Quasi-matter bounce and inflation in the light of the CSL model
aa r X i v : . [ g r- q c ] J u l Quasi-matter bounce and inflation in the light of the CSL model
Gabriel Le´on, ∗ Gabriel R. Bengochea, † and Susana J. Landau ‡ Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, Ciudad Universitaria - Pab.I, 1428 Buenos Aires, Argentina Instituto de Astronom´ıa y F´ısica del Espacio (IAFE),CONICET - Universidad de Buenos Aires, 1428 Buenos Aires, Argentina Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires and IFIBA, CONICET,Ciudad Universitaria - Pab.I, 1428 Buenos Aires, Argentina
The Continuous Spontaneous Localization (CSL) model has been proposed as a possible solutionto the quantum measurement problem by modifying the Schr¨odinger equation. In this work, weapply the CSL model to two cosmological models of the early Universe: the matter bounce scenarioand slow roll inflation. In particular, we focus on the generation of the classical primordial inhomo-geneities and anisotropies that arise from the dynamical evolution, provided by the CSL mechanism,of the quantum state associated to the quantum fields. In each case, we obtained a prediction forthe shape and the parameters characterizing the primordial spectra (scalar and tensor), i.e. the am-plitude, the spectral index and the tensor-to-scalar ratio. We found that there exist CSL parametervalues, allowed by other non-cosmological experiments, for which our predictions for the angularpower spectrum of the CMB temperature anisotropy are consistent with the best fit canonical modelto the latest data released by the Planck Collaboration.
I. INTRODUCTION
After approximately three decades since the cosmological inflationary paradigm was conceived [1–4], all of its genericpredictions have withstood the confrontation with observational data, in particular, those coming from the CosmicMicrowave Background (CMB) radiation [5–7]. That has led a large group of cosmologists to consider inflation as awell established theory of the early Universe. Inflation was originally proposed to provide a solution to the puzzlesof the hot Big Bang theory (e.g. the horizon and flatness problems). However, the modern success of inflation isthat, allegedly, it can offer us an explanation about the origin of the primordial inhomogeneities [8–11]. The standardargument is also rather pictorial: the quantum fluctuations of the vacuum associated to the inflaton are stretched outto cosmological scales due to the accelerated expansion of the spacetime; those fluctuations are considered the seedsof all large scale structures observed in the Universe. Furthermore, in Ref. [12] is investigated the detectability ofpossible traces of the quantum nature regarding the primordial perturbations.On the other hand, proponents of alternative scenarios to inflation argue that even if it is the most fashionablemodel of the early Universe, that does not mean it is necessarily true [13, 14]. Furthermore, another feature that wouldmake alternative models worthwhile of study is that they might avoid some long standing puzzles of the inflationaryparadigm. Among those issues, we can mention: the subject of eternal inflation, a feature that is present in almostevery model of inflation [15] and which also leads to the controversial topic of the multiverse; the initial singularityproblem and the trans-Planckian problem for primordial perturbations [16], which are related by the fact that oneis interpolating the solutions provided by General Relativity in regimes where it may no longer be valid; and finally,it has been argued that the potentials associated to the inflaton, that best fit the latest observed data, need to befine-tuned [17, 18]. Although the aforementioned problems are not considered real problems by some scientists [19, 20],others seem to disagree [18, 21]. However, we think that if other alternative models can reproduce the main resultslinked to inflation, we should make use of the observational data available to test them.One of the alternative models to inflation that seems to be consistent with the latest data is the so called matterbounce scenario (MBS) [21–28]. In this cosmological model, the initial singularity of the standard model is replacedby a non-singular bounce. That is, instead of an ever-expanding Universe, it assumes an early contracting matter-dominated Universe, which continues to evolve towards a bouncing phase and, later, enters into the expanding-phaseof standard cosmology. The Universe described by the MBS relies on a single scalar field satisfying an equation of state ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] that mimics that of a dust-like fluid. Additionally, in order to describe successfully a bouncing phase with a singlescalar field, one needs to use cosmologies beyond the realm of General Relativity, such as, loop quantum cosmologies,teleparallel F ( T ) gravity or F ( R ) gravity. Proponents of the MBS claim that the potential associated to the scalarfield is less fine-tuned than that of inflation, and also solves the historical problem of requiring very special initialconditions for the Big Bang model [22, 23], which originally motivated the development of the inflationary framework.However, the MBS is not exactly problem free. A complete assessment of the present conceptual issues is providedin Ref. [23]. In spite of not being completely finished from a theoretical point of view, the MBS is quite simple in itstreatment of the primordial perturbations. That makes it an interesting case of study for the purpose of this article.In particular, the generation of the primordial perturbations is depicted during the contracting phase, i.e. in a regimewhere gravity is well described by General Relativity, and the perturbations correspond to inhomogeneities of a singlescalar field.In addition to the prior puzzles and successes of inflation and the MBS, there remains an important question: whatis the precise physical mechanism that converts quantum fluctuations of the vacuum into classical perturbations ofthe spacetime? This question has been the subject of numerous works in the past and the consensus seems to favorthe decoherence framework [29–33]. Nevertheless, decoherence cannot address that question by itself. In otherwords, even if one would choose (or not) to embrace the decoherence program, a particular interpretation of QuantumMechanics must be selected (implicitly or explicitly). The Copenhagen–orthodox–interpretation requires to identify anotion of observer that performs a measurement on the system; which, in the decoherence framework, is equivalent toidentify the unobservables or external degrees of freedom of the system. It is not clear how to do such identifications ifthe system is the early Universe. Other interpretations such as many-worlds, consistent histories and hidden variablesformulations, might be adopted with varying degrees of success (see for instance [38–40]).In the present article, in order to address the quantum-to-classical transition of the primordial perturbations, wewill choose to work with the Continuous Spontaneous Localization (CSL) model. The CSL model belongs to a largeclass of models known as objective dynamical reduction models or simply called collapse models. Collapse modelsattempt to provide a solution to the measurement problem of Quantum Mechanics [41–45]. The proponents of thesemodels state that the measurement problem originates from the linear character of the quantum dynamics encoded inthe Schr¨odinger equation. The common idea shared in these collapse models is to introduce some nonlinear stochasticcorrections to the Schr¨odinger equation that breaks its linearity. According to the collapse models, a noise field couplesnonlinearly with the system (usually with the spatial degree of freedom of the system), inducing a spontaneous randomlocalization of the wave function in a sufficiently small region of the space. Suitably chosen collapse parameters makesure that micro-systems evolve essentially (but not exactly) following the dynamics provided by the Schr¨odingerequation, while macro-systems are extremely sensible to the nonlinear effects resulting in a perfectly localization ofthe wave function. Furthermore, there is no need to mention or to introduce a notion of an observer/measurementdevice as in the Copenhagen interpretation, which is a desired feature in the context of the early Universe andcosmology in general.The CSL model has been applied before to the inflationary Universe in an attempt to explain the quantum-to-classical transition of the primordial perturbations [46–50]. Also, recently a new effective collapse mechanism,independent of the CSL model, has been proposed to deal with the measurement problem during the inflationaryera [51]. However, among those works, the key role played by the collapse mechanism varies and also yields differentpredictions for the primordial power spectrum, some of which might be consistent with the observational data. In thepresent article, we will subscribe to the conceptual point of view first presented in [47, 52], which was developed withinthe semiclassical gravity framework, and latter in [50, 53] was extended to the standard quantization procedure of theprimordial perturbations using the so called Mukhanov-Sasaki variable [8, 54]. The main role that we advocate for thedynamical reduction mechanism of the state vector, modeled in this paper by the CSL model, is to directly generatethe primordial curvature perturbations. Specifically, the initial state of the quantum field–the vacuum state–evolvesdynamically according to the modified Schr¨odinger equation provided by the CSL model. This evolution leads to afinal state that does not share the initial symmetry of the vacuum, i.e. it is not homogeneous and isotropic. In thisway, the collapse mechanism generates the inhomogeneities and anisotropies of the matter fields. These asymmetriesare codified in the evolved quantum state and, thus, are responsible for generating the perturbations of the spacetime. Note that the previous prescription, regarding our approach to address the birth of the primordial perturbations,does not require the inclusion of an exponential expansion phase in the Universe that “stretches out” the quantum Although for the reasons exposed in Refs. [34, 35] we do not find such posture satisfactory. See comments by Mukhanov on pages 347-348 of Ref. [36] and by Weinberg on page 476 of Ref. [37]. For a formal proof of this statement see Appendix A of Ref. [35] and Appendix A of Ref. [50]. We encourage the reader to consult Refs. [35, 50, 52] for a detailed exposition of the concepts involved in our approach and its relationwith the collapse of the wave function. fluctuations of the vacuum (or the squeezing of the field variables as usually argued). Therefore, in principle, it shouldbe possible to extend our picture to alternative scenarios dealing with the origin of the cosmological perturbations.Moreover, since the cosmic observations are well constrained, it should also be feasible to test the predictions thatresult from applying our framework in those alternative cosmological models. In the present article, we focus on theimplementation of the CSL model within the framework of the MBS and, in parallel, we present the same appliance ofthe CSL model to the slow roll inflationary model of the early Universe. In this way, we can appreciate more clearlywhere the CSL model enters into the picture; particularly at the moment when computing the theoretical predictions.The main motivation behind the present work is that if the CSL model can be truly considered as a physical model ofthe quantum world, which also avoids the standard quantum measurement problem, then it should also be possible touse it in different contexts from the traditional laboratory settings. The cosmological context provides a rich avenue toexplore such foundational issues and, more important, there exist sufficient precise data to test the initial hypotheses.As a consequence, we will analyze the predictions resulting from implementing the CSL model in the MBS and in theinflationary model of the early Universe, and we will compare the corresponding results with the one provided by thebest fit standard cosmological model. Additionally, we will focus on the range of values allowed for the parameters ofthe CSL model, experimentally tested [55, 56] in non-cosmological frameworks.The paper is organized as follows: in Sect. II, we provide a very brief synopsis of the main features of the CSLmodel, with particular emphasis on those that will be useful for the next sections. In Sect. III, we present thecharacterization of the primordial perturbations within the two cosmological models that we are considering, i.e. theMBS and standard slow roll inflation. In Sect. IV, we show the connection between the observational quantities andthe theoretical predictions that result from adopting our conceptual point of view concerning the CSL model. InSect. V, we explicitly show the implementation of the CSL model to the MBS and inflation, and we also present thepredictions for the primordial power spectra (scalar and tensor) in each case. In Sect. VI, we discuss the implicationsof the results obtained; additionally, we compare the predicted scalar power spectra with the standard one. In Sect.VII, we analyze the viability of the CSL model using the data extracted from the CMB when considering the bestfit cosmological model. Finally, in Sect. VIII, we end with our conclusions. We include an Appendix containing thecomputational details that led to the results presented in Sect. V.
