Generation of inflationary perturbations in the continuous spontaneous localizationmodel: The second order power spectrum

Gabriel Leon∗ and Marıa Pıa Piccirilli†

Grupo de Astrofısica, Relatividad y Cosmologıa, Facultad de Ciencias Astronomicas y Geofısicas,Universidad Nacional de La Plata, Paseo del Bosque S/N 1900 La Plata, Argentina.CONICET, Godoy Cruz 2290, 1425 Ciudad Autonoma de Buenos Aires, Argentina.

Cosmic inflation, which describes an accelerated expansion of the early Universe, yields themost successful predictions regarding temperature anisotropies in the cosmic microwave background(CMB). Nevertheless, the precise origin of the primordial perturbations and their quantum-to-classical transition is still an open issue. The continuous spontaneous localization model (CSL), inthe cosmological context, might be used to provide a solution to the mentioned puzzles by consider-ing an objective reduction of the inflaton wave function. In this work, we calculate the primordialpower spectrum at the next leading order in the Hubble flow functions that results from applyingthe CSL model to slow roll inflation within the semiclassical gravity framework. We employ themethod known as uniform approximation along with a second order expansion in the Hubble flowfunctions. We analyze some features in the CMB temperature and primordial power spectra thatcould help to distinguish between the standard prediction and our approach.

I. INTRODUCTION

The most recent observational data obtained from theCosmological Microwave Background (CMB) are consis-tent with the hypothesis that the early Universe under-went an accelerated expansion [1]. The model to de-scribe that epoch, known as inflation, is now consideredas an essential part of the concordance ΛCDM cosmo-logical model. The success of the inflationary scenariois based on its predictive power to yield the initial con-ditions for all the observed cosmic structure, which arecommonly referred to as primordial perturbations [2].

In the most simple inflationary model, the origin ofprimordial perturbations is substantially related to quan-tum vacuum fluctuations of the scalar field driving theaccelerated expansion. Here a subtle question arises:How exactly do these quantum fluctuations become ac-tual (classical) inhomogeneities/anistropies? And in par-ticular, How does the standard inflationary model ac-counts for the transition from the initially homogeneousand isotropic quantum state (i.e. the vacuum) into astate lacking such symmetries? It is fair to say that theanswer to these questions have not been completely set-tled, and a large amount of literature has been devotedto this subject [3–13].

The main reason why this debate continues is becauseit touches on another controversial issue, i.e. the quan-tum measurement problem. Specifically, in the stan-dard Copenhagen interpretation of Quantum Mechanics(QM), it is an essential requirement to define (or iden-tify) an observer who performs a measurement with somekind of device. However, in the early Universe there areno such entities, and the measurement problem becomes

∗ gleon@fcaglp.unlp.edu.ar† mpp@fcaglp.unlp.edu.ar

exacerbated [3, 6, 14–16].1 One of the first attempts todeal with the aforementioned issue was by invoking thedecoherence framework [8, 9]. Although, decoherence canprovide a partial understanding of the issue, it does notfully addresses the problem mainly because decoherencedoes not solve the quantum measurement problem [17].We will not dwell in all the conceptual aspects regardingthe appeal of decoherence during inflation; instead, werefer the interested reader to Refs. [6, 18] for a more indepth analysis.

There are many approaches to the subject of Founda-tions of Quantum Theory and in particular to the quan-tum measurement problem, but a good method to clas-sify them is provided by the result of [19]. There, one canfind a particularly useful way to state the measurementproblem, which consists in a list of three statements thatcannot be all true at the same time:

A. The physical description given by the quantumstate is complete.

B. Quantum evolution is always unitary.

C. Measurements always yield definite results.

The need to forsake (at least) one of the above forcesone toward a specific conceptual path depending on thechoice one makes. Concretely speaking, forsaking (A)leads naturally to hidden variable theories, such as deBroglie-Bohm or “pilot wave” theory [20, 21]. Forsaking(B), one is naturally led to collapse theories and which forthe cosmological case seem to leave no option but those

1 Sometimes it is argued that it is us–humans–who are the ob-servers with our own astronomical observations. This argumentis rebuked, because if that is the case, then the Universe washomogeneous and isotropic until our astronomers started mak-ing observations; however, that is impossible because a Universethat is homogeneous and isotropic contains no astronomers.

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of the spontaneous kind, such as the Ghirardi-Rimini-Weber or Continuous Spontaneous Localization models[22–24]. The reason is that there is clearly no role forconscious observers or measuring devices that might bemeaningfully brought to bear to the situation at hand.Finally, forsaking (C) seems to be the starting point ofapproaches such as the Everettian type of interpreta-tions [25]. The latter, again, seem quite difficult to besuitably implemented in the context at hand, simply be-cause “observers”, “minds”, and such notions, that playan important role in most attempts to characterize theworld branching structure in those approaches, can onlybe accounted for within a Universe in which structurehas already developed, well before the emergence of thesaid entities.

All of those approaches have been followed to inves-tigate the generation of primordial perturbations duringinflation [3–5, 7, 10, 12, 26–28]. In the present work,we will focus on the Continuous Spontaneous Localiza-tion (CSL) model applied to the standard slow roll in-flationary scenario, and just for notation comfort, fromnow on we will refer to this idea as the CSL inflationarymodel (CSLIM). Other applications of the CSL modelto cosmology have been analyzed recently, for instanceto account for the late-time accelerated expansion of theUniverse [29, 30].

Several aspects of the CSLIM have been studied be-fore. The first implementation, based on the semiclas-sical gravity framework2, was done in [5]. Afterwards,using observational data it was possible to statisticallyconstrain the cosmological parameters of the model; alsoa Bayesian analysis was performed in order to comparethe model performance within the standard cosmologicalmodel [26].

Moreover, in [32–34], and working in the context ofsemiclassical gravity, it was shown that the CSLIM pre-dicts a strong suppression of primordial B-modes, i.e. thepredicted amplitude of the tensor power spectrum is verysmall generically (undetectable by current experiments).Also in [35] it was found that, when enforcing the CSLIM,the condition for eternal inflation can be bypassed.

One of the main features of the CSLIM is that it mod-ifies the standard primordial power spectrum through acharacteristic k dependence [26]; specifically, the spec-trum is of the form P (k) ∝ (k/k)

ns−1C(k), where C(k)is a new function of the model’s parameters (and k isthe pivot scale). The predicted spectral index ns is givenin terms (as in the traditional approach) of the slow rollparameters or equivalently in terms of the Hubble flowfunctions (HFF). At this point, we introduce the mainmotivation for the present work; our purpose is to answerthe question: How can one distinguish the k dependenceintroduced by the CSLIM from a “simple” running of the

2 It is worthwhile to mention that the CSL model has also beenapplied to inflation using the Mukhanov-Sasaki variable, whichquantizes both the metric and inflaton perturbations [3, 4, 7, 31].

spectral index? and Is it possible to use observationaldata (recent or future) to answer that question? Herewe remind the reader that the running of the spectral in-dex is traditionally interpreted as an extra k dependenceinduced, in the power spectrum, by the spectral indexns(k). In single field slow roll inflation, one immediatelyrealizes that an attempt to answer those questions re-quires first a calculation of the power spectrum at secondorder in the HFF within the CSLIM. In the present pa-per, we present the result and computational details forsuch calculation. Furthermore, we perform a compari-son between our prediction and the second order powerspectrum given in the traditional approach [36–42]. Alsowe perform a preliminary analysis of the observationalconsequences for each model. Our calculations made useof the uniform approximation method [36, 43]; these aresupplemented in two Appendices, where one can also findour prediction for ns and αs at higher order in the HFF.

We can further motivate the significance of the soughtresult in this paper. Recent data from Planck collabora-tion seem to indicate that a scale dependence of the scalarspectral index is still allowed by observations [1]. As wehave mentioned, this scale dependence of ns is knownas the running of the spectral index αs. The currentdata from Planck indicates that αs = −0.0045 ± 0.0067at 68% CL and αs = −0.005 ± 0.013 at 95% CL (whenthe running of the running of the spectral index is setto zero). Although these values are consistent with azero running, future experiments may detect a non-zerovalue of αs. The relevant issue here would be the orderof magnitude of αs.

Let us recall that at the lowest order in the HFF,the standard prediction from slow roll inflation yields:ns− 1 = −2ε1− ε2, r = 16ε1 (known as the tensor-to-scalar ratio) and αs = −2ε1ε2 − ε2ε3, where εj de-notes the HFF evaluated at the pivot scale; consequently,αs = (ns − 1 + r/8)(r/8 + ε3). Furthermore, as moretight constraints on r are obtained by future collabora-tions [see e.g. [44]], a plausible scenario could ensue: Itmay be the case that r would remain undetected, de-creasing the order of magnitude of ε1 allowed by thedata. In that case, a conservative estimate for the mag-nitude of the running would be |αs| ' |ns− 1||r/8 + ε3|.However, assuming also a detection of the running of or-der |αs| ' 10−3, and taking into account that currentdata indicate |ns − 1| ' 10−2, then we would have theestimate |ε3| ' 10−1. That result can be puzzling forthe traditional slow roll inflationary paradigm, becauseone would have |ε3| > |ε2| > |ε1|. In other words, theso called hierarchy of the HFF [39] would be lost, sug-gesting a possible inconsistency with the single field slowroll inflationary model [45]. Note that |αs| ' 10−3 is notan unrealistic estimate based on the current 1σ,2σ CLreported by Planck [1] and by future observations [46].

Moreover, a recent theoretical motivated proposal,known as the Trans-Planckian Censorship Conjecture(TCC) [47], leads to the prediction of a negligible am-plitude of primordial gravitational waves, that is |ε1| <

3

10−31 [48]. The TCC simply put states that in an ex-panding Universe sub-Planckian quantum fluctuationsshould remain quantum and can never become largerthan the Hubble horizon and classically freeze.3 Fur-thermore, it has been found [52] that a large value of thesecond slow-roll parameter and a small ε1 is essentiallypreferred not only by the TCC, but also by the so called“swampland conjecture,” which is more general. While,we will left for future work how exactly the TCC couldbe implemented in the CSLIM, the implications of theTCC do serve to highlight that it is not quite improb-able that predictions and observations in standard slowroll inflation might face some issues in the future.

