PHYSICAL REVIEW D 88, 023526 (2013) Quantum origin of the primordial fluctuation spectrum and its statistics CORE Metadata, citation and similar papers at core.ac.uk Provided by CONICET Digital Susana Landau, 1, * Gabriel Leo´n, 2,3, † and Daniel Sudarsky 3,4, ‡ 1Instituto de Fı´sica de Buenos Aires, CONICET-UBA, Ciudad Universitaria—Pab. 1, 1428 Buenos Aires, Argentina 2Dipartimento di Fisica, Universita` di Trieste, Strada Costiera 11, 34151 Trieste, Italy 3Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de Me´xico, Me´xico D.F. 04510, Mexico 4Instituto de Astronomı´a y Fı´sica del Espacio, Casilla de Correos 67, Sucursal 28, 1428 Buenos Aires, Argentina (Received 3 July 2012; revised manuscript received 23 April 2013; published 26 July 2013) The usual account for the origin of cosmic structure during inflation is not fully satisfactory, as it lacks a physical mechanism capable of generating the inhomogeneity and anisotropy of our Universe, from an exactly homogeneous and isotropic initial stateassociated withthe early inflationary regime. The proposalin[A. Perez, H. Sahlmann, and D. Sudarsky, Classical Quantum Gravity 23 , 2317 (2006)] considers the spontaneous dynamical collapse of the wave function as a possible answer to that problem. In this work, we review briefly the difficulties facing the standard approach, as well as the answers provided by the above proposal and explore their relevance to the investigations concerning the characterization of the primordial spectrum and other statistical aspects of the cosmic microwave background and large-scale matter distribution. We will see that the new approach leads to novel ways of considering some of the relevant questions, and, in particular, to distinct characterizations of the non-Gaussianities that might have left imprints on the available data.

DOI: 10.1103/PhysRevD.88.023526 PACS numbers: 98.80.Cq, 98.80. k

I. INTRODUCTION of anisotropies and inhomogeneities that characterize our late Universe. This issue is sometimes characterized as the At what point in the cosmic evolution do the actual ‘‘transition from the quantum regime to the classical re- primordial inhomogeneities arise? In other words, when gime,’’ but we find this a bit misleading: most people would does our Universe depart from the exceedingly high homo- agree that there exist no distinct and separated classical and 1 geneity and isotropy that is thought to result from the first quantum regimes. The fundamental description ought to be stages of inflation? This is a question that one might expect always quantum mechanical; the so-called classical regimes should be addressed, at least, in principle, by any theory that are those in which certain quantities can be described to a deals with the emergence of cosmic structure. Yet in the sufficient accuracy by their classical counterparts represent- standard inflationary account [ 1], which is nowadays re- ing the corresponding quantum expectation values. The garded as a remarkable success, the context in which such paradigmatic example of this classical regime is provided issues can be addressed seems to be simply absent [ 2]. That by the coherent states of a harmonic oscillator, which cor- is, within the orthodox accounts, one cannot identify the respond to minimal wave packets with expectation values of physical process responsible for the generation of those position and momentum that follow the classical equations features in our Universe. In fact, according to the infla- of motion (Ehrenfest theorem). In any case, it seems clear tionary paradigm, from a relatively wide initial set of pos- that from a situation corresponding to a H&I background, sibilities marking the end of the mysterious quantum gravity and quantum fields characterized by a H&I state, one cannot era, the accelerated inflationary burst leads to a homoge- end up—in the absence of something else, which in other neous and isotropic (H&I) Universe where the quantum circumstances would be identified as a measurement, but fields are all characterized by the equally homogeneous which clearly cannot be invoked in the present setting 3—in a and isotropic vacuum states (usually taken specifically to situation that is characterized, at any level, as containing be the so-called Bunch-Davies vacuum). From these con- actual inhomogeneities and anisotropies. It is clear that, in 2 ditions, it is usually argued, in a rather unclear although terms of the standard dynamics, such a transition cannot strongly image-evoking manner, that the ‘‘quantum fluctua- be accounted for by anything that relies just on the tions’’ present in such a quantum state morph into the seeds gravity/inflaton action, 4 which is known to preserve such

*[email protected] †[email protected] 3Observers and measuring apparatuses are only possible well ‡[email protected] after the H&I has been broken, so those can hardly be part of the 1The level of inhomogeneity that might still be present at any cause of the breakdown. point during inflation is expected to be of order eN, where N is 4In fact, even the interaction with other fields, controlled by the number of e-folds of inflation occurred up to that point. the usual symmetry preserving dynamics, cannot account for the 2Acknowledgments that this is an unclear aspect of the stan- emergence of inhomogeneities and anisotropies, since, accord- dard approach can be seen, for instance, in the book Cosmology ing to the inflationary paradigm, the state for all fields should by Weinberg [ 3], where the author explicitly states his view on correspond to a homogenous and isotropic state such as the the subject. Bunch-Davies vacuum.

1550-7998 =2013 =88(2) =023526(27) 023526-1 Ó 2013 American Physical Society SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) symmetries. Simply put, if the initial state is H&I and the the standard inflationary paradigm, which we have been Schro¨dinger evolution is tied to a Hamiltonian that preserves advocating in previous works [ 4,8–11 ]. The basis of that these symmetries, the resulting state cannot be anything but proposal is to modify the standard inflationary paradigm a H&I state (see Appendix A). Nonetheless, various types of with the incision of a modified quantum mechanics that arguments are often put forward in attempts to bypass the involves the spontaneous collapse of the wave function. above conclusion. Most cosmologists adopt a posture that We should note, however, that we cannot escape from attributes to decoherence the role of explaining the emer- the related problems, even if we choose to adopt a very gence of inhomogeneities and anisotropies from the H&I ‘‘pragmatic position’’: Assume, that one chooses to ignore state. This approach faces several problems: the shortcomings of the standard accounts [ 5] and accepts (i) The decoherence program is based on partitioning that, say, decoherence addresses somehow the issue at hand the degrees of freedom in two categories—The de- and that the mystery lies ‘‘only’’ in the question concerning grees of freedom of the environment and the degrees the precise mechanism that lies behind the fact that, from of freedom of the system of interest. In the cosmo- the options exhibited in those analyses (i.e., the options logical case however, in which one cannot evoke displayed in the diagonal reduced-density matrix; see observers or measurements, the way to do the sepa- Appendix D), one single realization seems to be selected ration of the degrees of freedom is rather ad hoc . [12 ] for our Universe. In adopting such a point of view, one (ii) In order to argue that the symmetry was broken, would be assuming that the initial symmetry has been lost one needs to assume that the world is suddenly (at least for practical purposes) in association with that represented by one of the elements appearing in particular ‘‘realization’’ or ‘‘actualization’’ (represented by the diagonal of the deciphering density matrix, a particular element in the density matrix). Thus, it seems and it is not clear how to argue for that. clear that, for the sake of self-consistency, one should (iii) Sometimes people evoke the many worlds interpre- consider, when studying aspects of the inhomogeneity tation of quantum theory in order to deal with the and anisotropies in the cosmic microwave background previous point but seem to ignore that, in order to (CMB) we observe, the state corresponding to such real- do that, one needs to choose a privileged basis ization or actualization, and not the complete vacuum state, which describes the H&I state of affairs previous to the associated with the world splittings, and that choice, 5 in practice, is tied to the notion of conscience, again actualization. In following such views, the discussion that a notion that cannot be invoked in the context at we are presenting in this paper would have to be taken to hand. Another popular posture is to rely on the represent the effective description corresponding to ‘‘our consistent histories approach, ignoring the problem- perceived Universe’’ (in a context in which one puts atic issues afflicting that proposal. In particular, we together something like the many worlds interpretation, with the arguments based on decoherence). Although we should note that the usage of the formalism requires definitely do not adhere to such a view for the reasons a choice of realm, a choice that in the current context explained in Ref. [ 5], it is clear that when accepting a seems completely arbitrary. In fact, one can make description, such as the one presented above, one would one such choice when one is led to the conclusion have to use the characteristics of the selected state in order that the Universe is, even today, perfectly homoge- to estimate the details of the inhomogeneities and anisot- nous and isotropic (see Appendix D). ropies in the cosmic structure and its imprints in the CMB. The extended discussion of the conceptual problems As we indicated, the purpose of this paper is to discuss inherent to quantum theory and those associated to its the manner in which the consideration of statistical aspects application to the cosmological situation at hand have of the CMB and the large-scale matter distributions should been presented in previous works by some of us and by others in Refs. [ 4–6]. The main message is that the problem 5 we face is tied with the so-called measurement problem of The reliance on a particular realization or actualization refers, of course, to the fact that, according to the standard arguments, quantum theory and that this problem becomes exacer- the resulting density matrix, after becoming essentially diagonal bated in the present case, in which we are dealing with due to decoherence, would be taken to represent an ensemble of cosmology, a field in which the standard ways to address universes, with our particular one corresponding to one of the such problems are simply unavailable [ 7]. In this work, we elements occurring in the diagonal of that matrix. That state reproduce all those arguments in detail, mentioning them should then be considered as somehow ‘‘selected by nature’’ to become realized (or to be the one we perceive). If one wanted to only briefly, as the main objective of the present manu- consider the issue at a deeper level, one would have to face the script is to focus in the statistical aspects that emerge in question of what such actualization represents at the theoretical this context (a slightly more detailed discussion of those is level and what is, if any, the physics that controls it. offered in Appendix D for the benefit of the reader). Alternatively, one might take the view (often referred as the many worlds interpretation) that these other universes are some- We will discuss a new way of looking at those issues, how also realized, and thus they exist in realms completely based on what we consider to be a conceptually more inaccessible to us. In that case, the actualization corresponds transparent picture that relies on a modified version of to that Universe in which we happen to exist.

023526-2 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) be modified when taking into account the modifications to the kind of statistics one considers at the observational needed to explain the emergence of inhomogeneities and level, and which kind of statistics one is dealing with. That anisotropies in terms of theories incorporating something is, in the standard approach, our specific Universe is not like the spontaneous collapse of the wave function. The described in any sense (not even in terms of unknown yet need to rely on a different approach to study things like explicitly identified quantities), and the randomness that the non-Gaussianities in the CMB, arises, in part, due to the one invokes, as characterizing the relation of theory and vastly larger space of possibilities for exotic effects, which observation, lies hidden in unspecified aspects associated opens in connection to the unknown dynamics of the with the vagueness of the interpretations. In other words, collapse processes. In other words, in the standard treat- one cannot identify the random variables; one does not ments, the spectrum would be determined by the inflationary know how many there are, and one cannot say how exactly theory (number of fields, kinetic terms, and interacting the various elements of the ensemble of Universes differ potentials), and the nature of the initial state, while in the from each other. One imagines an ensemble of universes approach we have been advocating a novel source of statis- and assumes that their collective departure from H&I is tical anomalies, is provided by the details of the modifica- somehow characterized by the H&I vacuum state or the tion of quantum theory by the dynamics of collapse. state that results form the unitary evolution thereof (despite One example of these novel possibilities is provided by such a state being homogeneous and isotropic). One then the study of the details of the mode by mode collapse considers that the ensemble is being described, while each within the semiclassical treatment of the problem as de- of the individual elements of the ensemble cannot be veloped in Ref. [ 13 ]. In that work, it was found the collapse described, or that its description is irrelevant. of a mode with comoving wave vector k~0 must be tied with Within such setting, one proceeds to make, either ex- the modification of the state of the field in the higher plicitly or implicitly, the assumption that statistics over harmonics of that mode. It was found, in particular, that such an ensemble correspond to the statistics, over time, the effect would be stronger for mode 2k~0. This, in turn, over space, or over orientation, in our particular Universe. leads to the consideration of the possibility of strong In fact, one assumes that they are all equivalent. It should correlations in the collapse parameters of the two modes, be clear that such assumptions are, therefore, taken to say an effect that would produce a particular type of exotic something about the individual element of the ensemble, correlations—it is unclear if they should be called and it is not completely clear what it is. If our Universe is non-Gaussianities as they involve modifications of the not described by the quantum state we use in our equations, two-point functions—something that would produce a what can we say about it? In order to look for justification particular kind of signature in the CMB [ 14 ]. and clarification of such identifications, we must turn to the The organization of this manuscript is as follows: In quantum theory from which one expects to extract the Sec. II , we offer a preliminary discussion of the posture we predictions. The problem is that, while quantum theory advocate regarding the emergence of structure and its has a clear and workable interpretation (even if not com- implications for the statistical analysis of the CMB and pletely satisfactory [ 15 ]) for dealing with laboratory ex- some aspects of the usual approach focusing on the aspects periments (the Copenhagen interpretation), for which the we consider to be conceptually unclear. In Sec. III , we measuring devices and observers can be taken as clearly review the standard picture for primordial non- identified, for the case of the cosmological problem at Gaussianities. In Sec. IV , we review the collapse models hand, we are faced with a situation deprived of such description for the inflationary origin of the seeds of the entities that normally provide an interpretation. cosmic structure. In Sec. V, we focus on the statistical Thus, the issue we will be addressing cannot be turned aspects as seen from our perspective of the primordial into one of ‘‘measurement,’’ while implicitly assuming that inhomogeneities, propose new characterizations of the such concept can be used in the delicate quantum mechani- non-Gaussianities, and discuss new measures to be asso- cal context examined in this paper. This is simply because ciated with the bispectrum. Finally, in Sec. VI , we discuss as we have already noted, cosmology needs to account for our findings. We use conventions in which the signature the emergence of the conditions 6 that make such things as of the space-time metric is ð ; þ; þ; þÞ and units where observers and apparatuses possible to start with. c ¼ 1 but will keep the gravitational constant G and ℏ In order to fully and satisfactorily address the problem at appearing explicitly throughout the paper. hand, it seems we need to be able to point out ‘‘what exactly is wrong with the argument leading to the conclu- II. SOME PRELIMINARIES ON THE sion drawn above. In other words, where does nature EMERGENCE OF FEATURES OF THE CMB AND STATISTICAL CONSIDERATIONS 6Primordial inhomogeneities are supposed to evolve into gal- axies and galaxy clusters, and within galaxies, stars and planets Let us start this section by noting that, in the usual are supposed to arise by gravitational collapse, and then life is accounts, it is hard to pinpoint where exactly the statistical supposed to arise in the appropriate circumstances on some aspects enter at the theoretical level, how that is connected planets, particularly on Earth.