II. A CONCISE SYNOPSIS OF THE CSL MODEL
In this section, we provide a brief summary of the relevant features of the CSL model; for a detailed review, werefer the reader to Refs. [43, 44].In the CSL model, the modification of the Schr¨odinger equation induces a collapse of the wave function towards oneof the possible eigenstates of an operator ˆ Θ , called the collapse operator, with certain rate λ . The self-induced collapseis due to the interaction of the system with a background noise W ( t ) that can be considered as a continuous-timestochastic process of the Wiener kind. The modified Schr¨odinger equation drives the time evolution of an initial stateas | Ψ , t i = ˆ T exp (cid:26) − Z tt dt ′ (cid:20) i ˆ H + 14 λ (cid:16) W ( t ′ ) − λ ˆ Θ (cid:17) (cid:21)(cid:27) | Ψ , t i , (1)with ˆ T the time-ordering operator. The probability associated with a particular realization of W ( t ) is, P [ W ( t ) , t ] D W ( t ) = h Ψ , t | Ψ , t i t Y t i = t d W ( t i ) p πλ/dt . (2)The norm of the state | Ψ , t i evolves dynamically, and Eq. (2) implies that the most probable state will be the onewith the largest norm. From Eqs. (1) and (2), it can be derived the evolution equation of the density matrix operatorˆ ρ . That is, d ˆ ρdt = − i [ ˆ H, ˆ ρ ] − λ Θ, [ ˆ Θ, ˆ ρ ]] . (3)The density matrix operator can be used to obtain the ensemble average of the expectation value of an operator h ˆ O i = Tr[ ˆ O ˆ ρ ]. Henceforth, from Eq. (3) it follows that ddt h ˆ O i = − i h [ ˆ O, ˆ H ] i − λ h [ ˆ Θ, [ ˆ Θ, ˆ O ]] i . (4)The average is over possible realizations of the noise W ( t ), each realization corresponding to a single outcome of thefinal state | Ψ , t i .One of the most important features of collapse models is the so-called amplification mechanism. That is, assumingthat the reduction (collapse) rates for the M constituents of a macroscopic object are equal ( λ i = λ ), it can be provedthat the reduction rate for the center of mass of an M -particle system is amplified by a factor of M with respect tothat of a single constituent [41, 57]; in other words, λ macro = M λ .The parameter λ sets the strength of the collapse process. In the original model, proposed by Ghirardi-Rimmini-Webber (GRW), the authors suggested a value of λ GRW ≃ − s − for r C ≃
100 nm. However, Adler suggested agreater value λ Adler ≃ − s − for r C ≃
100 nm [58] (the parameter r C is called the correlation length of the noiseand provides a measure for the spatial resolution of the collapse [41, 43, 57]). Recent experiments have been devisedto set bounds on the parameter λ [59, 60]. Furthermore, it is claimed that matter-wave interferometry provides themost generic way to experimentally test the collapse models [55, 56]. Those results suggest that the range between λ GRW and λ Adler is still viable for some variations of the original CSL model (e.g. by considering non-white noise).Henceforth, the main characteristics of the CSL model are: (1) The modification to the Schr¨odinger equation isnonlinear and leads to a breakdown of the superposition principle for macroscopic objects; (2) The random nature ofQuantum Mechanics is concealed in the noise W ( t ) and is consistent with Born’s rule; (3) An amplification mechanismexists, through the parameter λ which is related to the strength of the collapse. This strength is weak for microscopicobjects and strong for macroscopic bodies.Another main aspect of the collapse models is that the collapse mechanism injects energy into the system. In fact,previous works have performed a preliminary analysis using cosmological data to set bounds on the value of λ [61].The energy increase is minimal, e.g. for a particle of mass m = 10 − g, one obtains δE/t ≃ − eV s − [43]. Inother words, an increase of 10 − eV will take 10 years. However, even if the energy increase can be ignored at thephenomenological level, a more realistic model should remove this issue.Moreover, the increase of energy in the collapse models leads to difficulties when trying to formulate relativisticcollapse models. Additionally, the collapse mechanism occurs in such a way that is nonlocal. This implies that thecollapse of the wave function must be instantaneous or superluminal (but the nonlocal features cannot be exploited tosend signals at superluminal speed). Also, the nonlocality is necessary to ensure that the models are consistent withthe violation of Bell’s inequalities. Several relativistic models have been proposed so far [62–64], none of which canbe considered completely finished. In spite of the lack of a relativistic collapse model, we will apply the CSL model tothe primordial Universe, i.e. to inflation and the MBS, but in order to provide a more detailed picture, we need firstto establish the mathematical framework of the primordial Universe in the two approaches considered in this work. III. TWO APPROACHES: ACCELERATED EXPANSION OR QUASI-MATTER CONTRACTION
This section presents the details of the two cosmological approaches, describing the dynamics of the Universe, thatwe will be considering in the rest of the manuscript. In particular, we are going to work with the following twoscenarios:1. An accelerated expansion of the early Universe given by the simplest inflationary model, that is, a single scalarfield in the slow roll approximation with canonical kinetic term. Since such a model is probably very well knownfor most readers, we will not dwell into much detail here.2. The MBS [22–26], a cosmological model in which the Universe undertakes a quasi-matter contracting phase,then experiences a non-singular bounce and finally enters into the standard cosmological expansion. Since inthis model the primordial perturbations are born during the contracting stage of the Universe, we will focusexclusively on that cosmic stage. We will refer to such a stage as the quasi-matter contracting Universe (QMCU).
A. The background
The inflationary Universe and the QMCU are both described by Einstein equations G ab = 8 πGT ab ( c = 1), whilethe matter fields are characterized by a single scalar field. In the case of inflation the scalar field is the inflaton φ ,and in the QMCU the scalar field will be denoted by ϕ .As mentioned earlier, for inflation, we will consider standard slow roll inflation. In that case, the backgroundspacetime is described by a quasi-de Sitter Universe, characterized by H ≃ − / [ η (1 − ǫ )], with H ≡ a ′ /a the conformalexpansion rate, a being the scale factor and the slow roll parameter is defined as ǫ ≡ −H ′ / H ; a prime denotes partialderivative with respect to conformal time η . The energy density of the Universe is dominated by the potential of theinflaton V , and during slow roll inflation the condition ǫ ≃ M P / ∂ φ V /V ) ≪ M P ≡ (8 πG ) − thereduced Planck mass. Since we will work in a full quasi-de Sitter expansion, another useful parameter to characterizeslow roll inflation is the second slow roll parameter, i.e. δ ≡ ǫ − ǫ ′ / H ǫ ≪ ϕ is separated into an homogeneous part ϕ ( η ) plus small inhomogeneities δϕ ( x , η ).The homogeneous part satisfies H = a M P (cid:18) ϕ ′ a + W (cid:19) ; ϕ ′′ + 2 H ϕ ′ + a ∂ ϕ W = 0 , (5)where W is the potential associated to the field ϕ .In the QMCU, it is assumed that the equation of state associated to the scalar field almost mimics that of ordinarymatter, i.e. P = ωρ such that | ω | ≪
1; the latter implies ϕ ′ ≃ a W . Consequently, the scale factor (in conformaltime) evolves as a ( η ) ≃ η / | ǫ | ≪
1, which plays the same role as theslow roll parameter in inflation. The parameter ǫ is defined as (see e.g. [22]) ǫ ≡ − (cid:18)
12 + H ′ H (cid:19) ≃ (cid:18) ∂ ϕ WW (cid:19) − . (6)The case ǫ = 0 corresponds to an exact matter-dominated contracting phase (note that ǫ = ω ). Furthermore, for sakeof completeness we introduce another parameter δ ≡ ǫ ′ H ≃ − ∂ ϕ (cid:18) ∂ ϕ WW (cid:19) , (7)such that | δ | ≪ | ǫ | . The parameter | δ | is analog to the δ parameter of slow roll inflation and it is related to therunning of the spectral index in the QMCU model (see e.g. [22]).As is well known, it is not straightforward to accomplish a non-singular bounce within the framework of GeneralRelativity by considering a single canonical scalar field, since the null energy condition (NEC) is violated (see forinstance [14, 23]). As a consequence, one possible option is to work with cosmologies within the context of modifiedgravity theories. In the case of the QMCU presented in [22, 23], the authors worked within the framework of holonomycorrected loop quantum cosmology and teleparallel F ( T ) gravity.It is also important to note that even if a non-singular bounce cannot be achieved within General Relativity, theorigin of the primordial perturbations is assumed to take place during the contracting (pre-bounce) phase of theUniverse, where the curvature and energy scales are low enough to be described by General Relativity. On the otherhand, one must present the conditions that need to be fulfilled such that the shape of the primordial spectrum,associated to the perturbations, remains practically unchanged when passing through the bounce. We will discussthis subject in more detail in the next section. B. Perturbations
In the inflationary Universe and in the QMCU, one can separate the scalar field into an homogeneous part plussmall inhomogeneous perturbations. Moreover, the metric associated to the spacetime, in both cases, is described by aFLRW background metric plus perturbations; which are classified as scalar, vector and tensor types (in this paper wewill not consider vector perturbations). One useful quantity to describe the scalar (an also the tensor) perturbationsis the so called Mukhanov-Sasaki (MS) variable. During inflation, the MS variable is defined by v ( x , η ) ≡ a ( η ) (cid:20) δφ ( x , η ) + φ ′ ( η ) H ( η ) Φ( x , η ) (cid:21) , (8)with Φ the gauge invariant quantity known as the Bardeen potential [65], which, in the longitudinal gauge, correspondsto the curvature perturbation. A similar expression to Eq. (8) can be used in the QMCU by replacing the fields φ ′ and δφ with ϕ ′ and δϕ , respectively. The advantage of relying on the MS variable is that, when expanding the actionof a scalar field minimally coupled to gravity into second order scalar perturbations, one obtains δ (2) S = R dηd k L ,where L = v ′ k v ⋆ ′ k − k v k v ⋆ k − z ′ z (cid:16) v k v ⋆ ′ k + v ′ k v ⋆ k (cid:17) + (cid:18) z ′ z (cid:19) v k v ⋆ k , (9)with v k the Fourier modes associated to the MS variable, z = aφ ′ / H during inflation, and z = aϕ ′ / H when consideringthe QMCU. However, it is important to note that during the bouncing phase, the action given by the Lagrangianin Eq. (9) remains the same but the expression for z changes (see Ref. [25] for an explicit calculation within F ( T )theories). On the other hand, during the contraction phase, the quantity z ′ /z can be written explicitly in terms ofthe QMCU parameter ¯ ǫ , as in a similar fashion using the slow roll inflation parameters, i.e. z ′ ( η ) z ( η ) = βη ; where β ≡ ( − (1 + 2 ǫ − δ ) if assuming the inflationary universe2(1 − ǫ ) if assuming the QMCU (10)Note that, since | δ | ≪ | ǫ | , the δ parameter does not enter into the expression z ′ /z at first order for the QMCU.The CSL model is based on a nonlinear modification to the Schr¨odinger equation; consequently, it will be advanta-geous to perform the quantization of the perturbations in the Schr¨odinger picture, where the relevant physical objectsare the Hamiltonian and the wave functional. The Hamiltonian