Quasi-matter bounce and inflation in the light of the CSL model
The Continuous Spontaneous Localization (CSL) model has been proposed as a possible solution to the quantum measurement problem by modifying the Schrödinger equation. In this work, we apply the CSL model to two cosmological models of the early Universe: the matter bounce scenario and slow roll inflation. In particular, we focus on the generation of the classical primordial inhomogeneities 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 for the shape and the parameters characterizing the primordial spectra (scalar and tensor), i.e. the amplitude, the spectral index and the tensor-to-scalar ratio. We found that there exist CSL parameter values, allowed by other non-cosmological experiments, for which our predictions for the angular power spectrum of the CMB temperature anisotropy are consistent with the best fit canonical model to the latest data released by the Planck Collaboration.
After approximately three decades since the cosmological inflationary paradigm was conceived Guth (1981); Starobinsky (1980); Linde (1982); Albrecht and Steinhardt (1982), all of its generic predictions have withstood the confrontation with observational data, in particular, those coming from the Cosmic Microwave Background (CMB) radiation Ade et al. (2015a, b, c). That has led a large group of cosmologists to consider inflation as a well established theory of the early Universe. Inflation was originally proposed to provide a solution to the puzzles of the hot Big Bang theory (e.g. the horizon and flatness problems). However, the modern success of inflation is that, allegedly, it can offer us an explanation about the origin of the primordial inhomogeneities Mukhanov and Chibisov (1981); Starobinsky (1983); Hawking and Moss (1983); Bardeen et al. (1983). The standard argument is also rather pictorial: the quantum fluctuations of the vacuum associated to the inflaton are stretched out to cosmological scales due to the accelerated expansion of the spacetime; those fluctuations are considered the seeds of all large scale structures observed in the Universe. Furthermore, in Ref. Martin and Vennin (2016) is investigated the detectability of possible 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 fashionable model of the early Universe, that does not mean it is necessarily true Ijjas et al. (2013); Battefeld and Peter (2015). Furthermore, another feature that would make alternative models worthwhile of study is that they might avoid some long standing puzzles of the inflationary paradigm. Among those issues, we can mention: the subject of eternal inflation, a feature that is present in almost every model of inflation Vilenkin (1983) and which also leads to the controversial topic of the multiverse; the initial singularity problem and the trans-Planckian problem for primordial perturbations Martin and Brandenberger (2001), which are related by the fact that one is 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 be fine-tuned Ijjas et al. (2014); Ijjas and Steinhardt (2016). Although the aforementioned problems are not considered real problems by some scientists Linde (2015); Guth et al. (2014), others seem to disagree Brandenberger (2011); Ijjas and Steinhardt (2016). However, we think that if other alternative models can reproduce the main results linked 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 matter bounce scenario (MBS) Brandenberger (2011); Elizalde et al. (2015); de Haro and Cai (2015); de Haro and Amorós (2015); Haro (2013); Haro and Amoros (2014); Cai (2014); Cai et al. (2016). In this cosmological model, the initial singularity of the standard model is replaced by 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-phase of standard cosmology. The Universe described by the MBS relies on a single scalar field satisfying an equation of state that mimics that of a dust-like fluid. Additionally, in order to describe successfully a bouncing phase with a single scalar field, one needs to use cosmologies beyond the realm of General Relativity, such as, loop quantum cosmologies, teleparallel gravity or gravity. Proponents of the MBS claim that the potential associated to the scalar field is less fine-tuned than that of inflation, and also solves the historical problem of requiring very special initial conditions for the Big Bang model Elizalde et al. (2015); de Haro and Cai (2015), 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 provided in Ref. de Haro and Cai (2015). In spite of not being completely finished from a theoretical point of view, the MBS is quite simple in its treatment 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 regime where gravity is well described by General Relativity, and the perturbations correspond to inhomogeneities of a single scalar field.
