a typical rainfall time series has several properties that are difficult to capture in a simple statistical model , including a heavily skewed marginal distribution that is distinctly non - gaussian , a large proportion of zero values , and frequent large fluctuations .thus there is a need for specialised models for rainfall which can capture the unique characteristics of this type of process .the existing literature on modelling rainfall is large and spread over fields such as hydrology , atmospheric sciences , environmental risk analysis and statistics . classifies the different approaches into four categories : meteorological models based on large sets of differential equations , multi - scale models concerned with the spatial evolution of rainfall , statistical models that capture spatial and temporal trends , and finally stochastic process models that make simple assumptions in order to remain parsimonious . in the followingwe will focus on models of the last category .many of the early attempts at modelling rainfall use a simple model that represents the rainfall occurrence process as a two - state markov chain , and models the intensity of rainfall with a gamma distribution  .this model is easy to interpret and enables the direct use of likelihood methods for fitting .however , it makes several restrictive assumptions on the rainfall process , and may require a high - order markov chain with many parameters to capture observed temporal dependence .there is also a large literature on modelling rainfall for hydrological applications based on a form of the poisson - cluster model , first developed by and .this model is based on a hierarchical structure , with a primary poisson process controlling the arrival of storms and a secondary process generating cells from each storm , which then deposit rainfall .there have been numerous extensions of the poisson - cluster model that focus on fitting specific properties of the observed rainfall process .for example , considers an extended model that allows for different rainfall cell types , where each cell has a random duration and intensity depending on its type , allowing for the different types of precipitation that are observed in practice .  in terms of fitting specific properties of rainfall that are of interest for hydrological applications , extensions to the poisson - cluster modelgenerally perform very well .however , as remarked by , there is always a trade - off between the inclusion of more features and the mathematical tractability of the resulting models .poisson - cluster based models are usually fitted by the method of moments , which involves matching analytical expressions for properties such as the mean , variance , and proportion of dry intervals to their empirical equivalents . , remarks that the method of moments approach `` suffers from the disadvantage that the parameter estimates can vary greatly depending on the properties used in the fitting procedure '' , and proposes a spectral estimation method for estimating rainfall models based on point processes .the contribution of the present paper is twofold .first , we develop a parsimonious and analytically tractable model that captures the distributional features and autocorrelation structure of the observed rainfall time series .we relate the model to the framework of lvy - driven , continuous - time arma ( carma ) processes , and use this connection to develop a suitable fitting method , which is illustrated using empirical rainfall data .the model structure may be interpreted as a non - clustered poisson model with multiple cell types , cf .the main benefit of this model is its parsimonious formulation based on a stochastic integral , making it suitable for applications where mathematical tractability and fitting methodology are of primary importance .second , we derive a formula specifying the so - called risk - neutral distribution of a general class of lvy - driven stochastic processes , which includes our model as a special case .we then use this result to calculate prices for rainfall futures based on our model .rainfall derivatives were introduced at the chigago merchantile exchange ( cme ) in 2010 , as a recent addition to the class of weather - related products .these products have a large potential market in all economic sectors that depend on favourable weather conditions , such as farming and energy development .the literature on rainfall derivatives pricing is currently rather limited .because the underlying rainfall can not be traded directly , the rainfall derivative market is incomplete , and thus there is no single fixed price for the derivative . due to this incompleteness, there are several distinct methods that can be used for rainfall derivative pricing .it appears that most current approaches rely on either the utility indifference approach or risk - neutral pricing using the esscher transform .the utility indifference approach is used in . in this paper a modification of the poisson - cluster model is considered , which makes the rainfall intensity a markov jump process , thus enabling maximum likelihood estimation .this modification relies on the assumption that the data used , though inevitably discrete , approximates continuous - time observations of the rainfall intensity . also use the utility indifference approach for pricing hypothetical rainfall derivatives in kenya , based on a markov - gamma model with seasonality .a natural choice for pricing based on lvy process models is the esscher transform , as it is structure - preserving and moreover gives rise to a minimal entropy martingale measure . use the esscher transform for pricing based on a version of the daily rainfall model by .they fit simulated monthly rainfall totals to a normal inverse gaussian distribution , and use the esscher transform to obtain a risk - neutral distribution . also use the esscher transform for pricing , but base their underlying rainfall model on an independent increment process .in contrast to the markov - gamma and independent increment models that have been used for pricing rainfall derivatives , our proposed model has the advantage of not making assumptions about temporal independence or markovianity of the rainfall process increments . furthermore , the model and the fitting method used are based on the assumption that the available data represents accumulated rainfall , i.e. the instantaneous intensity can not be directly observed , which is typically the case for applications . by allowing for temporal dependence and considering carma processes of arbitrary order, we obtain a model with a flexible autocorrelation structure , which is particularly relevant for data on finer time scales .this flexibility is illustrated by fitting the model to hourly rainfall data .we also compare our model to that given in , which was used in for pricing rainfall futures