associated to L in Eq. (9) is H = R d k ( H R k + H I k ),with H R,I k = p R,I k p R,I k + z ′ z (cid:16) v R,I k p R,I k + v R,I k p R,I k (cid:17) + k v R,I k v R,I k , (11)where the indexes R, I denote the real and imaginary parts of v k and p k . The canonical conjugated momentumassociated to v k is p k = ∂ L /∂v ⋆ ′ k , i.e. p k = v ′ k − z ′ z v k . (12)Since v ( x , η ) is a real field, v ⋆ k = v − k .We promote v k and p k to quantum operators, by imposing canonical commutations relations [ˆ v R,I k , ˆ p R,I k ′ ] = iδ ( k − k ′ ).In the Schr¨odinger picture, the wave functional Ψ[ v ( x , η )] characterizes the state of the system. Furthermore, inFourier space, the wave functional can be factorized into modes components Ψ[ v ( x , η )] = Π k Ψ R k ( v R k )Ψ I k ( v I k ). Fromnow on, we will deal with each mode separately. Henceforth, each mode of the wave functional, associated to the realand imaginary parts of the canonical variables, satisfies the Schr¨odinger equation ˆ H R,I k Ψ R,I k = i∂ Ψ R,I k /∂η , with theHamiltonian provided by (11). Note that one can also choose to work with the wave functional in the momentumrepresentation, i.e. Ψ[ p ( x , η )] = Π k Ψ R k ( p R k )Ψ I k ( p I k ).The standard assumption is that, at an early conformal time τ → −∞ , the modes are in their adiabatic groundstate, which is a Gaussian centered at zero with certain spread. This applies to both, the inflationary Universe andthe QMCU. In addition, this ground state is commonly referred to as the Bunch-Davies vacuum. Thus, the conformaltime η is in the range [ τ, − ).Given that the initial quantum state is Gaussian, its shape will be preserved during its evolution. The explicitexpression of the Gaussian state, in the field representation, is:Ψ R,I ( η, v R,I k ) = exp h − A k ( η )( v R,I k ) + B k ( η ) v R,I k + C k ( η ) i , (13)and, equivalently, in the momentum representationΨ R,I ( η, p R,I k ) = exp h − ˜ A k ( η )( p R,I k ) + ˜ B k ( η ) p R,I k + ˜ C k ( η ) i . (14)Therefore, the wave functional evolves according to Schr¨odinger equation, with initial conditions given by A k ( τ ) = k/ , ˜ A k ( τ ) = 1 / k, B k ( τ ) = ˜ B k ( τ ) = C k ( τ ) = ˜ C k ( τ ) = 0 corresponding to the Bunch-Davies vacuum, which isperfectly homogeneous and isotropic in the sense of a vacuum state in quantum field theory. The fact that we areintroducing the wave functional in the field and momentum representations is related to the choice of the collapseoperator in the CSL model, i.e., since there is no physical reason to choose one over the other, both choices are equallyacceptable (at least from the phenomenological point of view). In the next section, we will show how to extract thephysical quantities from the theory to be compared with the observations. IV. THEORETICAL PREDICTIONS AND OBSERVATIONAL QUANTITIES
We begin this section by making some key remarks about the conceptual aspects of our approach and, then, weproceed to identify the relevant physical quantities that will be related with the observed data. We encourage thereader to consult Refs. [34, 35, 52] for a complete discussion regarding our full picture of the role played by thedynamical reduction of wave function in the cosmological setting. As a matter of fact, the relation between theobservables and the predictions from the theory, using the Mukhanov-Sasaki variable during inflation and the CSLmodel, has been previously exposed in [50]; however, in this section we reproduce the key arguments of such a referenceto make the present paper as self-contained as possible. Thus, there is no original work in the following of this section.The main role for invoking the collapse of the wave function is to find a physical mechanism for breaking the initialhomogeneity and isotropy associated to both, the quantum state and the spacetime. More specifically, we assumethat a nonlinear modification to the Schr¨odinger equation, which in the present work is provided by the CSL model,can break the homogeneity and isotropy associated to the vacuum state and, in turn, it can generate the metricperturbations, which correspond to the primordial curvature perturbation.Note that in the literature one can found statements suggesting that the vacuum fluctuations somehow becomeclassical when the proper wavelength associated to the perturbations becomes larger than the Hubble radius [29, 66].Nevertheless, there is nothing in the dynamics governed by the traditional Schr¨odinger equation that can changethe symmetry of the vacuum state, the symmetry being the homogeneity and isotropy. As a consequence, if thequantum state is perfectly symmetric and the Quantum Theory teaches us that the symmetries of a physical systemmust be encoded in the quantum state, then there is no clear way to describe the inhomogeneities and anisotropiesof the spacetime in the quantum sense. If the quantum state of the system is perfectly symmetric, then its classicaldescription must also be exactly symmetric. Thus, there is a lack of a proper explanation concerning the emergenceof the primordial inhomogeneities and anisotropies in the Universe. That is why some non-standard interpretations ofQuantum Mechanics, that make use of the Schr¨odinger equation (e.g. many-worlds, consistent histories, etc.), cannotprovide a satisfactory answer to the problem at hand. It is important to note that the previous discussion applies toboth cosmological models, the QMCU and inflation.The modified Schr¨odinger equation given by the CSL model can successfully change the symmetries of the vacuumstate and, at the same time, be responsible for the birth of the primordial curvature perturbation.Specifically, in the comoving gauge, the curvature perturbation R ( x , η ) and the MS variable v ( x , η ) are related by R ( x , η ) = v ( x , η ) /z ( η ). Thus, a quantization of v ( x , η ) implies a quantization of R ( x , η ). The question that arisesnow is: how to relate the quantum objects ˆ v ( x , η ) and ˆ R ( x , η )? Furthermore, one may wonder how to relate thephysical observables, such as the temperature anisotropies of the CMB, with the quantum objects that emerge fromthe quantum theory? The traditional answer relies on the quantum correlation functions, in particular, the two-pointquantum correlation function h | ˆ R ( x , η ) ˆ R ( x ′ , η ) | i and its relation with the two-point angular correlation function ofthe temperature anisotropies δT /T (ˆ n ) δT /T (ˆ n ), where the bar denotes an average over different directions in thecelestial sky and ˆ n and ˆ n are two unitary vectors denoting some particular directions. We do not find the previousanswer to be completely satisfactory, and for a detailed explanation we invite the reader to consult Refs. [34, 35].In order to illustrate our approach, we begin by focusing on the temperature anisotropies of the CMB observedtoday and its relation to the comoving classical curvature perturbation encoded in the quantity R . Such a relation isapproximately given by (i.e. for large angular scales) δTT ≃ − R . (15)On the other hand, the observational data are described in terms of the coefficients a lm of the multipolar seriesexpansion δT /T ( θ, ϕ ) = P lm a lm Y lm ( θ, ϕ ), i.e a lm = Z δTT ( θ, ϕ ) Y ∗ lm ( θ, ϕ ) d Ω (16)here θ and ϕ are the coordinates on the celestial two-sphere, with Y lm ( θ, ϕ ) the spherical harmonics.Given Eq. (15), the coefficients a lm can be further re-expressed in terms of the Fourier modes associated to R , i.e. a lm ≃ − πi l Z d k (2 π ) / j l ( kR D ) Y ∗ lm (ˆ k ) R k , (17)where R D is the comoving radius of the last scattering surface and j l ( kR D ) the spherical Bessel function of order l ofthe first kind.Finally, we can include the effects of late time physics that give rise to so called acoustic peaks. These effects areencoded in the transfer functions ∆ l ( k ), and thus the coefficients a lm are given by a lm = − πi l Z d k (2 π ) / ∆ l ( k ) Y ∗ lm (ˆ k ) R k , (18)where R k is the primordial comoving curvature perturbation. Also note that for large angular scales ∆ l ( k ) → j l ( kR D ).The next step is to relate R k with the quantum operator ˆ R k . Clearly, if one computes the vacuum expectation value h | ˆ R k | i and makes it exactly equal to R k , then one obtains precisely zero; while it is clear that for any given l, m , themeasured value of the quantity a lm is not zero. As matter of fact, the standard argument is that it is not the quantity a lm that is zero but the average a lm . However, the notion of average is subtle, since in the CMB one has an averageover different directions in the sky, while the average that one normally associates to the quantum expectation valueof an operator is related to an average over possible outcomes of repeatedly measurements of an observable associatedto an operator in the Hilbert space of the system (it is evident that concepts such as measurements, observers, etc.are not well defined in the early Universe).On the other hand, we will assume that the quantity R k , i.e. the classical value associated to the Fourier mode ofthe comoving curvature perturbation R ( x , η ), is an adequate description if the quantum state associated to each modeis sharply peaked around some particular value. In consequence, the classical value corresponds to the expectationvalue of ˆ R in that particular “peaked” state [53]. In other words, our assumption is that the CSL mechanism willlead to a final state such that the relation R k = h Ψ | ˆ R k | Ψ i = 1 z h Ψ | ˆ v k | Ψ i (19)holds.Therefore, in our approach, the coefficients a lm in Eq. (18), will be given by a lm = − πi l Z d k (2 π ) / Y ∗ lm (ˆ k )∆ l ( k ) h Ψ | ˆ R k | Ψ i , (20)where | Ψ i corresponds to the evolved state according to the non-unitary modification of the Schr¨odinger equationprovided by the CSL mechanism (see Refs. [46, 48, 49] for other ways to relate R k and ˆ R k using the CSL model,and [50] for a discussion on those approaches). Note also that | Ψ i does not share the same symmetries as the vacuumstate, i.e. the inhomogeneity and isotropy of the system is encoded in the quantum state | Ψ i . Furthermore, Eq. (20)shows how the expectation value of the quantum field ˆ R k in the state | Ψ i acts as a source for the coefficients a lm .A well known observational quantity is the angular power spectrum defined by C l ≡ l + 1 X m | a lm | . (21)We will assume that we can identify the observed value | a lm | with the most likely value of | a lm | ML obtained from thetheory and, in turn, assume that the most likely value coincides approximately with the average | a lm | . This averageis over possible realizations or outcomes of the state | Ψ i that results from the CSL evolution. Thus, the observed C obs. l approximately coincides with the theoretical prediction C l given in terms of the average | a lm | , i.e. C obs. l ≃ C l = 12 l + 1 X m | a lm | . (22)Using Eq. (20), the theoretical prediction for the angular power spectrum is C l = 12 l + 1 X m π Z d kd k ′ (2 π ) ∆ l ( k )∆ l ( k ′ ) Y ∗ lm (ˆ k ) Y lm (ˆ k ′ ) h ˆ R k ih ˆ R k ′ i ∗ . (23)Moreover, if the CSL evolution is such that there is no correlation between modes (which can be justified by the factthat we are working at linear order in cosmological perturbation theory), then h ˆ R k ih ˆ R k ′ i ∗ = (cid:16) h ˆ R R k i + h ˆ R I k i (cid:17) δ ( k − k ′ ) , (24)where ˆ R R,I k denotes the real and imaginary part of the field ˆ R k (also, we assume that there is no correlation betweenˆ R R k and ˆ R I k ). Therefore, C l = 12 l + 1 X m π Z d k (2 π ) | Y lm (ˆ k ) | ∆ l ( k ) (cid:16) h ˆ R R k i + h ˆ R I k i (cid:17) . (25)Performing the integral over the angular part of k and summing over m , we obtain C l = 225 π Z dk k ∆ l ( k ) (cid:16) h ˆ R R k i + h ˆ R I k i (cid:17) . (26)On the other hand, the standard relation between the primordial power spectrum and the C l is given by C l = 4 π Z dkk ∆ l ( k ) P s ( k ) , (27)where P s ( k ) is the dimensionless scalar power spectrum defined as R k R k ′ ≡ π k P s ( k ) δ ( k − k ′ ) (28)Henceforth, Eqs. (26) and (27) imply that the power spectrum in our approach is given by P s ( k ) = k π (cid:16) h ˆ R R k i + h ˆ R I k i (cid:17) . (29)Note that the definition of the power spectrum, Eq. (28), is the canonical definition when dealing with classicalrandom fields, where the average is over possible realizations of the random fields. In cosmology, the usual identificationof the two–point quantum correlation function h | ˆ R k ˆ R k ′ | i with R k R k is subtle and concepts such as ergodicity,decoherence and squeezing of the vacuum state are normally invoked.Thus, in terms of the MS variable, the scalar power spectrum in our approach is: P s ( k ) = k π z (cid:16) h Ψ | ˆ v R k | Ψ i + h Ψ | ˆ v I k | Ψ i (cid:17) . (30)Equation (30) is the key result from this section. It shows explicitly how to relate the quantities obtained from thequantum theory with the observed temperature anisotropies of the CMB. It also exhibits the difference between ourapproach and the traditional one. V. THE CSL MODEL IN QUASI-MATTER CONTRACTION AND INFLATION