In addition to the prior puzzles and successes of inflation and the MBS, there remains an important question: what is the precise physical mechanism that converts quantum fluctuations of the vacuum into classical perturbations of the spacetime? This question has been the subject of numerous works in the past and the consensus seems to favor the decoherence framework Kiefer and Polarski (2009); Halliwell (1989); Kiefer (2000); Polarski and Starobinsky (1996); Grishchuk and Sidorov (1990).111Although for the reasons exposed in Refs. Sudarsky (2011); Landau et al. (2013) we do not find such posture satisfactory. Nevertheless, decoherence cannot address that question by itself.222See comments by Mukhanov on pages 347-348 of Ref. Mukhanov (2005) and by Weinberg on page 476 of Ref. Weinberg (2008). In other words, even if one would choose (or not) to embrace the decoherence program, a particular interpretation of Quantum Mechanics must be selected (implicitly or explicitly). The Copenhagen–orthodox–interpretation requires to identify a notion of observer that performs a measurement on the system; which, in the decoherence framework, is equivalent to identify the unobservables or external degrees of freedom of the system. It is not clear how to do such identifications if the system is the early Universe. Other interpretations such as many-worlds, consistent histories and hidden variables formulations, might be adopted with varying degrees of success (see for instance Pinto-Neto et al. (2012); Valentini (2010); Goldstein et al. (2015)).
In the present article, in order to address the quantum-to-classical transition of the primordial perturbations, we will choose to work with the Continuous Spontaneous Localization (CSL) model. The CSL model belongs to a large class of models known as objective dynamical reduction models or simply called collapse models. Collapse models attempt to provide a solution to the measurement problem of Quantum Mechanics Ghirardi et al. (1986); Pearle (1989); Bassi and Ghirardi (2003); Bassi et al. (2013, 2012). The proponents of these models state that the measurement problem originates from the linear character of the quantum dynamics encoded in the Schrödinger equation. The common idea shared in these collapse models is to introduce some nonlinear stochastic corrections to the Schrödinger equation that breaks its linearity. According to the collapse models, a noise field couples nonlinearly with the system (usually with the spatial degree of freedom of the system), inducing a spontaneous random localization of the wave function in a sufficiently small region of the space. Suitably chosen collapse parameters make sure that micro-systems evolve essentially (but not exactly) following the dynamics provided by the Schrödinger equation, while macro-systems are extremely sensible to the nonlinear effects resulting in a perfectly localization of the wave function. Furthermore, there is no need to mention or to introduce a notion of an observer/measurement device as in the Copenhagen interpretation, which is a desired feature in the context of the early Universe and cosmology 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 Martin et al. (2012); Cañate et al. (2013); Das et al. (2013, 2014); León and Bengochea (2016). Also, recently a new effective collapse mechanism, independent of the CSL model, has been proposed to deal with the measurement problem during the inflationary era Alexander et al. (2016). However, among those works, the key role played by the collapse mechanism varies and also yields different predictions for the primordial power spectrum, some of which might be consistent with the observational data. In the present article, we will subscribe to the conceptual point of view first presented in Perez et al. (2006); Cañate et al. (2013), which was developed within the semiclassical gravity framework, and latter in Diez-Tejedor et al. (2012); León and Bengochea (2016) was extended to the standard quantization procedure of the primordial perturbations using the so called Mukhanov-Sasaki variable Mukhanov and Chibisov (1981); Kodama and Sasaki (1984). The main role that we advocate for the dynamical reduction mechanism of the state vector, modeled in this paper by the CSL model, is to directly generate the primordial curvature perturbations. Specifically, the initial state of the quantum field–the vacuum state–evolves dynamically according to the modified Schrödinger equation provided by the CSL model. This evolution leads to a final state that does not share the initial symmetry of the vacuum, i.e. it is not homogeneous and isotropic.333For a formal proof of this statement see Appendix A of Ref. Landau et al. (2013) and Appendix A of Ref. León and Bengochea (2016). In this way, the collapse mechanism generates the inhomogeneities and anisotropies of the matter fields. These asymmetries are codified in the evolved quantum state and, thus, are responsible for generating the perturbations of the spacetime.444We encourage the reader to consult Refs. Perez et al. (2006); Landau et al. (2013); León and Bengochea (2016) for a detailed exposition of the concepts involved in our approach and its relation with the collapse of the wave function.