023526-3 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) deviate from the theory leading to the erroneous conclusion Ref. [ 4]. We should emphasize, that although most of our that our Universe is, even today, at the fundamental work has centered on that rather simplistic collapse model quantum level, perfectly homogeneous and isotropic?’’ It developed specifically for the cosmological problem at follows that such explanation must indicate where the hand, the discussion of most of this paper would apply ordinary U evolution—with the symmetry preserving equally to more general models and, in particular, to Hamiltonian—breaks down. approaches based on exciting proposals like Ghirardi- We can easily see that none of the proposals to deal with Rimini-Weber [ 18 ] and continuous spontaneous localiza- the issue, and which are based on the standard paradigms, tion [ 23 ]. In fact, some recent works are devoted to the can single out any point where that breakdown might occur adaptation of the continuous spontaneous localization the- or, much less, point to a physical reason for that departure ory for its application to the problem of the emergence of from standard quantum theory (we turn the interested inhomogeneities and anisotropies in cosmology [ 24 ,25 ]. reader toward Appendix C, where we explore in more Here, we want to focus on the impact of such ideas on detail these issues and justify more precisely our point of the statistical study of the CMB. We will discuss some view). delicate interpretational aspects related to quantum theory, This has led us to take a view that ties this problem with its implicit usage in the standard approach to the study of the ideas advocated by L. Diosi and R. Penrose, which the CMB, and its characterization in terms of a spectra as argue [ 16 ,17 ] that quantum theory should itself suffer well as the accounts of the origin of cosmic structure. We modifications as a result of its combination with the fun- will briefly explore here, for the first time, some of the damental theory of space-time structure. 7 Among the as- basic differences associated with statistical considerations, pects of the theory that would be substantially affected between those tied to the usual approach and those appro- according to those views are those related to the reduction priate to our proposal. postulate (or R process) and its contrast with the unitary In order to make things a bit more explicit, let us start by evolution (or U process) controlled by Schro¨dinger’s equa- reminding the reader that in the standard approaches, the tion. In fact, the issue of dynamical quantum reduction has study of the statistical nature of the problem is based on the n received a lot of attention within the community working study of the statistical -point functions of the Newtonian . . . in foundational aspects of quantum theory, and there are, in potential, ðx1Þ ðxnÞ, with the overline denoting the the existing literature, several rather well-defined pro- average over an ensemble of universes. Having no access posals in this regard, such as those in Refs. [ 15 ,16 ,18 –20 ]. to such ensemble, one needs to face the issue of what n The proposal behind our work is based on the hypothesis the relationship between those -point functions and the that a dynamical collapse of the wave function lies behind quantities we actually measure is. Moreover, one needs the breakdown of the initial homogeneity and isotropy. In to consider how these quantities are connected with the n other words, a nonunitary ‘‘jump’’ in the quantum state of quantum -point functions. The usual approach [ 26 ,27 ] the system plays a role in transforming the inflaton vacuum relies on the identification into a quantum state that lacks the translational and rota- ^ ^ ðx1Þ . . . ðx Þ ¼ h 0jðx1Þ . . . ðx Þj 0i; (1) tional symmetries of the former state. n n It goes without saying that we cannot, at this stage, try or ^ ^ where h0jðx1Þ . . . ðx Þj 0i is a standard quantum me- hope to point out the precise physical origin of such n 8 chanical n-point function for the quantum field operators dynamical collapse. However, once one has accepted (corresponding to the vacuum state at hand). That is, one is that something of this sort is occurring, one can parame- making the identification of quantum and statistical n-point trize its basic characteristics and use the relevant observa- functions. As we said, the latter are naturally associated tional data to infer something about the nature of the novel with an ensemble of universes, all of which, even if real, physics that must lie behind such phenomena. This has are unaccessible to us. The usual line of argument contin- been the basic attitude behind the program started in ues by invoking ergodic arguments, to make a further connection between ensemble averages and time averages, 7This is what is often thought of as quantum gravity. We did with other vague arguments indicating one might replace not use that term because that often presupposes that one is the latter with spatial averages and often turning, in prac- considering the relevant theory to be simply the adaptation of tice, to orientation averages. On the other hand, at the general relativity to the standard quantum theory, while what one has in mind, when following Diosi and Penrose’s ideas, is quantum level, the interpretation is, as we noted before, something much more distant from known physics, involving, even more problematic. In the standard laboratory situ- as indicated, modifications of quantum theory itself. ations, one has an apparatus designed to measure a certain 8In particular, collapse theories are known to face, in principle, observable O, and quantum theory then indicates that, in serious difficulties with Lorentz and general covariance and each individual measurement, one would obtain an eigen- issues related to conservation laws. However, important advan- ces have been made in addressing both classes of issues value of the corresponding operator. Furthermore, imme- (Refs. [ 21 ,22 ]), even if we cannot say we have at our disposal diately after the measurement, the individual system is anything resembling a completely satisfactory theory. taken to be in the state corresponding to the resulting

023526-4 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) eigenvalue (as immediate repetition of the same T measurement in such an individual system gives, with ; ’ lm Ylm ; ’ ; T0 ð Þ ¼ Xlm ð Þ probability 1, the exact same value). The sudden change (5) in the state of the system is known as the state function T lm ; ’ Ylm ; ’ d; reduction or wave function collapse and is thought as being ¼ Z T0 ð Þ ð Þ brought up by the measurement (in fact, the interpretation is not fully satisfactory, but we have become used to the here, and ’ are the coordinates on the celestial two-sphere, with Y ; ’ as the spherical harmonics. fact that in laboratory situations it works). Moreover, the lm ð Þ quantum expectation value of the observable O in The value for the quantities lm are then given by h j j i the state (the system’s state before the measurement) l 3 j i 4i d k ^ should be equal to the average of the observed values of the lm 3 jl kR D Ylm k k ~ R ; (6) ¼ 3 Z 2 ð Þ ð Þ ð Þ kð Þ corresponding quantity, over a large enough ensemble of ð Þ identical systems. It is important to note here that such an with jl kR D as the spherical Bessel function of order l; R interpretational scheme works only as long as a clearly is the conformalð Þ time of reheating, which can be associated identified measurement is involved, as one essential aspect with the end of the inflationary regime, and RD is the of the nature of the quantum world is that one cannot comoving radius of the last scattering surface. We have consistently adopt a point of view advocating that the explicitly included the modifications associated with late- measurement simply served to reveal a preexisting value time physics encoded in the transfer functions k . of such a quantity (see, for instance, Ref. [ 28 ]). Now, the problem is that, if we compute the expectationð Þ Let us illustrate these and other related issues by value of the right-hand side (i.e., identifying ^ ) in considering the simplest place where one can appreciate the vacuum state 0 , we obtain 0, while it is clearh i ¼ that for the problematic aspects of such identifications: the case of any given l, m, thej measuredi value of this quantity is not 0. 9 the one-point function. Let us focus here on the standard That is, if we rely in this case on the one-point function and treatment that relies on the so-called Mukhanov-Sasaki the standard identification, we find a large conflict between variables, defined by expectation and observation. We might even be tempted to _ say that evidence of non-Gaussianity has already been a 0 u ; v a ; (2) observed in each measurement of a particular . This 4 _ þ H lm G 0 is, of course, not what one wants. Advocates of the stan- dard approach would indicate that 0 is not to be where is the metric perturbation known as the h lm i ¼ _ taken as ‘‘the prediction of the approach’’ regarding our Newtonian potential, 0 is the derivative of the back- ground inflaton with respect to conformal time , is Universe and that this would only hold for an ensemble of the perturbation in the inflaton field, a is the scale factor, universes. The issue, of course, is what precise interpreta- and H a_ (related to the standard Hubble parameter H tional posture regarding the theory can be used to justify a this, while at the same time justifying the positions taken through H aH ). Einstein’s equations then lead to ¼ _ vis-a`-vis the other quantities that emerge from the theory u z v and v 1 zu , where z a0 . Given the ¼ ðzÞ ¼ z ð Þ H (such as the higher n-point functions). A theory that de- equations of motion, the Newtonian potential can thus be pends on a case by case adaptation of an interpretational expressed in terms of the field v x;~ and its momentum rule is not a very satisfactory theory. However, this makes ð Þ conjugate v x;~ v_ x;~ . The expression for the clear that disentangling the various statistical aspects ð Þ ¼ ð Þ corresponding Fourier components is (ensemble statistics; space and time statistics, including orientation statistics; and, finally, the nature of the assumed p4G H z_ connection of quantum and statistical aspects) and making ~ ~ v ~ ; (3) kð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffik2 vkð Þ z kð Þ explicit the assumptions underlying the identifications, as well as the expected limitations, is paramount to avoid where is the so-called slow-roll parameter confusion and to allow the judging of a theory on its true _ 1 H =H 2. merits. We are interested in the temperature anisotropies of As a matter of fact, it seems clear that anything that can the CMB observed today on the celestial two-sphere, be considered as a satisfactory approach should enable which are related to the inhomogeneities in the Newtonian one to understand what exactly is wrong with the above potential on the last scattering surface, 9 T 1 We are ignoring the remote possibility that, just by coinci- ; ’ D; x~D : (4) dence, and for some specific l and m, the quantity lm would T0 ð Þ ¼ 3 ð Þ vanish within the observational margin of error. As can be seen in Sec. IV , according to our point of view, that would require a The data are described in terms of the coefficients lm of remarkable cancelation between terms determined by a large the multipolar series expansion collection of random numbers.

023526-5 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) argument. First, let us note that, just as the Fourier ‘‘measurement problem’’ is quite unsettling. However, in transform of a function is a weighted average (with weight the case at hand, the problem is exacerbated because we ik~ x~ e ), so are the spherical harmonic transforms of cannot even contemplate any physical observer or measur- functions. Thus, lm is a weighted average over the last ing device existing prior to the emergence of the seeds of scattering surface (cosmologists often refer to the average structure. Thus, we cannot even rely on our old battle tool: over the sky) because it is an integral over the celestial the Copenhagen interpretation, which explains the non- T vanishing of those quantities that predates both the growth two-sphere of T weighted with a given function, the Ylm . The common argument in the literature, as we have noted, of galaxies and the existence of ‘‘observers and measuring indicates that averaging over the sky justifies the identifi- devices.’’ cation of observations with quantum expectation values. Regarding the second issue, we would like to comment In other words, the argument indicates that the relevant the work of Armendariz-Picon [ 29 ], which starts to address prediction (obtained in terms of quantum expectation val- (albeit in a rather limited way, because the analysis is done ues) concerns the ensemble averages, and these should be for a very small number of values of l) that question. The equal to spatial averages and the latter to averages over the results of this work indicate that the l are small (1 order of sky. However, apparently, this should not hold for weighted magnitude smaller than the variance of the lm , that is, averages over the sky (otherwise, all the lm ’s would be 0). pCl), and that seems reassuring. But is this sufficient? Is If not, why not? There seems to be no clear answer. thatffiffiffiffiffi what we should expect according to our theory? Why? Namely, if we take the theoretical estimate as Should it not be zero up to the actual experimental errors 10 in the observations? Evidently, these are just rhetorical 3 4i l d k questions, raised only to show that it is easy to be confused th j kR Y k^ k 0 ^ 0 0 lm ¼ 3 Z 2 3 lð DÞ lm ð Þ ð Þh j k~ð RÞj i ¼ regarding the comparisons of theory and observations, if ð Þ (7) one accepts, without questioning, the usual arguments given by the standard approach. It seems evident that, in obs and compare it with the measured quantity lm , we would order to have a clear answer to those questions, one needs find a large discrepancy. The answer, within the standard to have a precise and unambiguous characterization of accounts, would need to be that, for some reason, in order what exactly the mapping between the theory and the to be allowed to make identifications, we should invoke a measured quantities is. Actually, it seems one would further averaging: the average over orientations. Only then need to consider those comparisons as tests of whether would we have any confidence that our estimates are the identifications one is making are or are not appropriate reliable. Now, let us ask ourselves the question of why ones. this should be; it seems completely unclear. Anyhow, the It is our view, as advocated in Refs. [ 4,5], that the point is that we would be asked to compute standard paradigm has no satisfactory answers to these issues. We hope this brief discussion serves to illustrate 1 ; (8) the problem we must face concerning the identification of l ¼ 2l 1 X lm þ m theoretical predictions and observations in the situation at hand. and we would then expect this quantity to be zero. We end this section by reminding the reader that, if one We need to confront the following issues: wants to consider the average value of any quantity, it is (i) Why is that so? Why can this average be expected to imperative to specify over which set the average is defined. yield zero but not each individual lm as in Eq. ( 7)? There are just no ‘‘averages’’ as absolute concepts. In the (ii) Empirically, does this hold? In other words, is the remainder of the manuscript, we will make an important actual average of observed complex quantities in differentiation between averages over ensembles of uni- Eq. ( 8), in fact, zero, or is it not? verses, averages over a spacelike hypersurface, averages Regarding the first question, it seems imperative to over the last scattering surface, and averages over orienta- choose a suitable interpretational framework in order to tions. The question we want to address is how we are able be able to decide a priori what the appropriate identifica- tions are and also to be able to evaluate whether or not we 10 have a good theoretical understanding. It appears that, in Here, we should be careful in considering the sources of error: As in any measurement, we have the systematic errors and the standard way of looking at the issue, there is really no the statistical errors associated with uncontrolled disturbances, justification to expect anything but the vanishing of each but we should not confuse statistics over several determinations lm . We must avoid getting confused with the notion that of a specific lm , say, with different experimental runs or with quantum theory involves uncertain predictions. The point different satellites, and the statistics for a fixed l over the is that the only part of quantum theory that involves such orientation number m. For a fixed value of l, the variability of lm with m should not, in our view, be taken as some statistical indeterminism is the measurement process, and we do not error but as truly valuable data containing valuable information want to call upon that in this particular situation. It is about the physics behind the emergence of the seeds of cosmic true that, even in ordinary laboratory situations, the structure.