In this section, we will focus on the specific details of implementing the CSL model to the QMCU and the inflationaryUniverse, and the main goal will be to obtain a prediction for the power spectra.We begin by noting that, in Eq. (30), the predictions related to the observational data are the objects h Ψ | ˆ v R,I k | Ψ i .Therefore, we will apply the CSL model to each mode of the field and to its real and imaginary parts. As a consequence,we will assume that the evolution of the state vector characterizing each mode of the field, written in conformal time,is given by | Ψ R,I k , η i = ˆ T exp (cid:26) − Z ητ dη ′ (cid:20) i ˆ H R,I k + 14 λ k (cid:16) W ( η ′ ) − λ k ˆ Θ R,I k (cid:17) (cid:21)(cid:27) | Ψ R,I k , τ i (31)with H R,I k given in (11). Note that the Hamiltonian ˆ H R,I k depends on the field ˆ v R,I k which is defined in terms of theinflaton perturbations, but also it can be defined analogously using the perturbations of the scalar field associatedto the QMCU [one has to take into account the change in z ( η )]. Furthermore, we will consider that Eq. (4) can beextrapolated to a generic quantum mode ˆ F k , that is, the real and imaginary parts of the mode ˆ F k satisfy: ddη h ˆ F R,I k i = − i [ ˆ F R,I k , ˆ H R,I k ] − λ k Θ R,I k , [ ˆ Θ R,I k , ˆ F R,I k ]] . (32)At this point, we have to make a choice regarding the collapse operator ˆ Θ R,I k . At first sight, the natural candidateis the MS variable, namely ˆ Θ R,I k = ˆ v R,I k . Nevertheless, we think that in absence of a full relativistic CSL model,there is no a priori choice and, thus, the canonical conjugated momentum ˆ p R,I k can also be considered as the collapseoperator. In fact, in Ref. [50], we have shown that in the framework of the inflationary Universe, the momentumoperator can be used as the collapse operator given that, in the longitudinal gauge, the momentum operator is directlyrelated with the curvature perturbation.Thus, we are going to consider four different cases:0(i) The collapse operator is ˆ v R,I k during slow roll inflation.(ii) The collapse operator is ˆ v R,I k during the QMCU.(iii) The collapse operator is ˆ p R,I k during slow roll inflation.(iv) The collapse operator is ˆ p R,I k during the QMCU.We stress that only the third case, that is, the implementation of the CSL model within the inflationary frameworkusing the field ˆ p R,I k as the collapse operator, was first developed in Ref. [50]. Nevertheless, we are including it in thepresent work for the sake of completeness. Note however that the analysis in Ref. [50] was done in the longitudinalgauge. In the present paper, we will work in the comoving gauge in all the four cases. The analysis of the threeremaining cases, and in particular the implementation of the CSL model during a contracting phase of the earlyUniverse, are presented here for the first time.For each of these four cases, we will obtain the scalar (and tensor) power spectrum.Furthermore, the calculation of the object h Ψ | ˆ v R k | Ψ i is identical to h Ψ | ˆ v I k | Ψ i . Consequently, we will omit fromnow on the indexes R, I unless it creates confusion.Using the Gaussian wave functions in the field representation, Eq. (13), and the probability associated to W ( η ) inEq. (2), it can be shown that [47], h ˆ v k i = h ˆ v k i − A k ( η )] . (33)The quantity (4Re[ A ( η )]) − is the standard deviation of the squared field variable ˆ v k . It is also the width of everypacket in Fourier’s space. In a similar manner, using the Gaussian wave function in the momentum representation,Eq. (14), along with Eq. (2), it follows that h ˆ v k i = h ˆ v k i − | ˜ A k ( η ) | Re[ ˜ A k ( η )] . (34)For cases (i) and (ii), it is convenient to work with Eq. (33); and for cases (iii) and (iv) with Eq. (34). Thus, tocalculate h ˆ v k i , we only need to find the two terms on the right hand side of (33) or (34), respectively. The secondterm on the right hand side of both equations can be found from the CSL evolution equation, Eq. (31), while the firstone by using Eq. (32) with the wave function in the corresponding representation. Also, in Eqs. (33) and (34), weconsider the regime − kη →
0, which correspond to the range of observational interest, that is, the regime for whichthe modes are larger than the Hubble radius.Once we have computed Eqs. (33) and (34), in the corresponding case, we can substitute it into Eq. (30) to givea specific prediction for the scalar power spectrum. The actual calculations are long, so we have included them inAppendix A for the interested reader. In the following, we will show only the main results.In case (i), our predicted scalar power spectrum during inflation is (at the lowest order in the slow roll parameter): P s ( k ) = H ( − kη ) − ν s +3 πǫM P F ( λ k , ν s ) , (35)with ν s ≡ / ǫ − δ and F ( λ k , ν s ) ≡ ν s − sin ( ν s π )Γ (1 − ν s ) (cid:20) − λ k τk + 3 λ k k sin( − kτ ) cos( − kτ ) (cid:21) − π " λ k ν s − k ( − kη ) − ν s +2 + ζ ν s k π sin( πν s + 2 ν s θ k )sin( πν s )2 ν s Γ ( ν s ) − . (36)For case (ii), we have P s ( k ) = 112 π (cid:18)Z η −∞ d ˜ ηz (cid:19) (cid:18) k | aH | (cid:19) − µ s +3 F ( λ k , µ s ) , (37)1with µ s ≡ / − ǫ and F ( λ k , µ s ) ≡ π sin ( µ s π )Γ (1 − µ s ) (cid:20) − λ k τk − λ k k sin( − kτ ) cos( − kτ ) (cid:21) − " µ s − λ k ( µ s − k ( − kη ) − µ s +2 + ζ µ s k π sin( πµ s + 2 µ s θ k )sin( πµ s )8Γ ( µ s ) − . (38)In both cases, (i) and (ii), we have also defined: ζ k ≡ (cid:18) λ k k (cid:19) / , θ k ≡ −
12 arctan (cid:18) λ k k (cid:19) . (39)The calculations for obtaining the tensor power spectra are very similar to the one used to obtain the scalar ones(see Appendix A for further details). In case (i), the formula obtained for the tensor power spectrum is P t ( k ) = H − kη ) − ν t +3 πM P F ( λ k , ν t ) , (40)where ν t ≡ / ǫ . Therefore, the tensor-to-scalar ratio r ≡ P t ( k ) /P s ( k ), at the lowest order in the slow rollparameter, is given by r = 16 ǫ (41)which is exactly the same prediction as in the standard inflationary slow roll scenario.Meanwhile, in case (ii), the tensor power spectrum is P t ( k ) = 29 π (cid:18)Z η −∞ d ˜ ηz T (cid:19) (cid:18) k | aH | (cid:19) − µ t +3 F ( λ k , µ t ) , (42)where µ t ≡ / − ǫ = µ s . Also, for very low energy densities and curvatures, z T = a (see Ref. [22]). Thetensor-to-scalar ratio is given by r = 83 R ∞−∞ dηz T ( η ) R ∞−∞ dηz ( η ) k = | aH | , (43)which is also the same as the one presented in Refs. [22, 23]. Note that we have evaluated the upper limit of theintegrals at η = ∞ . The motivation is essentially the same as the one given in Refs. [22, 23]. That is, one evaluatesthe scalar and power spectra at very late times corresponding to when the mode “re-enters the horizon”, or moreprecisely when k = | aH | during the expanding (post-bounce) phase.The previously presented cases (i) and (ii) correspond to selecting ˆ v R,I k as the collapse operator. Next, we focus onthe results for cases (iii) and (iv), which correspond to choose ˆ p R,I k as the collapse operator.For case (iii), we obtain: P s ( k ) = 2 ν s − H Γ ( ν s ) ǫM P π ( − kη ) − ν s +1 F ( λ k , ν s ) , (44)where we have defined ν s ≡ / ǫ − δ and F ( λ k , ν s ) ≡ − λ k kτ + λ k sin( − kτ ) cos( − kτ ) − sin( πν s )˜ ζ ν s sin(2 ν s ˜ θ k + πν s ) . (45)In case (iv), the corresponding expression results P s ( k ) = 112 π (cid:18)Z η −∞ d ˜ ηz (cid:19) (cid:18) k | aH | (cid:19) − µ s +3 F ( λ k , µ s ) (46)2with the definitions µ s ≡ / − ǫ and F ( λ k , µ s ) ≡ ( µ s ) π (cid:20) − λ k kτ − λ k sin( − kτ ) cos( − kτ ) (cid:21) + c − µ s +4 (1 + 4 λ k ) λ k ( − kη ) − µ s + c λ k ( − kη ) − µ s +2 + c ˜ ζ µ s k sin[2( µ s + 1)˜ θ k + πµ s ] . (47)The constants c , c and c are shown in Appendix A. In both cases, (iii) and (iv), we have the following definitions˜ ζ k ≡ (1 + 4 λ k ) / , ˜ θ k ≡ −
12 arctan(2 λ k ) . (48)The predictions for the tensor-to-scalar ratios are exactly the same as the ones presented in cases (i) and (ii) (seeAppendix A).We end this section by summarizing the main results. We have applied the CSL model to the inflationary Universeand to the QMCU. Moreover, in order to employ the CSL model, we need to choose the collapse operator. We havechosen to work with ˆ v R,I k and ˆ p R,I k as the collapse operators. Henceforth, we have obtained the scalar power spectra infour different cases Eqs. (35), (37), (44) and (46). On the other hand, introducing the CSL mechanism does not affectthe tensor-to-scalar ratio r . Specifically, if one works within the standard inflationary scenario, then the predictionfor r is equal to the standard one given by slow roll inflation; meanwhile, if one adopts the QMCU framework, thenthe predictions are equal to the ones presented in Refs. [22, 23]. VI. DISCUSSION ON THE CSL INSPIRED POWER SPECTRA
In this section, we will discuss the implications of the results obtained in the previous section. In particular, wewill compare our predicted scalar power spectra with the standard one.The scalar power spectrum predicted by slow roll inflation is traditionally expressed as [36, 67] P s ( k ) = A s (cid:18) kk (cid:19) n s − , (49)where k is a pivot scale, and the amplitude A s and the spectral index n s are given by A s = (cid:18) H π M P ǫ (cid:19) k = aH , n s − − ǫ + 2 δ. (50)On the other hand, we have four different expressions for the scalar power spectrum, corresponding to the fourcases mentioned at the beginning of Sect. V. In the following, we will analyze each one of them, but first we will makea few observations regarding the parameter λ k .The dependence on k in the parameter λ k encodes the “amplification mechanism”, which is characteristic of dy-namical reduction models (see Refs. [48, 49]). One possible way to determine the exact dependence on k , and perhapsthe simplest, is by dimensional analysis. That is, the main evolution equations are given in Eqs. (31) and (32);consequently, in order for those equations to be dimensionally consistent, the fundamental dimensions of λ k changedepending on the fundamental units associated to the collapse operator ˆ Θ R,I k . Moreover, we expect that λ k is directlyrelated to λ , i.e. the CSL parameter, which clearly must be the same in all physical situations (cosmological or oth-erwise). Moreover, taking into account that we are working in units in which ~ = c = 1, the fundamental dimensionof λ is [Length] − .Thus, in the case where the collapse operator is chosen to be ˆ v R,I k , the most natural expression of λ k , which isconsistent with the dimensions of all terms involving the dynamical equations, is λ k = λk (51)And in the case where the selected collapse operator is ˆ p R,I k , such an expression is λ k = λk , (52)where λ is the CSL parameter, with the same numerical value in all cases. From now on, we will assume that λ k takesthe form of Eqs. (51) and (52) depending on the chosen operator acting as the collapse operator.3 A. The CSL power spectra during inflation