The CSLIM also predicts a strong suppression ofprimordial gravity waves, but in this case the tensormodes are generated by second order scalar perturbations[33, 34]; in fact, an estimate for the tensor-to-scalar ratiohas been obtained in Ref. [33]: r = 10−7ε21. This resultmeans that in the CSLIM, r is no longer related at theleading order with ns−1 and αs, which contrasts with thestandard prediction. Moreover, since in the CSLIM thepredicted spectrum has an extra k dependence throughthe function C(k), then, in principle, it is possible thatC(k) acts as a “running effect” which does not dependentirely on αs. As a consequence, the supposed sce-nario above in the traditional approach, and which wouldlead to inconsistencies in the slow roll inflationary model,might be resolved within the CSLIM. In particular, anon-detection of r (with tightest constraints) and a suffi-ciently high detection of a running of the spectral indexcould be consistent within our proposed framework, butthe hierarchy of the HFF would not be violated (as wouldbe the case in the standard approach). These plausiblesequence of events, would also serve to show that theCSLIM is not “just a philosophically” motivated model(as sometimes is often dismissed) but that it can haveimportant observational consequences.

Thus, in the present work, we will make a first step inthat direction, obtaining a prediction for the primordialspectrum at second order in the HFF. This will allow usto analyze clearly the dependence on k of the primordialspectrum, i.e. to single out the contribution given by αsand C(k) in the predicted form of the power spectrum.Hopefully, future observations could be used to performa full data analysis using the result obtained here.

The paper is organized as follows: In Sec. II, wepresent the technical setting that, based on the semi-classical gravity framework, represents an adequate ap-plication of the collapse hypothesis to standard slow rollinflation; this is done at second order in the HFF, alsowe show how we can obtain a formula for the primordial

3 The TCC serves to address the trans-Planckian problem for cos-mological fluctuations [49–51]. In particular, it is conjecturedthat the trans-Planckian problem can never arise in a consistenttheory of quantum gravity and that all models which would leadto such issues are inconsistent and belong to the Swampland.

power spectrum with the previous considerations. In Sec.III the quantum treatment of inflaton is shown by tak-ing into account the CSL model, the novel feature here,with respect to previous works, is the second order equa-tions in the HFF. These equations enable us to obtainthe primordial spectrum at the next leading order. InSec. IV, we compare the primordial power spectrum ob-tained in the previous section with the phenomenologicalexpression from standard inflationary models. Specifi-cally, we plot the primordial power spectrum at secondorder for some particular parameterizations of the col-lapse parameter and compare it with the primordial spec-trum preferred by the data, which corresponds to thestandard prediction in slow roll inflation. Moreover, wepresent our prediction for the CMB temperature fluctu-ation spectrum and show that possible differences existwith respect to the best fit model obtained in traditionalslow roll inflation. The analysis presented in this sectiontakes into account the inflation parameters As, ns andαs. Finally, in Sec. V, we summarize the main results ofthe paper and present our conclusions.

We have included two appendixes with the aim to pro-vide supplementary material for the reader interested inall the computational details. Appendix A contains thetechnical steps required to solve of the CSL equationsat second order in the HFF, these are based on the uni-form approximation method. Employing those results, inAppendix B we provide the calculations used to obtainthe primordial power spectrum at second order, and wealso include the prediction for the spectral and runningspectral indexes at third and fourth order respectively.

Regarding notation and conventions, we will work withsignature (−,+,+,+) for the metric, and we will useunits where c = ~ = 1 but keep the gravitational constantG.

II. THE COLLAPSE PROPOSAL AND THEPRIMORDIAL POWER SPECTRUM

Before addressing in full detail the main equations ofour model, we present the framework that underlies ourdescription of the space-time metric and that of the infla-ton [6, 53–57]. The proposed model is based on the semi-classical gravity framework, in which gravity is treatedclassically and the matter fields are treated quantum me-chanically. This approach accepts that gravity is quan-tum mechanical at the fundamental level, but considersthat the characterization of gravity in terms of the metricis only meaningful when the space-time can be consideredclassical. Therefore, semiclassical gravity can be treatedas an effective description of quantum matter fields liv-ing on a classical space-time. Clearly this approach isvery different from the standard inflationary theory, inwhich the perturbations of both the metric and the mat-ter fields are treated in quantum mechanical terms. Theframework employed is thus based on semiclassical Ein-

4

stein’s equations (EE),

Gab = 8πG〈Tab〉. (1)

In our approach the initial state of the quantum field istaken to be the same as the standard one, i.e. the Bunch-Davies (BD) vacuum. Nonetheless, the self-induced col-lapse will spontaneously change this initial state into afinal one that does not need to share the symmetries ofthe BD vacuum. These symmetries are homogeneity andisotropy. Consequently, after the collapse, the expecta-tion value 〈Tab〉 will not have the symmetries of the BDvacuum, and this will led, through semiclassical EE, toa geometry that is no longer homogeneous and isotropicgenerically. The interested reader can consult Refs. [54–57]; in those works the formalism of the collapse proposalwithin the semiclassical gravity framework has been de-veloped. In the present paper, we will only make use ofthe most relevant equations.

A. Classical description of the perturbations

As in standard slow roll inflationary models, we con-sider the action of a single scalar field, minimally coupledto gravity, with an appropriate potential:

S[φ, gab] =

∫d4x√−g[

1

16πGR[g]−1

2∇aφ∇bφgab−V [φ]

].

(2)The background metric is described by a flat FRW space-time, with a(t) the scale factor. Meanwhile the mattersector can be modeled by a scalar field which can bedecomposed into a homogeneous part plus “small” per-turbations φ(x, t) = φ0(t) + δφ(x, t).

In order to describe slow roll (SR) inflation, it is con-venient to introduce the Hubble flow functions εi (HFF)[41], these are defined as

εn+1 ≡d ln εndN

, ε0 ≡Hini

H, (3)

where N ≡ ln(a/aini) is the number of e-folds from thebeginning of inflation; H ≡ a/a the Hubble parameterand the dot denotes derivative respect to cosmic time t.Inflation occurs if ε1 < 1 and the slow roll approxima-tion assumes that all these parameters are small duringinflation |εn| 1. Additionally, since dN = Hdt, it isstraightforward to obtain another useful expression forthe HFF, i.e.

εn = Hεnεn+1. (4)

In terms of the first two HFF, the dynamical equationsfor the homogeneous part of the model can be expressedas

H2 =V

M2P (3− ε1)

, (5)

3Hφ(

1− ε13

+ε26

)= −∂φV, (6)

where M2P ≡ 1/(8πG) is the reduced Planck’s mass. The

previous equations are exact.Let us now focus on the perturbations part of the

theory. We start by switching to conformal coordi-nates; thus, the components of the background metric

are g(0)µν = a(η)ηµν , with η the conformal cosmological

time; ηµν the components of the Minkowskian metric.We choose to work in the longitudinal gauge; in such

a gauge, and focusing on the scalar perturbations at firstorder, the line element associated to the metric is:

ds2 = a2(η)[−(1 + 2Φ)dη2 + (1− 2Ψ)δijdx

idxj], (7)

where Φ and Ψ are scalar fields, and i, j = 1, 2, 3. Ein-stein’s equations (EE) at first order in the perturbations,δG0

0 = 8πGδT 00 , δG0

i = 8πGδT 0i and δGij = 8πGδT ij , are

given respectively by

∇2Ψ− 3H(HΦ + Ψ′) = 4πG[−φ′20 Φ + φ′0δφ′ + ∂φV a

2δφ],(8)

∂i(HΦ + Ψ′) = 4πG∂i(φ′0δφ), (9)

[Ψ′′ +H(2Ψ + Φ)′ + (2H′ +H2)Φ +1

2∇2(Φ−Ψ)]δij

− 1

2∂i∂j(Φ−Ψ) = 4πG[φ′0δφ

′ − φ′20 Φ− ∂φV a2δφ]δij .

(10)

Equation (10) with components i 6= j lead to Ψ = Φ,from now on we will use this result and refer to Ψ as theNewtonian potential. Furthermore, in the longitudinalgauge Ψ represents the curvature perturbation (i.e. theintrinsic spatial curvature on hypersurfaces on constantconformal time for a flat Universe). Subtracting Eq. (8)from (10), together with (9) and the motion equationfor the homogeneous part of the scalar field a2∂φV =−φ′′0 − 2Hφ′0, one obtains

Ψ′′ −∇2Ψ + 2

(H− φ′′0

φ′0

)Ψ′ + 2

(H′ − Hφ

′′0

φ′0

)Ψ = 0.

(11)Regarding notation, primes denote derivative with re-spect to conformal time η, and H ≡ a′/a.

Switching to Fourier’s space4, in the super-Hubblelimit kη → 0, the solution to the above differential equa-tion is

Ψk(η) = CG(k)[ε1 + (ε21 + ε1ε2)] +O(ε3), (12)

4 We define the Fourier transform of a function f(x, η) as

f(x, η) =1

(2π)3/2

∫R3d3k eik·xfk(η)

5

where CG(k) is a constant fixed by the initial conditions.Also note that solution (12) is approximately constant.From (12) and (4), it follows that Ψ′k is order 2 at thelowest order in the HFF. In particular, we have that Ψ′k =CG(k)ε1ε2H+O(ε3); hence we approximate

Ψ′kH−1 ' ε2Ψk. (13)

This will be a useful result in the following, however notethat the approximation breakdowns at order 3 or higherin εn.

The collapse of the inflaton’s wave function, which isgoverned by the CSL mechanism, is the process that gen-erates the curvature perturbations. We will be more spe-cific in the next section, but for now let assume that theCSL process simply changes randomly the initial state ofthe field to a different one. This mechanism can be im-plemented in the early Universe using the semiclassicalgravity framework. The semiclassical EE at linear orderin the perturbations read δGab = 8πG〈δTab〉.

Therefore, the semiclassical version of Eq. (9) in

Fourier’s space, ki(Ψk + H−1Ψ′k) = 4πGkiH−1φ′0〈δφk〉together with (13), yields

Ψk + ε2Ψk ' 4πGH−1φ′0〈δφk〉, (14)

note that (13) comes from solving (11), that is equationsδG0

0 = 8πGδT 00 , δG0

i = 8πGδT 0i and δGij = 8πGδT ij have

been combined to solve for Ψk.We can rewrite Eq. (14) in terms of the HFF only.

Taking the derivative of Eq. (5) with respect to t and

combining it with: Eq. (6), the defintion ε1 = −H/H2

and ε1 = Hε1ε2; we can find that φ20 = M2PH

22ε1 orequivalently in conformal coordinates

φ′20 = M2PH22ε1, (15)

that relation is exact. Finally, substituting Eq. (15) into(14) leads to the following main equation for the metricperturbation:

Ψk '1

MP

√ε12

〈δφk〉(1 + ε2)

, (16)

the approximation is valid up to order 2 in εn.This is the main result of the present subsection. Equa-

tion (16) indicates that when the state is the vacuum, one

has 〈0|δφk|0〉 = 0, i.e. there are no perturbations at anyscale k; thus Ψk = 0. It is only after the collapse hastaken place |0〉 → |Ξ〉, that the expectation value satis-

fies 〈δφk〉 6= 0, and thus giving birth to the primordialperturbations.