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 fluctuations of the vacuum (or the squeezing of the field variables as usually argued). Therefore, in principle, it should be 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 that result from applying our framework in those alternative cosmological models. In the present article, we focus on the implementation of the CSL model within the framework of the MBS and, in parallel, we present the same appliance of the CSL model to the slow roll inflationary model of the early Universe. In this way, we can appreciate more clearly where 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 of the quantum world, which also avoids the standard quantum measurement problem, then it should also be possible to use it in different contexts from the traditional laboratory settings. The cosmological context provides a rich avenue to explore 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 the inflationary model of the early Universe, and we will compare the corresponding results with the one provided by the best fit standard cosmological model. Additionally, we will focus on the range of values allowed for the parameters of the CSL model, experimentally tested Toros and Bassi (2016a); Toros and Bassi (2016b) 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 CSL model, with particular emphasis on those that will be useful for the next sections. In Sect. III, we present the characterization of the primordial perturbations within the two cosmological models that we are considering, i.e. the MBS and standard slow roll inflation. In Sect. IV, we show the connection between the observational quantities and the theoretical predictions that result from adopting our conceptual point of view concerning the CSL model. In Sect. V, we explicitly show the implementation of the CSL model to the MBS and inflation, and we also present the predictions for the primordial power spectra (scalar and tensor) in each case. In Sect. VI, we discuss the implications of 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 best fit cosmological model. Finally, in Sect. VIII, we end with our conclusions. We include an Appendix containing the computational details that led to the results presented in Sect. V.
Ii A concise synopsis of the CSL model
In the CSL model, the modification of the Schrödinger equation induces a collapse of the wave function towards one of the possible eigenstates of an operator , called the collapse operator, with certain rate . The self-induced collapse is due to the interaction of the system with a background noise that can be considered as a continuous-time stochastic process of the Wiener kind. The modified Schrödinger equation drives the time evolution of an initial state as
with the time-ordering operator. The probability associated with a particular realization of is,
The norm of the state evolves dynamically, and Eq. (2) implies that the most probable state will be the one with the largest norm. From Eqs. (1) and (2), it can be derived the evolution equation of the density matrix operator . That is,
The density matrix operator can be used to obtain the ensemble average of the expectation value of an operator . Henceforth, from Eq. (3) it follows that
The average is over possible realizations of the noise , each realization corresponding to a single outcome of the final state .
One of the most important features of collapse models is the so-called amplification mechanism. That is, assuming that the reduction (collapse) rates for the constituents of a macroscopic object are equal (), it can be proved that the reduction rate for the center of mass of an -particle system is amplified by a factor of with respect to that of a single constituent Ghirardi et al. (1986, 1990); in other words, .