Let us begin the discussion by working within the framework of the inflationary Universe, analyzing cases (i) and(iii).The scalar power spectrum given in Eq. (35), corresponding to case (i), can be written in a similar form to theone showed in Eq. (49). As usual, the power spectrum can be evaluated at the conformal time where the pivot scale“crosses the horizon”; or more precisely, when − k η = 1 (i.e. k = aH ) during the inflationary epoch. Furthermore,the different coefficients that multiply each term of the function F ( λ k , ν s ) involve the quantity ν s . For these terms,we can approximate ν s ≃ / k involving ν s because these are directly related to the scalar spectral index n s , for which the value n s = 1 is ruled out. Furthermore, in order to provide a suitable normalization for the CSL power spectra, we multiplyand divide by the quantity λ | τ | . Thus, the power spectrum in Eq. (35) can be rewritten as: P s ( k ) = A s (cid:18) kk (cid:19) n s − C ( k ) (53)where A s = (cid:18) H λ | τ | π M P ǫ (cid:19) k = aH , n s − − ǫ + 2 δ, (54)and C ( k ) ≡ F ( λ k = λk, ν s ≃ / /λ | τ | ; that is, C ( k ) = 1 λ | τ | − λτ + 3 λk sin( − kτ ) cos( − kτ ) − " λk ( − kη ) n s − + ζ − n s k cos[(4 − n s ) θ k ]2 − (55)[expressions for ζ k and θ k are given in Eq. (39) with λ k = λk ].Within the inflationary framework, and with the same arguments followed to arrive to Eq. (53), we can write thepower spectrum Eq. (44), corresponding to case (iii), in the following form: P s ( k ) = A s (cid:18) kk (cid:19) n s − C ( k ) (56)where A s and n s are the same as in Eq. (54), and C ( k ) ≡ F ( λ k = λ/k, ν s ≃ / /λ | τ | . Thus, C ( k ) = 1 λ | τ | ( − λτ + λk sin( − kτ ) cos( − kτ ) − ζ − n s k cos[(2 − n s )˜ θ k ] ) (57)[expressions for ˜ ζ k and ˜ θ k are given in Eq. (48) with λ k = λ/k ]Let us make some remarks. Notice that the scalar index predicted by the CSL power spectra is exactly the same asthe standard one from slow roll inflation, but the amplitude is slightly different. The difference between the standardamplitude and the one using the CSL model is a factor of λ | τ | / / h ˆ v R k i and h ˆ v I k i . However, the factor 1 / λ | τ | , which comes from the normalization of C ( k ) and C ( k ), does modify the standardpredicted amplitude. A quantitative analysis will be done in the next section.A second remark has to do with the following. It is well known that there is a minimum number of e -foldings forinflation related to the solution of the “horizon problem”, and this minimum number depends on the characteristicenergy of inflation. A shared characteristic of the functions C ( k ) and C ( k ) is that they include the quantity τ ,which represents the conformal time at the beginning of inflation. This quantity depends on the energy scale at whichinflation ends, which is associated to the inflaton potential V at that time, and the number of e -foldings correspondingto the total duration of inflation.Third, note that another important feature of the CSL power spectra in inflation is that the function C ( k ),corresponding to the case in which the collapse operator is ˆ v R,I k , depends explicitly on the conformal time η , whilstthe function C ( k ), which corresponds to the case when ˆ p R,I k is the collapse operator, does not exhibit such timedependence. The time dependence on the power spectrum when the collapse operator is ˆ v R,I k has been previouslynoted by other authors [46, 48]. Nevertheless, the exact form of their predicted power spectrum is different from the4 -7 -6 -5 -4 -3 -2 -1 C ( k ) k [1/Mpc] V =10 -4 M P , N=65V =10 -3 M P , N=70V =10 -4 M P , N=68 Figure 1: The function C ( k ), corresponding to the power spectrum in the case when the collapse operator is ˆ v R,I k duringinflation. We have set the value λ GRW = 1 . × − Mpc − and n s = 0 .
96. The values of conformal time at the beginningof the inflationary regime τ corresponding to the three cases depicted (from top to bottom) are: − τ = 7803894 , C ( k ) at η = η f , and the values of η f corresponding to the three different cases considered(from top to bottom) are: − η f = 4 . × − , . × − and 4 . × − Mpc. M P denotes the reduced Planck mass M P ≃ GeV. one shown here. As a matter of fact, that difference is illustrated by considering the limiting case λ k = 0. In suchmentioned works, for λ = 0 (i.e. standard Schr¨odigner evolution), their predicted power spectrum is the same as thetraditional one. Contrarily, in our approach, if λ k = 0 then P s ( k ) = 0, which is consistent with our point of viewregarding the role played by the CSL model. In any case, even if the pictures used for the role of the CSL model aredifferent between our work and the one in Refs. [46, 48], the time dependence on the power spectrum is shared.In order to continue, we choose to evaluate the power spectrum (or equivalently the function C ( k )) at the conformaltime when inflation ends, which we denote by η f . We think it is consistent with the previous calculations in whichthe power spectrum was obtained in the limit − kη →
0, which is satisfied by the value η f . The precise value of η f depends mainly on the characteristic energy scale of inflation and the number of e -foldings assumed for the fullinflationary phase N ≡ ln[ a ( η f ) /a ( τ )].Readers familiar with previous works, can check that our expression for the scalar power spectrum Eq. (56), whichfeatures the function C ( k ), is essentially the same as the one obtained in Ref. [50]. The difference is that in thepresent paper we chose to work in the comoving gauge (where R represents the curvature perturbation), whilst inthe aforementioned reference we worked in the longitudinal gauge, where the Bardeen potential Φ corresponds to thecurvature perturbation. Therefore, we find reassuring that even having worked in different gauges, the expressionfor the power spectrum, when the collapse operator is the momentum associated to the Fourier’s mode of the MSvariable, is the same and it has the attractive feature that does not depends on the conformal time.Figures 1 and 2 show different plots for the functions C ( k ) and C ( k ), respectively. In both cases, we haveconsidered the value of the CSL parameter as λ GRW = 1 . × − Mpc − , which corresponds to a value favored byexperimental data [55, 56, 59, 60]. The various plots in each figure correspond to different values of the characteristicenergy of inflation V / , and the total e -foldings N that inflation is assumed to last, which also set the values of τ and η f . The values of k considered correspond to these of observational interest, i.e. we consider k in the range from10 − to 10 − Mpc − .As we can observe, the functions C ( k ) and C ( k ) exhibit an oscillatory behavior around the unit. For increasingvalues of k , the oscillations decrease in amplitude. However, we note that even for decreasing values of k the functions C ( k ) and C ( k ) are very close to 1. Consequently, for the chosen values of λ , V / and N , the functions C ( k ) ≃ C ( k ) ≃
1. That means that the shape of the angular power spectrum C l will not be very different from the standardone, but the amplitude could vary (a complete analysis will be presented in the next section).Additionally, the fact that C ( k ) depends on the conformal time does not seem to affect its behavior in a significantmanner. In fact, it is closely similar to the one of C ( k ), which does not depend on the conformal time. That meansthat the contribution from the time dependent term (i.e. the last term in Eq. (55)), to the total value of the function C ( k ) is negligible when − kη → -7 -6 -5 -4 -3 -2 -1 C ( k ) k [1/Mpc] V =10 -4 M P , N=65V =10 -3 M P , N=70V =10 -4 M P , N=68 Figure 2: The function C ( k ), corresponding to the power spectrum in the case when the collapse operator is ˆ p R,I k duringinflation. We have set the value λ GRW = 1 . × − Mpc − and n s = 0 .
96. The values of conformal time at the beginningof the inflationary regime τ corresponding to the three cases depicted (from top to bottom) are: − τ = 7803894 , M P denotes the reduced Planck mass M P ≃ GeV.
B. The CSL power spectra in the QMCU
We switch now the discussion to the framework of the QMCU, i.e. cases (ii) and (iv), which correspond to selectingˆ v R,I k or ˆ p R,I k as the collapse operator, respectively.The scalar power spectra given in both cases, i.e. Eqs. (37) and (46), can also be written in a manner similar to thestandard spectrum Eq. (49). Once again, following Refs. [22, 23], we choose to evaluate the spectrum at the conformaltime where the pivot scale “reenters the horizon” k = | aH | , which happens at late times during the expansion phaseof the Universe (consequently the upper limit of the integral is evaluated at η → ∞ ). We approximate (for thesame arguments as in the previous subsection) µ s ≃ / F ( λ k , µ s ) and F ( λ k , µ s ) (but not in the powers of k as these powers are directly related to the spectral index n s ). Additionally, theparameter λ k is assumed to be λ k = λk for case (ii) and λ k = λ/k in case (iv). Moreover, we multiply and divide bya factor of λ | τ | in order to properly normalize the expressions F and F .Henceforth, the scalar power spectrum for case (ii), Eq. (37), will be written as P ( k ) = A s (cid:18) kk (cid:19) n s − C ( k ) (58)where A s = λ | τ | π (cid:18)Z ∞−∞ dηz ( η ) (cid:19) , n s − ǫ (59)and C ( k ) ≡ F ( λ k = λk, µ s ≃ / /λ | τ | ; thus, C ( k ) = 1 λ | τ | (cid:20) − λτ − λk sin( − kτ ) cos( − kτ ) (cid:21) − " λk ( − kη ) n s − + ζ − n s k − n s ) θ k ] − . (60)On the other hand, the power spectrum for case (iv), Eq. (46), will be written as P ( k ) = A s (cid:18) kk (cid:19) n s − C ( k ) (61)6 -7 -6 -5 -4 -3 -2 -1 C ( k ) k [1/Mpc] N=65N=55N=50
Figure 3: The function C ( k ), corresponding to the power spectrum in the case when the collapse operator is ˆ v R,I k in the QMCUframework. We have set the value λ = 1 . × − Mpc − and n s = 0 .
96. The value of conformal time at the beginning of thequasi-matter dominated stage τ corresponding to the three cases depicted (from top to bottom) are: − τ = 7803894 , C ( k ) at η = η f , and the values of η f corresponding to the three different casesconsidered (from top to bottom) are: − η f = 5 . × − , . × − and 1 . × − Mpc. -7 -6 -5 -4 -3 -2 -1 C ( k ) k [1/Mpc] N=65N=55N=50
Figure 4: The function C ( k ), corresponding to the power spectrum in the case when the collapse operator is ˆ p R,I k in the QMCUframework. We have set the value λ = 1 . × − Mpc − and n s = 0 .