023526-6 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) to compare the theoretical estimates, based on quantum treated as standard quantum fields, 11 evolving in a classical expectation values for some quantities, with measured quasi-de Sitter background space-time. The quantity of values of related quantities. The approach we will be observational interest is called the power spectrum of the primarily focusing on is the one pioneered in Ref. [ 4] curvature perturbation Pðk; Þ. The power spectrum is and which seems to have more potential for dealing uni- obtained from vocally with such questions than the standard one. h0j^ ðx;~ Þ^ ðy;~ Þj 0i; (9) 0 III. THE STANDARD PICTURE FOR THE where j i is called the Bunch-Davies vacuum and repre- ^ PRIMORDIAL NON-GAUSSIANITIES sents the initial state of the field v, which is the Mukhanov- Sasaki field variable defined in Eq. ( 2) (for a discussion This section will briefly review the standard accounts on about the symmetric properties of the Bunch-Davies the primordial non-Gaussianities following closely vacuum, see Appendix A). Refs. [ 30 –32 ]. There is absolutely no original work in It is precisely at this step where a subtle issue arises, this section or any extensive discussion of our views namely, that, in the standard picture, one is given various (just a few relevant comments); we simply present here and distinct arguments (e.g., decoherence, horizon cross- the usual treatment on the subject following what is ing, many worlds interpretation of quantum mechanics, commonly found in the literature, in order to compare it etc.) to accept the identification with our own approach, and discuss the main differences. ^ ^ For more details and derivations, we refer the reader to the h0jðx;~ Þðy;~ Þj 0i ¼ ðx;~ Þðy;~ Þ; (10) comprehensive review by Komatsu [ 33 ], Bartolo et al. where ðx;~ Þ now stands as a classical stochastic field [34 ], and the references cited therein. and the overline denotes the average over an ensemble of Historically, non-Gaussianity, as a test of the accuracy of universes. In other words, the value of the field in each perturbation theory, was first suggested by Allen et al. [35 ]. point ðx;~ Þ varies from each one of the members of the However, most of its importance to date relies on the ensemble of ‘‘universes,’’ with a variance 2. Therefore, premise that it will play a leading role in furthering our the power spectrum Pðk; Þ is defined in terms of the understanding of two fundamental aspects of cosmology Fourier components of ðx;~ Þ by and astrophysics [ 36 ]: 3 0 (i) the physics of the very early Universe that created 0 2 ~ ~ k~ðÞ k~ ðÞ ð Þ ðk þ k ÞPðk; Þ: (11) the primordial seeds for large-scale structures, (ii) the subsequent growth of structures via gravitational Consequently, the power spectrum is related to the two- instability and gas physics at later times. point function through Within the standard approach, by non-Gaussianity, one 1 dk sin kr ðx;~ Þðy;~ Þ ¼ P ðk; Þ ; (12) refers to any small deviations in the observed fluctuations Z0 k kr from the random field of linear, Gaussian, curvature per- with r j x~ y~j, and we also used the definition turbations. The curvature perturbations, , generate the of the dimensionless power spectrum P ðk; Þ CMB anisotropy, T=T . The linear perturbation theory ð Þ 3 2 2 2 gives a linear relation between and T=T on large scales P k; k = . The variance is given by (where the Sachs-Wolfe effect dominates) at the decou- 2 1 dk ðx;~ Þ ¼ P ðk; Þ: (13) pling epoch, i.e., T=T ð 1=3Þ. It follows from the Z0 k / relation, T , that if is Gaussian, then T is The expression ( 13 ) diverges generically. In particular, Gaussian, but what exactly does one mean by Gaussian we know that the spectrum of the primordial curvature at the observational level? 3 perturbation is roughly Pðk; Þ / k . That is, P ðk; Þ One of the most important results of the inflationary is nearly constant (i.e., independent of k); therefore, paradigm is that the CMB anisotropy arises due to Eq. ( 13 ) diverges in a logarithmic way for k ! 0 and curvature perturbations, which, in turn, are produced k ! 1 . The way the standard pictures deal with this issue by quantum fluctuations. In the standard single-field [27 ] is to establish a kmax equal to the ‘‘horizon’’ and work slow-roll scenario, these fluctuations are due to fluctua- in a cubic box of physical size aL much larger than the tions of the inflaton field itself, when it slowly rolls Hubble radius. Thus, down its potential VðÞ. Within this approach, the pri- aH mordial perturbation is Gaussian; in other words, its 2 dk ln aL ðx;~ Þ ’ P ðÞ ¼ P ðÞ 1 : (14) Fourier components are uncorrelated and have random ZL1 k H phases. When inflation ends, the inflaton oscillates 2 about the minimum of its potential and decays, thereby That is, in order to avoid the divergence in , one is reheating the Universe. forced to introduce some particular values of k as cutoffs In the inflationary paradigm, the perturbations of the field and the perturbations of the curvature are 11 In fact, they are both part of a unified field v.

023526-7 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) (for a detailed discussion related to this fact, see The delta function in Eq. ( 17 ) enforces the triangle Appendix B). condition, that is, the constraint that the wave vectors in The question that arises now is how can we evaluate any Fourier space must close to form a closed triangle, i.e., average over an ensemble of universes if we have obser- k~1 þ k~2 þ k~3 ¼ 0. Different inflationary models predict a vational access to just one—our own—Universe. The ob- maximal signal for different triangle configurations. The vious answer is that we cannot. Normally, one is presented standard approach of the study of the structure of the with ergodic arguments indicating that averages over time bispectrum is usually done by plotting the magnitude of should be equated with ensemble averages. However, er- 2 2 Bðk~1; k~2; k~3Þð k2=k 1Þ ðk3=k 1Þ (with jk~ j k ) as a func- godicity relies on equilibrium, and the inflationary regime i i tion of k2=k 1 and k3=k 1 for a given k1, with a condition that is not one of equilibrium. Furthermore, ignoring that issue, k1 k2 k3 is satisfied. The usual classification of vari- we would need to find an argument justifying the identi- ous shapes of the triangles uses the following names: fication of time averages and spatial averages, presumably squeezed ( k1 ’ k2 k3), elongated ( k1 ¼ k2 þ k3), to be carried over the hypersurface corresponding to the folded ( k1 ¼ 2k2 ¼ 2k3), isosceles ( k2 ¼ k3), and equilat- time of decoupling. Then, we need to make sure our argu- eral ( k1 ¼ k2 ¼ k3). Within the cosmology community ment applies only to direct averages and not to weighted [37 –39 ], these shapes of non-Gaussianity are potentially averages, as we discussed in the introduction. And finally, a powerful probe of the mechanism that creates the as we do not have access (at least using the CMB) to that primordial perturbations. whole hypersurface, nor to any large open region within it, One of the first (and most popular) ways to parametrize but only to the portion of it that intersects our past light non-Gaussianity phenomenologically was via a small non- cone (the two-sphere known as the last scattering surface), linear correction to the linear Gaussian perturbation we must find some argument indicating we can replace [40 ,41 ], such spatial averages with averages over orientations. In the next subsection, we will show how our approach ðx;~ Þ ¼ Lðx;~ Þ þ NL ðx;~ Þ deals with these questions. In the remainder of this section, loc 2 2 we will accept the validity of Eq. ( 10 ) and ignore those Lðx;~ Þ þ fNL ½Lðx;~ Þ Lðx;~ Þ ; (18) issues. 12 If ðx;~ Þ is Gaussian, then the two-point correlation where Lðx;~ Þ denotes a linear Gaussian part of the 2 function ( 9) specifies all the statistical properties of perturbation, and the variance Lðx;~ Þ is implemented ðx;~ Þ, for the two-point correlation function is the only in the same sense as presented in Eq. ( 14 ). Henceforth, loc parameter in a Gaussian distribution. If it is not Gaussian, fNL is called the local nonlinear coupling parameter then we need higher-order correlation functions to deter- and determines the ‘‘strength’’ of the primordial non- mine the statistical properties. Gaussianity. This parametrization of non-Gaussianity is 13 For instance, a nonvanishing three-point function local in real space and, therefore, is called local non- Gaussianity . In this local model , the contributions from ðx;~ Þ ðy;~ Þ ð~z; Þ (16) ‘‘squeezed’’ triangles are dominant, that is, with, e.g., is an indicator of non-Gaussian features in the cosmologi- k3 k1, k2. Using Eqs. ( 18 ) and ( 17 ), the bispectrum of cal perturbations. The Fourier transform of the three-point local non-Gaussianity may be derived: 14 function is called the bispectrum and is defined as loc Bðk~1; k~2; k~3Þ ¼ 2fNL ½Pðk~1ÞPðk~2Þ þ Pðk~2ÞPðk~3Þ 3 ~ ~ ~ ð 2Þ ðk~1 þ k~2 þ k~3ÞBðk1; k 2; k 3Þ: k1 k2 k3 þ Pðk~3ÞPðk~1Þ : (19) (17) In the standard picture, the non-Gaussianity produced by The importance of the bispectrum comes from the fact that many single-field slow-roll models is considered small and it represents the lowest-order statistics able to distinguish likely unobservable. However, a relatively large, possibly non-Gaussian from Gaussian perturbations. detectable, amount of non-Gaussianity can be expected when any of the following conditions are violated 12 That is, there exists some physical mechanism for which the [34 ,36 ,42 ,43 ]: ^ quantum variable ðx;~ Þ becomes a classical stochastic field (i) Single Field —There was only one quantum field ðx;~ Þ with Gaussian distribution. responsible for driving inflation. 13 Just as in the case of the two-point correlation function, the standard approach relies on the identification (ii) Canonical Kinetic Energy —The kinetic energy of the quantum field is such that the speed of propaga- h0j^ ðx;~ Þ^ ðy;~ Þ^ ð~z; Þj 0i ¼ ðx;~ Þðy;~ Þð~z; Þ: (15) tion of fluctuations is equal to the speed of light. (iii) Slow Roll —The evolution of the field was always 14 In the following, we will not write the explicit dependance of very slow compared to the Hubble time during the conformal time unless it leads to possible confusion. inflation.

023526-8 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013)

(iv) Initial Vacuum State —The quantum field was in the l1l2l3 Bm1m2m3 l m l m l m : (21) preferred ‘‘Bunch-Davies vacuum’’ state. 1 1 2 2 3 3 At this point, the standard picture leads us to another A. Non-Gaussianity in the CMB subtle issue; that is, the overline in Eq. ( 21 ) denotes, in principle, an average over an ensemble of universes. In In this subsection, we present the standard connection reality, we cannot measure the ensemble average of the between the primordial bispectrum at the end of inflation angular harmonic spectrum, as we have access to just one and the observed bispectrum of CMB anisotropies. realization, say, the collection of complex numbers: f . . . g l1m1 ; l2m2 ; ; lnmn . In order to overcome this issue, 1. Theoretical predictions for the CMB bispectrum the standard approach relies on the ergodic assumption from inflation [27 ]. The ergodicity of a system refers to that property of As we mentioned in Sec. I, the temperature anisotropies a process by which the average value of a system’s char- are represented using the lm coefficients of a spherical acteristic, measured over time, is the same as the average harmonic decomposition of the celestial sphere, value measured over an appropriately constructed en- semble. If one accepts the common supposition that the T inflationary perturbation is indeed ergodic, then one ex- ð; ’ Þ ¼ lm Ylm ð; ’ Þ; (20) T0 lm pects the volume average of the fluctuations to behave like X the ensemble average: The universe may contain regions and the curvature perturbation is imprinted on the CMB where the fluctuation is atypical, but with high probability, multipoles lm by a convolution involving the called trans- most regions contain fluctuations with a root-mean-square fer functions ðkÞ representing the linear perturbation amplitude close to [44 ]. Therefore, the probability dis- evolution, through Eq. ( 6): tribution on the ensemble, which is encoded in Eq. ( 21 ), 4i l d3k translates to a probability distribution on smoothed regions ^ lm ¼ 3 jlðkR DÞY ðkÞ ðkÞ ~ðRÞ: of a determined size within our own Universe. 3 ð2Þ lm k Z After the above analysis, we continue with the calcula- The CMB bispectrum, also called the angular bispec- tion relating the primordial bispectrum with the angular trum , is defined as the three-point correlator of the lm : bispectrum. By substituting Eq. ( 6) in Eq. ( 21 ), one obtains

3 3 3 3 4 d k1 d k2 d k3 l1l2l3 l1þl2þl3 ^ Bm1m2m3 ¼ i ðk1Þ ðk2Þ ðk3Þ j ðk1R Þj ðk2R Þj ðk3R Þ ~ ~ ~ Y ðk1Þ 3 2 3 2 3 2 3 l1 D l2 D l3 D k1 k2 k3 l1m1 ð Þ ð Þ ð Þ ^ Z ^ Yl2m2 ðk2ÞYl3m3 ðk3Þ 2 3 ¼ dk dk dk ðk k k Þ2B ðk ; k ; k Þðk Þðk Þðk Þ j ðk R Þj ðk R Þj ðk R Þ 3 1 2 3 1 2 3 1 2 3 1 2 3 l1 1 D l2 2 D l3 3 D 1 Z 2 ^ ^ ^ dxx jl1 ðk1xÞjl2 ðk2xÞjl3 ðk3xÞ d x^Yl1m1 ðxÞYl2m2 ðxÞYl3m3 ðxÞ; (22) 0 Z Z where in the last line, we have integrated over the angular parts of the three ki and used the exponential integral form for the delta function that appears in the bispectrum definition ( 17 ). The last integral over the angular part of x~ is known as the Gaunt integral, which can be expressed in terms of Wigner 3- j symbols as

ð2l þ 1Þð 2l þ 1Þð 2l þ 1Þ l1 l2 l3 l1 l2 l3 m1m2m3 ð ^Þ ð ^Þ ð ^Þ ¼ 1 2 3 Gl1l2l3 d x^Yl1m1 x Yl2m2 x Yl3m3 x : (23) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 0 0 0 ! m1 m2 m3 ! Z

l1l2l3 l1l2l3 The fact that the bispectrum Bm1m2m3 consists of the Gaunt construct generic Bm1m2m3 satisfying rotational invariance; integral, Gm1m2m3 , implies that the bispectrum satisfies the l1l2l3 thus, in the literature, one encounters bl1l2l3 more frequently triangle conditions and parity invariance: m1 þ m2 þ l1l2l3 than Bm1m2m3 . The quantity bl1l2l3 is called the reduced m3 ¼ 0, l1 þ l2 þ l3 ¼ even, and jl l j l l þ l i j k i j bispectrum, as it contains all the physical information in for all permutations of indices. l1l2l3 One, thus, can write Bm1m2m3 . Since the reduced bispectrum does not contain the Wigner 3- j symbol, which merely ensures the triangle con- l1l2l3 ¼ m1m2m3 ditions and parity invariance, it is easier to calculate the Bm1m2m3 Gl1l2l3 bl1l2l3 ; (24) physical properties of the theoretical bispectrum. where bl1l2l3 is an arbitrary real symmetric function of l1, In the standard picture, one assumes that, if there is a l2, and l3. This form, Eq. ( 24 ), is necessary and sufficient to nontrivial bispectrum, then it has arisen through a physical