The parameter sets the strength of the collapse process. In the original model, proposed by Ghirardi-Rimmini-Webber (GRW), the authors suggested a value of s for nm. However, Adler suggested a greater value s for nm Adler (2007) (the parameter is called the correlation length of the noise and provides a measure for the spatial resolution of the collapse Ghirardi et al. (1986, 1990); Bassi and Ghirardi (2003)). Recent experiments have been devised to set bounds on the parameter Bahrami et al. (2014); Vinante et al. (2015). Furthermore, it is claimed that matter-wave interferometry provides the most generic way to experimentally test the collapse models Toros and Bassi (2016a); Toros and Bassi (2016b). Those results suggest that the range between and 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 Schrödinger equation is nonlinear and leads to a breakdown of the superposition principle for macroscopic objects; (2) The random nature of Quantum Mechanics is concealed in the noise and is consistent with Born’s rule; (3) An amplification mechanism exists, through the parameter which is related to the strength of the collapse. This strength is weak for microscopic objects 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 Lochan et al. (2012). The energy increase is minimal, e.g. for a particle of mass g, one obtains eV s Bassi and Ghirardi (2003). In other words, an increase of eV will take years. However, even if the energy increase can be ignored at the phenomenological 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 relativistic collapse models. Additionally, the collapse mechanism occurs in such a way that is nonlocal. This implies that the collapse of the wave function must be instantaneous or superluminal (but the nonlocal features cannot be exploited to send signals at superluminal speed). Also, the nonlocality is necessary to ensure that the models are consistent with the violation of Bell’s inequalities. Several relativistic models have been proposed so far Tumulka (2006); Bedingham et al. (2013); Pearle (2015), none of which can be considered completely finished. In spite of the lack of a relativistic collapse model, we will apply the CSL model to the primordial Universe, i.e. to inflation and the MBS, but in order to provide a more detailed picture, we need first to 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, that we will be considering in the rest of the manuscript. In particular, we are going to work with the following two scenarios:
An accelerated expansion of the early Universe given by the simplest inflationary model, that is, a single scalar field in the slow roll approximation with canonical kinetic term. Since such a model is probably very well known for most readers, we will not dwell into much detail here.
The MBS Elizalde et al. (2015); de Haro and Cai (2015); de Haro and Amorós (2015); Haro (2013); Haro and Amoros (2014), 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 in this model the primordial perturbations are born during the contracting stage of the Universe, we will focus exclusively on that cosmic stage. We will refer to such a stage as the quasi-matter contracting Universe (QMCU).
iii.1 The background
The inflationary Universe and the QMCU are both described by Einstein equations (), while the matter fields are characterized by a single scalar field. In the case of inflation the scalar field is the inflaton , and in the QMCU the scalar field will be denoted by .
As mentioned earlier, for inflation, we will consider standard slow roll inflation. In that case, the background spacetime is described by a quasi-de Sitter Universe, characterized by , with the conformal expansion rate, being the scale factor and the slow roll parameter is defined as ; a prime denotes partial derivative with respect to conformal time . The energy density of the Universe is dominated by the potential of the inflaton , and during slow roll inflation the condition is satisfied, with the reduced Planck mass. Since we will work in a full quasi-de Sitter expansion, another useful parameter to characterize slow roll inflation is the second slow roll parameter, i.e. .
In the case of the QMCU, the starting point is also a flat FLRW geometry that leads to the Friedmann and conservation equations. The field is separated into an homogeneous part plus small inhomogeneities . The homogeneous part satisfies
where is the potential associated to the field .
In the QMCU, it is assumed that the equation of state associated to the scalar field almost mimics that of ordinary matter, i.e. such that ; the latter implies . Consequently, the scale factor (in conformal time) evolves as .
The quasi-matter contraction is characterized with a small parameter , which plays the same role as the slow roll parameter in inflation. The parameter is defined as (see e.g. Elizalde et al. (2015))
The case corresponds to an exact matter-dominated contracting phase (note that ). Furthermore, for sake of completeness we introduce another parameter
such that . The parameter is analog to the parameter of slow roll inflation and it is related to the running of the spectral index in the QMCU model (see e.g. Elizalde et al. (2015)).
As is well known, it is not straightforward to accomplish a non-singular bounce within the framework of General Relativity by considering a single canonical scalar field, since the null energy condition (NEC) is violated (see for instance Battefeld and Peter (2015); de Haro and Cai (2015)). As a consequence, one possible option is to work with cosmologies within the context of modified gravity theories. In the case of the QMCU presented in Elizalde et al. (2015); de Haro and Cai (2015), the authors worked within the framework of holonomy corrected loop quantum cosmology and teleparallel gravity.