based on daily data from detroit .this paper is structured as follows : section 2 discusses characteristic features of the rainfall process in light of data from different locations and time scales .section 3 presents the rainfall model and shows how it fits into the continuous - time arma ( carma ) model framework .section 4 gives details on the fitting method , and section 5 investigates the model performance using empirical data . in section 6we derive a method for pricing rainfall derivatives based on our model .in this section we motivate the structure of our model by illustrating some of the characterising features of rainfall time series .we base this illustration on two data sets which will be used throughout this paper : the first consists of hourly accumulated rainfall amounts at heathrow ( uk ) over the years 1980 - 2012 , provided by the .the second data set gives daily accumulated rainfall amounts in detroit ( us ) over the years 1980 - 2010 , provided by bloomberg professional service .figure [ detroit_heathrow_3yrs ] shows the rainfall time series for both locations over the years 2008 - 2010 .these plots illustrate that the rainfall process is subject to sudden transitions between periods with little or no rain and periods of higher intensity , causing the large spikes in the graph .there is also a large amount of zero values in both data sets , as shown in table [ zerotable ] .because the data is rounded to the nearest unit of measurement ( mm for the heathrow data and inch for the detroit data ) , these zero values represent all data points with a value smaller than this unit .the proportion of zero values is dependent on the measurement time scale , with fewer zero values for the daily data .this is because periods with no rainfall must persist throughout the measurement time interval in order to induce a value of zero in the data ..percentage of rainfall measurements equal to zero . [ cols="<,<,<,<",options="header " , ]we have introduced a new class of continuous - time stochastic processes , driven by the hougaard lvy process , and shown how it can be used to construct a parsimonious and analytically tractable model for rainfall . by generalising the ornstein - uhlenbeck process representing rainfall intensity to a continuous - time arma ( carma ) process, we obtain a model with a very flexible autocorrelation structure .we presented a general fitting method for this class which exploits a correspondence between integrated carma and arma processes .we showed that the model fits the marginal distribution of the rainfall very well on both hourly and daily time scales .in particular , the marginal fit for daily rainfall is better than that of the standard model described in , and we also obtain an adequate fit to the mean and marginal distribution on a month - by - month basis .  by virtue of the carma generalisation , the extended model manages to accurately reproduce the autocorrelation structure of the observed rainfall , a characterising feature of the process which becomes increasingly significant for smaller time scales .the last part of the paper gives a result specifying the risk - neutral expectation of a function of the rainfall process , which can be used for pricing general derivatives written on a precipitation index . to construct a risk - neutral measure we use the esscher transform , with a time - dependent parameter representing the risk premium .we state the result for a general moving average process , a class which includes our model as a special case .the pricing methodology constitutes an important generalisation of the independent increment model considered in , which does not allow for price dynamics of derivatives .we illustrated the pricing method by calculating futures prices based on empirical daily rainfall data from detroit , and showed how they can be calibrated to observed prices .given a sufficient amount of price data , one can study the properties of the market price of risk ( mpr ) that is implied by this calibration . discuss the market price of weather risk , focusing on temperature derivatives , and provides an example of modelling the mpr of temperature derivatives as a deterministic function of the seasonal temperature variation .as the market for rainfall derivatives matures , one could conduct a similar study for the mpr of rainfall derivatives .[ [ acknowledgements . ] ] acknowledgements .+ + + + + + + + + + + + + + + + +  we thank the associate editor and two anonymous referees for constructive suggestions that led to significant improvements of the paper .r. c. noven gratefully acknowledges financial support from the grantham institute for climate change , imperial college london .we thank the uk meteorological office and the british atmospheric data centre for providing the data used .in the following we present the proofs of our theoretical results .[ intlem ] let \rightarrow \mathbb{r} ] ensures that the terms in the above equation are well - defined .now we consider the lvy - khintchine formula for subordinators , which takes the form where is the lvy measure associated with .we can analytically continue this formula to complex arguments , and so we get that }-1 ) \nu(dy ) \ , dv \right\}. \end{aligned}\ ] ] we note that this expression can also be written as \ , dl_q(v ) \right\ } \right],\end{aligned}\ ] ] where is now a non - stationary stochastic process with jump measure depending on time , namely   thus we see that conditioning with respect to the measure has the effect of exponentially tilting the jump measure of at time according to , so the jumps of at times will be weighted more or less in the expectation depending on the sign of .   * hougaard process . * the hougaard process has lvy measure given by  in terms of the tweedie parameterisation .we also have that when is the hougaard process , the function defined in takes the form .\ ] ]    barndorff - nielsen , o. e. shephard , n. 2001 , ` non - gaussian ornstein - uhlenbeck - based models and some of their uses in financial economics ' , _ journal of the royal statistical society - series b : statistical methodology _ * 63*(2 ) , 167241 .onof , c. , chandler , r. e. , kakou , a. , northrop , p. , wheater , h. s. isham , v. 2000 , ` rainfall modelling using poisson - cluster processes : a review of developments ' , _ stochastic environmental research and risk assessment _ * 14*(6 ) , 384 411 .rodriguez - iturbe , i. , cox , d. r. isham , v. 1987 , ` some models for rainfall based on stochastic point processes ' , _ proceedings of the royal society a : mathematical , physical and engineering sciences _ * 410*(1839 ) , 269288 .2012 , ` met office integrated data archive system ( midas ) land and marine surface stations data ( 1853-current ) ' .ncas british atmospheric data centre .available from http://badc.nerc.ac.uk/view/badc.nerc.ac.uk__atom__dataent_ukmo-midas .