96. The value of conformal time at the beginning of thequasi-matter dominated stage τ corresponding to the three cases depicted (from top to bottom) are: − τ = 7803894 , C ( k ) at η = η f , and the values of η f corresponding to the three different casesconsidered (from top to bottom) are: − η f = 5 . × − , . × − and 1 . × − Mpc. with A s and n s the same as in Eq. (59), and C ( k ) ≡ F ( λ k = λ/k, µ s ≃ / /λ | τ | . Hence, C ( k ) = 1 λ | τ | (cid:26)(cid:20) − λτ − λk sin( − kτ ) cos( − kτ ) (cid:21) − (1 + 4 λ/k ) λ/k )( − kη ) n s − + 4( λ/k )( − kη ) n s − + ˜ ζ − n s k cos[˜ θ k (6 − n s )] (cid:27) . (62)As in the case of the inflationary Universe, the predicted value of the scalar spectral index n s is not affected bythe CSL model. In fact, it has the same expression as that of the QMCU original models presented in Refs. [22, 23].7Nevertheless, as can be seen in Eq. (59), the amplitude of the spectrum is modified by an extra factor of λ | τ | / A s = A orig. s × λ | τ | τ corresponds to the beginning of the quasi-matter dominated period. Regarding the amplitude of thespectrum in the QMCU model, when the background evolution is driven by a matter dominated Universe, it can beobtained analytically working within F ( T ) gravity or LQC. In the teleparallel F ( T ) case, the original amplitude is A orig. s = 13 π (cid:18)Z ∞−∞ dηz ( η ) (cid:19) = π ρ c ρ P (64)while in the LQC case, the original amplitude is A orig. s = 13 π (cid:18)Z ∞−∞ dηz ( η ) (cid:19) = 169 ρ c ρ P C (65)where ρ P is the Planck energy density, C ≃ . ρ c is called the critical density, whichcorresponds to the energy density at which the Universe bounces, both expressions for the amplitude can be consultedin Refs. [22, 23].Thus, in order to obtain an amplitude in both cases (the teleparallel gravity case and the LQC case) that isconsistent with that obtained from the CMB data (i.e. A s ≃ − ), and taking into account the extra factor of λ | τ | / ρ c ≃ − ρ P λ | τ | . (66)Generically λ | τ | ≫
1; hence, the CSL model introduces an extra constriction to the QMCU, that is, ρ c ≪ ρ P . Inthe next section we will perform a more quantitative analysis.The functions C ( k ) and C ( k ) share a characteristic feature, namely they depend explicitly on − kη , which comesfrom a series expansion around − kη →
0. Consequently, we choose to evaluate η = η f corresponding to the endof the quasi-matter domination stage or the onset of the bouncing phase. One can also define the total number ofe-foldings N ≡ ln[ a ( τ ) /a ( η f )] for the duration of the quasi-matter dominated phase; however, notice that in this case,since there is no horizon problem, there is no minimum value of N . Another important aspect is that if λ = 0 then C ( k ) = C ( k ) = 0. In other words, if the evolution of the state vector is completely unitary, then there are noperturbations of the spacetime at all and the state vector continues being perfectly symmetric, which is consistentwith our conceptual framework.In Figs. 3 and 4, we show different plots for the functions C ( k ) and C ( k ), respectively. In both cases, we haveconsidered the value λ GRW = 1 . × − Mpc − . The various plots in each figure correspond to different values of τ and η f . The values of k considered correspond to the values of observational interest, hence, we consider k in therange from 10 − to 10 − Mpc − . As we can see, the functions C ( k ) and C ( k ) exhibit the same oscillatory behavioraround the unity as its counterparts during inflation. Also, the amplitude of each oscillation decreases for increasingvalues of k .We end this section with a few comments regarding the dependence of the power spectrum on η in cases (i), (ii) and(iv). In case (i), which corresponds to selecting the MS variable as the collapse operator during inflation, the termcontaining the η dependence is the last one of C ( k ) in Eq. (55). On the other hand, since the amplitude associatedto the modes R k is “frozen” on super-Hubble scales, the behavior of C ( k ) will not change for super-horizon modes.As a matter of fact, the plots in Fig. 1 show that C ( k ) is essentially a constant in the limit − kη →
0, which meansthat the term containing the η dependence is sub-dominant in such a limit. In cases (ii) and (iv), corresponding tothe framework of the QMCU, the behavior of the functions C ( k ) and C ( k ) are very similar to that of C ( k ) (seeFigs. 3 and 4). That is, they are practically a constant in the limit − kη →
0, which means that the terms involving η , i.e. the last terms of Eqs. (60) and (62), are sub-dominant in the super-Hubble limit. Nevertheless, since in cases(ii) and (iv) the Universe approaches a non-singular bounce, it might be the case that, when the mode “reenters thehorizon” ( k ≫ | aH | ) during the bouncing phase, a modification of the dynamical evolution of the functions C ( k )and C ( k ) would occur. However, if the duration of the bouncing phase is short enough then one could intuitivelyconsider that the spectrum is left unchanged (although counterexamples exist in the literature [68]). Therefore, onecould perform a full analysis regarding the CSL model during the bounce within the QMCU. Nonetheless, we will8take a pragmatical approach and assume that the shape of the spectrum, provided by the functions C ( k ) and C ( k ),survives the bouncing phase and, then, we will use the observational data to further constraint or completely discardthe predicted spectra. In case that the predicted spectra are consistent with observations, one can proceed to performthe full-fledged analysis of implementing the CSL model to the QMCU during the bouncing phase and study thepossible corrections that may arise from passing the perturbations through the bounce. This subject, however, willnot be explored in the present paper.In the next section, we will explore the implications of the predicted spectra using the observational data. VII. EFFECTS ON THE CMB TEMPERATURE SPECTRUM AND ITS IMPLICATIONS ON THECOSMOLOGICAL PARAMETERS
The aim of this section is to analyze the viability of the CSL model by comparing the corresponding predictionswith the ones coming from the best fit canonical model to the CMB data. In particular, we will focus on the powerspectra obtained using the CSL model and its effect on the angular power spectrum.In order to perform our analysis, we start by setting the cosmological parameters of our fiducial model, which willbe used as a reference to compare with the CSL inspired spectra. The fiducial cosmology will be the best fitting flatΛCDM model from Planck data, with the following cosmological parameters and values: baryon density in units ofthe critical density Ω B h = 0 . CDM h = 0 . H = 67 .
74 in units of km s − Mpc − , reionization optical depth T = 0 . n s = 0 .
96 and a pivot scale of k = 0 .
05 Mpc − . These values can be found in the Table 4 presented by the latestPlanck Collaboration [5].Furthermore, we recall that the primordial power spectrum and the angular power spectrum are related by Eq.(27), i.e. C l = 4 π Z dkk ∆ l ( k ) P s ( k ) . (67)Hence, we will use the CSL predicted power spectra P s ( k ) = A s ( k/k ) n s − C i ( k ) with i = 1 , , ,
4, which correspondto the four different cases that we have considered so far. We will focus first on the inflationary model of the earlyUniverse and, then, on the QMCU model. Also, note that the fiducial model corresponds to: P s ( k ) = A s ( k/k ) n s − .The precise prediction for the angular power spectrum will be obtained by using the Boltzmann code CAMB [69],with the aforementioned cosmological parameters. A. The angular power spectrum and the CSL model during inflation
During inflation, the CSL power spectra is characterized by the functions C ( k ) and C ( k ), Eqs. (55) and (57),with standard spectral index n s − − ǫ + 2 δ and amplitude A s = (cid:18) V λ | τ | π M P ǫ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) k = aH (68)The output of the CAMB code, that is, the temperature autocorrelation power spectrum of the fiducial model andthe one provided by the CSL model during inflation are indistinguishable; thus, we have decided not to show theplots. Instead, we present the the relative difference, which we define as S ( l ) ≡ | C CSL l − C fiducial l | C fiducial l (69)Figure 5 shows the relative difference between both predictions. On the left, we have chosen ˆ v R,I k as the collapseoperator while on the right we have chosen ˆ p R,I k . In both cases, we have set an energy scale of 10 − M P for the energyat which inflation ends, and a total amount of inflation corresponding to 65 e -foldings.We observe that the relative difference between the fiducial spectrum and the one predicted using the CSL modelwith, for instance λ GRW , is practically null (the highest difference is around 0 . λ values listed in Table I (not shown in the figure).We have also checked that the essentially null relative difference between the fiducial model and the CSL modelduring inflation is also present in the E polarization autocorrelation power spectrum C EE l and the temperaturepolarization cross correlation power spectrum C TE l .9 × -5 × -5 × -5 × -5 × -5 × -4
0 500 1000 1500 2000 2500 S ( l ) l λ GRW × -5 × -5 × -5 × -5 × -5 × -4
0 500 1000 1500 2000 2500 S ( l ) l λ GRW
Figure 5: The relative difference S ( l ) [defined in Eq. (69)] between the angular power spectrum predicted using the fiducialmodel and the one provided by the CSL model during inflation. Left: ˆ v R,I k as the collapse operator. Right: ˆ p R,I k as the collapseoperator. For the CSL model, we have chosen the value of λ GRW = 1 . × − Mpc − . Also, we have assumed that inflationends at an energy scale of 10 − M P and a total amount of inflation corresponding to N = 65, i.e. τ = 7 . × Mpc and η f = 4 . × − Mpc. Other values of λ were plotted (not shown) achieving an excellent fit, but they are observationally ruledout by their ǫ and r values predicted. See Table I and text for details.Table I: Estimation of ǫ from Eq. (71). We have used | τ | = 7 . × Mpc, which corresponds to V / = 10 GeV and N = 65,and the four values of λ shown below. Also, we have estimated the tensor-to-scalar ratio using r = 16 ǫ . Note that only the λ GRW case is compatible with the latest observations from Planck Collaboration [5]. λ type λ [s − ] λ [Mpc − ] ǫ rλ GRW − . × − . × − λ − λ − λ Adler − On the other hand, the amplitude of the power spectrum A s consistent with the CMB data is A s ≃ − [5].Henceforth, the amplitude obtained using the CSL model, as shown in Eq. (68), must satisfy10 − ≃ (cid:18) V λ | τ | π M P ǫ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) k = aH (70)Clearly, different values of λ will have an effect on the amplitude of the spectrum.Assuming that the pivot scale k crosses the Hubble radius at an energy scale of V / = 10 − M P (i.e. one orderof magnitude less than the presumed energy at which inflation ends), an estimate for ǫ can be calculated. Therefore,the above equation leads to ǫ ≃ λ | τ | − π . (71)Table I shows the different values of ǫ obtained by considering several λ values. Also, in the same table, we providean estimate for the tensor-to-scalar ratio r (recall that the CSL model predicts the same relation as standard inflation,i.e r = 16 ǫ ). From Table I, it can be seen that only the value corresponding to λ GRW is consistent with both, theobserved shape and amplitude of the spectrum. In particular, assuming a characteristic energy scale of inflation of10 − M P ≃ GeV, a total amount of inflation corresponding to N = 65, and the value of λ GRW , we obtain anangular spectrum with a shape and an amplitude that is indistinguishable from the fiducial model, which we knowis consistent with the observational data. The amplitude of the spectrum for this particular set of values leads toan estimate for the slow roll parameter and the tensor-to-scalar ratio of ǫ ≃ − and r ≃ − , respectively. Thosevalues of ǫ and r are consistent with the ones presented by the latest results of the Planck Collaboration [6].It is also instructive to mention that, if future observations confirm the results of the BICEP2 Collaboration [70],i.e. r ≃ .