023526-9 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) process that is statistically isotropic, 15 so we can employ one concludes that no significant detection of the given the angle-averaged bispectrum Bl1l2l3 without loss of infor- shape is produced by the data, but one still determines model mation, that is [ 33 ,34 ], important constraints on the allowed range of fNL . Note that, ideally, one would like to do more than just constrain l l l B ¼ 1 2 3 : (25) the overall amplitude and reconstruct the entire shape from l1l2l3 m m m l1m1 l2m2 l3m3 mi 1 2 3 ! the data by measuring configurations of the bispectrum. X However, the expected primordial signal is too small to We now can obtain a relation between the averaged bis- allow the signal from a single bispectrum triangle to pectrum, B , and the reduced bispectrum, b , by l1l2l3 l1l1l2 emerge over the noise. For this reason, one studies the substituting Eq. ( 24 ) into Eq. ( 25 ), cumulative signal from all the configurations that are model sensitive to fNL . ð2l1 þ 1Þð 2l2 þ 1Þð 2l3 þ 1Þ l1 l2 l3 Given the above analysis, the standard picture then Bl l l ¼ bl l l ; 1 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 0 0 0 ! 1 2 3 makes use of estimation theory to extract an estimate for (26) fNL from the CMB data set. An unbiased bispectrum-based minimum variance estimator for the nonlinearity parame- where the identity ter can be written as [ 45 ,46 ]

l1 l2 l3 th m1m2m3 1 l1 l2 l3 Bl l l Gl l l f^ ¼ 1 2 3 m m m ! 1 2 3 NL all m 1 2 3 N m1 m2 m3 ! ðCl Cl Cl Þobs X lXimi 1 2 3 ð2l1 þ 1Þð 2l2 þ 1Þð 2l3 þ 1Þ l1 l2 l3 ð Þ ; (30) ¼ (27) l1m1 l2m2 l3m3 obs sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 0 0 0 ! where Bth is the angle-averaged theoretical CMB bis- l1l2l3 was used. The reduced bispectrum obtained from Eq. ( 22 ) th pectrum for the model in consideration, with f ¼ 1; Cl then takes the much simpler form NL is the observed angular spectrum; and lm are the multi- 2 3 poles of the observed CMB temperature fluctuations. The b ¼ dk dk dk ðk k k Þ2B ðk ; k ; k Þðk Þ l1l2l3 3 1 2 3 1 2 3 1 2 3 1 normalization N is calculated requiring the estimator to be Z ‘‘unbiased,’’ i.e., the averaged value is equal to the ‘‘true’’ ðk Þðk Þ j ðk R Þj ðk R Þj ðk R Þ 2 3 l1 1 D l2 2 D l3 3 D ^ value of the parameter, hfNL i ¼ fNL . If the bispectrum 1 2 B is calculated for f ¼ 1, then the normalization ð Þ ð Þ ð Þ l1l2l3 NL dxx jl1 k1x jl2 k2x jl3 k3x : (28) 0 takes the following form: Z This is the main equation for this section, since it ex- ðB Þ2 plicitly relates the primordial bispectrum, predicted by the N ¼ l1l2l3 : (31) Cl Cl Cl standard inflationary theories, to the averaged bispectrum Xli 1 2 3 [through Eq. ( 26 )] obtained from the CMB angular bispec- The estimator for non-Gaussianity ( 30 ) is then simpli- trum l1m1 l2m2 l3m3 . This formula is entirely analogous to the well-known relation linking the primordial power spec- fied using Eqs. ( 28 ) and ( 26 ) to yield trum PðkÞ and the CMB angular power spectrum Cl, i.e., 1 f^ ¼ d Y ðx^ÞY ðx^ÞY ðx^Þ 2 NL x^ l1m1 l2m2 l3m3 2 2 2 N l m Cl ¼ k PðkÞ ðkÞjl ðkR DÞdk: (29) Xi i Z 9 1 Z 2 ð Þ ð Þ ð Þð 1 1 1Þ x dxj l1 k1x jl2 k2x jl3 k3x Cl Cl Cl obs 0 1 2 3 Z 2. Measuring primordial non-Gaussianity from the CMB 2 3 dk dk dk ðk k k Þ2Bðk ; k ; k Þðk Þ As we mentioned before, in most inflationary models, 1 2 3 1 2 3 1 2 3 1 Z the parameter characterizing primordial non-Gaussianity is ð Þ ð Þ ð Þ ð Þ ð Þ k2 k3 jl1 k1RD jl2 k2RD jl3 k3RD fNL . Thus, the next task within the standard picture is to ð Þ estimate fNL from the CMB data set. That is, one chooses l1m1 l2m2 l3m3 obs ; (32) the primordial model that one wants to test, characterizing it through its bispectrum shape. One then proceeds to where Bðk1; k 2; k 3Þ is the primordial bispectrum obtained model from the three-point function, as defined in Eq. ( 17 ). In this estimate the corresponding amplitude fNL from the model manner, the sought constraints are obtained. The best data. If the final estimate is consistent with fNL ¼ 0, results, corresponding to the so-called, local, equilateral,

15 and orthogonal shape of non-Gaussianities using the Although, it would be interesting, and possibly a more local realistic approach to the problem, to proceed in the analysis WMAP 7 year data [ 47 ], yield fNL ¼ 32 21 (1), equil orthog without this assumption. fNL ¼ 26 140 (1), and fNL ¼ 202 104 (1).

023526-10 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) 1 IV. THE COLLAPSE MODEL ACCOUNT FOR S d4xp g R g 1=2 g ab V ; Z a b THE INFLATIONARY ORIGIN OF ¼ 16 G ½ r r ð Þ COSMIC STRUCTURE ffiffiffiffiffiffiffi (33) Before proceeding, it seems worthwhile to briefly where stands for the inflaton and V stands for the explain the view we take regarding quantum physics inflaton’s potential. One then splits both metric and scalar and Einstein’s gravity. The framework we adopt is fields into a spatially homogeneous part (‘‘background’’) based on a description of the problem that allows, at and an inhomogeneous part (‘‘fluctuation’’), i.e., g g ¼ 0 þ the same time, the quantum treatment of other fields g , 0 . and a classical treatment of gravitation, that is, the The¼ backgroundþ is taken to be the spatially flat realm of semiclassical gravity , together with quantum Friedmann-Robertson universe with line element ds 2 ¼ field theory in curved space-time. We will assume that a 2 d 2 dx idx j and the homogeneous scalar ð Þ ½ þ ij to be a valid approximation most of the time, with the field 0 . The evolution equations for this background exception associated precisely with the dynamical are scalarð fieldÞ equations, collapse, as we will explain below. Such a description a_ of gravitation in interaction with quantum fields is € 2 _ a2@ V 0; 0 þ a 0 þ ½ ¼ characterized by the semiclassical Einstein equation: 2 (34) ^ a_ 2 2 R 1=2 g R 8G T , whereas the other fields, 3 4G _ 2a V : ð Þ ¼ h i 2 0 0 including the inflaton, are treated in the standard quan- a ¼ ð þ ½ Þ tum field theory fashion. It seems clear that this ap- The scale factor corresponding to the inflationary regime, proximated description would break down in association written in terms of the conformal time, is: a ð Þ ¼ with the quantum mechanical collapses or state jumps, 1= H2 1 with H2 8= 3 GV . The slow roll ½ I ð Þ I ’ ð Þ which we are considering to be part of the underlying parameter 1 H_ =H 2 is considered to be very small quantum theory containing gravitation. The reason for 1 during the inflationary stage. The Hubble factor H I this breakdown is simply that the left-hand side of the is approximately constant, and the scalar 0 field is in the _ 3 equation above has zero divergence ( G 0), while slow roll regime, i.e., 0 a =3a_ V0. According to the r ¼ ¼ ð Þ the divergence of the right-hand side, T^ , will be standard inflationary scenario, this era is followed by a nonvanishing (even discontinuous) inr connectionh i with reheating period in which the Universe is repopulated with the jumps of the quantum state (such a jump is how we ordinary matter fields, a regime that then evolves toward a are describing here the self-induced collapse of the standard hot big bang cosmology regime leading up to the wave function). present cosmological time. The functional form of a ð Þ In this setting, we start from the assumption that, in during these latter periods changes, but we will ignore accordance with the standard inflationary accounts, and those details because most of the change in the value as mentioned before, the state of the Universe before the of a occurs during the inflationary regime. We will set time at which the seeds of structure emerge is described by a 1 at the ‘‘present cosmological time’’ and assume ¼ the H&I Bunch-Davies vacuum state for the matter degrees that the inflationary regime ends at a value of of freedom (DOF) and the corresponding H&I classical 0, negative and very small in absolute terms ( 0 10¼ 22 Mpc ). ’ Robertson-Walker space-time. Then, we assume that, at a later stage, the quantum state Next, we turn to consider the perturbations. We shall of the matter fields reaches a stage whereby the corre- focus in this work on the scalar perturbations and ignore, sponding state for the gravitational DOF is forbidden, for simplicity, the tensor perturbations or gravitational and a quantum collapse of the matter field wave function waves. Working in the so-called longitudinal gauge, the is triggered by some unknown physical mechanism. In this perturbed metric is written as manner, the state resulting from the collapse of the quan- ds 2 a 2 1 2 d 2 1 2 dx idx j ; (35) tum state of the matter fields does not need to share the ¼ ð Þ ½ð þ Þ þð Þ ij symmetries of the initial state. After the collapse, the where stands for the scalar perturbation usually known gravitational DOF are assumed to be, once more, accu- as the Newtonian potential . rately described by Einstein’s semiclassical equation. The perturbation of the scalar field is related to a per- However, as T^ for the new state does not need to turbation of the energy-momentum tensor and reflected h i have the symmetries of the precollapse state, we are led into Einstein’s equations, which, at the lowest order, lead to a geometry that, generically, will no longer be homoge- to the following constraint equation for the Newtonian neous and isotropic. potential: The starting point of the specific analysis is the same as 2 4G _ _ s ;_ (36) the standard picture, i.e., the action of a scalar field coupled r ¼ 0 ¼ to gravity: where we introduced the abbreviation s 4G _ . 0

023526-11 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) Now, we consider in some detail the quantum theory state defined by a^ R;I 0 0 is 100% translational and k~ j i ¼ of the field . It is convenient to work with the rescaled rotationally invariant (see Appendix A). field variable y ¼ a and its conjugate momentum For the next step, we must specify in more detail the ¼ y_ ya=a_ . For simplicity, we set the problem in a modeling of the collapse. Then, we must take into account finite box of side L, which can be taken to at the end 1 that, after the collapse has taken place, one should consider of all calculations. We decompose the field and momentum the continuing evolution of the expectation values of the operators as field variables until the end of inflation and eventually up to the hypersurface of decoupling. In fact, if we wanted to 1 y^ ; x~ eik~ x~y^ ; actually compare our analysis directly with observations, ð Þ ¼ L3 k~ð Þ k~ we would need evolve the perturbations both through the X (37) 1 reheating period and through the decoupling era. This, ^ ; x~ eik~ x~^ ; however, is normally taken into account through the use ð Þ ¼ L3 k~ð Þ Xk~ of appropriate transfer functions, and we will assume that the same procedure could be implemented in the context where the sum is over the wave vectors k~ satisfying of the present analysis, but we will not consider it further in k L 2n for i 1, 2, 3, with n integer, and where the present manuscript. i ¼ i ¼ i y^ y a^ y a^y and ^ g a^ We will further assume that the collapse is somehow k~ k k~ k k~ k~ k k~ ð Þ ð Þ þ ð Þ ð Þ ð Þ þ analogous to an imprecise measurement 16 of the operators g a^y with the usual choice of modes: k k~ R;I R;I ð Þ y^ and ^ . Now, we will specify the rules accord- k~ ð Þ k~ ð Þ 1 i ing to which collapse happens. Again, at this point, our y ik ; k 1 exp criteria will be simplicity and naturalness. What we have to ð Þ ¼ p2k k ð Þ (38) describe is the state after the collapse. ffiffiffiffiffi k It seems naturalj toi assume (taking the view that a g i exp ik ; kð Þ ¼ sffiffiffi2 ð Þ collapse effect on a state is analogous to some sort of approximate measurement) that after the collapse, the which leads to what is known as the Bunch-Davies expectation values of the field and momentum operators vacuum. in each mode will be related to the uncertainties of the Note that, according to the point of view we discussed at precollapse state (recall that the expectation values in the the beginning of this section and having, at this point, the vacuum state are zero). In the vacuum state, y^k~ and ^ k~ quantum theory for the relevant matter fields, the effects of individually are distributed according to Gaussian wave the quantum fields on the geometrical variables are codi- functions centered at 0 with spread y^ 2 and ^ 2, ð k~Þ0 ð k~Þ0 fied in the semiclassical Einstein equations. Thus, Eq. ( 36 ) respectively. must be replaced by We might consider various possibilities for the detailed form of this collapse. Thus, for their generic form, asso- 2 4G _ _ s _ s=a ^ : (39) r ¼ 0 ¼ h i ¼ ð Þh i ciated with the ideas above, we write At this point, one can clearly observe that, if the state of the y^R;I c xR;I y^R;I 2 xR;I y c ℏL3=2; quantum field is in the vacuum state, the metric perturba- k~ k 1 k;~ 1 k~ 0 1 k;~ 1 k k h ð Þi ¼ ð Þ ¼ j ð Þj qffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions vanish, and thus the space-time is homogeneous and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (40) isotropic. As already mentioned, our proposal is based on the R;I c R;I R;I 2 R;I c ℏ 3 consideration of a self-induced collapse, which we take ^ 2x ^ ; 2x gk L =2; h k~ ð kÞi ¼ k;~ 2 ð k~ Þ0 ¼ k;~ 2 j ð kÞj to operate in close analogy with a ‘‘measurement’’ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (41) (but, evidently, with no external measuring apparatus or observer involved). This leads us to want to work with where xR;I , xR;I have been assumed, in our previous works, Hermitian operators, which, in ordinary quantum mechan- k;~ 1 k;~ 2 to be selected randomly from within a Gaussian distribu- ics, are the ones susceptible to direct measurement. c tion centered at zero with spread one, and ~ represents the Therefore, we must separate both y^k~ and ^ k~ into k ð Þ R ð Þ I their real and imaginary parts y^k~ y^ ~ iy^ ~ ð Þ ¼ k ð Þ þ k ð Þ 16 and ^ ^ R i^ I so that the operators An imprecise measurement of an observable is one in which k~ð Þ ¼ k~ ð Þ þ k~ ð Þ one does not end up with an exact eigenstate of that observable y^R;I and ^ R;I are Hermitian operators. but rather with a state that is only peaked around the eigenvalue. k~ ð Þ k~ ð Þ So far, we have proceeded in a manner similar to the Thus, we could consider measuring a certain particle’s position and momentum so as to end up with a state that is a wave packet standard one, except in that we are treating at the quantum with both position and momentum defined to a limited extent and level only the scalar field and not the metric fluctuation. At which, of course, does not entail a conflict with Heisenberg’s this point, it is worthwhile to emphasize that the vacuum uncertainty bound.