Abstract: we propose a parsimonious stochastic model for characterising the distributional and temporal properties of rainfall . the model is based on an integrated ornstein - uhlenbeck process driven by the hougaard lvy process . we derive properties of this process and propose an extended model which generalises the ornstein - uhlenbeck process to the class of continuous - time arma ( carma ) processes . the model is illustrated by fitting it to empirical rainfall data on both daily and hourly time scales . it is shown that the model is sufficiently flexible to capture important features of the rainfall process across locations and time scales . finally we study an application to the pricing of rainfall derivatives which introduces the market price of risk via the esscher transform . we first give a result specifying the risk - neutral expectation of a general moving average process . then we illustrate the pricing method by calculating futures prices based on empirical daily rainfall data , where the rainfall process is specified by our model .


reading in wallace s paper that  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` the plain truth is that there are currently no hidden - variable theories which are generally accepted to reproduce the empirical predictions of any interacting quantum field theory .this is a separate matter to the conceptual problems with such strategies we do not even have qft versions of these theories to have conceptual problems with . '' , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  i was very surprised . in my opinon , the problem of _ existence _ of a realistic theory which allows to recover the _empirical _ predictions of relativistic quantum field theory has been solved with bell s proposal for `` beables for quantum field theory '' , and that the ongoing search for better pilot wave theories ( for example in ) in this domain is motivated by metaphysical rather than empirical problems .wallace s justification for this claim was even more surprising for me :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` it is debatable whether field - based modificatory strategies will actually succeed in reproducing the predictions of qm . for recall: as i argued , it is crucial for these strategies that they are compatible with decoherence : that is , that the preferred observable is also decoherence - preferred . a hidden - variable theory whose hidden - variablesare not decoherence - preferred will fail to recover effective quasiclassical dynamics . and in qft ( at least where fermions are concerned ) the pointer - basis states are states of definite particle number , which in general are not diagonal in the field observables . '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  instead , i have considered it to be the job of decoherence to explain why we observe particles in a situation where the fundamental pilot wave beables are fields or even something more fundamental .following wallace , decoherence is not only unable to do this job , but causes even problems with empirical viability .this was sufficient motivation to evaluate this thesis in more detail .once empirical viability is questioned , the natural place to look for is the equivalence proof between pilot wave theory and quantum theory . considering this proof in section [ sec : proof ] , we find it necessary to include the state of the observer into the picture .the measurement device is not enough once there exists doubts if the observer really observes the state of the measurement device , one has to consider the full picture . as a consequencewe need only the actual , unobserved observer state .the equivalence theorem in this form would remain valid even if the beables themself would be unobservable .the only loophole which could prevent the recovery of quantum predictions is the question if different macroscopic states can be distinguished by the beables , or , in other words , if these states have negligible overlap as functions of the beables . for the particular case of field beables ,it has been shown in that the overlaps become negligible for large particle numbers .but field theory is only a particular example .one should expect that more fundamental physical theories give other , more fundamental candidates for the beables strings , loops , or cells as proposed in  which give the observable fields only in the large distance limit .their configuration variables ( assuming some lagrange formalism ) are natural candidates for pilot wave beables .we obviously have to expect that their connection to the decoherence - preferred observables is different from that in field theory .this leads to some questions : what is the possible relation between pilot wave beables and decoherence in such a more general situation ?while the decoherence - preferred particle variables differ from the field variables , there exist , nonetheless , some nontrivial connection between them .what if there is no such connection ?if there is a connection , how does it appear ?if it appears in some more or less natural way , what are the consequences ? does this connection have an influence on the ability of pilot wave theories to recover quantum predictions ?these are the questions we want to study in this paper .the answers we find are quite positive for pilot wave theories with more fundamental beables : if there is absolutely no connection between beables and decoherence , we can even prove a theorem that the overlap becomes insignificant .on the other hand , we find that a connection has to be expected : decoherence starts from a predefined decomposition into systems , and such a decomposition can be obtained in a reasonable way only from the beables .this leads to a non - trivial connection , where different systems have independent sets of beables .we find that this leads to some preference for product states .while this makes it impossible to apply our general theorem , it appears plausible that in this case the overlap descreases approximately exponentially with an increasing number of system .wallace s thesis claims that pilot wave beables have to be decoherence - preferred to recover quantum predictions . to reject it , it is sufficient to consider one equivalence proof which does not depend on this assumption . in our case ,a quite standard version of the proof seems sufficient . on the other hand , in anotherequally standard , but inferior variant wallace s thesis looks quite natural .the version of the equivalence proof which does not depend on it is enough to falsify wallace s thesis , but the other variant gives us some interesting hint about the origin of this thesis .a nice summary of the main lesson of our proof has been given by struyve :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` saunders expressed a worry for using fields as beables .he expressed some doubts whether localized macroscopic bodies are represented by localized field beables .if not , he claimed , it would be unclear how a pilot - wave model with field beables reproduces the quantum predictions .however , although it is true that a pilot - wave model in which localized macroscopic bodies are represented by localized fields reproduces the quantum predictions , it is by no means a necessary requirement .as long as wavefunctionals corresponding to macroscopically distinct states are non - overlapping in the configuration space of fields , the field beables will display the outcomes of measurements . '' , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  if you do not doubt this and are not interested to understand the origin of wallace s thesis you can skip the remaining part of this section .  