2, then the value of λ GRW would not be compatible with the values of V and N used in Table I. In fact, the0 S ( l ) l λ Adler S ( l ) l λ Adler
Figure 6: The relative difference S ( l ) [defined in Eq. (69)] between the angular power spectrum predicted using the fiducialmodel and the one provided by the CSL model during the quasi-matter contracting phase. Left: ˆ v R,I k as the collapse operator.Right: ˆ p R,I k as the collapse operator. For the CSL model, we have chosen, as an illustrative example, the value of λ Adler =1029378 Mpc − . Also, we have assumed that the quasi-matter contracting phase begins at conformal time | τ | = 1 . × Mpcand lasts a total amount of N = 50 e -foldings, with − η f = 1 . × − Mpc. Other values of λ listed in Table II also achievean excellent fit (not shown), but the analysis done in this work does not allow prefer one value over another. λ value that might be compatible would be one such that λ ≪ λ GRW . That would open a new range of parameterspace to explore in addition to considering other experimental setups different from cosmological ones [55, 56].
B. The angular power spectrum and the CSL model during the quasi-matter contracting phase
In the QMCU framework, the power spectra are characterized by the functions C ( k ) and C ( k ), i.e. Eqs. (60) and(62), respectively. The predicted scalar spectral index is n s − ǫ , and the amplitude is given by A s ≃ ρ c ρ P λ | τ | . (72)Notice we have approximated the integral that appears in the amplitude, corresponding to Eq. (59), by ρ c /ρ P [seeEqs. (64) and (65)].The output of the CAMB code, that is, the temperature autocorrelation power spectrum of the fiducial model andthe one provided by the CSL model during the QMCU are also indistinguishable. Thus, we present again the relativedifference defined in Eq. (69) where now C CSL l corresponds to the angular power spectrum during the QMCU.Figure 6 shows the relative difference between both predictions. On the left, we have chosen ˆ v R,I k as the collapseoperator while on the right we have chosen ˆ p R,I k . In both cases, we have assumed a total duration of 50 e -foldingsfor the quasi-matter contracting phase and a conformal time | τ | ≃ Mpc corresponding to the beginning of thecontracting stage. We show only the plot of S ( l ) corresponding to the value of λ Adler merely as an illustrative example;the plots for the values corresponding to λ , λ , λ GRW follow the exactly same behavior as the one shown in Fig. 6.We found no difference between the fiducial spectrum and the one provided by the CSL model for the four valuesof λ listed in Table II; the highest relative difference is around 0 . E polarization autocorrelation power spectrum C EE l andthe temperature polarization cross correlation power spectrum C TE l .On the other hand, the amplitude of the power spectrum A s consistent with the CMB data is A s ≃ − [5].Therefore, the ratio ρ c /ρ P , obtained using the CSL model, as shown in Eq. (72), must satisfy ρ c ρ P ≃ κ where κ ≡ λ | τ | × − . (73)Consequently, an estimate for the energy scale of the critical energy E c is E c ≃ κ M P where κ ≡
14 log κ . (74)1 Table II: Estimation of κ and κ defined in Eqs. (73) and (74) respectively. We have used | τ | ≃ Mpc and the four valuesof λ shown below. Also we have estimated the quantities λ | τ | . λ type λ [s − ] λ [Mpc − ] λ | τ | κ κ λ GRW − . × − . × − -3.75 λ − . × − -4.75 λ − . × − -5.25 λ Adler − . × − -5.75 In Table II, we show the different values of κ and κ by using the chosen λ values, with | τ | = 1 . × Mpc. Weinfer that the four values of λ considered are consistent with a critical energy scale in the range (10 − M P , − M P ).It is worthwhile to mention that, in the QMCU, the spectral index n s and the tensor-to-scalar ratio r are notrelated each other as in the standard inflationary paradigm [see Eqs. (43) and (59)]. However, the spectral index,along with the running of the spectral index α s ≡ dn s /d ln k , are the two main parameters of the QMCU modelused to compare with the observational data [22–24]. The CSL model applied to the QMCU does not affect thoseparameters. The only observable affected by the CSL model is the amplitude of the spectrum (which is not related tothe parameter r as in the standard spectrum). Consequently, in order to put an upper bound to the energy scale atwhich the bouncing phase begins, and which would be equivalent to set a constraint on the parameter λ , one shouldconsider a specific theoretical model of the QMCU (i.e. to choose a specific dynamics and a potential of the field ϕ ).Therefore, in the QMCU, and with the same degree of accuracy of past works dealing with the same model, all of thefour values of the CSL parameter λ considered here yield consistent predictions with observational data; specifically,the predictions regarding the shape of the spectrum, the scalar spectral index and the running of the spectral index.On the other hand, note that the information that could discriminate among different values of λ is codified in theamplitude of the spectrum. The predicted amplitude of the spectrum depends on the critical energy density ρ c (thevalue of the energy density at the bouncing time), which is model dependent. VIII. CONCLUSIONS
The CSL model is a physical mechanism that attempts to provide a solution to the measurement problem ofQuantum Mechanics by modifying the Schr¨odinger equation. The CSL model can be referred as an objective reductionmechanism or “effective collapse” of the wave function, and one of the main elements of this model is the collapseoperator, i.e. the operator whose eigenstates correspond to the evolved states by the collapse mechanism. Also, inprinciple, it is possible to apply such a mechanism to any physical system.In this work, we have applied the CSL model to the early Universe by considering two cosmological models: thematter bounce scenario (MBS) and standard slow roll inflation. Additionally, we have considered two differentcollapse schemes, one in which the field variable (given in terms of the Mukhanov-Sasaki variable) serves as thecollapse operator, and other scheme where the collapse operator is the conjugated momentum.In all cases, we have found a prediction for the primordial power spectrum, which is a function of the standardparameters of each cosmological model, and also of the CSL parameter λ . Although the exact expressions for theprimordial power spectra are different in each case, there are features that are essentially the same as its standard–non-collapse–counterparts. Specifically, the predictions for the scalar spectral index and the tensor-to-scalar ratio areexactly the same as the ones given in the MBS and slow roll inflation without collapse. On the other hand, in eachcase, the shape of the spectrum is modified by a function of the wave number k , associated to the modes of the field,and by the inclusion of the λ parameter. However, for a suitable choice of values corresponding to the parameters ofthe cosmological models, there is no significant change in the prediction for the CMB angular power spectrum (i.e.the C l ’s) that can be distinguished from the canonical flat ΛCDM model.Meanwhile, the prediction for the amplitude of the spectrum is modified directly by the parameter λ . We haveempirically explored the range of values of λ , from the originally value suggested by Ghirardi-Rimini-Weber (GRW) λ GRW ≃ − s − [41], to the one given by Adler λ Adler ≃ − s − [58]. In the case of slow roll inflation, we havefound that for a characteristic energy scale of 10 GeV and a total amount of inflation of 65 e -folds, only the valuesuggested by GRW is compatible with the observational bound of the amplitude; other values of λ greater than λ GRW ,e.g. λ Adler , cannot be made compatible with the observed amplitude (because that would require values for the slowroll parameter such that ǫ > λ parameter, causes that the critical energy density ρ c , i.e. the energy density at whichthe bouncing phase begins, to be several orders of magnitude less than the Planck energy density ρ P . The precisenumber of orders of magnitude varies according to the value of λ . For instance, by assuming λ GRW and a total amount2of ∼ e -folds for the matter dominated contracting phase, we have ρ c ≃ − ρ P . The latter relation is obtainedby requiring the compatibility between the predicted amplitude of the scalar power spectrum and the one from thePlanck CMB data A s ∼ − .In conclusion, it was possible to incorporate the CSL model into the cosmological context again, in particular whendealing with the quantum-to-classical transition of the primordial inhomogeneities. Moreover, it is remarkable thatour implementation of the CSL model yields predictions that are also in agreement with experiments in the regimesso far investigated empirically. Those experiments involve values of the CSL parameter λ that have been testedin laboratory settings, quite disengaged from the cosmological framework. We acknowledge that at this stage, theapplication of CSL model to the early Universe, as done in this manuscript, can be seen as an ad hoc employment.However, the fact that the predictions can be empirically tested make us hopeful that future studies will overcomethe perceived shortcomings. Acknowledgments
G. L.’s research was funded by Consejo Nacional de Ciencia y Tecnolog´ıa, CONACYT (Mexico). G. R. B. acknowl-edges support from CONICET (Argentina) Grant PIP 112-2012-0100540. S. J. L. and G. L. are supported by PIP11220120100504 CONICET (Argentina).
Appendix A: Calculations of Section V
In this Appendix, we provide a sketch of the computational steps that led to the results presented in Sect. V. Inthe first half of this Appendix, we will focus on cases (i) and (iii), which correspond to selecting the MS variable asthe collapse operator during inflation and the QMCU, respectively. The second half will contain the details accordingto cases (ii) and (iv), corresponding to selecting ˆ p R,I k as the collapse operator during inflation and the QMCU,respectively.