023526-12 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) time of collapse for each mode. Here, 1 and 2 are With this information at hand, we can now compute the parameters taking the values 0 or 1 that allow us to specify perturbations of the metric after the collapse of all the the kind of collapse proposal we want to consider. (The modes. 17 main ones we have considered in Refs. [ 4,8,9] are 1 ¼ 2 1 for the symmetric collapse and 1 0, 2 1 for A. Connection to observations ¼Newtonian collapse ¼ ¼ the ). At this point, we must emphasize Now, we must put together our semiclassical description that our Universe corresponds to a single realization of of the gravitational DOF and the quantum mechanics these random variables, and, thus, each of these quantities description of the inflaton field. We recall that this entails xR;I , xR;I has a single specific value. The fact that we can k;~ 1 k;~ 2 the semiclassical version of the perturbed Einstein equa- represent the specific details of the first inhomogeneities tion that, in our case, leads to Eq. ( 39 ). The Fourier and anisotropies, the seeds of cosmic structure, is some- components at the conformal time are given by thing that has no counterpart on the standard treatments. It H ℏ is clear that one can now investigate how the different ðÞ ¼ I h^ ðÞi ; (44) k~ rffiffiffi2 2 k~ specific proposals for the process of collapse could affect MPk the statistics of the xR;I , xR;I . One could now inquire about k;~ 1 k;~ 2 where we have used the fact that, during inflation, s ¼ both, the statistics of these quantities in some imaginary = 2ðaH I=M PÞ, with MP as the reduced Planck mass ensemble of possible universes as well as the statistics of 2 ℏ2 8 pMffiffiffiffiffiffiffiffiP =ð G Þ. The expectation value depends on the such quantities for the particular Universe we inhabit. state of the quantum field; therefore, as we already noted, Regarding the collapse models, it should be clear that 0 prior to the collapse, we have k~ðÞ ¼ , and the space- there are many other possibilities that we have not even time is still homogeneous and isotropic at the correspond- thought about and that might require drastically modified ing scale. However, after the collapse takes place, the state formalisms. In fact, in a recent work [ 13 ], grounds were of the field is a different state with new expectation values found that suggest a correlation between the xR;I , xR;I of k;~ 1 k;~ 2 that generically will not vanish, indicating that, after this any mode with those of their higher harmonics (something time, the Universe becomes anisotropic and inhomogene- reminiscent of the so-called parametric resonances found ous at the corresponding scale. We now can reconstruct the in quantum optics in materials with nonlinear response space-time value of the Newtonian potential using functions [ 48 ]). As we noted in Ref. [ 14 ] that particular 1 ð; x~Þ ¼ eik~x~ ðÞ; (45) types of correlations, in turn, would lead to a very specific L3 k~ signature, which might be looked for in the statistical Xk~ features of the CMB. to extract the quantities of observational interest. Returning to the specific models we have described In order to connect with the observations, we shall relate above, we need to compute the relevant expectation values the expression ( 44 ) for the evolution of the Newtonian of the field operators in the post-collapse state at the potential during the early phase of accelerated expansion relevant times. For each specific model, we do thisj byi using to the small anisotropies observed in the temperature of the Eqs. ( 40 ) and ( 41 ) above and the evolution equations for cosmic microwave background radiation, T ð; ’ Þ=T 0 the expectation values (i.e., using Ehrenfest’s theorem). with T0 2:725 K as the temperature average. They are Thus, one obtains hy^R;I ðÞi and h^ R;I ðÞi for the state considered the fingerprints of the small perturbations per- k~ k~ that resulted from the collapse, for all later times. The vading the Universe at the time of decoupling, and un- R;I R;I doubtedly any model for the origin of the seeds of cosmic explicit expressions for the hy^ ðÞi and h^ ðÞi are k~ k~ structure should account for them. As already mentioned in cos 1 1 sin 1 Sec. I, these data can be described in terms of the coef- R;I Dk Dk hy^ ðÞi ¼ þ þ1 ficients of the multipolar series expansion, i.e., Eq. ( 5). k~ k k z k kz lm k k The different multipole numbers l correspond to different sin R;I c Dk R;I c angular scales: low l to large scales and high l to small h ^ ð Þi þ cos Dk hy^ ð Þi ; k~ k~ k k~ k~ scales. At large angular scales ( l & 20 ), the Sachs-Wolfe (42) effect is the dominant source for the anisotropies in the CMB. That effect relates the anisotropies in the tempera- ture observed today on the celestial sphere to the inhomo- sin R;I Dk R;I c geneities in the Newtonian potential on the last scattering h^ ðÞi ¼ cos D þ h^ ð Þi k~ k k~ k~ zk surface, sin ^R;I c k Dkhy ~ ð ~Þi ; (43) k k 17 In fact, we need only be concerned with the relevant modes, where D k z and z k c. This calculation is those that affect the observational quantities in a relevant way. k k k k~ Modes that have wavelengths that are either too large or too explicitly done in Refs. [ 4,11 ]. small are irrelevant in this sense.

023526-13 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) T 1 hidden in a maze of often unspecified assumptions and ð; ’ Þ ¼ ðD; x~DÞ: (46) T0 3 unjustified identifications [ 5]. Thus, according to Eq. ( 48 ), all the modes contribute to Here, D is the conformal time of decoupling that lm , with a complex number. If we had the outcomes lies in the matter-dominated epoch, and x~D ¼ characterizing each of the individual collapses, we would RDðsin sin ’; sin cos ’; cos Þ, with RD as the radius be able to predict the exact value of each of these individ- of the last scattering surface. Furthermore, using Eq. ( 45 ) ual quantities. However, we have, at this point, no other ~ ^ and eik ~xD 4 ilj kR Y ; ’ Y k , the expres- access to such information than the observational quanti- ¼ lm lð DÞ lm ð Þ lm ð Þ sion ( 5) for lm Pcan be rewritten in the form ( 6). The ties lm themselves. transfer function k represents the evolution of the We hope to be able to say something about these, but ð Þ Newtonian potential from the end of inflation R to doing so requires the consideration of further hypothesis the time of decoupling , i.e., k . regarding the statistical aspects of the physics behind the D k~ð DÞ ¼ ð Þ k~ð RÞ Substituting Eq. ( 43 ) in Eq. ( 44 ) and using Eqs. ( 40 ) and collapse as well as the conditions previous to them. (41 ) gives As is generally the case with random walks, one cannot hope to estimate the direction of the final displacement. ℏ 3=2 sin L pHI Dk R I However, one might say something about its estimated ð Þ 2 cos D x ix k~ R 3=2 k k;~ 2 k;~ 2 magnitude. It is for that reason that we will be focusing ð Þ¼ 2p2M k ffiffiffi þ zk ð þ Þ P on estimating the most likely value of the magnitude: ffiffiffi 1 1=2 R I 1 sin D 1 x ix : (47) k 2 k;~ 1 k;~ 1 2 þz ð þ Þ 2 16 ^ k k k j kR j k R Y k^ Y k lm 9 6 0 l D l 0 D lm lm 0 j j ¼ L X~ ~ ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ Finally, using Eq. ( 47 ) in Eq. ( 6) yields k; k0 : lℏ3=2 2 k~ R ~ R (49) i p HI ^ ð Þ k0 ð Þ k j kR Y k lm ¼ 3 Lk 3=2ffiffiffiffiffiffiM ð Þ lð DÞ lm ð Þ ð Þ P Xk~ Note, however, that although, in our approach, each of sin the quantities ~ R has, in principle, a particular nu- Dk R I k 2 cos D x ix ð Þ k þ z ð k;~ 2 þ k;~ 2Þ merical value, the fact that such value is the result of a k quantum collapse characterized by random numbers 1 1=2 R I 1 sin D 1 x ix ; (48) indicates we cannot make a definite prediction for it. We k 2 k;~ 1 k;~ 1 þ zk ð þ Þ believe that our approach has, among others, the advantage of offering a clear way to express the prediction for the note that in Eqs. ( 47 ) and ( 48 ), Dk is evaluated at R, observable quantities, in a manner in which the aspects that i.e., D k z . kð RÞ ¼ R k are controlled by randomness are clearly identified. This It is worthwhile to mention that the relation of lm with allows, in principle, the consideration in a separate way of the Newtonian potential, as obtained in Eq. ( 48 ), within the each of the hypotheses and identifications one is interested collapse framework has no analogue in the usual treat- in making. Our inability of predicting the specific values ments of the subject. It provides us with a clear identifica- for the quantities lm , characterizing our observations, is tion of the aspects of the analysis where the ‘‘randomness’’ then clearly identifiedj j and located in the particular random is located. In this case, it resides in the randomly selected variables introduced in the collapse hypothesis. But, of values xR;I, xR;I that appear in the expressions of the k;~ 1 k;~ 2 course, we want to make predictions. So further consider- collapses associated with each of the modes. Here, we ations become necessary, but the point is that these are also find a clarification of how, in spite of the intrinsic clearly identifiable. We will see below what these hypoth- randomness, we can make any prediction at all. The indi- eses are and how they lead to more specific predictions. vidual complex quantities lm correspond to large sums of One of the advantages we have is that one is able, in complex contributions, each one having a certain random- principle, to consider removing or modifying each one of ness but leading in combination [i.e., the sum of contribu- those hypotheses. In this case, we can make progress, for tions appearing in Eq. ( 48 )] to a characteristic value in just instance, by making the assumption that we can regard the the same way as a random walk made of multiple steps. In specific outcomes characterizing our Universe as a typical other words, the justification for the use of statistics in our member of some hypothetical ensemble of universes. approach is that the quantity lm is the sum of contribu- For example, we are interested in estimating the most 2 tions from the collection of modes, each contribution being likely value of the magnitude of lm above, and, in such a a random number leading to what is, in effect, a sort of hypothetical ensemble, we mightj hopej that it comes very ‘‘two-dimensional random walk,’’ for which the total dis- close to our single sample. It is worth emphasizing that, for placement corresponds to the observational quantity. each l and m, we have one single complex number charac- Nothing like this can be found in the most popular ac- terizing the actual observations (and, thus, the real Universe counts, in which the issues we have been focussing on are we inhabit). For a given l, for instance, we should avoid

023526-14 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) confusing ensemble averages with averages of such quanti- the ensemble average) and identifying it with the most likely ties over the 2l þ 1 values of m. The other universes in the value of the quantity, and it is needless to say that these two ensemble are just figments of our imagination, and there is notions are not exactly equal for many types of ensembles. nothing in our theories that would indicate that they are real. However, let us, for the moment, ignore this issue and We can simplify things even further by taking the en- assume the identity of those two values and look at the 2 j j2 semble average jlm j (the bar indicates that we are taking ensemble average of the quantity lm , which is given by

16 2 2 0 0 ^ ^0 jlm j ¼ ðkÞðk ÞjlðkR DÞjlðk RDÞY ðkÞYlm ðk Þ ~ðRÞ ðRÞ: (50) 9L6 lm k k~0 Xk;~ k~0 One can, for instance, assume that collapsing events are all uncorrelated (Something not always justified, as exemplified in the analysis of Ref. [ 13 ]) and then consider estimating the most likely value; thus, 16 2 2 0 0 ^ ^0 j jML ¼ ðkÞðk Þj ðkR Þj ðk R ÞY ðkÞY ðk Þ ~ð Þ ð Þ: (51) lm 9L6 l D l D lm lm k R k~0 R Xk;~ k~0 Under the assumption of the validity of such an approxi- explicit form of CðkÞ for the class of collapse schemes mation and the additional assumption that the random considered here is R I R I variables x ~ , x ~ , x ~ , x ~ are all uncorrelated, we obtain 1 sin 2 k; 1 k; 1 k; 2 k; 2 2 1 sin 2 2 cos zk that all the information regarding the ‘‘self-collapsing’’ CðkÞ ¼ 1 þ 2 zk þ 2 zk : (54) z zk model will be codified in the quantity k As we have noted in previous works, this quantity becomes ð Þ ð Þ: (52) k~ R k~0 R a simple constant if the collapse time happens to follow a particular pattern in which the time of collapse of the mode Generally, one expects this term to be proportional to 0 , k~k~ k~ is given by c ¼ Z=k with Z as a constant. In fact, the but alternatives cannot be ruled out. In fact, a case in which k standard answer would correspond to CðkÞ ¼ constant this assumption is relaxed was explored in Ref. [ 14 ]. (which can be thought as an equivalent ‘‘nearly scale- Furthermore, we will take the limit k ! 0 in Eq. ( 52 ), R invariant power spectrum’’). Thus, the result obtained for which can be expected to be appropriate when restricting the relation between the time of collapse and the mode’s interest to the modes that are ‘‘outside the horizon’’ at the frequency, i.e., ck ¼ constant, is a rather strong conclu- end of inflation (including the modes that give a major k~ contribution to the observationally relevant quantities). sion that could represent relevant information about what- Then, with the help of Eq. ( 47 ) and after taking the ever the mechanism of collapse is. continuum limit ( L ! 1 ), we obtain It is quite clear that if the time of collapse of each mode does not adjust exactly to the pattern k c ¼ Z, then the 3 2 k ℏ H dk collapse schemes under consideration (characterized by j j2 ¼ I 2ðkÞj2ðkR ÞCðkÞ; (53) lm ML 36 2 l D M P k the values of 1, 2), or some other one resulting in a Z nontrivial function CðkÞ, would lead to different predic- where some of the information regarding that a collapse tions for the exact form of the spectrum, and comparing 18 has occurred is contained in the function CðkÞ. The these predictions with the observations can help us to discriminate between the distinct collapse schemes. An 18 The standard amplitude for the spectrum is usually presented as analyses of these issues have been presented in Refs. [ 8,9]. 4 2 2 / V= ðM PÞ / HI =ðMPÞ. The fact that is in the denominator We end this section by noting that the treatment of the leads, in the standard picture, to a constraint scale for V. However, statistical aspects in the collapse proposal is quite different in Eq. ( 53 ), is in the numerator. This is because we have not used from the standard inflationary paradigm. We will deepen (and, in fact, we will not) explicitly the transfer function ðkÞ. In the standard literature, it is common to find the power spectrum for this discussion in the next section. However, at this point, the quantity ðxÞ, a field representing the curvature perturbation the differences should be evident. In the standard accounts, in the comoving gauge. This quantity is constant for modes one is going from quantum correlation functions to classi- ‘‘outside the horizon’’ (irrespectively of the cosmological epoch); cal n-point functions averaged over an ensemble of uni- thus, it avoids the use of the transfer function. The quantity can verses; then, one goes to n-point correlation functions be defined in terms of the Newtonian potential as averaged over different regions of our own Universe, þ ð 2=3Þð H 1_ þ Þ=ð1 þ !Þ, with ! p= . For large- 2 3 1 1 1 and, finally, one relates this last quantity with the observ- scale modes k~ ’ k~½ð = Þð þ!Þ þ , and during inflation, 2 1 2 3 able jlm j . These series of steps are not at all direct, and þ ! ¼ ð = Þ. For these modes, k~ ’ k~= , and the power 2 2 2 4 they involve a lot of subtle issues that the standard picture spectrum is P ðkÞ ¼ P ðkÞ= / HI =ðMPÞ / V= ðM PÞ, which contains the correct amplitude. For a detailed discussion regarding does not provide in a transparent way. On the other hand, the amplitude within the collapse framework, see Ref. [ 10 ]. within the collapse approach to the subject, the observable