else, let s consider the general scheme of a quantum measurement in pilot wave theory .we restrict ourself in this paper to the simplest case of a measurement with two discrete eigenstates .this seems sufficient : as human beings , we can distinguish only finite numbers of different states , and , given that the number of particles in macroscopic states is much larger than the number of states we can distinguish , the number of the discrete states does not really matter .so assume we have a quantum system in the initial state , with .the measurement is an interaction with the remaining part of the universe which measures the projection operator  an operator with two eigenvalues , for and for all other states .the interaction hamiltonian may be something like , where is the momentum operator of some unspecified pointer variable .for the system - internal hamilton operator we simply set .as the initial state of the wave function of the universe we assume a product state where describes the remaining part of the universe .after the interaction , we obtain some superpositional state where and are already different states .all the time , the conditional wave function obtained by putting the actual value of into the wave function of the universe is well - defined and correctly defines the guiding equation for .but in general it does not follow the effective schrdinger equation of the system ( which would be trivial for ) : during the measurement , it evolves in dependence of and .this is the effective collapse process .the collapse is finished if the two wave functions no longer overlap , and if this property is stable in time .after this , the conditional wave function no longer collapses and becomes an effective wave function , that means , it follows the ( in our case trivial ) schrdinger equation of the system . if ( for fixed ) the do not overlap , the conditions define a decomposition into two sets with for the effective wave function of the system appears to be .if the wave function of the universe is in quantum equilibrium , this happens with probability .all this corresponds nicely with the description given by quantum theory .the only difference is the following : for a quantum measurement being finished , the should be in macroscopically different states . in pilot wave theory , we have to require that the do not overlap , and that this condition remains stable in time .all we have to prove is that pilot wave theory recovers quantum predictions , thus , we need only one direction .moreover we can assume that macroscopically different states remain to be macroscopically different in time .therefore , all we need to recover quantum predictions is the hypothesis that _ macroscopically different states do not overlap as functions of the pilot wave beable .  note that the configuration , as the configuration of the rest of the universe , contains also the configuration of the observer himself .the condition that the observer has seen the result is of course different , more restrictive than the condition that the measurement has been finished .but it is quite close : all we need is that the configurations of the observer are different for different measurement results .thus , there should be a decomposition into two sets with in other words , should be already sufficient to distinguish different measurement results .it is also worth to note that the whole consideration is based on really existing objects , not on observed objects . according to pilot wave theory ,the wave function of the universe , as well as the configuration , really exist .thus , the conditional wave function we have constructed based on these ingredients also really exists .as well , the quantum equilibrium distribution is a distribution for the actual , real values of .last but not least , describes the real configuration of the observer , who , therefore , is in really different states if for different . none of these objects is observed in some way , nor is there any necessity for them of being observed.  this seems different if we , instead of including the observer into the pilot wave picture , follow the copenhagen quantum scheme , where the observer is located in the classical part , outside the quantum domain .everything else seems similar , with being replaced by the configuration of some measurement device .but , once we have not included the observer into the picture , we can not consider the observers real configuration . as a consequence , lot s of unnecessary questionsappear : the observer has to observe , somehow , the result of the measurement. what does he observe if he looks at the device ?the wave function of the measurement device or , instead , the actual value of ?is it justified to put the actual value into the wave function , or should we use , instead , some observed value of ? maybe the observation is fooled like the particle detectors in ? in this incomplete picture , these questions seem unavoidable and difficult to answer . examples of arguments of this type have been given by wallace and brown :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` suppose we accept that it is the entered wavepacket that determines the outcome of the measurement .is it trivial that the observer will confirm this result when he or she `` looks at the apparatus '' ?no , though one reason fothe effective wave function of the system appears to be the nontriviality of the issue has only become clear relatively recently . the striking discovery in 1992 of the possibility ( in principle ) of `` fooling '' a detector in debroglie - bohm theory should warn us that it can not be a mere definitional matter within the theory that the perceived measurement result corresponds to the `` outcome '' selected by the hidden corpuscles.'' ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  in the objective picture , the observer is not obliged to `` confirm '' the result .it is simply an objective fact that for the effective wave function of the system appears to be .  __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` that the possibility of fooling detectors casts doubt on the claim by maudlin p. 483that the so - called effective ( post - measurement ) wavefunction of the object system is _ defined _ ( in part ) by the positions of the corpuscles associated with the apparatus '' . __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  it is exactly reverse : because the conditional wave function is _ defined _ by , and not by any results of observations , the possibility of fooling detectors does not matter at all .it is this incomplete consideration of the measurement process which can be easily used to motivate wallace s thesis : the incomplete consideration can be interpreted as a reduction of general measurements to measurements of the beables .how does this fit with decoherence ? to reduce a decoherence - preferred measurement to a non - decoherence - preferred one seems awkward , in conflict with decoherence .thus , it seems necessary to require that the measurements of beables have to be decoherence - preferred .but once we do not have to consider any measurement of , this argument fails .nonetheless , let s consider in more detail if anything can go wrong with the classical limit .last but not