B. The scalar power spectrum

In this subsection, we want to find an expression forthe scalar power spectrum in terms of the metric pertur-bation equation (16). We begin by recalling a well-known

quantity defined as

R ≡ Ψ +

(2ρ

3

)(H−1Ψ′ + Ψ

ρ+ P

), (17)

where ρ and P are the energy and pressure densities asso-ciated to the type of matter driving the expansion of theUniverse. The importance of the quantity R is that, foradiabatic perturbations, it is conserved for super-Hubblescales, irrespective of the cosmological epoch one is con-sidering. The type of cosmological epoch is characterizedby the equation of state P = ωρ. For a matter domi-nated epoch ω ' 0, and for a radiation dominated epochω ' 1/3. The Newtonian potential Ψ, is also a con-served quantity for super-Hubble scales, but its ampli-tude changes between epoch transitions; on the contrary,the amplitude of R does not change during the transi-tions. The amplitude variation of Ψ during the transi-tion from radiation to matter dominated epoch is notvery significant, |Ψmatt.| ' (9/10)|Ψrad.|. Nevertheless,the amplitude variation between inflation and radiationera does changes significantly; let us see this explicitly.

During inflation ρ + P = φ′20 /a2 = M2

PH22ε1/a2, and

because of Friedmann’s equation H2 = a2ρ/3M2P , we

have

R = Ψ +1

ε1

(H−1Ψ′ + Ψ

). (18)

The above equation is exact. However, using approxima-tion (13) for the Fourier components results in

Rk 'Ψk

ε1(1 + ε1 + ε2). (19)

On the other hand, during the radiation dominatedepoch Rk = (3/2)Ψrad.

k . Since R is a conserved quantity,hence, we can obtain the change in the amplitude of theNewtonian potential from the inflationary epoch to theradiation dominated epoch,

|Ψrad.k | = 2(1 + ε1 + ε2)

3ε1|Ψk|. (20)

Thus, in the radiation epoch, the amplitude of the New-tonian potential during inflation is amplified by a factorof 1/ε1.

Another important aspect of the quantity R is that inthe comoving gauge, it represents the curvature pertur-bation. In fact, the primordial power spectrum usuallyshown in the literature is associated to R. The scalarpower spectrum (associated to the curvature perturba-tion in the comoving gauge) in Fourier space is definedas

RkR∗q ≡2π2

k3Ps(k)δ(k− q), (21)

where Ps(k) is the dimensionless power spectrum. Thebar appearing in (21) denotes an ensemble average overpossible realizations of the stochastic field Rk. In the

6

CSLIM each realization will be associated to a particularrealization of the stochastic process characterizing thecollapse process.

On the other hand, our main equation from the lastsubsection (16), was obtained in the longitudinal gauge.Fortunately, Eq. (17) relates Ψ and R exactly; in otherwords, we can compute the curvature perturbation in thelongitudinal gauge using the CSLIM, and then switch tothe comoving gauge in order to compare the primordialspectrum obtained in our model with the standard one.Furthermore, during inflation, we can use approximation(19) to compute the scalar power spectrum, associatedto Rk, that results from our main equation (16). This is,

RkR∗q =1

2M2P ε1

(1 + ε1 + ε2)2

(1 + ε2)2〈δφk〉〈δφq〉∗. (22)

Therefore, we can identify the scalar power spectrumas

Ps(k)δ(k− q) =k3

4π2M2P ε1

(1 + ε1 + ε2)2

(1 + ε2)2〈δφk〉〈δφq〉∗.

(23)

The quantity 〈δφk〉〈δφq〉∗, must be evaluated in thesuper-Hubble regime kη → 0. In the next section, wewill focus on that quantity.

III. QUANTUM TREATMENT OF THEPERTURBATIONS: THE CSL APPROACH

We now proceed to describe the quantum theory of theperturbations. Our treatment is based on the QFT ofδφ(x, η) in a curved background described by a quasi–deSitter spacetime. Expanding the action (2) up to sec-ond order in the perturbations, one can find the actionassociated to the matter perturbations. Given that weare working within the semiclassical gravity framework,we are only interested in quantize the matter degrees offreedom. Introducing the rescaled field variable y = aδφ,the second order action is δ(2)S =

∫d4xδ(2)L, where

δ(2)L =1

2

[y′2 − (∇y)2 − y2a2V,φφ +

a′′

ay2]

+ a[4φ′0Ψ′y − 2a2V,φΨy] (24)

and V,φ indicates partial derivative with respect of φ.

Note that in δ(2)L there are terms containing metric per-turbations. In the vacuum state, according to our ap-proach, Ψ = Ψ′ = 0. However, since the CSL mechanismis a continuous collapse process, the quantum state char-acterizing the system will change from |0〉 to a new finalstate |Ξ〉. As a consequence, the metric perturbations(which are always classical) will be changing from zeroto a non-vanishing value in a continuous manner. Thus,including the terms containing Ψ and Ψ′ in the actioncan be considered as a backreaction effect of the CSLmodel, and as we will see this effect is of second order inthe HFF.

We next switch to Fourier space. This is justified bythe fact that we work with a linear theory and, hence,all the modes evolve independently. We define the field’smodes as

y(x, η) =1

(2π)3/2

∫R3

d3k yk(η)eik·x, (25)

Ψ(x, η) =1

(2π)3/2

∫R3

d3k Ψk(η)eik·x, (26)

with y−k = y∗k and Ψ−k = Ψ∗k because y(x, η) andΨ(x, η) are real. Substituting the Fourier expansions intoLagrangian (24), the resulting action is δ(2)S =

∫dηL(2),

with L(2) ≡∫R3+ d

3kL(2),

L(2) ≡ y′ky∗′

k − (k2 − a′′

a+ a2V,φφ)yky

∗k

+ 4aφ′0(Ψ′ky∗k + Ψ∗

′

k yk)− 2a3V,φ(Ψky∗k + Ψ∗kyk).

(27)

Note that we are defining L(2) by integrating the functionL(2) over the k+ half-space.

The CSL model is based on a non-unitary modifica-tion to the Schrodinger equation; consequently, it will beadvantageous to perform the quantization of the pertur-bations in the Schrodinger picture, where the relevantphysical objects are the Hamiltonian and the wave func-tional.

We first define the canonical conjugated momentumassociated to yk is pk ≡ ∂L(2)/∂y?

′

k , that is pk = y′k.

The Hamiltonian associated to Lagrangian L(2), can befound as H(2) =

∫R3+ d

3k(y∗′

k pk+y′

kp∗k)−L(2). Therefore,

H(2) =∫R3+ d

3kH(2), with

H(2) ≡ p∗kpk + y∗kyk

(k2 − a′′

a+ a2V,φφ

)− 4aφ′0(Ψ′ky

∗k + Ψ∗

′

k yk) + 2a3V,φ(Ψky∗k + Ψ∗kyk).

(28)

From the Hamiltonian above we can find the equationof motion for yk and pk. That is, using that

p′k = −∂H(2)

∂y∗k, y∗

′

k =∂H(2)

∂pk, (29)

the field’s mode equation of motion is

y′′k +

(k2 − a′′

a+ a2V,φφ

)yk − 4aφ′0Ψ′k + 2a3V,φΨk = 0.

(30)The previous equation coincides with the evolution equa-tion for δφk usually found in the literature [58], thus itserves as a self-consistency check.

Given that we are carrying out the quantization in theSchrodinger picture, it will be more convenient to work

7

with real variables, which later can be associated to Her-mitian operators. Therefore, we introduce the followingdefinitions

yk ≡1√2

(yRk + iyIk), pk ≡1√2

(pRk + ipIk), (31)

and also

Ψk ≡1√2

(ΨRk + iΨI

k). (32)

In the Schrodinger approach, the quantum state of thesystem is described by a wave functional, Φ[y(x, η)]. InFourier space (and since the theory is still free in thesense that it does not contain terms with power higherthan two in the Lagrangian), the wave functional can alsobe factorized into mode components as

Φ[y(x, η)] =∏k

Φk(yRk , yIk) =

∏k

ΦRk (yRk )ΦI

k(yIk). (33)

Quantization is achieved by promoting yk and pk toquantum operators, yk and pk, and by requiring thecanonical commutation relations,

[yR,Ik , pR,I

q ] = iδ(k− q). (34)

In the field representation, the operators would take theform:

yR,Ik Φ = yR,I

k Φ, pR,Ik Φ = −i ∂Φ

∂yR,Ik

. (35)

For the moment let us put aside the CSL mechanism,and analyze the standard evolution of the wave function.The wave functional Φ obeys the Schrodinger equationwhich, in this context, is a functional differential equa-tion. However, since each mode evolves independently,this functional differential equation can be reduced toan infinite number of differential equations for each Φk.Concretely, we have

i∂ΦR,I

k

∂η= HR,I

k ΦR,Ik , (36)

where the Hamiltonian densities HR,Ik , are related to the

Hamiltonian as H(2) =∫R3+ d

3k(HRk + HI

k), with the fol-lowing definitions

HR,Ik =

(pR,Ik )2

2+

(yR,Ik )2

2

(k2 − a′′

a+ a2V,φφ

)− 4aφ′0Ψ

′R,Ik yR,I

k + 2a3V,φΨR,Ik yR,I

k . (37)

The standard assumption is that, at an early conformaltime τ → −∞, the modes are in their adiabatic groundstate, which is a Gaussian centered at zero with certainspread. In addition, this ground state is commonly re-ferred to as the Bunch-Davies (BD) vacuum. Thus, theconformal time η is in the range [τ, 0−).

Since the initial quantum state is Gaussian and theHamiltonian (as well as the collapse Hamiltonian, see Eq.

(44)) is quadratic in yR,Ik and pR,I

k , the form of the statevector in the field basis at any time is

ΦR,I(η, yR,Ik ) = exp[−Ak(η)(yR,Ik )2 +Bk(η)yR,Ik +Ck(η)].(38)

Therefore, the wave functional evolves according toSchrodinger equation, with initial conditions given by

Ak(τ) =k

2, Bk(τ) = Ck(τ) = 0. (39)

Those initial conditions correspond to the BD vacuum,which is perfectly homogeneous and isotropic in the senseof a vacuum state in quantum field theory.

After the identification of the Hamiltonian that resultsin Schrodinger’s equation, which from now on we refer toas the “free Hamiltonian,” we now incorporate the CSLcollapse mechanism.

The main physical idea of the CSL model is that anobjective reduction of the wave function occurs all thetime for all kind of particles. The reduction or collapseis spontaneous and random. The collapse occurs whetherthe particles are isolated or interacting and whether theparticles constitute a macroscopic, mesoscopic or micro-scopic system.