023526-15 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) 2 ^ jlm j is related to the random variables, xk, through a processes were associated with the observables yk~ and their two-dimensional (i.e., the result is the sum of individual conjugate momenta according to Eqs. ( 40 ) and ( 41 ). It is complex numbers) random walk. As we mentioned, the clearly conceivable that the elementary process might have 2 value of jlm j corresponds then to the ‘‘length’’ of the been associated, instead, with other observables. One sim- random walk. This random walk is associated to a particu- ple possibility for those alternative observables is the vari- lar realization of a physical quantum process (i.e., the ous options offered by linear combinations of the former. collapse of the inflaton’s wave function), and as we have only access to one realization—the random walk corre- A. The new outlook on non-Gaussianities sponding to our own Universe—the most natural assump- tion (but certainly not the only one) is that the average In this section, we discuss the aspects that need modifi- value of the length of the possible random walks, which cation in the study of primordial non-Gaussianities, in view of the approach we have been discussing to the origin of corresponds to j j2, is equal to the most likely value, i.e., lm the primordial fluctuations. to j j2 , and this, in turn, is associated with the j j2 of lm ML lm The first point we should stress is that, from the two our observable Universe. aspects of cosmology mentioned in Sec. I, we have seen that we have had to modify the first, namely, the nature of V. FURTHER STATISTICAL ASPECTS the quantum state, in order to be compatible with the The first thing we should now note is that there are existence, at the fundamental (quantum) level, of the in- several statistical issues at play and that, within our ap- homogeneities and anisotropies that are behind the emer- proach, various novel ones emerge. One central aspect is gence of structure and, thus, of everything—including the exact nature of the state previous to all collapses, i.e., observers—in our Universe. the state characterizing the first stages of the inflationary In other words, the standard physics of the very early regime, and normally taken to be the Bunch-Davies vac- Universe had to be supplemented with the collapse hy- uum. There are various possibilities that might modify the pothesis in order to fully account for the process that nature of that state: For instance, if the field is not truly a created the primordial seeds for large-scale structure. free field, and self-interactions are important, one might Otherwise, we could not really identify the process by find correlations between the various modes of the field. which the inhomogeneity and anisotropies emerged from These effects could be manifest, for instance, by nonvan- the initial vacuum. 0 ^ ^ 0 ishing values of quantities like h jyk~yk~0 j i (as argued in the As in the standard approach, we take the curvature case studied in Ref. [ 14 ]). However, we should be aware perturbations to be the generators of the CMB anisot- not only of the inherent problems of accessing those sta- ropy, T=T . However, in our approach, the observed fluc- tistical signatures associated with the fact that we have at tuations are determined, not just by the initial vacuum our disposal a single Universe but also that our Universe, state, which is and remains homogeneous and isotropic, including the relevant perturbations, is not characterized by but also by the characteristics of the collapse process, the vacuum state but rather by the state that results after the besides, of course, by the effects of the late-time physics. collapses of all the modes, and it is quite clear that the In this more precise and detailed approach, it is clear collapse process itself can be a source of unexpected that, even if the primordial state can be considered as correlations. These would manifest themselves, for ex- Gaussian, in the sense that the corresponding n-point ample, in correlations between the values taken by the functions are completely determined by the two-point xk~s appearing in the collapse process and which we have functions—and, thus, the odd n-point functions vanish— so far assumed were different and independent quantities it might still be possible for the collapse processes to for each mode. drastically affect and modify this. In other words, there Moreover, we have to note that the quantities that are exists, in principle, the possibility that the collapse process more or less directly accessible to observational investiga- itself introduces non-Gaussian characteristics into the ^ state. We will not discuss this possibility here, but only tion are not the h jyk~j i, and the n-point functions, in general, for the post-collapse state, but the various lm s, point it out as something to have in mind and a topic for and the latter are related to the former, as can be seen in future research. Eq. ( 48 ) in a nontrivial way. In fact, as we saw, each As we have argued, the quantity of observational interest lm ^ ^ corresponds to a sort of two-dimensional random walk is not really h0jðx;~ Þðy;~ Þj 0i, as the argument to (i.e., a sum of complex quantities), and each of the steps justify that in the standard approach depends not only ^ 0 ^ ^ 0 is related to h jyk~j i. It is, thus, clear that there might be on accepting the identification h j ðx;~ Þ ðy;~ Þj i ¼ correlations between the various lm s simply due to the ðx;~ Þðy;~ Þ, where ðx;~ Þ is taken to be a classical fact that they arise from different combinations of the same stochastic field and the overline denotes the average over random variables. Of course, we should note that the an ensemble of universes, but also on a series of arguments version of the collapse proposal we have presented here indicating one can replace the ensemble averages with is based on the assumption that the elementary collapse suitable spatial averages of quantities in our Universe.

023526-16 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013)

As a matter of fact, a clear example of how a careless disposal not only the average quantities Cl but also, for every approach to the statistics at hand can lead to wrong con- value of l and m, the individual quantities lm . Each one of clusions is brought by the variance 2. We mentioned in those corresponds, in our approach, to different random Sec. III that 2 diverges generically if one does not intro- walks. It could prove very interesting to study the distribu- duce an ad hoc cutoff for k. Therefore, if we consider the tion of the pair of real quantities that constitute the complex temperature fluctuations in a particular point x~0 of the last number lm : namely, we can look at the plot of, say, the real Re Im scattering surface, and we estimate it in terms of and imaginary part of lm , i.e., ðlm Þ and ðlm Þ, for a 2 given value of l. This set of 2l þ 1 numbers for each one of 0 ^ x~0; Þj 0i, we obtain a divergent quantity. Note that hwej areð not saying that the temperature anisotropy is diver- the real and imaginary parts can naturally be expected to gent, but only that h0j^ 2ðxÞj 0i, during the inflationary display a Gaussian shape (which, in turn, would make the period, is divergent (see Appendix B). This divergence at distribution of jlm j a Rayleigh distribution). an early state would invalidate any subsequent analysis This seems to be a particularly relevant analysis, and we based on perturbation theory, which works under the as- do not know of anything like that which has been studied in sumption that the metric perturbations are small in every the literature. It seems to us that the traditional approach point. However, we know from the observational data that does not naturally lead to the consideration of that issue. these fluctuations of the mean temperature, in any particular Looking at the distribution of the corresponding phases point, are rather small 10 5 K. On the other hand, in the should be equally enlightening. Moreover, as we mentioned collapse proposal, these issues become much less problem- in the discussion around the Eq. ( 8), it would be interesting atic because the scheme indicates which variable we should to evaluate the quantity l defined there and compare the focus on: the variables subjected to the collapse are not result with any of the natural estimates for its value, par- ^ ^ ^ ^ ticularly, the expected ensemble average of its magnitude. yðx;~ Þ, ðx;~ Þ but the field modes yk~ðÞ, k~ðÞ, i.e., the collapse does not occur in the position space, and an inde- Another point worth revisiting is that it is usually believed that a large detectable amount of non-Gaussianity can pendent collapse is assumed for each mode k~. The quantities be expected when the initial state of the quantum of observational interest, namely, the j j2s, depend on the lm field is not the preferred Bunch-Davies vacuum state. expectation values hy^ ðÞi , h^ ðÞi , in the state ji after k~ k~ Nevertheless, in the collapse proposal, the quantum state the collapse, and, as we have shown, these can be estimated of the field after the collapse, is ji Þ j0i (the analysis of a directly in terms of the values of the random variables. particular characterization of the post-collapse state has been As we saw in the introduction, if we really took ðx;~ Þ done in Ref. [ 11 ]). Therefore, the curvature perturbation to be Gaussian and allowed the identification of its n-point responsible for the temperature anisotropies in the CMB is functions with the observations, we would have to accept due to the expectation values hy^ i and h^ i, which, in that such identification holds, in particular, for the one- k~ k~ point function, and that would lead us to a clear conflict principle, could generate detectable non-Gaussianities. between theory and observation. These quantities are never considered in the standard ac- Similarly, one must be careful when considering the counts, and it is clear that a further exploration of these ideas identification of the quantum three-point function: would be required for a serious assessment of their value. The other delicate issue related to the statistical aspects h0j^ ðx;~ Þ^ ðy;~ Þ^ ð~z; Þj 0i; (55) of the traditional approach is related to the ergodicity assumption. As we already saw in Sec. III , the CMB with the average over an ensemble of Universes bispectrum was defined as the three-point correlator of ðx;~ Þðy;~ Þð~z; Þ, and finally, the identification of l1l2l3 the lm through Bm1m2m3 l1m1 l2m2 l3m3 . The standard the latter with the measured quantities as an indicator of picture forces us to deal with the issue that the rhs repre- non-Gaussian features in the cosmological perturbations. sents an average over an ensemble of universes, while we 3 ~ As we saw, the bispectrum ~ ~ ~ ¼ ð 2Þ ðk1 þ f . . . g k1 k2 k3 have but one realization l1m1 ; l2m2 ; ; lnmn . To over- k~2 þ k~3ÞBðk1; k 2; k 3Þ is usually said to represent the lowest- come this issue, the standard approach relies on an ergo- order statistics able to distinguish non-Gaussian from dicity assumption, which identifies the average value of a Gaussian perturbations because Gaussianity is identified certain quantity in a process measured over time with the with the requirement that all statistical information is con- average value measured over the ensemble. tained in the two-point functions and, thus, implicitly, with There are various issues that lead one to be concerned the vanishing of all n-point functions with n odd. However, about this assumption and the application to the situation at the lowest odd integer is not 3 but 1, and, as we have already hand. The first thing we must be aware of is that ergodicity seen, there is a serious issue that arises when considering the is a property of systems in equilibrium, and it is rather one-point function. This, we believe, forces us to question unclear why this should be valid regarding the conditions and reconsider some of the standard arguments. associated with the inflationary regime. In fact, looking anew at the quantities normally associ- Next, as already mentioned, the ergodicity assumption is ated with the one-point function, we see that we have at our translated, in the case at hand, into the notion that the

023526-17 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) volume average of the fluctuations behaves like the en- spectrum, we see these are indeed orientation averages. For semble average; ‘‘the universe may contain regions where instance, the observational quantity Cobs 1 j j2 l ¼ 2lþ1 m lm the fluctuation is atypical, but with high probability most is just the orientation average value of the magnitudeP of the regions contain fluctuations with root-mean-square ampli- lm s for a fixed value of l. In the same way, we see that the tude close to ,’’ and, thus, one argues that the probability angle-averaged bispectrum Bl1;l 2;l 3 (25 ) is an orientation distribution on the ensemble translates to a probability average for fixed ls, and, as for the same reason as the one- distribution on smoothed regions of a determined size point function, it is quite unclear how to identify orienta- within our own Universe [ 44 ]. tion averages with ensemble averages. Thus, the statistical There are at least three issues that arise here: analysis would be more transparent if one would focus on (i) How do we go from the arguments supporting l1l2l3 the distribution of the quantities Bm1m2m3 . ergodicity in time averages to the corresponding As we saw, it is customary to take as an estimator for arguments for spatial averages? ^ the nonlinearity parameter the quantity fNL defined in (ii) Regarding the CMB, we, in fact, do not have access Eq. ( 32 ). This seems a bit problematic, as it involves a to the spatial sections that would allow us to inves- mixture of theoretical and observational quantities. tigate the space averages. We only have access to Ideally, one would like to have the two aspects rather the particular intersection of our past light come well separated. In fact, even within the standard approach, with the 3D hypersurface of decoupling. That is, for the case of the two-point functions, we have on one to a two-sphere that we see as the source of the hand the theoretical quantity, CMB photons that reach us today: the surface of last scattering. How do we go from spatial averages to th 2 2 2 2 19 C ¼ k PðkÞ ðkÞj ðkR Þdk; (56) averages over that two-sphere? l l D (iii) Each one of the quantities of interest, lm , is itself Z already a weighted average over the CMB two- and on the other hand the observational quantity, sphere [with the weight function given by the cor- 1 responding Y ð; ’ Þ]. Therefore, what would be Cobs j j2: (57) lm l ¼ 2l þ 1 lm the role of a new average over the ms? Why do we Xm need to perform any additional average? In other This independence of the definitions allows one to words, if one is willing to accept that the ensemble cleanly compare theory and observation. It, thus, seems averages should coincide with averages over the that one would want to consider studying the aspects tied to two-sphere, why would one not also accept that the non-Gaussianity using a quantity that can be equally sus- weighted averages over the two sphere should co- ceptible to theoretical and observational determination. incide with the equally weighted average over en- Here, we would like to propose, based on the considera- sembles? If we were to accept this, we would tions we have been discussing, the option we present conclude that the weighted average (with weight below. Y and fixed l and m) of T=T over the surface of lm First, motivated by the quantity defined in Eq. ( 25 ), let us last scattering for our Universe should coincide introduce the definition of the observed bispectrum as the with the corresponding weighted average of that orientation average, quantity over the ensemble of universes, without any further averaging over m. The problem is that l1 l2 l3 obs ð Þ the latter would be zero, but the former is just lm , Bl1l2l3 l1m1 l2m2 l3m3 obs ; (58) m1 m2 m3 ! which, empirically, is not zero. Thus, there must be Xmi something wrong with our arguments and assump- and the definition of the normalized observational reduced tions. One should then consider what it is, and why. bispectrum as the quantity Let us leave that rhetorical question based on a position we are rejecting and consider again the issue of averaging over 1 obs ð2 þ1Þð 2 þ1Þð 2 þ1Þ l1 l2 l3 obs m. It seems clear that what we are dealing with here are b~ l1 l2 l3 B ; l1l2l3 2 4 3 l1l2l3 orientation averages: The different lm would mix among sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 ! themselves if we were to redefine the orientation of the 5 4 (59) coordinate chart used to describe the CMB two-sphere. Thus, when we look at the averages that are actually and, finally, let us define the magnitude of the bispectral performed in connection with the study of the primordial fluctuations as

obs 1 obs F l1l2l3 jð l1m1 l2m2 l3m3 Þ 19 ð2l1 þ1Þð 2l2 þ1Þð 2l3 þ1Þ We note, in relation to this point, that there are intrinsic mi problems in considering ergodicity of processes within a two- X m1m2m3 ~obs j2 sphere as discussed in Ref. [ 49 ]. Gl1l2l3 bl1l2l3 : (60)