least , wallace focusses in his consideration of the quasiclassical domain :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` a hidden - variable theory whose hidden - variables are not decoherence - preferred will fail to recover effective quasiclassical dynamics . '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  and the classial limit works in pilot wave theory in a very different way than in usual quantum mechanics , a way which is centered around the beables . indeed , assume we have a wave function with .looking at the quantum potential , we find it being zero . as a consequence , for the phase and the trajectory the classical hamilton - jacobi equation holds .thus , and is a solution of the classical hamilton equations , and we are in a completely classical situation . once all our configuration is defined by , we can not distinguish our world from a purely classial world , and , in particular , can not observe nor nor the wave function .this defines an essential difference between pilot wave theory and usual quantum theory : in usual quantum theory , a state with is a widely distributed , non - localized quasiclassical state , very different from a wave - packet localized in and we would need for the classical limit in pure quantum theory .is there any conflict between this pilot - wave - specific picture of the classical limit and the one provided by decoherence ?the limit is described in terms of the beable variables , not in terms of decoherence - preferred observables , so , at least in principle , one could expect some problems .but i see no base for any conflict , because in this limit we do no longer have to choose between incompatible measurements : in the classical limit , all observables commute , that means , in particular , that a measurement of the decoherence - preferred observables , whatever they are , is no longer incompatible with other observations .as we have seen , in the more complete pilot wave picture , which includes the observer as well , we do not have to consider observations of beables .so we avoid all problems related with unobservable trajectories or fooled detectors .the picture is completely based on objective entities as .we obtain an objective collapse picture described by the conditional wave function , which with the correct born probability _ actually becomes _ one of the eigenstates . what remains to checkis only one assumption : that _ macroscopically different states do not overlap . _  of course, not overlapping at all is only an idealization .what we need is that the overlap is not significant . to evaluate the significance of overlaps , we need a precise definition . we have proposed and used such a definition in : the overlap has been defined there as a relation between two wave functions and : this definition has a simple interpretation as the probability that a particle guided by appears to be in a region where dominates .if all relevant overlaps are sufficiently small compared with , one can replace the by non - overlapping approximations so that the probability that this approximation becomes relevant can be expressed in terms of the overlaps , and , therefore , appears to be sufficiently small .if the overlap between macroscopically different states becomes insignificant or not depends on the particular pilot wave theory on it s hamilton operator as well as on it s choice of the beables .decoherence plays an important role in the process of macroscopic amplification , therefore it can have a nontrivial influence on the overlap .  despite thisit appears possible to prove a quite general theorem about the overlap between two states .this theorem assumes that there is no connection at all between decoherence and the pilot wave beables : the result of the decoherence - governed amplification process are simply two different states , and whatever the connection between them from point of view of the decoherence process , from point of view of the pilot wave beables they have no connection at all .thus , they have to be simply two independent random states in a sufficiently large - dimensional hilbert space . in this case, we can compute the expectation value of the overlap and consider it s dependence on the dimension of the hilbert space .it appears that in this limit the overlap becomes insignificant .such a general theorem is important even for those cases where we can not apply it it allows to play the scientific game known as `` burden tennis '' : what happens if there is no connection at all is a reasonable `` null hypothesis '' .if one thinks that in a particular case the situation is different , one has the burden of the argumentation . to show only that the general theorem is not immediately applicable seems not sufficient, it has to be at least plausible that the resulting overlaps will become significant .monte carlo simulation results for overlap between two random states in -dimensional space .the slowing down of the initial decrease rate is explained by the integral of , which fastly approaches it s limit .,scaledwidth=90.0% ]  for simplicity , we start with the consideration of a real n - dimensional hilbert space. we want to compute the following expectation value for the overlap between two states , , which are independent and randomly distributed on the unit sphere :    for , this integral can be taken explicitly and gives . for small values of one can use monte carlo simulation to compute the integral .the result , as presented in figure [ fig : points ] , does not look promising : after an initial decrease down to , the rate of decrease dramatically slows down , suggesting the possibility of a nontrivial lower bound ( see ) .a cross - check with another way to compute the integral has given an even worse result : an approach to a fixed limit .this was caused by a program error , but a remarkable and helpful one : it has shown that the integral of equation below behaves very well for large .this observation was a good starting point to find the proof below .[ th : real ] for the limit of the expectation value for the overlap in is zero .  to estimate the integrals of type we replace them by integrals over the unit cube for the function .this requires the introduction of a weight factor .the weight is the inverse of the volume of the infinitesimal conus , which becomes projected on a given surface element .thus , we obtain the following rule for replacement of the integrals : given the form of the weigth factor , it seems useful to split the integral into two parts a `` localized part '' containing states with for above states , and a remaining part containing everything else .let s at first evaluate the remaining part . here , in one of the two integrals we can use the estimate ,while for the other part the estimate holds ( for every , and , therefore , for the maximum as well , we have ) .this gives with but because , , and the remaining integrals are probability measures on the unit cube , we immediately obtain .in fact , it can be proven as well that this integral has the limit .it fastly approaches this limit , as can be seen in figure [ fig : points ] .but is sufficient to give it remains to estimate the localized part .the reason for naming it that way is that the states in this part are , for small , sharply localized around their maximal value .indeed , it follows from that now , the whole integral splits into parts defined by the maximal coordinates of the two wave functions , .the function we