In order to apply the CSL model to the inflationarysetting, we will consider a particular version of the CSLmodel in which the nonlinear aspects of the CSL modelare shifted to the probability law. Specifically, the evolu-tion equation is linear, which is similar to Schrodinger’sequation; however, the law of probability for the realiza-tion of a specific random function, becomes dependentof the state that results from such evolution. In otherwords, the theory can be characterized in terms of twoequations:

The first is a modified Schrodinger equation, whosesolution is

|ψ, t〉 = T e−∫ t0dt′[iH+ 1

4λ [w(t′)−2λA]2]|ψ, 0〉. (40)

T is the time-ordering operator. The modifiedSchrodinger’s equation given by (40), induces the col-lapse of the wave function towards one of the possibleeigenstates of A, which is called the collapse generatingoperator. The parameter λ is the universal CSL param-eter that sets the strength of the collapse. In particular,λ serves to characterize the rate at which the wave func-tion increases its “localizations” in the eigen-basis of thecollapse operator. In laboratory situations, the collapseoperator is usually chosen to be the position operatorand λ is assumed proportional to the mass of the particle[24, 59]; in this model, the collapse rate, which has di-mensions of [Time]−1, is given by λa2 (here a is a secondparameter that sets the correlation length above whichspatial superpositions are reduced).

The function w(t) describes a stochastic process (i.e. isa random classical function of time) of white noise type.

8

In other words, CSL regards the state vector undergoingsome kind of Brownian motion. The probability for w(t)is given by the second equation, the Probability Rule

P [w(t)]Dw(t) ≡ 〈ψ, t|ψ, t〉t∏

ti=0

dw(ti)√2πλ/dt

. (41)

The norm of 〈ψ, t|ψ, t〉 evolves dynamically, i.e. does notequal 1. Hence, Eq. (41) implies that state vectors withlargest norm are most probable. Furthermore, the to-tal probability satisfies

∫PDw(t) = 1. The stochastic

term also prevents a wave-packet from spreading indefi-nitely, and causes the width of the packet to reach a finiteasymptotic value [59].

In the case of multiple identical particles in three di-mensions, the CSL theory would contain one stochasticfunction for each independent degree of freedom wi(t),but only one parameter λ. In the case of several speciesof particles, the theory would naturally involve a param-eter λi for each particle species. In fact, there is strongphenomenological preference for a λi that depends on theparticle’s mass mi [22, 23].

Returning to the inflationary context, in Ref. [5] itis shown that with the appropriate selection of the fieldcollapse operators and using the corresponding CSL evo-lution law one obtains collapse in the relevant operatorscorresponding to the Fourier components of the field andthe momentum conjugate of the field.5 We further as-sume that the reduction mechanism acts on each modeof the field independently. Also, it is suitable to chooseyk as the collapse operator because our main equation(16) suggests that 〈yk〉 can act as a source of the New-tonian potential. Therefore, the evolution of the statevector characterizing the inflaton as given by the CSLtheory is assumed to be:

|ΦR,Ik , η〉 = T exp

−∫ η

τ

dη′[iHR,I

k

+1

4λk(W(k, η′)− 2λky

R,Ik )2

]|ΦR,I

k , τ〉,

(42)

T is the time-ordering operator, and recall that τ denotesthe conformal time at the beginning of inflation.

The parameter λk is a phenomenological generalizationof the CSL parameter, and now it depends on the modek. From the point of view of pure dimensional analy-sis, λk must have dimensions of [Length]−2. Hence, thesimplest parameterization we can assume is λk = λ0k,which is also the same parameterization considered e.g.

5 We also acknowledge at this point that there is no complete ver-sion of the CSL theory that is applicable universally, rangingfrom the laboratory setting to the cosmological one. However,we adopt the point of view that proposing educated guesses, incombination with phenomenological models applicable to partic-ular situations, allow us to advance in our program.

in [5, 26]. At first glance, one can postulate that λ0should coincide with the empirical bounds obtained fromlaboratory experiments for the collapse rate. For exam-ple, the value λGRW = 10−16s−1 was originally proposedby Ghirardi, Rimini and Weber [22] and later adopted byPearle [23] for his CSL theory, as providing sound behav-ior when applied to laboratory contexts. However, as ar-gued in [60] there is no particular reason why one shouldexpect that the collapse rate associated to the parameterλ, utilized in applications of the CSL model at presentday laboratory situations (and whose values are probablytied to underlying atomic structure that did not exist ininflationary times), should necessarily, or even naturally,be the ones utilized in modeling the inflationary regime,i.e. λ0. In Sec. IV, we will say more about λk and theparticular value(s) of λ0 considered in the present work.

Additionally, we postulate that the white noise w(t),which appears in Eqs. (40) (41), is now a stochastic fieldthat depends on k and the conformal time. That is, sincewe are applying the CSL collapse dynamics to each modeof the field, it is natural to introduce a stochastic functionfor each independent degree of freedom. Henceforth, thestochastic field W(k, η) might be regarded as a Fouriertransform on a stochastic spacetime field W(~x, η). Herewe would like to mention that the generalization fromw(t) to w(x, t) is in fact considered in standard treat-ments of non-relativistic CSL models [61]. For instance,the generalization of the CSL model of a single particlein one dimension to a single particle in three dimensions,implies to consider a joint basis of operators Ai, whichcommute [Ai, Aj ] = 0, instead of single collapse oper-

ator A. This change requires one white noise functionwi(t) for each Ai. A further generalization is to consider

a “continuum” collapse operator A(x) (e.g. the massdensity operator smeared over a spherical volume), thisrequires a random noise field w(x, t) instead of a set ofrandom functions [23, 24, 59, 62].

Continuing with the calculations, we can take the timederivative of (42) (see [23]), obtaining

∂

∂η|ΦR,I

k , η〉 = −iHR,Ik + HR,I

k CSL|ΦR,Ik , η〉, (43)

with

HR,Ik CSL ≡ −

W(k, η)2

4λk+W(k, η)yR,I

k − (yR,Ik )2λk. (44)

Next, taking into account that our main goal is to ob-tain the primordial power spectrum, see Eq. (23), we

turn our attention to compute the quantity 〈yk〉〈yq〉∗.The expectation values of course will be evaluated at theevolved state provided by (42).

In terms of the real and imaginary parts, we have

〈yk〉〈yq〉∗ =(〈yRk 〉2 + 〈yIk〉2

)δ(k− q). (45)

Note that we have assumed that the CSL model does notinduce modes correlations. Also from (45), it is clear that

9

we are interested in computing the quantities 〈yR,Ik 〉2. In

fact, the calculation of the real and imaginary part areexactly the same, so we will only focus on one of them.Additionally, we simplify the notation by omitting theindexes R,I from now on.

Using the Gaussian wave function (38), and the CSLevolution equations, it can be shown [5] that

〈yk〉2 = 〈y2k〉 −1

4Re[Ak(η)]. (46)

Therefore, in order to obtain a prediction for the powerspectrum, we need to calculate the two terms on the righthand side of (46). Explicit computation of Eq. (46) im-plies solving the corresponding CSL equations. At thispoint we would like to mention that the actual calcula-tions are long and cumbersome, but we have include themin Appendix A for the interest reader. The final resultcorresponding to the quantity 〈yk〉2 is given in (A32).

Another final remark regarding Eq. (46) is that theright hand side is technically easier to handle than todirectly solve the main CSL evolution equation for thestate vector (43). The latter procedure is difficult be-

cause the Hamiltonian Hk given by (37), contains termsthat involve Ψ′k and Ψk. These terms in turn dependon the state vector through the expectation value 〈yk〉as shown in the main equation (16). On the other hand,given that we are only interested in computing the powerspectrum, the computation of the right side of (46) allowus to bypass such a direct calculation. The details canbe found in Appendix A, but here we can mention, thatfor instance, the evolution equation for Ak(η) decouplesfrom the other quantities Bk(η) and Ck(η) which do in-volve a more elaborated method to solve. Moreover, the

evolution equation for 〈y2k〉 does not involve the linearterm 〈yk〉 only [see Eqs. (A4) and Eq. (A8)].

We acknowledge that this is a pragmatic way to pro-ceed and that formally one would have to perform thefull quantization using Hamiltonian (43), which meansthe formal characterization of the collapse process withinthe semiclassical treatment. Some advances in this di-rection have been made [54, 55]. The main idea devel-oped in those works is that any sudden change in thequantum state (i.e. a collapse) will result, generically,in a sudden modification in the expectation value of theenergy-momentum tensor, and thus to a different space-time metric. Nonetheless, such modification would ingeneral, require also a change in the quantum field the-ory construction, and consequently a new Hilbert spaceto which the state can belong. In this way, one wouldhave a QFT and a space-time metric corresponding to

the initial state, and a different QFT/metric for the post-collapse state. Then the two different space-times mustbe glued together in a consistent way. While this is thecorrect framework to adopt, it is beyond the scope of thepresent paper. Instead we will focus on computing theright hand side of (46) using two reasonble assumptions:(i) the collapse generating operator is yk; therefore, weexpect that the modified Schrodinger equation (42) drivethe initial Gaussian state to a final state that is verysimilar to an eigenvector of yk. That is to say, the finalwave function can be approximated by a Dirac functionδ(yk − Yk), where Yk ≡ 〈yk〉 evaluated in the final state.(ii) we will assume that the localization process is fastenough compared to the total duration of inflation, inconformal time this is λ0|τ | > 1; so the time evolutionof Ψ(x, η) is deterministic [in fact given by Eq. (11)]. InAppendix A we present the computational details of Eq.(46) using those two assumptions [in particular to obtainEqs. (A9) and (A24)].

IV. ANALYSIS OF THE PRIMORDIAL ANDANGULAR POWER SPECTRA

Given the solutions of the CSL equations, we can ob-tain the power spectrum. Clearly, this allow us to com-pare the predictions between the standard model and ourproposal.

The path is straightforward: we substitute 〈yk〉2[whose explicit form is shown in (A32)] into (23), thisyields the power spectrum. The detailed calculations canbe found in Appendix B, and the resulting expression ofPs(k) is given in (B26). Such an expression representsthe primordial power spectrum at second order in theHFF, and can also be used to obtain ns and αs at thirdand fourth order in the HFF respectively [see AppendixB, Eqs. (B29) and (B30)].