It is also important to note that even if a non-singular bounce cannot be achieved within General Relativity, the origin of the primordial perturbations is assumed to take place during the contracting (pre-bounce) phase of the Universe, where the curvature and energy scales are low enough to be described by General Relativity. On the other hand, 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 discuss this subject in more detail in the next section.
In the inflationary Universe and in the QMCU, one can separate the scalar field into an homogeneous part plus small inhomogeneous perturbations. Moreover, the metric associated to the spacetime, in both cases, is described by a FLRW background metric plus perturbations; which are classified as scalar, vector and tensor types (in this paper we will not consider vector perturbations). One useful quantity to describe the scalar (an also the tensor) perturbations is the so called Mukhanov-Sasaki (MS) variable. During inflation, the MS variable is defined by
with the gauge invariant quantity known as the Bardeen potential Bardeen (1980), which, in the longitudinal gauge, corresponds to the curvature perturbation. A similar expression to Eq. (8) can be used in the QMCU by replacing the fields and with and , respectively. The advantage of relying on the MS variable is that, when expanding the action of a scalar field minimally coupled to gravity into second order scalar perturbations, one obtains , where
with the Fourier modes associated to the MS variable, during inflation, and when considering the QMCU. However, it is important to note that during the bouncing phase, the action given by the Lagrangian in Eq. (9) remains the same but the expression for changes (see Ref. Haro (2013) for an explicit calculation within theories). On the other hand, during the contraction phase, the quantity can be written explicitly in terms of the QMCU parameter , as in a similar fashion using the slow roll inflation parameters, i.e.
Note that, since , the parameter does not enter into the expression at first order for the QMCU.
The CSL model is based on a nonlinear modification to the Schrödinger equation; consequently, it will be advantageous to perform the quantization of the perturbations in the Schrödinger picture, where the relevant physical objects are the Hamiltonian and the wave functional. The Hamiltonian associated to in Eq. (9) is , with
where the indexes denote the real and imaginary parts of and . The canonical conjugated momentum associated to is , i.e.
Since is a real field, .
We promote and to quantum operators, by imposing canonical commutations relations .
In the Schrödinger picture, the wave functional characterizes the state of the system. Furthermore, in Fourier space, the wave functional can be factorized into modes components . From now on, we will deal with each mode separately. Henceforth, each mode of the wave functional, associated to the real and imaginary parts of the canonical variables, satisfies the Schrödinger equation , with the Hamiltonian provided by (11). Note that one can also choose to work with the wave functional in the momentum representation, i.e. .
The standard assumption is that, at an early conformal time , the modes are in their adiabatic ground state, which is a Gaussian centered at zero with certain spread. This applies to both, the inflationary Universe and the QMCU. In addition, this ground state is commonly referred to as the Bunch-Davies vacuum. Thus, the conformal time is in the range .
Given that the initial quantum state is Gaussian, its shape will be preserved during its evolution. The explicit expression of the Gaussian state, in the field representation, is:
and, equivalently, in the momentum representation
Therefore, the wave functional evolves according to Schrödinger equation, with initial conditions given by corresponding to the Bunch-Davies vacuum, which is perfectly homogeneous and isotropic in the sense of a vacuum state in quantum field theory. The fact that we are introducing the wave functional in the field and momentum representations is related to the choice of the collapse operator in the CSL model, i.e., since there is no physical reason to choose one over the other, both choices are equally acceptable (at least from the phenomenological point of view). In the next section, we will show how to extract the physical 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, we proceed to identify the relevant physical quantities that will be related with the observed data. We encourage the reader to consult Refs. Perez et al. (2006); Sudarsky (2011); Landau et al. (2013) for a complete discussion regarding our full picture of the role played by the dynamical reduction of wave function in the cosmological setting. As a matter of fact, the relation between the observables and the predictions from the theory, using the Mukhanov-Sasaki variable during inflation and the CSL model, has been previously exposed in León and Bengochea (2016); however, in this section we reproduce the key arguments of such a reference to 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 initial homogeneity and isotropy associated to both, the quantum state and the spacetime. More specifically, we assume that a nonlinear modification to the Schrödinger 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 metric perturbations, which correspond to the primordial curvature perturbation.