have to integrate in can be estimated for the diagonal parts by , and , because of , for the non - diagonal parts , by .if we , instead , replace this function with , we obtain an integral over a probability probability measure on some subset , which gives something . in thisintegral , all parts appear in a symmetric way , thus , the integrals over each of them should not depend on and .thus , each part can give at most .this gives the following estimate for the local part of the integral : thus , for every given we can choose so that for sufficiently large   it seems worth to note that in this proof we have not even required that the states , should be orthogonal .what changes if we consider , instead , the usual complex hilbert space ? not much .we have to double the dimension of space . the invariant measure on be obtained as the image of the invariant measure on from the standard hopf projection , so that we can simply use the same measure on as the probability measure .the function we have to integrate changes only in a minor way : instead of one - dimensional condition , we have to use the complex analogon , and to multiply it with : for this integral , the following theorem holds :  [ th : complex ] for the limit of the expectation value for the overlap in is zero .the only place which requires modification in comparison with the previous proof is that we obtain an overlap of order for of the localized pairs .everything else remains unchanged .thus , we obtain with the same conclusion .thus , if there is no connection at all , everything is fine .but this is an assumption we can not rely on . at the other extreme, we have an explicit counterexample where the overlap never becomes negligible , so that the standard equivalence proof fails .this example is one - particle theory , artificially described with a field ontology . in this case , in agreement with the results of struyve , we have found in that all states have an overlap of at least  the universal overlap for orthogonal one - particle states in field theory .what are the characteristic properties of this example which lead to this failure of the equivalence proof ?we can find several very special properties , which , in combination , allow to explain this failure :  * first , there is an additional non - trivial conservation law the conservation of particle number . without this , the counterexample would not work .it would be possible to obtain states with higher number of particles , which already have negligible overlap .in particular , an approximate conservation of particle number is not sufficient : it only increases the decoherence time . *second , we have an extremal choice of the initial value only one particle . with a much larger number of particles ,say , , the problem would disappear as well . *last but not least , we have a special relation between the field ontology and the decoherence - preferred observables : the decoherence - preferred states with fixed particle number are localized in a certain environment of the vacuum state in the field variables . as well , without this special relation , the problem would disappear , as a consequence of theorem [ th : complex ].  given the thesis of wallace that the pilot wave variables have to be decoherence - preferred , there is a certain irony in the observation that some non - trivial connection between pilot wave variables and decoherence - preferred variables is necessary for the things going wrong .unfortunately , in the physically interesting cases , in particular in field theory , we can not apply our theorems [ th : real ] , [ th : complex ] , because we have some nontrivial connection between decoherence - preferred observables and beables .even if , in the case of field theory , there is no identity , they are far away from being completely unrelated , so that one can not assume that the macroscopic states generated by decoherence are independent random states .the question we want to consider now is if there is some natural connection between them .  in this contextit becomes important to observe that the hamilton operator taken alone does not allow to define `` the '' decoherence - preferred basis .one needs more , namely a `` decomposition into systems '' , to define a decoherence - preferred basis . where does this `` decomposition into systems '' come from ?decoherence theory remains silent , it takes the decomposition as given . the particular subdivision into the various systems we see around us stars , planets , stones , cats , and human beings can be defined only in some environment of the actual state of the universe . for other possible states of the universe for example , states where the earth does not even exist the system which defines , for example , a given cat simply does not make sense .thus , in interpretations like many worlds , which do not have some well - defined actual configuration of the universe as pilot wave interpretations , we have also no natural subdivision around this state into systems .  leaving this problem to these interpretations ( see for more ) let s continue with the situation in pilot wave theory . herewe have some basis for a derivation of a decomposition into subsystems we need for decoherence .indeed , pilot wave theory defines , at every moment of time , some well - defined configuration of the universe .this configuration can be used to develop some linear theory in the tangent space of the configuration space at .the various systems we see around us may be identified with linear subspaces of .this is , of course , not the place to consider possible details of such a construction .the only point we want to make here is that the tangent space at is a very natural ecological niche for the definition of all these subsystems .if we use this place to define the subsystems , the decomposition of the tangent space gives a corresponding decomposition of the hilbert space , which approximates in the environment of .the notion of `` approximation '' used here is also a specific property of pilot wave theories : the guiding equation for some finite time interval depends only on the wave function in the relevant environment of , so that the restriction of the global wave function to some environment can be considered as a meaningful approximation an argumentation which is not valid in other interpretations .now , the decomposition already defines a natural decomposition of into systems which can be used to start decoherence considerations .if we use a construction of this type to obtain the decomposition into systems , we obtain , automatically , a natural connection between the pilot wave beables , which describe the configuration space , and the decoherence - preferred observables derived starting with the decomposition : for each of the subsystems we have , by construction , separate pilot wave beables , that means , the decomposition into systems coinsides with a decomposition of the pilot wave beables .  