Using expressions for Ps, ns and αs at second order inthe HFF allow us to parameterize the primordial powerspectrum in terms of the scalar spectral index and itsrunning, this is

Ps(k) = As

(k

k

)ns−1+αs2 ln k

k

C(k) (47)

with

As =H2

π2M2P ε1

, (48)

and the function C(k) expressed in terms of ns (scalarspectral index) and αs (running of the spectral index) is

10

C(k) = 1 +λk|kτ |k2

+λk2k2

cos(2|kτ |)

− exp

[− 4 + ns + αs ln

2k

k

]ln ζk −

αs2

(ln2 ζk − θ2k

)×[

cos

[−4 + ns + αs ln

2k

k

]θk − αs

(−∆N +

2

3+D + ln

k

2k

)θk ln ζk

]−1, (49)

where k = 0.05 Mpc−1 is a pivot scale, D ≡ 1/3 −ln 3 and ∆N is the number of e-folds from the horizoncrossing of the pivot scale to the end of inflation, typically∆N ∼ 60. The quantities θk and ζk are defined as:

ζk ≡(

1 +4λ2kk4

)1/4

, θk ≡ −1

2arctan

(2λkk2

). (50)

We note that if λk = 0, which means ζk = 1 and θk =0, one can check that C(k) = 0, hence P(k) = 0. Thisis consistent with our model in which the collapse of thewave function, given by the CSL mechanism, is the sourceof the metric perturbations. Therefore, in our approachif there is no collapse, and the vacuum state remainsunchanged there are no primordial perturbations, thus,P(k) = 0 because Ψk = 0 at all scales.

On the other hand, if λk/k2 1 then ζk 1 and

θk ' −π/4. This means that the evolution of the wavefunction is certainly being affected by the extra termsadded by the CSL evolution equation. We recall that inthe previous section we proposed the parameterization,

λk = λ0k, (51)

where λ0 can be related to the universal CSL rate pa-rameter, which has units of [Time]−1. Therefore, λ−10

provides us with a localization time scale for the wavefunction associated to each mode of the field. For thepurpose of our analysis, we set the numerical value ofthe CSL parameter as λ0 = 10−14 s−1, or equivalently1.029 Mpc−1 in the units chosen for the present work.This value is two orders of magnitude greater than thehistorical value λGRW = 10−16 s−1 suggested by GRW[22], and also consistent with empirical constraints ob-tained from different experimental data such as: spon-taneous X-ray emission [63], matter-wave interferometry[64], gravitational wave detectors [65] and neutron stars[66]. Also, according to assumption (ii) mentioned at theend of Sec. III, we have chosen the value |τ | = 7803894Mpc, so λ0|τ | = 8 × 106 > 1. Our proposed param-eterization in Eq. (51) is the most simple one for a kdependence in the λk parameter, although it is not theonly possibility. As we shall see in the following analy-sis, the CSLIM induces oscillatory features in Ps(k) thatremain at some scales, showing that the effect of the col-lapse cannot be “turned off”. This can be explained bynoting first that, since the modes k are infinite, there willbe some modes k such that the condition λk/k

2 1 fails.However, for the chosen value λ0 = 1.029 Mpc−1 and at

least for the range of modes of observational interest 10−6

Mpc−1 ≤ k ≤ 10−1 Mpc−1, i.e. the ones that contributethe most to the CMB angular spectrum, the conditionλk/k

2 = λ0/k 1 is fulfilled. It is for these range of kthat we will analyze the features in Ps(k) induced by theCSLIM.

In order to analyze those novel features, we plot ex-pression (47). The resulting plot is shown in Fig. 1,together with the prediction corresponding to the stan-dard model, the latter being essentially Eq. (47) withC(k) = 1. The values of the inflationary parameters wehave used are: ns = 0.9641 and αs = −0.0045. Oscilla-tions appear at low scales while no difference at all canbe found for k > 0.0001 Mpc−1. In fact the oscillations,induced by the CSLIM around the standard spectrum,show a decrease in amplitude at the higher end of scales.

2.0×10−9

2.2×10−9

2.4×10−9

2.6×10−9

10−6

10−5

10−4

10−3

10−2

10−1 1

Ps(k

)

k

CSLIMcanonical model

Figure 1. Comparison between the CSLIM power spectrumand the canonical model. The wave number k is given inMpc−1. A good agreement is shown at high scales, whilefor k < 0.0001 Mpc−1 oscillatory features introduced by theCSLIM become evident.

The next step in our analysis is to investigate whetherthe oscillations shown in the primordial power spectrumhave any incidence in the observational predictions. How-ever, we want to stress that, in this paper, we will onlyperform a preliminary analysis of the CMB angular powerspectrum (also known as the Cl in the literature [67])predicted by the CSLIM taking into account our sec-ond order power spectrum. A complete data analysis,including statistical analysis, is left for future work. Fur-thermore, we will limit ourselves to the analysis of the

11

temperature auto-correlation spectrum; however, from aprevious analysis of similar models [26] we might expectthat the E-mode polarization and Temperature-E-modecross correlation will also be modified as a consequenceof the collapse hypothesis.

In order to perform our analysis, we have modifiedthe Code for Anisotropies in the Microwave Background(CAMB) [68] as to include the CSLIM predictions, whichonly affect the inflationary part of the ΛCDM canonicalmodel. The rest of the cosmological parameters remainunchanged. Let us define the cosmological parameters ofthe canonical model: baryon density in units of the criti-cal density ΩBh

2 = 0.02237, dark matter density in unitsof the critical density ΩCDMh

2 = 0.12, Hubble constantin units of Mpc−1 km s−1 H0 = 67.36. We also include inthat set the aforementioned values of ns, αs and k-pivot;all represent the best-fit values presented by the Planckcollaboration [67]. The value of As is 2.1×10−9 for bothCSLIM and canonical model.

Figure 2 shows the temperature auto-correlation (TT)spectrum for both CSL and canonical models, showingno difference between them. As can be seen there, oscil-lating features at low k in our predicted power spectrumdo not translate into any peculiar features in the theoret-ical predictions of the Cl coefficients characterizing theangular temperature anisotropies. In this way, parame-terization (51) represents a good choice to set a basis forcomparison with the canonical model, and in a sense alsoserves as a consistency check.

0

1000

2000

3000

4000

5000

6000

1 10 100 1000

l(l+

1)C

l/(2

π)

l

canonical modelCSLIM

Figure 2. Temperature auto-correlation (TT) spectrum com-parison between the canonical model (boxes) and the CSLIM(blue solid line), the latter using the parameterization λk =λ0k. No difference is shown among them. Oscillations at lowvalues of k in the primordial power spectrum, as shown inFig. 1, are wiped off in the TT spectrum.

At this point of the analysis we have learned thatλk = λ0k yields an indistinguishable prediction from thecanonical model. However, there is no reason to expectan exact k dependence of λk. As a consequence, we pro-ceed to explore possible effects of the CSLIM that can be

reflected in the observational data by introducing a newparameter B through the parameterization of λk. Therole of B will be to imprint a slight departure from thecanonical model shape. The new proposal to parameter-ize λk is

λk = λ0(k +B), (52)

where B has units of Mpc−1 and conforms a new param-eter of the model that needs to be estimated with recentobservational data, this will be left for future work. Inthe rest of the present section, we will be interested inanalyzing the consequences of varying B on the predictedspectrum.

The effect of considering different values of B on thepower spectrum is shown in Fig. 3, where the same plotof Fig. 1 has been included as the case B = 0, and servesas a reference. For negative B (green line) the CSLIMpower spectrum seems to approach to the canonical onefrom below, showing significant differences for k < 10−4

Mpc−1. Meanwhile, for positive B the CSLIM powerspectrum approaches from above (black and red lines).The differences in the predicted spectrum between theCSLIM and the canonical seem to dissolve progressivelyas B approaches zero, remaining only a small differencesat low k due to the oscillations. Also, it is worthwhileto mention that oscillations present in the B = 0 casecannot be significantly appreciated in the rest of cases.Figure 3 suggests that observational predictions in theCSLIM may be distinguished from the ones of the canon-ical model. In the next final part, we analyze whetherthese departures (from the canonical model) have observ-able consequences on the CMB fluctuation spectrum.

2.0×10−9

2.2×10−9

2.4×10−9

2.6×10−9

10−6

10−5

10−4

10−3

10−2

10−1 1

Ps(k

)

k

canonical model

B=0

B=10−3

B=10−4

B=−10−6

Figure 3. Here we appreciate small departures from thecanonical power spectrum. The departures are parameterizedby B. The set of B values considered show effects properlyattributed to the CSLIM, and become explicitly manifest inthe primordial power spectrum. Power suppression is seen atlow scales for negative values of B, whereas positive valuesimply an upper departure from the canonical model. In thisfigure, B and k are given in Mpc−1.

12

Figure 4 shows our prediction for the CMB tempera-ture fluctuation spectrum and the canonical one, wherewe used the same values for the cosmological parame-ters as before. From Fig. 4, it can be inferred an esti-mated upper limit for the B parameter, i.e. for B = 10−3

Mpc−1 the first peak is shifted upwards which is totallyincompatible with the latest high precision observationalmeasurements. The negative B value tested does not in-duce any significant difference in parameter estimationwhen compared with the canonical model. In the caseof B = 10−4 Mpc−1, a small departure from the canon-ical prediction is seen at low multipoles. Whether thischange is favored by the data or simply lost in the cosmicvariance uncertainty will be addressed in future research.Nonetheless, from this analysis we can infer that in orderfor our predicted power spectrum to be consistent withthe best fit temperature auto-correlation spectrum, andat the same time, to manifest departures from the canon-ical model, the B parameter must be then constrainedbetween B > 0 and B < 10−3 Mpc−1.

0

1000

2000

3000

4000

5000

6000

1 10 100 1000

l(l+

1)C

l/(2

π)

l

canonical model

B=0

B=10-3

B=10-4

B=-10-6

Figure 4. The temperature auto-correlation (TT) spectrumincluding the B parameter, the values of the cosmologicalparameters considered are the same as in Fig. 2. The negativeB value does not exhibit any difference at all with respect tothe canonical model (boxes). On the other hand, for positiveB values there is a progressive departure from the canonicalmodel at low l (big angular scales). The caseB = 10−3 Mpc−1

can be discarded in advance as it modifies substantially theposition of the first peak, which is constrained at a high degreeof accuracy by current data.

A. Consequences of varying λ0

In this final part of the present section, we would like tomake some remarks about how our previous results wouldbe affected by considering different values of λ0. As wehave argued, λ−10 may be used to set a localization timescale for the wave function associated to each mode ofthe field, so varying λ0 means to change the localizationtime scale.

The criteria to select appropriate values of λ0 is basedon the condition λk/k

2 1. If that condition issatisfied then the collapse occurs successfully; particu-larly, one would require that said condition is met forall modes k within the range of interest 10−6 Mpc−1

≤ k ≤ 10−1 Mpc−1. Moreover, we consider the parame-terization λk = λ0(k + B) constrained within the range0 ≤ B < 10−3 Mpc−1. Therefore, we note first that thecondition λk/k

2 = λ0(1 +B/k)/k 1 is fulfilled for theprevious chosen value λ0 = 1.029 Mpc−1 (with B and kin the aforementioned ranges). Second, it is clear thatin order to satisfy the condition λk/k

2 1 for differentvalues of the parameter, we must consider λ0 ≥ 1.029Mpc−1.