Note that in the literature one can found statements suggesting that the vacuum fluctuations somehow become classical when the proper wavelength associated to the perturbations becomes larger than the Hubble radius Kiefer and Polarski (2009); Riotto (2002). Nevertheless, there is nothing in the dynamics governed by the traditional Schrödinger equation that can change the symmetry of the vacuum state, the symmetry being the homogeneity and isotropy. As a consequence, if the quantum state is perfectly symmetric and the Quantum Theory teaches us that the symmetries of a physical system must be encoded in the quantum state, then there is no clear way to describe the inhomogeneities and anisotropies of the spacetime in the quantum sense. If the quantum state of the system is perfectly symmetric, then its classical description must also be exactly symmetric. Thus, there is a lack of a proper explanation concerning the emergence of the primordial inhomogeneities and anisotropies in the Universe. That is why some non-standard interpretations of Quantum Mechanics, that make use of the Schrödinger equation (e.g. many-worlds, consistent histories, etc.), cannot provide a satisfactory answer to the problem at hand. It is important to note that the previous discussion applies to both cosmological models, the QMCU and inflation.
The modified Schrödinger equation given by the CSL model can successfully change the symmetries of the vacuum state and, at the same time, be responsible for the birth of the primordial curvature perturbation.
Specifically, in the comoving gauge, the curvature perturbation and the MS variable are related by . Thus, a quantization of implies a quantization of . The question that arises now is: how to relate the quantum objects and ? Furthermore, one may wonder how to relate the physical observables, such as the temperature anisotropies of the CMB, with the quantum objects that emerge from the quantum theory? The traditional answer relies on the quantum correlation functions, in particular, the two-point quantum correlation function and its relation with the two-point angular correlation function of the temperature anisotropies , where the bar denotes an average over different directions in the celestial sky and and are two unitary vectors denoting some particular directions. We do not find the previous answer to be completely satisfactory, and for a detailed explanation we invite the reader to consult Refs. Sudarsky (2011); Landau et al. (2013).
In order to illustrate our approach, we begin by focusing on the temperature anisotropies of the CMB observed today and its relation to the comoving classical curvature perturbation encoded in the quantity . Such a relation is approximately given by (i.e. for large angular scales)
On the other hand, the observational data are described in terms of the coefficients of the multipolar series expansion , i.e
here and are the coordinates on the celestial two-sphere, with the spherical harmonics.
Given Eq. (15), the coefficients can be further re-expressed in terms of the Fourier modes associated to , i.e.
where is the comoving radius of the last scattering surface and the spherical Bessel function of order of the first kind.
Finally, we can include the effects of late time physics that give rise to so called acoustic peaks. These effects are encoded in the transfer functions , and thus the coefficients are given by
where is the primordial comoving curvature perturbation. Also note that for large angular scales .
The next step is to relate with the quantum operator . Clearly, if one computes the vacuum expectation value and makes it exactly equal to , then one obtains precisely zero; while it is clear that for any given , the measured value of the quantity is not zero. As matter of fact, the standard argument is that it is not the quantity that is zero but the average . However, the notion of average is subtle, since in the CMB one has an average over different directions in the sky, while the average that one normally associates to the quantum expectation value of an operator is related to an average over possible outcomes of repeatedly measurements of an observable associated to 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 , i.e. the classical value associated to the Fourier mode of the comoving curvature perturbation , is an adequate description if the quantum state associated to each mode is sharply peaked around some particular value. In consequence, the classical value corresponds to the expectation value of in that particular “peaked” state Diez-Tejedor et al. (2012). In other words, our assumption is that the CSL mechanism will lead to a final state such that the relation
Therefore, in our approach, the coefficients in Eq. (18), will be given by
where corresponds to the evolved state according to the non-unitary modification of the Schrödinger equation provided by the CSL mechanism (see Refs. Martin et al. (2012); Das et al. (2013, 2014) for other ways to relate and using the CSL model, and León and Bengochea (2016) for a discussion on those approaches). Note also that does not share the same symmetries as the vacuum state, i.e. the inhomogeneity and isotropy of the system is encoded in the quantum state . Furthermore, Eq. (20) shows how the expectation value of the quantum field in the state acts as a source for the coefficients .
