆ω 2 = 0 the modifications in the power spectrum are suppressed by the slow-roll conditions (8). Among the different contributions induced in the (gauge invariant) curvature perturbation R k , the most relevant one is associated 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] with the primes denoting derivatives with respect to the conformal time τ , and Although studying the resonant band structure of this quasi-periodic equation might be interesting on its own, we will restrict ourselves to the simplest perturbative treatment, 7 postponing the non-perturbative analysis to a future publication. In particular, we will expand the Mukhanov-Sasaki mode functions as u k = u k,0 +u k,1 +... with u k,0 given by the underlying inflationary model and u k,1 a small perturbation satisfying The total curvature power spectrum is then given by with and For the purposes of this paper it would be enough to approximate u k,0 by the standard de Sitter mode functions leaving aside a small correction of the order of slow-roll parameters. The solution of Eq. (21) can be computed using the Green function method and yields [63] u k,1 (τ ) = iu k,0 (τ ) where we have assumed homogeneous boundary conditions at the onset of the oscillations. Note, however, that the exact boundary conditions used do not play a central role in the final result. Indeed, the integrals in this expression are dominated by the time at which frequency of the de Sitter mode function u k,0 coincides with that of the oscillating field χ. In other words, the resonant behaviour at allows to safely evaluate the integral at that time using the saddle-point approximation.
At this point we have to specify the inflationary model in order to analytically compute the corrections to the primordial power spectrum. For this purpose, and in order to facilitate the comparison with the existing literature, we consider an illustrative (power-law) inflation scenario with runaway potential V (φ) = V 0 e −λφ [14]. For this particular choice, the equation of motion of the background field φ admits a slow-roll solution with For simplicity we will also consider constant rate variations of the theory defining functions f and m, namely δ m , δ f = constant. This corresponds to dilatonic-like couplings m 2 ∼ e λδmφ and f 2 ∼ e λδ f φ , ubiquitously appearing in non-minimally coupled theories when written in the Einstein-frame [43,64]. Under these assumptions, the oscillatory correction to the power spectrum can be written as where k r ≡ 2a r m eff,r is the momentum associated with the first resonant mode and we have defined an amplitude tilt with For m = f (or equivalently δ m = δ f or γ = 1) the effective mass in Eq. (13) is (approximately) constant and our results reduce to those in Ref. [17]. Note also that there exists a "singular" scenario with δ f − δ m = 1 in which the 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 nextto-leading order computation is required. We will not be interested in this particular setting in what follows.

B. Non slow-roll suppressed interactions
If ∆ω 2 = 0 the direct kinetic coupling ω 2 (∂φ) 2 in Eq. (2) can easily dominate over the other interaction terms, since, contrary to them, it is not suppressed by the slow-roll conditions (8).
Since we want the backreaction of the χ field on the background dynamics to stay small during the whole observable window, we will assume the conditions (10) and (11) to hold and consider a simple perturbation [39] with Λ a given energy scale. When this coupling is taken into account the variable z in Eq. (20) is effectively replaced by z ω, leading to an additional mass term proportional to χ in the Mukhanov-Sasaki equation (18). Taking into account Eq. (15) we obtain with ∆ T (τ ) = ∆(τ ) cos (2Ω + θ) + ∆ ω (τ ) sin (Ω + θ/2) , and ∆ ω (τ ) Performing a similar computation to that in the ∆ω 2 = 0, we obtain an oscillatory correction to the power spectrum displaying the same frequency dependence 8 but a different amplitude tilt, namely This scenario is particularly interesting since for m = f 2 (or equivalently δ m = 2δ f ) the signal mimics that of a model with power-law indexp ≡ p/γ. This correspondence is expected since the equation of motion for the χ field effectively reproduces in this case that of a FLRW universe with scale factorā = f a [65,66]. Note indeed that the frequency of the oscillations in the curvature perturbation is determined by the ratio of two time scales: the effective mass m eff and the expansion rate H. For a constant mass, the change in µ = m eff /H is solely due to the Hubble rate H and therefore the frequency probes directly the expansion rate of the Universe. When the mass gets a time dependence we see a combination of both. The time dependence of the mass can be, however, absorbed into a new redefined Hubble rateH, namely When m = f 2 the quantityH coincides with the expansion rate of a universe with scale factorā = f a. That is the reason why a time dependent spectator field mass is able to mimic the frequency of the oscillating features above. Note that this intuitive argument is not linked to matter domination. In particular, an exponential/dilatonic choice of the theory defining functions f (φ) and m(φ) is able to mimic any oscillatory feature generated by standard spectator fields in cosmologies with power-law scale factor evolutions, as, for instance, ekpyrotic scenarios with 0 < p 1.