023526-18 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) One can then compare this pure observational quantity We have seen in previous works that, despite the fact that with the corresponding theoretical estimation characterizing the motivation for such considerations seems to be purely a suitable ensemble average, where each element of the philosophical and tied to issues like the measurement ensemble is specified by a concrete choice of the random problem in quantum mechanics, the analysis has led us to numbers xk~ that we have used to represent the collapses. expect certain departures that could potentially be of ob- That is, one can carry out a Monte Carlo simulation leading servational significance. to an ensemble of possible CMB skies characterized by In previous works, we have focused on two main ob- possible choices of xk~s and then characterize each one of servationally related issues: the shape of the spectrum and obs the question of tensor modes. We have argued previously those in terms of the corresponding value of F l1l2l3 . Finally, one would analyze the degree to which our own real sky is that it would be very unlikely that one could find a scheme generic when characterized in that manner. It seems clear in which the function CðkÞ would be exactly a constant and that this kind of theoretical calculation or simulation cannot that some dependence on k is likely to remain in any be carried out in the standard approach, as there is no place reasonable collapse scheme, simply because we do not c there for the concrete randomness (characterized, in our expect those collapses to follow exactly the k ¼ Z=k rule for the time of collapse for each mode. Any remaining approach, by the numbers xk~), which would be produced in a simulation of our collapse proposal. dependency of CðkÞ on k will lead to slight deviations from Thus, the study of the quantity displayed in Eq. ( 60 ) the standard form of the predicted spectrum. In fact, analy- seems to offer an approach to study the issue at hand that ses of this issue have been carried out in Refs. [ 8,9], indeed has the advantage of allowing a direct comparison confirming these expectations. These have been used to between the purely observational quantities, untainted by set the first bounds emerging from the CMB observational theoretical models, and the quantities that are purely de- data on these kinds of theories. fined in terms of such theoretical analysis. This, in fact, We have also stated, in earlier works, that the most clear seems to share some of the spirit of the analyses made in prediction of the novel paradigm we have been proposing Refs. [ 30 ,31 ], although our proposal provides a clear op- is the absence of tensor modes, or at least their very strong tion to compute the observational and theoretical quantities suppression. The reason for this can be understood by in complete separation, and that seems not to be available considering the semiclassical version of Einstein’s equa- in the former. The reason for this seems easy to understand: tions and its role in describing the manner in which the The fact that we maintain a clear distinction between inhomogeneities and anisotropies arise in the metric. As ensemble averages and orientation averages avoids the we have explained in our approach, the metric is taken to possibility of the confusion associated with the simple be an effective description of the gravitational DOF, in the observation that the ensemble average of the quantity classical regime, and not as the fundamental DOF suscep- tible to be described at the quantum level. It is, thus, the ð Þobs m1m2m3 ~obs l1m1 l2m2 l3m3 Gl1l2l3 bl1l2l3 ; (61) matter degrees of freedom (which in the present context are represented by the inflaton field), the ones that are de- appearing in Eq. ( 60 ), vanishes identically. scribed quantum mechanically and which, as a result of a The detailed analysis of estimators like this will be fundamental aspect of gravitation at the quantum level, carried out in future works, but we wanted to present it undergo effective quantum collapse (the reader should as an example of the type of studies that could be motivated recall that our point of view is that gravitation at the by our approach to the whole question of the emergence of quantum level will be drastically different from standard structure from quantum fluctuations in the inflationary quantum theories, and, in particular, it will not involve early Universe. universal unitary evolution). This collapse of the quantum state of the inflaton field leads to a nontrivial value for hT^ , which then generates the metric fluctuations. The VI. PREDICTIONS AND DISCUSSION i point is that the energy momentum tensor contains linear Focusing on trying to understand the essence of the and quadratic terms in the expectation values of the quan- emergence of inhomogeneous and anisotropic features tum matter field fluctuations, which are the source terms from a quantum state, that is, homogeneous and isotropic 20 determining the geometric perturbations. In the case of the and in the absence of a measurement process, has led us scalar perturbations, we have first-order contributions pro- to consider modifying the standard approach through the _ ^_ incorporation of the collapse hypothesis. portional to 0 , while no similar first-order terms appear as sourceh ofi the tensor perturbations (i.e., of the gravitational waves). Of course, it is possible that the 20 In the early Universe, there were no observers or measuring collapse scheme works at the level of the simultaneously devices, and, in fact, the conditions for their emergence is the result of the breakdown of such symmetries, so it would seem quantized matter and metric fluctuations, as has been pre- very odd if one takes a view that they are part of the cause of that sented in Ref. [ 50 ], although, as explained there, we would breakdown. find it much harder to reconcile that with the broad general

023526-19 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) picture that underlies our current understanding of physical the hospitality of the IAFE. The work of S. J. L. is sup- theories. ported by PIP N 0152/10 from Consejo Nacional de In the present work, we have focused on the modified Investigaciones Cientı´ficas y Te´cnicas, Argentina. statistical considerations associated with this novel para- digm. We have argued that the collapse process itself could APPENDIX A: THE BUNCH-DAVIES VACUUM IS be the source of non-Gaussian features. We discussed some HOMOGENEOUS AND ISOTROPIC, difficulties associated with the usual identification of mea- CORRELATIONS NOT WITHSTANDING suring quantities with the quantum n-point functions and particularly found that extending the standard arguments to Theorem.— The Bunch-Davies vacuum state (this, by the the one-point functions lead to disastrous disagreements way, is also valid for the Minkowski vacuum state) is with observations. homogeneous and isotropic. We have shown that our approach provides expressions Proof.— The vacuum state is defined by a^ 0 0. This k~j i ¼ that have no parallel in the standard formulations and that is supposed to represent the state of the quantum field after allow a precise identification of the location of the random- a few e-folds of inflation (up to negligible corrections of N ness, as exemplified by our theoretical formula ( 48 ) for lm order e , with N as the number of e-folds), i.e., the in terms of the random numbers characterizing the collap- exponential expansion of the Universe takes the metric ses, namely, the quantities xR , xI , xR , xI . This kind of k~;1 k~;1 k~;2 k~;2 and all fields to a very simple state, which, in particular, expression facilitates all resulting statistical considera- is highly symmetric. It is easy to see that the state 0 is ^ j i tions, and, in particular, it is the basis for the theoretical ~ ~ ^ ^ H&I. The generator of spatial translations is P k~ kay~ ak~. estimation of the quantity ( 60 ). ¼ k So a translation by D~ leaves the state unchanged,P We have proposed various novel ways to look into the ^ eiD~ P~ 0 0 , and, thus, the state is homogeneous. One statistical aspects of the problem: j i ¼ j i (i) We indicate the importance of exploring the true can equally check that it is isotropic considering the be- nature of the one-point function by studying the havior of the state under rotations. Q.E.D. degree of deviation from zero of the complex quan- Furthermore, this is clearly not in contradiction with the obs 1 obs existence of quantum correlations, as they do not imply a tity l (i.e., expanding and refining l ¼ 2lþ1 lm breakdown of the symmetry. This can be easily seen in a the analysis of Ref.P [ 29 ]). (ii) We have argued that it is worthwhile to study the Einstein-Podolsky-Rosen setup. Consider a state of two specific form of the distribution of the values of the spin- 1=2 particles that result from the decay of a spin 0 obs particle. observed quantities lm for each fixed l. (iii) We have proposedj newj characterizations of the Let us assume that the decay occurring along the x axis (the particles’ momenta are P~ Px^ with x^ the unit quantities normally associated with the bispectrum ¼ and the quantum three-point functions, which can vector in the x~ direction). Using the ~z polarization states be computed both in purely theoretically and in as a basis for the Hilbert space of each of the spin- 1=2 a completely observational fashion. This is the particles, the state of system after the decay is quantity defined in Eq. ( 60 ). 1 1 2 2 1 It is clear that this work represents only the first step in ð Þ ð Þ ð Þ ð Þ : (A1) the study of the statistical aspects of the cosmic structure j i ¼ p2 ðjþi ji þ jþi ji Þ and its generating process during inflation, within the ffiffiffi context of the new paradigm that centers on the collapse The state can be seen to be invariant under rotations hypothesis. Much more work remains to be done, but we around the x axis (simply because it is an eigenstate with hope this can become a research avenue of great richness zero angular momentum along that axis). It is, neverthe- and one that would lead to important insights, with pos- less, straightforward to compute the correlations between ^ 1 ^ 1 ^ 1 ^ 2 ^ 2 ^ 2 sible implications not only for the generation of structure the operators Sð Þ ~ð Þ N~ ð Þ and Sð Þ ~ð Þ N~ ð Þ cor- ¼ ¼ itself but for the modification of quantum theory, which responding to the projectors of the spin along the vectors ^ 1 ^ 2 would underlie the collapse mechanism and which, as has (taken to be on the y z plane) N~ ð Þ and N~ ð Þ, respectively. been argued before, might have a deeper origin at the The result, as is well known from the studies of Bell’s quantum/gravity interface [ 5,16 ,17 ,51 ]. inequalities, is proportional to cos , where is the angle ^ 1 ^ 2 between N~ ð Þ and N~ ð Þ. Thus, the existence of these corre- ACKNOWLEDGMENTS lations is in no conflict, whatsoever, with the rotational The work of G. L. and D. S. is supported by CONACyT invariance of the state . It seems that the belief that there Grant No. 101712 and PAPIIT Project No. IN107412-3. is something in the correlationsj i that indicates the break- G. L. acknowledges financial support by a CONACyT down of the symmetry is tied to some implicit intuition postdoctoral grant. D. S. was supported in part by sabbati- suggesting that each one of the particles is in a definite state cal fellowships from CONACyT and DGAPA-UNAM and of spin, even before there are any measurements involved.

023526-20 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) We, of course, know that such notions are in strong conflict The issue we are facing is, of course, related to the so- with both quantum theory and the experimental facts. called measurement problem in quantum mechanics [ 52 ]. Any reasonably complete discussion of this question and 2 APPENDIX B: DIVERGENCE OF 0 ^ x;~ Þj 0i the broader one concerning the interpretation of quantum h j ð mechanics would require much more space than what can In order to illustrate a common misconception about 2 reasonably be accommodated here, so we point the reader the quantity h0j^ ðx;~ Þj 0i, let us consider the following to some of the literature [ 53 ]. Here, we will merely present argument: The gravitational potential that gives tempera- our version of the status of the general problem, touching, ture anisotropy is not ^ ðk~Þ from the primordial era but ^ when appropriate, on the particular instance that concerns ðk~ÞðkÞ, where ðkÞ is the transfer function. Since us here: the cosmological setting. That issue has not re- 2 ^ 2 ðkÞ / ln ðkÞ=k for large k, h0j j0i is convergent in the ceived to much attention from the physics community, with UV regime. notable exceptions represented by Penrose [ 17 ], Hartle [ 7], The previous statement is not correct. The transfer func- and others. tions characterize the physics that is relevant after the end (i) Quantum physics as a theory of statistical physics — of inflation (i.e., the physics characterizing the behavior of A point of view indicating that quantum mechan- the radiation and particles that emerge as the result of ics acquires meaning only as it is applied to an reheating, including, for instance, the famous plasma os- ensemble of identically prepared systems [ 54 ]. cillations). That is why they are called transfer functions. Thus according to this view, a single atom, in They indicate how the perturbations that were present isolation, is not described by quantum mechanics. during inflation (to be exact, at its end point) evolve during We must avoid getting confused by the correct, but the radiation-dominated era into the perturbations that are simply distracting, argument that atoms in isola- present at the time of decoupling, which is the relevant one tion do not exist. The issue is whether, to the for what we see in the CMB. The transfer functions are, of extent to which one can neglect its interactions course, not relevant at all during the inflationary era itself, with other systems, quantum mechanics is appli- which is the era we are focusing on (and the one in which cable to the description of a single atom. One we argue the collapse should occur). The issue, related to might wonder about the meaning of that question, 0 ^ 2 0 the divergence of h j ðx;~ Þj i, clearly refers to the infla- given that we can not say anything about the atom tionary era. The (rhetorical) question we are posing is the without making it interact with a measuring de- ^ following: If we do not take the expectation value of k to vice. The question is simply whether or not apply- be the inflationary prediction for the k, as that would be ing the formalism of quantum mechanics to treat zero, and we are instead instructed to compute the vacuum the single isolated atom can be expected to yield ^ ^ expectation value for quantity k k and to use it in order correct results regarding subsequent measure- to make our estimates of k, then, why would it be ments. It is sometimes argued that this is a non- incorrect to compute the vacuum expectation value of ^ ^ sensical question, as these results are always ðxÞðxÞ and take it as an estimate of the value of ðxÞ statistical in nature. However, in fact, this state- (during the inflationary era)? The issue is that such an ment is not accurate; for instance, if the atom (e.g., estimate would be infinite, and then the whole scheme of of hydrogen) was known to have been prepared in perturbation theory on which the treatment is based would its ground state, the probability of detecting it in be invalid. We would, therefore, not be able to rely on it, some other energy eigentstate is zero. Thus, there either to consider the study of the predictions regarding the must be something empirical in the description of radiation-dominated era or the CMB. that single atom by its usual quantum mechanical state. The notion that quantum mechanics is not APPENDIX C: ON THE INTERPRETATION applicable to a single system [ 55 ] is, thus, simply OF QUANTUM THEORY AND THE incorrect. However, the most important point in COSMOLOGICAL CONTEXT relation with the issue that concerns us in this Here, we will briefly consider, for the convenience of the article is that taking a posture like this about reader, some of the most common views we have found quantum physics, would be admitting from the among colleagues regarding the interpretation of quantum beginning that we would have no justification in physics and their implications for its application to the employing such a theory in addressing questions cosmological problem at hand, and, at the same time, we concerning the unique Universe. Note that the present our basic perspective on such views. A more de- situation would not be helped if we assumed that tailed discussion of these issues has been presented in there exists, in some sense of the word, an en- Ref. [ 5], and the reader is advised to turn to that reference semble of universes, as we would, in principle, for a thorough analysis of the alternative postures taken by have access just to one: ours. Advocates of the researchers in the field. standard accounts of inflation usually invoke some

023526-21 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) sort of identification of the statistics with an en- descriptions of the object that cannot be cast in the semble of universes and the statistics within one quantum mechanical state vector. On the other hand, single Universe. However, as we have argued if the answer is ‘‘yes’’, we would be implying that throughout this work, it is paramount to avoid the state vector says something about the object in confusion between those types of statistical mea- itself. Independently of these issues, it seems rather sures as a matter of principle, even if one would clear that if we follow the above described view, we later want to argue they might be identified in would have abandoned the possibility to consider some special cases. It should be clear that in order questions about the evolution of the Universe in the to be able to argue for any such identification, one absence of sapient beings or to consider the emer- must be in a position to say something about the gence, in that Universe, of the conditions that are individual system. In fact, we can imagine consider- necessary for the eventual evolution of humans, ing any individual system whatsoever, say, a cloud of while making use of our quantum theory. gas, and constructing an ensemble of similar systems (iii) Quantum physics as a noncompletable description by performing say ‘‘all possible rotations and trans- of the world —Within this class we consider any lations of the system.’’ It is clear that the resulting posture that effectively, if not explicitly, states, ensemble is, by construction, homogeneous and iso- ‘‘The theory is incomplete, and no complete theory tropic. Now, can this be used to say anything about the containing it exists or will ever do.’’ Such a view original system? Clearly, it cannot, simply because our includes any posture indicating we should use starting point was a completely arbitrary system. Thus, quantum mechanics ‘‘as we all know how’’ and if we negate from the start that our theory could say supported by the observation that no violation of 22 anything about an individual system, there is no way quantum mechanics has ever been observed. At we can apply it to our Universe. Furthermore, going this point we must note that although this is a back to the general case, if a quantum state serves only literally correct statement, the prescription refers, to represent an ensemble, how is each element of the in fact, to the Copenhagen interpretation, which, as ensemble to be described? Perhaps, it cannot be de- discussed above raises severe interpretational is- scribed at all. Then, how are we supposed to make sues that become insurmountable once we leave statistics over the attributes of such systems? the laboratory and attempt to apply quantum theory (ii) Quantum physics as a theory of human knowl- to something like the Universe itself. edge —According to this view, the state of a quan- The fact is that, in situations in which one cannot tum system is not considered as reflecting anything identify the system and the measuring apparatus, the ‘‘objective’’ about the system in question but just observables that are to be measured, the entity provides a characterization of ‘‘what we know about carrying out those measurements, and the time at the system’’. 21 This attitude, naturally rises the which the measurements take place, the theory does question of what there is to be known about the not provide any clearly defined scheme specifying system if not something that pertains to the system. how to make the desired predictions. Thus applies, Advocates of this point of view often answer in in particular, to the questions pertaining to the early terms of correlations between the system and the Universe. However, according to such pragmatic measuring devices. This leads us to consider the approach we should be satisfied with the fact that question of the significance of these correlations. the predictions have, in fact, been made and that Note that the meaning of the word ‘‘correlation’’ they do seem to agree with observations. The prob- implies some coincidence of certain conditions per- lem is, that in the absence of a well-defined set of taining to one object, with some other conditions rules, it becomes quite unclear whether or not such pertaining to a second object. Therefore, if a quan- ‘‘predictions’’ follow or not from the theory without tum state describes such a correlation, there must be the use of extraneous and ad hoc , but convenient hypothesis suitably introduced in connection to the some meaning to the conditions pertaining to each particular problem at hand. Especially suspicious one of the objects: the quantum state and the system. are, of course, those predictions that are, in fact, Are not these, then, the very same conditions that retrodictions, and, on this point, we should recall are described by the quantum mechanical state and that, long before inflation was invented, Harrison those that correspond to the object? If the answer and Zel’dovich [ 59 ] had already concluded what is ‘‘no’’, it must mean that there are further should be the form of the primordial spectrum, based on quite general observations about the nature 21 One can find statements in this sense in well-known books, of the large-scale structure of our Universe. for instance, ‘quantum theory is not a theory about reality, it is a prescription for making the best possible predictions about the future, if we have certain information about the past’’ [ 56 ]. See 22 In practice, this view is essentially indistinguishable from the also Ref. [ 57 ]. so-called for all practical purposes approach to the matter [ 58 ].