We have reproduced Figs. 1, 2, 3 and 4 for the val-ues λ0 = 10.29 Mpc−1 and λ0 = 102.9 Mpc−1 obtain-ing exactly the same plots as the ones corresponding tothe original value λ0 = 1.029 Mpc−1. In particular, theshape of the spectra is exactly equal. However, to achievea similar amplitude, we had to adjust the combinationV |τ |/M4

P ε1, we remind the reader that τ is the confor-mal time at which inflation begins.

In order to attain a better understanding of this result,we focus on our prediction for Ps(k), Eq. (47). To makethings simple and without loss of generality, we assumens = 1, αs = 0 and we use Friedmann’s equation H2 'V/3M2

P . Consequently with these assumptions Eq. (47)is approximately

Ps(k) ' V

3π2M4P ε1

(1 +

λk|kτ |k2

+λk2k2

cos(2|kτ |)

− 1

ζ3k cos 3θk

). (53)

From definitions (50), λk/k2 1 implies that ζk 1

and θk ' −π/4. Let us consider the parameterizationλk ' λ0k; hence, if condition λk/k

2 1 is met, then Eq.(53) can be approximated by6

Ps(k) ' V

3π2M4P ε1

λ0|τ |. (54)

Note that assumption (ii) mentioned at the end of Sec.III and Eq. (54) are consistent, this is λ0|τ | > 1. Withresult (54) at hand, we can now conclude that increasingλ0 requires to decrease the combination V |τ |/M4

P ε1, suchthat Ps(k) would be consistent with the observed ampli-tude Ps(k) ' 10−9. For example, if one increases thevalue of λ0 but ε1, τ remain fixed, then the characteristicenergy scale of inflation V 1/4 must decrease. Also, notethat using Eq. (53) and the parameterization λk ' λ0k

6 In approximation (54), we also used the fact that k|τ | > 1. Re-calling that τ is the conformal time at which inflation begins,k|τ | > 1 is essentially satisfied by all the modes because suchcondition means that said modes begun in the Bunch-Daviesvacuum.

13

lead to a scale invariant power spectrum (independentof k), Eq. (54). This was expected since we consideredns = 1 and αs = 0, but the main point is that varying λ0does not affect the shape of the spectrum. On the con-trary, varying the parameterization of λk would certainlyalter the scale dependence of the spectrum, and Figs. 1,2, 3 and 4 would also change substantially.

V. SUMMARY AND CONCLUSIONS

In this work, we have calculated the primordial powerspectrum for a single scalar field during slow roll in-flation. The calculation considered the application ofthe Continuous Spontaneous Localization (CSL) objec-tive reduction model to the inflaton wave function, withinthe semiclassical gravity setting. The novel aspect in thispaper was to consider the second order approximation inthe Hubble flow functions (HFF), and solve the corre-sponding CSL equations using the uniform approxima-tion method in slow roll inflation [36, 43].

The implementation of the CSL model to slow roll in-flation or CSL inflationary model (CSLIM) for short, in-duced a modification of the standard scalar power spec-

trum (PS) of the form Ps(k) = Askns−1+αs

2 ln kk C(k).

One of the main features uncovered here is that the func-tion C(k) depends on the inflationary parameters ns, αsas well as the collapse parameter λk, see Eq. (49).

We have chosen the most simple parameterization forthe collapse parameter, this is λk = λ0(k + B), whereλ0 is the fundamental CSL parameter, representing thecollapse rate, and B is a new parameter. We have setλ0 = 10−14 s−1 or 1.029 Mpc−1 (which is two ordersof magnitude greater than the historical value suggestedfor the collapse rate [22]), and varied B from B = −10−6

Mpc−1 to B = +10−3 Mpc−1; these values gave rise tosignificative departures from the standard PS, see Fig.3, mostly at the lower range of k. Next, we have shownthe effects of the CSLIM on the CMB temperature fluc-tuation spectrum, Fig. 4. For this preliminary analysis,the proposed parameterization of λk seems to be in goodagreement with the present data of the CMB fluctuationspectrum. In particular, within the range B = −10−6

Mpc−1 and B = 0 there are no differences between theprediction of the CSLIM and the standard inflationarymodel, in spite of the evident variations in the PS. How-ever, between B > 0 and B < 10−3 Mpc−1 there areimportant departures from the standard model predic-tion in the temperature fluctuation spectrum but at thesame time could be consistent with the best fit tempera-ture auto-correlation spectrum. We have also shown thatvalues B ≥ 10−3 Mpc−1 could be discarded without per-forming any statistical analysis. Finally, we have arguedthat increasing λ0 will not have an effect on the shape ofthe spectra but it can have theoretical consequences inthe parameters characterizing the spectrum’s amplitude,e.g. the characteristic energy scale of inflation.

Our result Ps(k) = Askns−1+αs

2 ln kk C(k), with C(k)

depending explicitly on λk, αs and ns, allow us to iden-tify exactly the dependence on k attributed to: the CSLmodel, the spectral index and the running of the spectralindex. We think this is an important result because ofthe following. Our predicted PS allows departures fromthe traditional inflationary approach that can be testedexperimentally. As we have argued in the Introduction,if future experiments detect a significant value of therunning of the spectral index, i.e. of order |αs| ' 10−3

and the tensor-to-scalar ratio r remains undetected, thenthe hierarchy of the HFF would be broken and the stan-dard slow roll inflationary model would be in some sensejeopardized. On the other hand, the CSLIM genericallypredicts a strong suppression of tensor modes, that isr ' ε2110−12 [33, 34]. And, since the function C(k) in-troduces an extra k dependence on the PS, the situa-tion described previously, in principle, could not yield aninconsistency between the CSLIM and hierarchy of theHFF. Specifically, what in the standard approach mightbe identified as a running of the spectral index, whichis essentially a particular dependence on k of the PS, inthe CSLIM the same effect could be attributed to C(k)through the parameterization of λk, and in particular tothe B parameter. In other words, in the CSLIM, the hi-erarchy |ε1| > |ε2| > |ε3| could be satisfied and still beconsistent with observations, namely a non-detection ofprimordial gravity waves and a particular shape of thePS characterized by a “running of the spectral index” inthe standard approach.

Evidently, to test if the above conjecture is true, werequire to perform a complete statistical analysis usingthe most recent (and future) observational data from theCMB. In particular, we would be able to constrain thevalue of B as well as ns and αs within our model. Nev-ertheless, our main conclusion is that the CSLIM possessobservational consequences, different from the standardinflationary paradigm. In fact, some particular obser-vations that would cause some issues in the traditionalmodel, could be potentially resolved within our approach.A final important lesson to be drawn from this analysisis that it displays how, at least in applications to cosmol-ogy, considerations regarding the quantum measurementproblem can lead to striking alterations concerning ob-servational issues. This contributes to oppose a posturethat claims such questions as of mere philosophical in-terest and dismisses their relevance regarding physicalpredictions.

ACKNOWLEDGMENTS

G.L. and M. P. P are supported by CONICET (Ar-gentina) and the National Agency for the Promotion ofScience and Technology (ANPCYT) of Argentina grantPICT-2016-0081. M. P. P is also supported by grantsG140 and G157 from Universidad Nacional de La Plata(UNLP).

14

Appendix A: Solving the CSL equations

We begin by writing some useful expressions involvingV , ∂φV and ∂2φφV in terms of the Hubble flow functions

(HFF). Therefore, one has the following quantities [37,39]

M2P

2

(∂φV

V

)2

= ε1

(1 +

ε22(3− ε1)

)2

, (A1)

M2P

∂2φφV

V=

6ε1 − 3ε2/2− 2ε21 − ε22/4 + 5ε1ε2/2− ε2ε3/23− ε1

.

(A2)There are no approximations in the previous equations.

Next, we focus on the first term of the right hand side

of (46), i.e 〈y2k〉. We define the quantities Q ≡ 〈y2k〉, R ≡〈p2k〉 and S ≡ 〈pkyk + ykpk〉. The evolution equations ofQ, R, and S, can be obtained from the CSL evolutionequation (43). In fact, for any operator any operator Oone has

d

dη〈Ok〉 = −i〈[Ok, Hk]〉 − λk

2〈[yk, [yk, Ok]]〉, (A3)

which is the evolution equation of the ensemble averageof the expectation value of any operator O. Thus, theevolution equations of Q, R and S obtained from (A3)are:

Q′ = S, (A4a)

R′ = −m1(η)S − 2m2(η)〈pk〉+ λk, (A4b)

S′ = 2R− 2Qm1(η)− 2m2(η)〈yk〉, (A4c)

where

m1(η) ≡ k2 − a′′

a+ a2V,φφ, (A5)

and

m2(η) ≡ −4aφ′0Ψ′k + 2a3V,φΨk. (A6)

At this point it is important to point out that in ourapproach the metric perturbation, characterized by Ψ, issourced by 〈yk〉. In particular, that relation is given byour equation (16), which can be rewritten as

Ψk =φ′0

2aM2PH

〈yk〉(1 + ε2)

. (A7)

Therefore, the term m2(η) can be considered as a sort of“backreaction” effect of the collapse, since m2 containsexplicitly the terms Ψ′, Ψ. Moreover, by using approx-imation (13), i.e. Ψ′k ' Hε2Ψk, together with (A7), wereexpress m2 as

m2(η) ' 〈yk〉H2(−6ε1 + 2ε21 + ε1ε2). (A8)

From Eq. (A8), we see that Ψ′ and Ψ induce terms oforder 2 in the HFF. Using assumptions (i) and (ii) men-tioned at the end of Sec. III and Eq. (A8), we rewritethe evolution equations (A4) as

Q′ = S, (A9a)

R′ = −[k2 −M(η)]S + λk, (A9b)

S′ = 2R− 2Q[k2 −M(η)], (A9c)

where

M(η) ≡ H2

[2− ε1 +

3

2ε2 +

ε222− 7

2ε1ε2 +

ε2ε32

].

(A10)The solutions to Eqs. (A9), are

Q = C1y21 + C2y

22 + C3y1y2 +Qp, (A11a)

R = C1y′21 + C2y

′22 + C3y

′1y′2 +Rp, (A11b)

S = C12y1y′1 + C2y2y

′2 + C3(y′1y2 + y1y

′2) + Sp, (A11c)

where y1 and y2 are two linearly independent solutionsof

y′′1,2 +[k2 −M(η)

]y1,2 = 0. (A12)

The functions Qp, Rp and Sp are particular solutionsof the system (A9). The constants Ci, with i = 1, 2, 3are determined by imposing the initial conditions corre-sponding to the Bunch-Davies vacuum state. The func-tion Q(η) is the quantity that we are interested. Weproceed to solve (A12).