from point of view of pure theory ,this relation is problematic : our theorem [ th : complex ] , which would be sufficient for our purpose , is not applicable once something very special happens we can not rely on the assumption that the states created by decoherence are independent random states .is there a special property of the states which could prevent us from applying theorem [ th : complex ] ?such a property should be stable in time else , it could be rejected as unimportant . butdecompositions into subsystems are useful exactly because the subsystems develop almost independently in time . in particular states which are product statesinitially remain product states with high probability .moreover , local interactions of some systems with their local environments tend to destroy superpositions between such product states . in pilot wave theories, this works in the following way : if a system in a superpositional state with some far away system interacts with its local environment , this gives some superposition of the local environment to obtain an effective wave function for the two systems .if is not in the overlap of the two functions , this gives or , or , destroying the superposition between the pure product states .giving these two effects stability of product states because of independent evolution of the systems , and reduction to product states because of independent interactions of particular systems with their particular environments we have to expect that the states obtained by decoherence have a tendency to be product states . given the connection between decoherence and pilot wave theory , products of states of subsystems appear to be product states in terms of the corresponding pilot wave beables : thus , the theorem [ th : complex ] is not sufficient in a situation where the systems used by decoherence are subsystems of beables , because the probability distribution of the states created by decoherence is not random , but has a preference for product states .we can not present a comparable proof for this general situation .nonetheless , it appears sufficiently plausible that nothing dangerous happens .the situation is similar to the case of orthogonal particles we have considered in : the n - particle states considered there have been also product states of one - particle states .as in this case , we have no simple product rule the product of the overlaps gives only a lower bound of the overlap of the product states .nonetheless , there are good reasons to expect that the decrease is sufficiently fast : indeed , it seems not unreasonable to expect that the overlap of the product states decreases in a similar way as the overlap of the n - particle states . in this case, the decrease would be approximately exponential in the number of different degrees systems , which is much more than we need to distinguish macroscopically different states .the plausibility arguments we have used in for the n - particle product state can be applied in the case of general product states : the distance between the maximal values of of the two product functions can be easily computed . indeed , whatever the functions which participate in products of positive - valued functions , the maximum will be reached in the points where every function reaches it s maximum .therefore , the distance between the maxima of the products can be obtained from the corresponding distances for the factors by the simple formula .thus , it is increasing with the number of factors .some plausible expectations about the behaviour of the overlap can be obtained if we consider the case of equal factor functions for all . in this case , the distance between the maxima increases as .let s also look at the line connecting the maxima .assume , the two functions have only one maximum at and decrease with distance from the maximum .then , in the one - dimensional case , there can be only one local maximum $ ] of the overlap function the point defined by .then , on the line connecting the two maxima in the n - dimensional case , the overlap function reaches it s maximum at the point , and this maximal value is .now , given that the integrals of functions with maxima give , we can expect the order of the corresponding overlap integral as being roughly proportional to the relation of the maxima .this relation is , thus , it decreases exponentially with .this gives a plausibility argument that the overlap integral will decrease approximately exponentially with the number of systems .an extremal case where we can compute the overlap exactly is the case where the wave functions can have only two values : and . in these cases ,the overlap for each system consists of the normalized area of the configuration space where above functions are . for product states of this type , the overlap is the product of the normalized areas of the subsystems , thus , the product of the probabilities of overlap for the subsystems. if all subsystems are of the same type , thus , give the same overlap proability , we obtain as the overlap probability for systems .this gives , again , the expectation that the overlap decreases exponentially with the number of systems .in we have made some numerical computations of overlap integrals for product states for some other functions .the results also support the thesis that the overlap decreases approximately exponentially with the number of systems .all these are , of course , only plausibility arguments . as well , we have considered here only the two extremal situations : independent random states , and pure product states. the general situation will be some mixture .  nonetheless , with these plausibility argumentsthe advantage in burden tennis is already quite large : we have not only the general theorem as the default assumption .we have also found lot s of arguments for the most plausible derivation from the default assumption product states instead of random states . additionally there is a large gap between the fast , exponential decrease which seems plausible and the minimal decrease which would be sufficient to preserve the phenomena , given the large numbers of particles in macroscopic states .given that the apparatus of quantum theory depends on such uncertain notions as macroscopic states , we should not expect much more .thus , the current situation seems close enough to the ideal one . designing new pilot wave theories, we are free to introduce whatever beables we prefer . while there is some theoretical possibility that the equivalence theorem fails for the resulting theory , it is sufficiently plausible that in general it does not .motivated by wallace s thesis that pilot wave beables have to be decoherence - preferred to allow a recovery of quantum theory predictions , we have studied the connection between pilot wave beables and decoherence .wallace s thesis has to be rejected : on the contrary , if we have no connection at all between decoherence and pilot wave beables , we can prove that the overlaps between macroscopic states become insignificant , which is sufficient to recover quantum predictions .on the other hand , we have found that we have to expect a nontrivial connection between decoherence and pilot wave beables .the point is that decoherence depends on the choice of a decomposition into systems , and we have found that the natural environmental niche for such a decomposition is closely connected with the beables . in this case, we have to expect that the states created by decoherence are not random , but have some preference for product states .fortunately , product states do not seem problematic as well .we have presented plausibility arguments that the overlap between product states decreases exponentially with the number of systems .that means , in above extremal cases random states as well as pure product states the overlap decreases sufficiently fast . thus , there is no reason to expect something wrong happens in the general case .  