023526-22 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) (iv) Quantum physics as part of a more complete ‘‘times of measurements,’’ and so forth. In other description of the world —Completing the theory words, concerning a specific measurement situation, would require something that removes the need for and the corresponding description within the Many invoking any sort of a priori distinct notions of Wolds Interpretation, the issues would be the follow- external measurement apparatus, an external ob- ing: When does a world splitting occur? Why, and server, etc. One proposal of this kind is provided under what circumstances does it occur? What con- by D. Bohm’s ‘‘pilot wave theory’’ [ 60 ], and, in stitutes a trigger? And, finally, what selects the basis particular, we note a specific proposal to apply in which such splittings takes place? The ideas based such ideas to the cosmological problem at hand on a decoherence type solution and its shortcomings [61 ,62 ]. As we have mentioned, there are other will be discussed in some detail in Appendix D. proposals invoking something like the dynamical APPENDIX D: SHORTCOMINGS THE USUAL reduction models proposed by Pearle [ 23 ] and/or EXPLANATIONS OF THE EMERGENCE Ghirardi et al. [18 ] and the ideas of R. Penrose OF PRIMORDIAL INHOMOGENEITIES about the role of gravitation in modifying quantum AND ANISOTROPIES mechanics in the merging of the two aspects of nature [ 17 ] (see also Ref. [ 63 ]). In the We offer here a very brief version of the discussion context of inflationary cosmology, the works presented in Ref. [ 5] of why we find the most widely (Refs. [ 4,8–10 ,14 ]) are an example in which the held views on the way of addressing our problem as issues are faced directly. Those works represent lacking. In our experience, these are the ‘‘decoherence the position we favor, and which is inspired, in approach’’ (perhaps supplemented by the many worlds part, by the arguments made in Refs. [ 17 ,18 ,20 ]. interpretation of quantum theory) and the ‘‘consistent or (v) Quantum physics as a complete description of the decohering histories approach.’’ world —Here we refer to any of the postures indicat- We will, thus, offer some considerations regarding the ing that quantum mechanics faces no open issues degree to which these two proposals do, in general, offer a and that, in particular, the measurement problem has ‘‘solution’’ to the measurement problem and, particularly, been solved. The advocates of this position fall into of their applicability in the present context. Again, we groups identified with one of the the main currents: suggest turning to Ref. [ 5] for a more exhaustive discussion those that subscribe to ideas along the so-called of all these issues. ‘‘many world interpretation of quantum mechan- 1. Decoherence ics,’’ and consider this to be a solution to the mea- surement problem, and those that consider that this Decoherence is a direct prediction of quantum mechan- problem has been solved by the various considera- ics, with very important implications in many experimen- tions involving ‘‘decoherence.’’ We consider that the tal situations. The central observation is that, in the many world interpretation does very little to ameli- general experimental setting involving a quantum me- orate the measurement problem, as there is a map- chanical system, one should take into account the fact ping between what in that approach would be called that generally the system becomes entangled with the the ‘‘splittings of worlds’’ 23 and what would be environment, consisting of degrees of freedom that are called measurements in the Copenhagen interpreta- not under the control of the experimentalist and which tion. In fact one can see that almost every issue that are, moreover, uninteresting from the point of view of can be raised against in the context of the latter has a what one is interested in measuring. On the other hand corresponding one in the many worlds interpretation many colleagues seem to think that it has implications choice of basis or context in dealing with measure- that go well beyond that and which represents a complete ment problems correspond to the selection of basis and satisfactory solution of the measurement problem in for the ‘‘world splittings,’’ time of such splittings quantum mechanics. This is not the case, and the inter- would need to include those that one takes as the ested reader is directed to consult the literature on the matter (see, for instance, Ref. [ 64 ]).

23 Here, we will limit ourselves to indicating the postures It is often claimed that there is no splitting of the worlds in the that, in this regard, are held by several people that have many worlds interpretation, but the fact of the matter is that, whenever people make use of it, they cannot avoid talking about considered the issue at length, in order to contrast them with things such as ‘‘our branch,’’ ‘‘the realms we perceive,’’ or other the prevalent notions among inflationary cosmologists. notions that implicitly make use of a notion that is essentially just Let us start with the explicit conclusion by that of a splitting of the world. One can see this in each specific A. Neumaier [ 65 ]: application of the idea, by focusing on the complete description of what one would take as ‘‘the relevant state describing our reality’’ and following it in time backward and forward. In the ‘‘Many physicist nowadays think that decoherence inflationary situation at hand, this is easily done by focusing on provides a fully satisfying answer to the measure- the symmetry of the state describing the quantum fields. ment problem. But this is an illusion.’’

023526-23 SUSANA LANDAU, GABRIEL LEO ´ N, AND DANIEL SUDARSKY PHYSICAL REVIEW D 88, 023526 (2013) Also worthwhile is the warning by M. Schlosshauer [ 66 ] associated with the system possessing a value of certain against misinterpretations: physical property in a given range at a given time. 24 That is, each one of the projector operators is associated with ‘‘...note that the formal identification of the reduced a certain range within the spectrum of a given observ- density matrix with a mixed state density matrix is able. A given family F of such projectors, is called self- easily misinterpreted as implying that the state of consistent if the resulting histories do not interfere the system can be viewed as mixed too.... the total among themselves. In that case, one may consistently composite system is still described by a superposi- assign probabilities to each individual ‘‘coarse-grained tion, it follows from the rules of quantum mechan- history’’ within the family. 25 ics that no individual definite state can be attributed The probability assigned to one particular coarse- to one of (the parts) of the system...’’ grained history within a consistent family is given by P ¼ Tr ðP^ ðt ÞUðt ;t ÞP^ Uðt ;t Þ...... P^ Uðt ;t Þ and the explicit refutation by E. Joos [ 67 ]: n n n n1 n1 n1 n2 2 2 1 ^ y P1Uðt1;t 0ÞU^ ðtn;t 0Þ Þ; (D1) ‘‘Does decoherence solve the measurement prob- 0 lem? Clearly not. What decoherence tells us is that where Uðt; t Þ stands for the standard unitary evolution certain objects appear classical when observed, but operators connecting the two times. This approach, what is an observation? At some stage we still have apparently, has many followers within the cosmology to apply the usual probability rules of Quantum community, even though it has received some strong Theory.’’ criticisms in the foundational physics community [ 71 ]. The issue is that, although the scheme works fine once Thus, the decoherence ideas, even if taken together with one has selected a particular decoherent family F, there the many worlds interpretation, clearly fail to offer a exist, in principle, an infinitude of other such decoherent 0 satisfactory resolution of the matter in general [ 68 ], and, families F , which are, however, mutually inconsistent, 0 in particular, it fails to do so in connection for the situation (i.e., there are elements of F and F that do interfere, 0 we face here. and, thus, fFg [ f F g is not a decohering set of histories). Let us end by noting that even W. Zurek, one of the most This problem, which is well known to the advocates of well-known researchers in the field of decoherance, states this approach should according to them be addressed by unequivocally that [ 69 ]: the ‘‘single family rule,’’ which indicates one should never consider, simultaneously, more than one family. ‘‘The interpretation based on the ideas of decoher- Moreover, according to this approach, we should never ence and ein-selection has not really been spelled out make any logical inferences while considering together to date in any detail. I have made a few half-hearted two inconstant sets, as they can produce logical contra- attempts in this direction, but, frankly, I was hoping dictions (see, for instance, the example discussed in to postpone this task, since the ultimate questions Sec. 3 of Ref. [ 72 ]. As noted in Ref. [ 73 ], it is unclear tend to involve such ‘‘anthropic’’ attributes of the what would justify this rule within a reasonable onto- ‘‘observership’’ as ‘‘perception’’, ‘‘awareness,’’ or logical view of what the theory is describing. ‘‘consciousness,’’ which, at present, cannot be mod- The issue becomes then how should we select a eled with a desirable degree of rigor.’’ particular family to be that from which the particular history that represents ‘‘the actual one’’ is to be chosen. The point is that in the context of inflationary cosmol- (It seems very reasonable that the fact that one assigns ogy, in which we want to explain the emergence of the probabilities within a family indicates that the interpre- seeds of cosmic structure, and, thus, the emergence of the tation must be that one of the histories in that family is conditions that would eventually create the conditions for ‘‘actualized’’ in our world. Otherwise, one must wonder the emergence of galaxies, stars, biology, and intelligent what these probabilities refer to [i.e., the probabilities life, we cannot even hope to rely on any of those anthropic assigned according to Eq. ( D1 ) are probabilities of what? (see, however, Ref. [ 72 ])]. Let us emphasize notions. Thus, decoherence does not represent an adequate once more that we do not want to take the view that solution to the problem at hand.

24 2. Consistent Histories In the cosmological setting, one must use a subtler relational time approach [ 7], in which one of the dynamical variables is The general scheme as described in Ref. [ 70 ] is based used as an effective time parameter. The cosmological scale on the consideration, given a quantum state of the sys- factor is a popular choice. 25 The characterization of the histories as coarse-grained is tem ji, or, more generally, a density matrix ^, for the ^ meant to reelect the fact that the projection operators PnðtnÞ system at time t0, of families of histories characterized are generically associated with a finite range of eigenvalues ^ by a set of projection operators fPnðtnÞg , each of which is rather than a single eigenvalue.

023526-24 QUANTUM ORIGIN OF THE PRIMORDIAL FLUCTUATION . . . PHYSICAL REVIEW D 88, 023526 (2013) they are probabilities of ‘‘observing a certain value of projector. Take the initial state for the quantum fluctuations a physical quantity when that quantity is measured’’ (usually called the vacuum) 0 , and note that it is homo- because the whole point of this program is to have an geneous and isotropic. j i interpretational framework for quantum theory that avoids The next step is to consider an arbitrary collection of using concepts like measurements or ‘‘observations’’ in the values for time ftig and construct the family associated with discussion (otherwise, one might as well have retained the that initial state and the two projector operators PHI and Copenhagen interpretation). Pnon at each of those times. One can easily check that this The fundamental problem is that there is in principle, no procedure defines a family of consistent histories, simply clear way to single out one specific family without relying because the dynamics (characterized by the operators U) on an a priori given set of questions one interested on— preserves the symmetries (homogeneity and isotropy). those associated with the quantities whose spectral char- Consider now the question of what the probability is acteristics one chooses to construct the family—and this that (at a given time, characterized in the appropriate ambiguity leads to serious interpretational difficulties [ 73 ]. relational way) the Universe is isotropic and homogene- In a specific situation, we might be guided in making the ous. Evaluating this using the formula ( D1 ) (and starting ‘‘appropriate’’ choice, by the questions the experimental with the vacuum state), we find that any history contain- setup is ‘‘asking’’ (in fact, this has a close analogy with the ing the orthogonal projector at any time Pnon has zero use of Bohr’s rule in a given experiment). Nevertheless the probability, while the history containing only the opera- fact remains that, in general, without such an common sense, tors PHI has probability one. We seem to be led to con- or practical guidance, there is no well defined procedure clude that, at any time, the Universe is homogeneous and indication how to select the family. We must here emphasize isotropic. It, thus, can have no inhomogeneities or anisot- that one is not asking how to select a particular history within ropies at all. We would then have to face not only such a the family but how to select a particular family of constant problematic prediction but also the fact that the approach histories from within the collection of all possible decoherent we followed has led us to two conflicting conclusions: this families. The problem here is: what justifies considering that latter one and the one obtained in, say, Ref. [ 74 ]. the experimental setup corresponds to asking a particular In fact, this problem is similar to those considered in question; this seems to implicitly assume that the measuring Ref. [ 71 ] and that discussed in Sec. 3 of Ref. [ 72 ]. The apparatus is always in a state of definite value for the mea- posture advocated in Ref. [ 72 ] is that one should accept all sured quantity and never in the superposition of states of that the different families and use only the appropriate one in type. This seems very close to what one does in adopting the connection with the particular question one is asking in Copenhagen interpretation. order to make ‘‘bets about the future,’’ while at the same Returning to our specific problem, of describing the time renouncing the idea that there is a single objective evolution of the very early universe, there is simply no reality. As discussed in Ref. [ 73 ], such a posture seems recipe provided by the theory, that would dictate the se- unsustainable in addressing the fact that we humans seem lection of the appropriate projector operators and, thus, of to coincide regarding our appreciation of the ‘‘world out the appropriate family (if we require a description that does there.’’ not make use of the fact of our own existence and our own Apparently, if choosing to accept the consistent histories asking of certain questions as part of the input). approach to the quantum theory, in general, one would Let us see a clear manifestation of this problem in the have to adopt a rather problematic position close to that cosmological situation of interest: Consider the family of of posture b) in Appendix C, with the difficulties already projector operators as is done in Ref. [ 74 ], and obtain their mentioned there. It seems that this is not a satisfactory results, but then note that, alternatively, we might consider situation regarding something that ought to serve as a the family in what follows. We next define PHI to be the fundamental theory and, in particular, to help us deal projector into the intersection of the kernels of the gener- with the quantum aspects of the early Universe. The inter- ators of translations and rotations (note that it is the pro- ested reader is referred to the literature, particularly, to jector onto the space of homogeneous and isotropic states). the works referred to above, for much more extensive Let us further define Pnon I PHI the orthogonal discussions on the matter.

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