IV. MIMICKING ALTERNATIVE SCENARIOS
The typical behaviours of the oscillatory signals generated by a free spectator field during a matter contraction era and during an inflationary stage are illustrated in Fig. 1. Let us discuss how this picture is modified in 8 Strictly speaking, the frequency is halved with respect to the previous case since the corrections depend now on χ instead of the presence of non-trivial interactions. To this end, note that by taking one can always adjust the scale dependence of the frequency to that of a universe with a power-law evolution a ∼ tp. For a fixed wavenumber k r , the frequency rescales as well, mimicking that of a field with effective The simultaneous choice of δ m and δ f allows therefore to fix the frequency and tilt of the oscillations in order to imitate alternative signals. However, if ∆ω 2 = 0 the scale dependence of the amplitude breaks the degeneracy since both ν 1 andp = p/γ depend only on the combination δ m − δ f (cf. Fig. 2). On the other hand, if the term proportional to ∆ω 2 dominates, the dependence of tilt on δ m and δ f changes and we can choose δ m = 2δ f such that the amplitude of the oscillations mimics also that a universe with power-law indexp, i.e. ν 1 = −3/2 + 1/(2p) (cf. Fig 3). Incidentally, we have found a degeneracy between the gravitational signal from a matter contraction and a direct signal from inflation with p ≈ 57, δ m ≈ −2.98 and δ f ≈ −0.017. As we will see in the next section, this accidental degeneracy does not hold when higher-order correlations are taken into account. Note also that if δ f > δ m the effective mass of the field χ decreases with time, such that it might eventually dominate the background evolution. We assume that such regime is never reached. Before ending this section, we would like to point out a particular scenario, only present for non-standard clocks like the ones under consideration. That would be an oscillatory signal with γ = 1 (i.e. with a frequency that could be interpreted as that of an standard clock during inflation) but with an increasing amplitude. This will happen whenever δ f < (1 − 3p)/2p (cf. Fig. (4)).

V. HIGHER-ORDER CORRELATIONS
The above discussion illustrates how the combination of sudden turns and time-dependent masses can easily produce varying-frequency oscillatory patterns in the power spectrum mimicking the scale dependence of alternative inflationary scenarios. Given this result, it is important to check if higher-order correlation functions could allow to break the degeneracy among these scenarios.
The overall scale dependence of the three-point correlation function can be studied by looking at the equilateral configuration. As before, we consider two cases according to the behaviour of the scalar interactions during slow-roll. For ∆ω 2 = 0, the leading contribution to the bispectrum reads [67][68][69] with and η the first and second slow-roll parameters.
Using the in-in formalism, we find that, up to a shape function, the associated non-gaussianity is given by 9 9 Note that if m = f the effective mass is constant and our results coincide with those in Refs. [17] and [21].
with K ≡ k 1 + k 2 + k 3 and On the other hand, if ∆ω 2 = 0 the leading contribution to the three-point correlation function is rather given by [40] In this case the scale dependence of the amplitude is modified to We see therefore that even in the case m = f 2 there is a correction to the scale dependence of the amplitude of the non-gaussianities as compared to that in an scenario with power-law indexp. This implies that, at least in the simple case we studied, the degeneracy would be broken by an eventual measurement of the scale dependence of non-gaussinities. Independently of their origin, the generated nongaussianities display the same frequency dependence as the power spectrum. This result is due to the existence of a single oscillation frequency and can be easily extended to higher-order correlation functions. We conclude therefore that, if the existence of oscillatory features in the power spectrum would be eventually established, one would need the full information of the amplitudes and the frequency in order to probe the evolution of the scale factor.

VI. CONCLUSIONS AND DISCUSSION
The frequency and momentum dependence of potential oscillatory features in the primordial density fluctuations has been advocated as a way to distinguish the inflationary paradigm from alternatives scenarios such as bounces or matter contraction eras [16][17][18][19][20][21][22][23][24][25][26]. These features are typically imprinted by the oscillations of a spectator field, provided that this is excited at some point in the primordial Universe.
In this work, we have shown that if the mass of a spectator field receives an explicit time dependence during inflation-due for instance to non-trivial interaction with other fields-the oscillatory signal in the power spectrum and other higher-order correlation functions can easily mimick the one appearing in alternatives scenarios to inflation, both in its frequency and momentum dependence. To illustrate this result we considered two particular examples, one with no dominant kinetic mixing among the spectator field and the inflaton kinetic term and one involving a relevant kinetic coupling χ(∂φ) 2 . Using a WKB approximation we showed that the oscilla-tions in the power spectrum take the generic form with C and θ constants, p the power-law index of the scale factor and γ a parameter related to the time-dependent spectator field mass and its kinetic term normalization. Interestingly, only the combination p/γ is probed by the frequency, but not just the power-law index p. The scale dependence of the amplitude in Eq. (44) is given by with δ m and δ f free parameters of the model related to the change rate of the mass and the kinetic coefficient of the χ field, respectively. In the former case (∆ω 2 = 0), one can choose the interaction among fields in such a way that the frequency matches that of an alternative matter contraction scenario, but the scale dependence of the amplitude breaks the degeneracy. In the second case (∆ω 2 = 0), the amplitude of the power spectrum can also follow that of the alternative matter contraction scenario for a particular choice of the interactions. Regarding the frequency, this result holds even when one considers higher order correlations functions, since the single oscillation frequency of the spectator field gets imprinted in the same way in all of them. However, the degeneracy is broken by the scale dependence of the amplitude of higher-order correlations functions. We illustrated this result by explicitly computing the bispectrum in the equilateral configuration, which takes the schematic form with The mimicking mechanism proposed in this paper does not require special fine-tunings or large number of parameters. Although we focused for concreteness on matter contraction scenarios leading to an almost scale invariant primordial power spectrum, our results can be easily extended to other alternative scenarios by properly choosing the form of the interactions. Our findings soften the claim that oscillatory features from spectators fields during inflation can probe inflation and its alternatives without a full knowledge of the n-point correlation functions. To do so, one would need the complete information of the frequency and amplitudes of, at least, the 2-and 3-point correlation functions.