A well known observational quantity is the angular power spectrum defined by
We will assume that we can identify the observed value with the most likely value of obtained from the theory and, in turn, assume that the most likely value coincides approximately with the average . This average is over possible realizations or outcomes of the state that results from the CSL evolution. Thus, the observed approximately coincides with the theoretical prediction given in terms of the average , i.e.
Using Eq. (20), the theoretical prediction for the angular power spectrum is
Moreover, if the CSL evolution is such that there is no correlation between modes (which can be justified by the fact that we are working at linear order in cosmological perturbation theory), then
where denotes the real and imaginary part of the field (also, we assume that there is no correlation between and ). Therefore,
Performing the integral over the angular part of k and summing over , we obtain
On the other hand, the standard relation between the primordial power spectrum and the is given by
where is the dimensionless scalar power spectrum defined as
Note that the definition of the power spectrum, Eq. (28), is the canonical definition when dealing with classical random fields, where the average is over possible realizations of the random fields. In cosmology, the usual identification of the two–point quantum correlation function with 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:
Equation (30) is the key result from this section. It shows explicitly how to relate the quantities obtained from the quantum theory with the observed temperature anisotropies of the CMB. It also exhibits the difference between our approach 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 inflationary Universe, 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 . 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
with given in (11). Note that the Hamiltonian depends on the field which is defined in terms of the inflaton perturbations, but also it can be defined analogously using the perturbations of the scalar field associated to the QMCU [one has to take into account the change in ]. Furthermore, we will consider that Eq. (4) can be extrapolated to a generic quantum mode , that is, the real and imaginary parts of the mode satisfy:
At this point, we have to make a choice regarding the collapse operator . At first sight, the natural candidate is the MS variable, namely . Nevertheless, we think that in absence of a full relativistic CSL model, there is no a priori choice and, thus, the canonical conjugated momentum can also be considered as the collapse operator. In fact, in Ref. León and Bengochea (2016), we have shown that in the framework of the inflationary Universe, the momentum operator can be used as the collapse operator given that, in the longitudinal gauge, the momentum operator is directly related with the curvature perturbation.
Thus, we are going to consider four different cases:
The collapse operator is during slow roll inflation.
The collapse operator is during the QMCU.
The collapse operator is during slow roll inflation.
The collapse operator is during the QMCU.
We stress that only the third case, that is, the implementation of the CSL model within the inflationary framework using the field as the collapse operator, was first developed in Ref. León and Bengochea (2016). Nevertheless, we are including it in the present work for the sake of completeness. Note however that the analysis in Ref. León and Bengochea (2016) was done in the longitudinal gauge. In the present paper, we will work in the comoving gauge in all the four cases. The analysis of the three remaining cases, and in particular the implementation of the CSL model during a contracting phase of the early Universe, 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 is identical to . Consequently, we will omit from now on the indexes unless it creates confusion.