1. The field ˆ v k as the collapse operator We will proceed the calculation of the scalar power spectrum by dealing simultaneously with the inflationary andthe QMCU frameworks; and finally, we will argue that the computation is similar for the tensor power spectrum.In order to obtain the scalar power spectrum, we will use Eq. (33). Let us focus first on the second term on theright hand side in that equation, which can be obtained by using the CSL evolution equation (31). Recall that, sincein this case ˆ Θ k = ˆ v k , it will be convenient to work with the wave function in the field representation Eq. (13). TheCSL evolution equation leads to the following equation of motion: A ′ k = ik λ k − βη A k − iA k . (A1)The previous equation is solved by performing the change of variable A k ( η ) ≡ f ′ k ( η ) / [2 if k ( η )], resulting in a Besseldifferential equation for f k ( η ). After solving such an equation, and returning to the original variable A k , we obtain A k ( η ) = q i (cid:20) − J m − ( − qη ) − e − iπm J − m ( − qη ) J m ( − qη ) − e − iπm J − m ( − qη ) (cid:21) , (A2)being q ≡ k (1 − iλ k /k ), and where the initial condition for the Bunch-Davies vacuum A k ( τ ) = k/ τ corresponds to the conformal time at the beginning of the contraction phase or theexponential expansion). The function J m is a Bessel function of the first kind of order m ≡ / − β [recall that β isdefined in Eq. (10)].We now expand A k ( η ) in the limit where the proper wavelength associated to the modes becomes larger than theHubble radius H − , that is, in the limit − qη → λ k ≪ β varies accordingwhether inflation or a contracting phase is assumed. For inflation β ≃ − β ≃
2. Thisimplies that the dominant modes correspond to m ≃ / m ≃ − /
2. Furthermore, we are only interested in the expansion on the real part of A k ( η ). Therefore, for case (i) the expansion of the real part of A k ( η ) isRe A k ( η ) ≃ λ k k ( m −
1) ( − kη ) + ζ mk sin( πm + 2 mθ k )sin( πm ) kπ m Γ( m ) ( − kη ) m − , (A3)3and for case (ii) the corresponding expansion isRe A k ( η ) ≃ λ k k ( − m −
1) ( − kη ) + ζ − mk sin( πm + 2 mθ k )sin( πm ) kπ − m Γ( − m ) ( − kη ) − m − . (A4)In both cases, ζ k and θ k are given in Eq. (39).Next, we focus on the first term of Eq. (33), i.e. h ˆ v k i . It will be useful to define the following quantities: Q ≡ h ˆ v k i , R ≡ h ˆ p k i and S ≡ h ˆ p k ˆ v k + ˆ v k ˆ p k i . (A5)The equations of evolution for Q, R and S are obtained using Eq. (32), with ˆ Θ k = ˆ v k . That is, Q ′ = S + 2 βη Q, R ′ = − k S − βη R + λ k , S ′ = 2 R − k Q. (A6)Therefore, we have a linear system of coupled differential equations, whose general solution is a particular solutionto the system plus a solution to the homogeneous equation (with λ k = 0). After a long series of calculations we find: Q ( η ) = ( − kη )[ C J n ( − kη ) + C J − n ( − kη ) + C J n ( − kη ) J − n ( − kη )] + λ k η k , (A7)with n = 32 + 23 α ; and α ≡ ( ǫ − δ if assuming the inflationary Universe − ǫ if assuming the QMCU (A8)and the constants C , C and C are found by imposing the initial conditions corresponding to the Bunch-Daviesvacuum state: Q ( τ ) = 1 / (2 k ) , R ( τ ) = k/ S ( τ ) = 0. Equation (A7) is exact; expanding it again around − kη → Q ( η ) ≃ π k sin ( nπ ) (cid:26) k − λ k τ mλ k k sin ∆ cos ∆ (cid:27) n Γ (1 − n ) ( − kη ) − n +1 , (A9)where ∆ = − kτ − nπ − π . (A10)Using the above results, we can compute the quantity h ˆ v k i using Eq. (33). In case (i), we substitute Eqs. (A3)and (A9) into Eq. (33), obtaining h ˆ v k i = Q ( η ) − A k ( η ) ≃ π k sin ( nπ ) (cid:26) k − λ k τ mλ k k sin ∆ cos ∆ (cid:27) n Γ (1 − n ) ( − kη ) − n +1 − (cid:20) λ k k ( m −
1) ( − kη ) + ζ mk sin( πm + 2 mθ k )sin( πm ) kπ m Γ( m ) ( − kη ) m − (cid:21) − . (A11)With the expression in (A11) at hand (which is valid for h ˆ v R k i and h ˆ v I k i ), and using Eq. (30), our predicted scalarpower spectrum during inflation (at the lowest order in the slow roll parameter) is given in Eq. (35). (we have alsoused that during inflation z ( η ) ≃ ǫM P / ( H η ) and m = n = 3 / ǫ − δ ).Analogously, in case (ii), substituting Eqs. (A4) and (A9) into (33) yields h ˆ v k i = Q ( η ) − A k ( η ) ≃ π k sin ( nπ ) (cid:26) k − λ k τ mλ k k sin ∆ cos ∆ (cid:27) n Γ (1 − n ) ( − kη ) − n +1 − (cid:20) λ k k ( − m −
1) ( − kη ) + ζ − mk sin( πm + 2 mθ k )sin( πm ) kπ − m Γ( − m ) ( − kη ) − m − (cid:21) − . (A12)4Hence, substituting the above expression in Eq. (30) yields: P s ( k ) = ( − kη ) − µ s +3 π z η (cid:26) µ s π sin ( µ s π )Γ (1 − µ s ) (cid:20) − λ k τk − λ k k sin( − kτ ) cos( − kτ ) (cid:21) − " λ k µ s − k ( − kη ) − µ s +2 + ζ µ s k π sin( πµ s + 2 µ s θ k )sin( πµ s )2 µ s Γ ( µ s ) − (cid:27) , µ s ≡ − ǫ (A13)where we have used m = − n = 3 / − ǫ . As argued in Refs. [22, 23], during the quasi-matter contracting phase z ≃ η / (3 √
3) and | aH | = − /η , which implies that Eq. (A13) can be written as in the final form of the powerspectrum presented in Eq. (37)Let us focus now on the tensor modes. The action for the tensor perturbations is obtained from the Einstein-Hilbertaction by expanding the tensor perturbations h ij ( x , η ) up to second-order [67]. The resulting action for the tensorfield h ij ( x , η ) can be expressed in terms of its Fourier modes h ij ( k , η ) = h k ( η ) e ij ( k ), with e ij ( k ) representing atime-independent polarization tensor. Performing the change of variable h k ( η ) ≡ M P ( e ij e ji ) / v k ( η ) a ( η ) , (A14)the action can be written as δ (2) S h = R dη d k L h , where L h = v ′ k v ⋆ ′ k − k v k v ⋆ k − a ′ a (cid:16) v k v ⋆ ′ k + v ′ k v ⋆ k (cid:17) + (cid:18) a ′ a (cid:19) v k v ⋆ k . (A15)The Lagrangian in Eq. (9), and the one in Eq. (A15), share the same structure. In particular, if one replaces z ′ /z → a ′ /a in Eq. (9), one obtains Eq. (A15). Thus, the preceding calculations can be directly employed to obtainthe tensor power spectrum by replacing z ′ /z → a ′ /a in the whole procedure. The explicit form of this last quantity is a ′ ( η ) a ( η ) = ˜ βη ; where ˜ β ≡ ( − (1 + ǫ ) if assuming the inflationary Universe2(1 − ǫ ) if assuming the QMCU . (A16)Consequently, the replacement β → ˜ β , in the equations of the present subsection, allows us to obtain the tensor powerspectra shown in Eqs. (40) and (42).
2. The momentum ˆ p k as the collapse operator In the rest of this Appendix, we will present the mathematical details for obtaining the scalar and tensor powerspectra, using the CSL model, when the collapse operator is ˆ p k . As in the previous subsection, we will use theframework provided by the inflationary Universe and the QMCU simultaneously. This will complete the computationof the power spectra for the last two cases mentioned at the beginning of Sect. V, i.e. cases (iii) and (iv).Since in this subsection we are considering that the collapse operator is the momentum operator, ˆ Θ k = ˆ p R,I k , it isconvenient to work with the wave function in the momentum representation, Eq. (14). Moreover, as argued at thebeginning of this section, in that representation, the quantities of interest, namely h ˆ v R k i and h ˆ v I k i , can be calculatedusing Eq. (34). Furthermore, in spite of the collapse operator being the momentum operator, the calculations ofthe previous subsection serve as a blueprint for the computations in this subsection. Once again, we will proceed byomitting the indexes R , I .Let us focus first on the second term of the right hand side of Eq. (34), i.e. | ˜ A k ( η ) | / Re[ ˜ A k ( η )]. The motionequation for ˜ A k ( η ) is obtained from the CSL Eq. (31), which leads to˜ A ′ k ( η ) = i λ k + 2 βη ˜ A k − ik ˜ A k ( η ) . (A17)Performing the change of variable ˜ A k ≡ g ′ k ( η ) / [2 ik g k ( η )] in the last equation, results in a Bessel differential equationfor g k ( η ). Now, using such a solution and returning to the original variable, we have˜ A k ( η ) = q ik (cid:20) J m +1 ( − qη ) + e − imπ J − m − ( − qη ) J m ( − qη ) − e − imπ J − m ( − qη ) (cid:21) , m ≡ − − β, (A18)5where q ≡ k (1 − iλ k ). Also, we have used the initial condition provided by the Bunch-Davies vacuum, which is˜ A k ( τ ) = 1 / k with τ → −∞ . Do not confuse the m of Eq. (A18) with the one of the previous subsection.The next step is to perform the expansion for − qη →
0. Note that if we consider the inflationary Universe, then β ≃ −
1, which means that m ≃ /
2. On the other hand, for the QMCU case β ≃
2, which implies that m ≃ − / β is defined in Eq. (10)].In other words, for case (iii), we have the following expansion | ˜ A k ( η ) | Re[ A k ( η )] ≃ sin( πm )Γ ( m + 1)2 m ˜ ζ − mk ( − kη ) − m − k sin(2 m ˜ θ k + πm ) π . (A19)And in case (iv), the corresponding expansion results, | ˜ A k ( η ) | Re[ A k ( η )] ≃ c ˜ ζ k k " c ζ k cos(2˜ θ k )( − kη ) + c ζ k cos[2( m + 1)˜ θ k + πm ]( − kη ) − m − sin(2˜ θ k )( − kη ) m +2 + c ˜ ζ k sin(4˜ θ k )( − kη ) m +4 − c ˜ ζ − m − k sin[2 m ˜ θ k + πm ] ( − kη ) m +3 (A20)where c ≡ m + 1) , c ≡ ( m + 1)( m + 2) , c ≡ m +2 Γ( m + 2)Γ( − m ) . (A21)In both cases, the definitions of the quantities ˜ ζ k and ˜ θ k are given in Eq. (48).Now, we have to obtain the first term of the right hand side of Eq. (34), that is, h ˆ v k i . We will employ the sameprocedure as in the previous subsection. We use the previous definitions for the quantities Q ( η ), R ( η ) and S ( η ) Eq.(A5) and Eq. (32) but taking into account that Θ R,I k = ˆ p R,I k . Thus, the evolution equations are: Q ′ = S + 2 βη Q + λ k , R ′ = − k S − βη R, S ′ = 2 R − k Q. (A22)Those equations are solved using the initial conditions provided by Q ( τ ) = 1 / (2 k ), R ( τ ) = k/ S ( τ ) = 0.We are mainly interested in the solution for Q ( η ) ≡ h ˆ v k i [which is the first term on the right hand side of Eq. (34)].Then, performing the series expansion to the lowest order around − kη → Q ( η ) ≃ π k sin ( nπ ) (cid:26) k − k λ k τ mkλ k sin ∆ cos ∆ (cid:27) n Γ (1 − n ) ( − kη ) − n +1 (A23)where ∆ = − kτ − nπ − π and n is exactly the same as the one defined in Eq. (A8).We are now in position to compute the scalar power spectrum. For case (iii), substituting Eqs. (A19) and (A23)into Eq. (34) yields h ˆ v k i = Q ( η ) − | A k ( η ) | Re A k ( η ) ≃ π k sin ( nπ ) (cid:26) k − k λ k τ mkλ k sin ∆ cos ∆ (cid:27) n Γ (1 − n ) ( − kη ) − n +1 − sin( πm )Γ ( m + 1)2 m ζ − mk ( − kη ) − m − k sin(2 mθ k + πm ) π . (A24)In addition, by noting that m = 1 / ǫ − δ and n = 3 / ǫ − δ , we substitute Eq. (A24) in Eq. (30), which resultsin the final expression for the power spectrum shown in Eq. (44).For case (iv), we substitute Eq. (A20) and Eq. (A23) into Eq. (34) which results in h ˆ v k i = Q ( η ) − | A k ( η ) | Re A k ( η ) ≃ π k sin ( nπ ) (cid:26) k − k λ k τ mkλ k sin ∆ cos ∆ (cid:27) n Γ (1 − n ) ( − kη ) − n +1 − c ζ k k (cid:20) c ζ k cos(2 θ k )( − kη ) + c ζ k cos[2( m + 1) θ k + πm ]( − kη ) − m − sin(2 θ k )( − kη ) m +2 + c ζ k sin(4 θ k )( − kη ) m +4 − c ζ − m − k sin[2 mθ k + πm ] (cid:21) ( − kη ) m +3 . (A25)6Since in this case m = − / ǫ and n = 3 / − ǫ , and considering only the first dominant term in the expansionaround − kη →
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