At first order in the HFF, equation (A12) is solvedexactly in terms of Bessel functions. However, at secondorder we require new techniques. Here we choose to usethe uniform approximation technique [43]. The idea is torewrite the term M(η) as

M(η) =ν(η)2 − 1/4

η2, (A13)

where the former equation should be understood as thedefinition of the function ν(η). Then two new functionsare introduced:

g(η) ≡ ν2

η2− k2, f(η) ≡ |η − η∗|

η − η∗

∣∣∣∣32∫ η

η∗

dη√g(η)

∣∣∣∣2/3.(A14)

The time η∗ is defined by the condition g(η∗) = 0 and iscalled the turning point, i.e. η∗ ≡ −ν(η∗)/k. Accordingto the uniform approximation, the two linearly indepen-dent solutions of (A12) are

y1(η) =

(f

g

)1/4

Ai(f), y2(η) =

(f

g

)1/4

Bi(f),

(A15)

15

where Ai and Bi denote the Airy functions of first andsecond kind respectively. One advantage of the Airy func-tions is that their asymptotic behavior is quite familiar.

At the onset of inflation, i.e. when η = τ → −∞, wehave

g1/4 =√keiπ/4, f = −

∣∣∣∣32kτ∣∣∣∣2/3. (A16)

In this regime, the Airy functions oscillate. Specifically,if x→ +∞, then

Ai(−x) 'sin(23x

3/2 + π4

)√πx1/4

, Bi(−x) 'cos(23x

3/2 + π4

)√πx1/4

.

(A17)Thus, at the beginning of inflation we approximate thesolutions

y1(τ) ' 1√kπ

sin(|kτ |+ π

4

), (A18a)

y2(τ) ' 1√kπ

cos(|kτ |+ π

4

). (A18b)

On the other hand, the Airy functions exhibit expo-nential behavior for large and positive arguments. Thatis, for x→ +∞, the Airy functions are approximated by

Ai(x) ' 1

2√πx−1/4 exp

(−2

3x3/2

), (A19a)

Bi(x) ' 1√πx−1/4 exp

(2

3x3/2

). (A19b)

Therefore, in the super-Hubble regime, that is, when|kη| → 0, the approximated solutions are

y1(η) ' 1

2√πg−1/4 exp

(−2

3f3/2

), (A20a)

y2(η) ' 1√πg−1/4 exp

(2

3f3/2

). (A20b)

Note that in this regime f → +∞.Taking into account that the power spectrum is evalu-

ated in the super-Hubble regime, and by considering theexponential solutions (A18), together with g1/2 ' −ν/η,we conclude that the term C2y

22 dominates over the rest

of the terms in (A11a). The quantity of interest is then

Q(η) ≡ 〈y2k〉 'C2

π

(−η)

νe2F , (A21)

with

F ≡ 2

3f3/2. (A22)

The constants Ci are found by imposing the initial condi-tions Q(τ) = 1/2k, R(τ) = k/2, S(τ) = 0 and using theapproximated solutions (A18) in the system of equations(A11). One also has to take into account the solutionsQp, Rp and Sp. In particular, in the sub-Hubble regime,Qp ' λkτ/2k2. The constant C2 of (A21) obtained is

C2 =π

2+λkπ|kτ |

2k2+λkπ

4k2cos(2|kτ |). (A23)

This completes the calculation of Q = 〈y2k〉. Now let usfocus on the second term on the right hand side of (46),i.e. the term [4Re(Ak)]−1.

We apply the CSL evolution operator as characterizedby Eq. (43) to the wave function (38), and regroup termsof order y2, y1 and y0; the evolution equations corre-sponding to these terms are thus decoupled. Fortunately,the evolution equation corresponding to y2 only containsAk(η), which is the function we are interested in. Theevolution equation is then

A′k =i

2

[k2 −M(η)

]+ λk − 2iA2

k, (A24)

where once again we have assumed that the Newtonianpotential is sourced by the expectation value 〈yk〉, andthe assumptions (i) and (ii) mentioned at the end ofSec. III. By performing the change of variable Ak ≡u′k/(2iuk), the evolution equation of Ak is equivalent to

u′′k +[q2 −M(η)

]uk = 0, (A25)

where we have introduced

q2 ≡ k2(

1− 2iλkk2

). (A26)

Equation (A25) is of the same form as (A12). Thegeneral solution is thus

u = c1

(f

g

)1/4

Ai(f) + c2

(f

g

)1/4

Bi(f), (A27)

the definitions of g and f given by (A14) hold as before,with the replacement k2 → q2 in (A14). Henceforth, gand f are complex functions in this case.

The constants c1,2 are found by imposing the ini-tial conditions associated to the Bunch-Davies vacuum:Ak(τ) = k/2. Therefore, by using the asymptotic behav-ior of the Airy functions when η = τ → −∞ given by(A17), we find that

c1 =

√π

2e−iπ/4, c2 = ic1. (A28)

It is straightforward to check that,

Re(Ak) =Wk

|uk|24i, (A29)

where, Wk is the Wronskian of (A25), i.e. Wk = u′ku∗k −

u∗′

k uk. We now proceed to evaluate Re(Ak) in the regime

16

of observational interest, that is, when −kη → 0. Asbefore, in this regime, the Airy functions can be approx-imated by (A19). Consequently, the solution uk is

uk(η) ' e−iπ4√2g1/4

[1

2exp

(−2

3f3/2

)+ i exp

(2

3f3/2

)].

(A30)After a long series of calculations using (A29), (A30) andg1/2 ' −ν/η, we find

1

4Re(Ak)' −η

2ν

e2Re(F)

cos[2Im(F)], (A31)

where F ≡ 23f

3/2. In principle F 6= F , although theirdefinition in terms of f is the same (see (A22)), the quan-tity F is real and F is complex.

Putting together Q given in (A21), and [4Re(Ak)]−1

obtained in (A31), we can finally obtain 〈yk〉2 = Q −[4Re(Ak)]−1, which is

〈yk〉2 =−η2ν

[(1 +

λk|kτ |k2

+λk2k2

cos(2|kτ |))e2F

− e2Re(F)

cos[2Im(F)]

]. (A32)

The last equation is the main result of this Appendix.

Appendix B: Calculation of the scalar powerspectrum at second order

In this Appendix, we proceed to compute the explicitform of the scalar power spectrum at second order in theHFF.

Using our previous main results, Eqs. (23), (45) and(A32), the full power spectrum is

Ps =k3

4π2M2P

(1 + ε1 + ε2)2

(1 + ε2)2|η|e2F

a2ε1ν

×[(

1 +λk|kτ |k2

+λk2k2

cos(2|kτ |))

− exp2[Re(F)− F ]cos[2Im(F)]

]. (B1)

In the former expression, there are functions that de-pend on η, these are: a2(η), ε1,2(η), ν(η), F (η) and F(η).However as we will show in the following, when thesefunctions are expressed explicitly as a function of η, thePs remains a constant, i.e. independent of η. Further-more, we will express all of these functions at secondorder in the HFF, and finally exhibit explicitly the k de-pendence that for now remains implicit in some termsof (B1). This latter step is required to identify the socalled spectral index, and running of the spectral index.In fact, we will make use of some the results obtained in[36] and [38].

We begin by recalling our definition of ν, (A13), whichis explicitly given by

ν =

[1

4+ η2H2(2− ε1 +

3

2ε2 +

ε222− 7

2ε1ε2 +

ε2ε32

)

]1/2.

(B2)In that equation, the functions H2, ε1,2 depend on η,however the second order terms involving ε1,2,3 can bealready considered to be constant.

The explicit dependence on the linear terms ε1,2, canbe found by expanding around N∗. We remind the readerthe definition N ≡ ln(a/aini) and that η∗ represents theturning point i.e. it is the time at which g(η∗) = 0, see(A14). Thus, N∗ is evaluated at a(η∗). The expansionyields

ε1 = ε1∗ +dε1dN

∣∣∣∣∗(N −N∗) +

1

2

d2ε1dN2

∣∣∣∣∗(N −N∗)2 + . . .

= ε1∗ + ε1∗ε2∗ ln

(a

a∗

)+

1

2

(ε1∗ε

22∗ + ε1∗ε2∗ε3∗

)ln2

(a

a∗

)+ . . .

= ε1∗ − ε1∗ε2∗ ln

(η

η∗

)+

1

2

(ε1∗ε

22∗ + ε1∗ε2∗ε3∗

)ln2

(η

η∗

)+ . . . (B3)

where in the second line the definition of the HFF (3) wasused, and in the third line we used that a(η) ∝ 1/η. Aswe see from (B3), the function ε1 now exhibits explicitits η dependence. A similar procedure is used to obtain

ε2 = ε2∗ − ε2∗ε3∗ ln

(η

η∗

)+

1

2

(ε2∗ε

23∗ + ε2∗ε3∗ε4∗

)ln2

(η

η∗

)+ . . . (B4)

Using the expansions (B3) and (B4), one can find theexpression for H up to second order in HFF, this is [36]:

H =−1

η

(1 + ε1∗ + ε21∗ + ε1∗ε2∗

)+ε1∗ε2∗

1

ηln

(η

η∗

)+O(ε3).

(B5)Moreover, from the last equation one can find an expres-sion for a expanded at second order in HFF [36]

a(η) ' −1

H∗η

[1 + ε1∗ + ε21∗ + ε1∗ε2∗

−(ε1∗ + 2ε21∗ + ε1∗ε2∗

)ln

(η

η∗

)+

1

2

(ε21∗ + ε1∗ε2∗

)ln2

(η

η∗

)]. (B6)

Therefore, using expansions (B3), (B4) and (B5), theexpression corresponding to ν (B2) expanded up to sec-ond order in HFF is

ν(η) = ν∗ −(ε1∗ε2∗ +

1

2ε2∗ε3∗

)ln

(η

η∗

)+O(ε3) (B7)

17

with

ν∗ ≡3

2+ ε1∗ + ε21∗ +

1

2ε2∗ +

5

6ε1∗ε2∗ +

1

6ε2∗ε3∗. (B8)

At this point, we have found the explicit dependencein the η variable corresponding to the functions: ε1,2(η),a(η) and ν(η). But we still require to calculate the func-tions F and F to obtain the complete expression for Ps(B1). This will be done by solving the correspondingintegrals.

Let us focus on F . From the definition F ≡ 23f

3/2 andf ,g, defined in (A14), we have

F =

∫ η

η∗

dη

√ν2(η)

η2− q2. (B9)

Using the definition of q2 (A26), we can check that ifλk = 0, i.e. if there is no collapse of the wave function,then q2 = k2. Thus, F = F when λk = 0 (recall F isdefined in (A22), and that F is real while F is complex).Therefore, we can obtain F and F from the same integral,i.e. solving integral (B9), automatically yields F , and bysetting λk = 0 in that result, we can obtain also F .