The quantity is the standard deviation of the squared field variable . It is also the width of every packet 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
For cases (i) and (ii), it is convenient to work with Eq. (33); and for cases (iii) and (iv) with Eq. (34). Thus, to calculate , we only need to find the two terms on the right hand side of (33) or (34), respectively. The second term on the right hand side of both equations can be found from the CSL evolution equation, Eq. (31), while the first one by using Eq. (32) with the wave function in the corresponding representation. Also, in Eqs. (33) and (34), we consider the regime , which correspond to the range of observational interest, that is, the regime for which the 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 give a specific prediction for the scalar power spectrum. The actual calculations are long, so we have included them in Appendix 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):
For case (ii), we have
In both cases, (i) and (ii), we have also defined:
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
where . Therefore, the tensor-to-scalar ratio , at the lowest order in the slow roll parameter, is given by
which is exactly the same prediction as in the standard inflationary slow roll scenario.
Meanwhile, in case (ii), the tensor power spectrum is
where . Also, for very low energy densities and curvatures, (see Ref. Elizalde et al. (2015)). The tensor-to-scalar ratio is given by
which is also the same as the one presented in Refs. Elizalde et al. (2015); de Haro and Cai (2015). Note that we have evaluated the upper limit of the integrals at . The motivation is essentially the same as the one given in Refs. Elizalde et al. (2015); de Haro and Cai (2015). That is, one evaluates the scalar and power spectra at very late times corresponding to when the mode “re-enters the horizon”, or more precisely when during the expanding (post-bounce) phase.
The previously presented cases (i) and (ii) correspond to selecting as the collapse operator. Next, we focus on the results for cases (iii) and (iv), which correspond to choose as the collapse operator.
For case (iii), we obtain:
where we have defined and
In case (iv), the corresponding expression results
with the definitions and
The constants and are shown in Appendix A. In both cases, (iii) and (iv), we have the following definitions
The predictions for the tensor-to-scalar ratios are exactly the same as the ones presented in cases (i) and (ii) (see Appendix A).
We end this section by summarizing the main results. We have applied the CSL model to the inflationary Universe and to the QMCU. Moreover, in order to employ the CSL model, we need to choose the collapse operator. We have chosen to work with and as the collapse operators. Henceforth, we have obtained the scalar power spectra in four different cases Eqs. (35), (37), (44) and (46). On the other hand, introducing the CSL mechanism does not affect the tensor-to-scalar ratio . Specifically, if one works within the standard inflationary scenario, then the prediction for is equal to the standard one given by slow roll inflation; meanwhile, if one adopts the QMCU framework, then the predictions are equal to the ones presented in Refs. Elizalde et al. (2015); de Haro and Cai (2015).
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, we will compare our predicted scalar power spectra with the standard one.
where is a pivot scale, and the amplitude and the spectral index are given by
On the other hand, we have four different expressions for the scalar power spectrum, corresponding to the four cases mentioned at the beginning of Sect. V. In the following, we will analyze each one of them, but first we will make a few observations regarding the parameter .
The dependence on in the parameter encodes the “amplification mechanism”, which is characteristic of dynamical reduction models (see Refs. Das et al. (2013, 2014)). One possible way to determine the exact dependence on , and perhaps the 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 change depending on the fundamental units associated to the collapse operator . Moreover, we expect that is directly related to , i.e. the CSL parameter, which clearly must be the same in all physical situations (cosmological or otherwise). Moreover, taking into account that we are working in units in which , the fundamental dimension of is [Length].
Thus, in the case where the collapse operator is chosen to be , the most natural expression of , which is consistent with the dimensions of all terms involving the dynamical equations, is
And in the case where the selected collapse operator is , such an expression is
where is the CSL parameter, with the same numerical value in all cases. From now on, we will assume that takes the form of Eqs. (51) and (52) depending on the chosen operator acting as the collapse operator.
vi.1 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 the one 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 (i.e. ) during the inflationary epoch. Furthermore, the different coefficients that multiply each term of the function involve the quantity . For these terms, we can approximate without loss of generality. However, note that such approximation cannot be done to the powers of involving because these are directly related to the scalar spectral index , for which the value is ruled out. Furthermore, in order to provide a suitable normalization for the CSL power spectra, we multiply and divide by the quantity . Thus, the power spectrum in Eq. (35) can be rewritten as:
and ; that is,