Inserting (B7) into the previous formula and expanding

everything to second order, the integrand in (B9) reads√ν2(η)

η2− q2 ' −ν∗

η

(1− q2η2

ν2∗

)1/2

+3

2ν∗η

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)×(

1− q2η2

ν2∗

)−1/2ln

(η

η∗

). (B10)

Therefore, we have two different integrals to calculate inorder to evaluate the term F . In the following we write,

F ≡ F1 + F2, (B11)

and calculate each of the F1,2 separately. These integralscan be solved analytically [36], for our model, the resultis

lim|η|→0

F1 = −ν∗[1 + ln

∣∣∣∣ ηη∗∣∣∣∣− ln 2 + ln ζk + iθk

],

(B12)where we define

ζk ≡(

1 +4λ2kk4

)1/4

, θk ≡ −1

2arctan

(2λkk2

)(B13)

and

lim|η|→0

F2 =3

16ν∗

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)(4 ln2

∣∣∣∣ ηη∗∣∣∣∣− 4 ln2 2 +

π2

3

)+

12

16ν∗

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)[ln2 ζk − θ2k

]+

24i

16ν∗

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)(θk ln ζk)

[1 + ln

∣∣∣∣ ηη∗∣∣∣∣] . (B14)

The explicit dependence on the collapse parameter λkis now manifested in the previous equations through ζkand θk. We notice that F1 and F2 contain terms that arelogarithmically divergent in the limit |η| → 0. We willsee that this is not a serious problem, the final expressionof Ps(k) will not have any divergent terms.

Equations (B12) and (B14) enable us to calculateF ≡ F1 + F2 and F . The latter, as we have indicatedpreviously, is obtained by setting λk = 0, i.e. ζk = 1 and

θk = 0, yielding

F = −ν∗[1 + ln

∣∣∣∣ ηη∗∣∣∣∣− ln 2

]+

3

16ν∗

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)×(

4 ln2

∣∣∣∣ ηη∗∣∣∣∣− 4 ln2 2 +

π2

3

). (B15)

Additionally, from the resulting expression of F1 +F2 = F we have,

2Im(F) = −2ν∗θk +3

ν∗

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)(θk ln ζk)

×[1 + ln

∣∣∣∣ ηη∗∣∣∣∣] , (B16)

and

2[Re(F)− F ] = −2ν∗ ln ζk +3

2ν∗

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)[ln2 ζk − θ2k

]. (B17)

18

We have all the expressions needed to give an expression of the Ps in terms of: the collapse parameter and thesecond order HFF. Therefore, collecting Eqs. (B15), (B16), (B17), as well as the corresponding ones to ε1,2(η), a(η)and ν(η) [this is Eqs. (B3), (B4), (B6) and (B7)], it is straightforward, although lengthy, to obtain the power spectrumfrom (B1). The final expression is

Ps '18e−3H2

∗π2M2

P ε1∗

1 + ε1∗

(−2

3+ 2 ln 2

)+ ε2∗

(−1

3+ ln 2

)+ ε21∗

(−26

9+

2

3ln 2 + 2 ln2 2

)+ ε22∗

(− 1

18− 1

3ln 2 +

1

2ln2 2

)+ ε1∗ε2∗

(−43

9+π2

12+

1

3ln 2 + ln2 2

)+ ε2∗ε3∗

(−1

9+π2

24+

1

3ln 2− 1

2ln2 2

)C(k), (B18)

where we have defined

C(k) ≡(

1 +λk|kτ |k2

+λk2k2

cos(2|kτ |))− exp2[Re(F)− F ]

cos[2Im(F)], (B19)

with

exp2[Re(F)− F ] = ζ−3k exp

[−2

(ε1∗ + ε21∗ +

1

2ε2∗ +

5

6ε1∗ε2∗ +

1

6ε2∗ε3∗

)ln ζk +

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)(ln2 ζk − θ2k

)](B20)

and

cos[2Im(F)] = cos

(3 + 2ε1∗ + 2ε21∗ + ε2∗ +

5

3ε1∗ε2∗ +

1

3ε2∗ε3∗

)θk

− 2

(ε1∗ε2∗ +

1

2ε2∗ε3∗

)(θk ln ζk)

[1 + ln

∣∣∣∣ ηη∗∣∣∣∣]. (B21)

At this point a few comments are in order. First, asdiscussed in Refs. [36, 38, 69], the presence of the factor18e−3 ' 0.896 is typical for the uniform approximation,and from now on, we will simply set this factor equal toone. Second, the divergent logarithmic term appears onlyin C(k) but as an argument of a cosine function, whichin turn appears in the denominator in the definition ofC(k); thus it represents no problem at all. In fact, we

can set ln∣∣∣ ηη∗ ∣∣∣ ' ln

∣∣a∗a

∣∣ = ∆N∗, i.e. is the number of

e-folds from η∗ to the end of inflation.

Our expression of P(k) depends on η∗, and the HFF,as well as H∗ all evaluated at η∗, which is the turningpoint of g, this means η∗ ≡ −ν(η∗)/k. Thus, there isa k dependence that remains hidden in those quantities.In order to uncover the k dependence, we define a pivotwave number k and expand all those terms around anunique conformal time η. It is customary to set this ηas the time of “horizon crossing,” which is defined as

− kη = 1. (B22)

Technically, this means that, for instance, the Hubbleparameter H∗ must be rewritten as an expansion around

η,

H∗ = H

[1 +

(ε21 + ε1

)lnη∗η

+1

2

(ε21 + ε1ε2

)ln2 η∗

η

](B23)

and the k dependence is thus uncovered by the relation

η∗η

=kkν∗. (B24)

Expanding the previous equation at second order in HFF,we obtain

ln

(η∗η

)=

(ln

3

2+ ln

kk

)(1− 2

3ε1ε2 −

1

3ε2ε3

)+

2

3ε1 +

1

3ε2 +

4

9ε21 −

1

18ε22

+1

9ε2ε3 +

1

3ε1ε2. (B25)

Hence, substituting (B25) into (B23), will exhibit explic-itly the k dependence in the Hubble factor H∗.

Applying this same technique to the HFF εi∗ and ν∗lead us to our main expression. This is, the scalar powerspectrum at second order in the HFF given by the CSLmodel is

19

Ps =H2

π2M2P ε1

1−Dε1 −Dε2 +

(−10

9− 2D + 2D2

)ε21 +

(2

9+D2

2

)ε22

+

(−29

9−D +D2 +

π2

12

)ε1ε2 +

(π2

24− 1

18− D2

2

)ε2ε3

+[−2ε1 − ε2 + 2(−1 + 2D)ε21 − (1− 2D)ε1ε2 +Dε22 −Dε2ε3

]ln

(k

k

)+

[2ε21 + ε1ε2 +

1

2ε22 −

1

2ε2ε3

]ln2

(k

k

)C(k), (B26)

where D ≡ 1/3 − ln 3 and in C(k) (defined in (B19)) we have the following expressions for the exp2[Re(F) −F ]/ cos[2Im(F)] term:

exp2[Re(F)− F ] = ζ−3k exp

[− 2ε1 − ε2 − 2ε21 +

(−5

3+ 2 ln

3

2− 2 ln

k

k

)ε1ε2

+

(−1

3+ ln

3

2− ln

k

k

)ε2ε3

]ln ζk +

(ε1ε2 +

1

2ε2ε3

)(ln2 ζk − θ2k

)(B27)

and

cos[2Im(F)] = cos

(− 3− 2ε1 − ε2 − 2ε21 +

(−5

3+ 2 ln

3

2− 2 ln

k

k

)ε1ε2

+

(−1

3+ ln

3

2− ln

k

k

)ε2ε3

)θk

+ 2

(ε1ε2 +

1

2ε2ε3

)(θk ln ζk)

[1−∆N − ln

3

2+ ln

k

k

]. (B28)

Notice that the former divergent logarithmic term, has now transformed into ∆N which is the number of e-foldsfrom the horizon crossing of the pivot scale k to the end of inflation. Typically ∆N ∼ 60. Equation (B26), is ourfinal expression for the PS, within the CSLIM, written in terms of the HFF and the collapse parameter λk.

In the standard approach for the predicted power spectrum (PS), the k dependence can be parameterized by theso called scalar spectral index ns and the running of the spectral index αs. The parameters ns and αs are of interestsince they are used to constrain the shape of the PS consistent with the observational data. On the other hand,in our main result (B26), we can see that the CSL model induces an extra k dependence on the PS through C(k)as expected. Consequently, it would be helpful to identify the parameters ns and αs, and then including them, ifnecessary, in the function C(k). This will allow us to compare directly the observational consequences between ourapproach and the standard inflationary model. That preliminary analysis is done in Sec. 4.

Thus, in order to deduce an expression for ns and αs in terms of the HFF, we set C(k) = 1 and follow the methodin [38]. Additionally, since the amplitude of the PS was computed up to second order in HFF, the expression for nsis valid up to third order and αs up to fourth order. Hence, the scalar spectral index is given by

ns − 1 = −2ε1 − ε2 − 2ε21 − (1 +D)ε1ε2 −Dε2ε3 − 2ε31 +

(−47

9− 5D + 3D2

)ε21ε2

+4

9ε22ε3 +

(− 1

18− D2

2+π2

24

)ε2ε

23 +

(− 1

18− D2

2+π2

24

)ε2ε3ε4

+

(−29

9− 2D +

π2

12

)ε1ε2ε3 +

(−38

9−D +

π2

12

)ε1ε

22 (B29)

20

and the running of the spectral index yields

αs = −2ε1ε2 − ε2ε3 − 6ε2ε21 − 12ε2ε

31 −Dε2ε23 −Dε2ε3ε4 +

8

9ε22ε

23 +

4

9ε22ε3ε4

+

(6D2 − 11D − 121

9

)ε22ε

21 +

(3D2 − 6D − 65

9

)ε2ε3ε

21 +

(−D

2

2+π2

24− 1

18

)ε2ε

33

+

(−D

2

2+π2

24− 1

18

)ε2ε3ε

24 +

(−3D2

2+π2

8− 1

6

)ε2ε

23ε4 +

(−D

2

2+π2

24− 1

18

)ε2ε3ε4ε5

+

(−D +

π2

12− 38

9

)ε32ε1 + (−D − 1)ε22ε1 +

(−3D +

π2

12− 29

9

)ε2ε

23ε1 +

(−4D +

π2

4− 38

3

)ε22ε3ε1

− (D + 2)ε2ε3ε1 +

(−3D +

π2

12− 29

9

)ε2ε3ε4ε1. (B30)

We note that at the lowest order in the HFF, ns and αs coincides with the standard expressions.

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