in special circumstances , the overlap may always remain significant : in the counterexample of one - particle theory described with field beables, we have an additional conservation law ( particle number ) and a very special initial value ( one ) for the conserved value as particular circumstances which force the overlap to remain significant . in the general case, it seems quite implausible that such things happen .thus , given the results of this paper , one can leave to burden of argumentation to those who doubt that a particular pilot wave theory is viable .the rejection of wallace s thesis allows to use decoherence as a useful , important tool in pilot wave theories .the tangential space at the actual configuration of the universe gives an ecological niche for decompositions of the degrees of freedom of the universe into systems .starting with such a decomposition into systems , we can apply decoherence to find out which are the most stable , most accessible observables .these observables are not necessarily the beables themself instead , as in the example of field theories , some sort of particles or quasiparticles may be decoherence - preferred , while the fundamental beables are not . in such a context , decoherence appears to be a useful tool to find out what will be observable .thus , the connection between decoherence and pilot wave theory is symbiotic : the actual configuration and its environment defines a place for the decomposition into systems , which is a necessary prerequisite for decoherence . in return, decoherence explains what we observe , even if we , instead of the fundamental beables , observe something different like some sorts of quasiparticles .y. aharonov , l. vaidman , about position measurements which do not show the bohmian particle position , in cushing , j. t. , fine , a. , goldstein , s. ( eds . ) : bohmian mechanics and quantum theory : an appraisal , dordrecht .kluwer academic publishers , pp . 141154 .( 1996 ) , http://arxiv.org/abs/arxiv:quant-ph/9511005[arxiv:quant-ph/9511005 ]  brown , h.r ., wallace , d. : solving the measurement problem : de broglie - bohm loses out to everett , foundations of physics , vol .4 , 517 ( 2005 ) http://arxiv.org/abs/arxiv:quant-ph/0403094[arxiv:quant-ph/0403094 ]         schmelzer , i. : a condensed matter interpretation of sm fermions and gauge fields , foundations of physics , vol .39 , 1 , p. 73( 2009 ) , also http://ilja-schmelzer.de/papers/clm.pdf[ilja-schmelzer.de/papers/clm.pdf ]

Abstract: motivated by wallace s thesis that pilot wave beables should be decoherence - preferred to recover quantum predictions , we consider the relation between pilot wave beables and decoherence .   we prove that without any connection between beables and decoherence the overlap between macrcopic states becomes negligible . this is sufficient to recover quantum predictions , so that wallace s thesis has to be rejected .   a natural connection between decoherence and beables appears if the decomposition into systems used by decoherence is based on the beables . while our first result becomes inapplicable in this case , we present evidence that the overlap becomes negligible too .


determination of the power spectrum of dark matter fluctuations in the observed hierarchical large - scale structures of the universe remains perhaps the compelling problem in cosmology .weak gravitational lensing due to the large - scale structure is recognized as a powerful probe of solving this problem as well as of constraining the cosmological parameters ( ; ; ; ) , because it can fully avoid uncertainties associated with the biasing problem .recently , several independent groups have reported significant detections of coherent gravitational distortions of distant galactic images ( ; ; ; ; ) . on the other hand ,the temperature anisotropies in the cosmic microwave background ( cmb ) can be the most powerful probe of our universe , especially of fundamental cosmological parameters ( e.g. , ) .the weak lensing similarly induces distortions in the pattern of the cmb anisotropies , and the lensing signatures will provide a wealth of information on inhomogeneous matter distribution and evolutionary history of dark matter fluctuations between the last scattering surface and present .we then expect that the cosmological implications provided from the measurements of lensing effects on the cmb will be very precise , because there is no ambiguity in theoretical understanding of the primary cmb physics and about the distance of the source plane . however , it is concluded that the weak lensing effects on the cmb angular power spectrum is small ( e.g. see and references therein ) , although the detailed cmb analyses need to also take into account the lensing contribution .recently , seljak & zaldarriaga ( 1999 ) ( see also ) developed a new method for a direct reconstruction of the projected matter power spectrum from the observed cmb map , and showed that it could be successfully achieved if there is no sufficient small scale power of intrinsic cmb anisotropies . in this method ,the lensing signals can be extracted by averaging quadratic combinations of the cmb derivative fields over many independent cmb patches like the analysis to extract the distortion effect on distant galactic images , even though the reconstruction maps have a low signal to noise ratio on individual patches .excitingly , the high - precision data from the boomerang ( ; ) and maxima-1 ( ; ) have revealed that the measured angular power spectrum is fairly consistent with that predicted by the inflation - motivated adiabatic cold dark matter models ( also see ; ) .the standard inflationary scenarios also predict that the primordial fluctuations are homogeneous and isotropic gaussian ( ) , and then statistical properties of any cmb fields can be completely determined from the two - point correlation function or equivalently based on the gaussian random theory ( hereafter bbks ; hereafter be ) . taking advantage of this predictability , various statistical methods to extract the non - gaussian signatures induced by the weak lensing have been proposed .bernardeau ( 1998 ) found that the lensing alters a specific shape of the probability distribution function ( pdf ) of ellipticity parameter for field point or peak for the gaussian case as a result of an excess of elongated structures in the observed ( lensed ) map .although the method could be a powerful probe to measure the matter fluctuations around the characteristic curvature scale of cmb , the beam smearing effect of a telescope is crucial for the detection because it again tends to circularize the deformed structures .van waerbeke , bernardeau & benabed ( 2000 ) then investigated that a statistically correlated alignment between the cmb and distant galactic ellipticities could be detected with a higher signal to noise ratio , provided that a galaxy survey follow - up can be done on a sufficiently large area .we have quantitatively investigated the weak lensing effect on the two - point correlation function of local maxima or minima in the cmb map , and it can potentially probe the lensing signatures on large angular scales such as that corresponds to the matter fluctuations with wavelength modes around ( ; ) . recently , using numerical simulations , it was shown that the lensing effect causes a change of normalization factors for three morphological descriptions of the cmb map , the so - called minkowski functionals , against their gaussian predictions ( ) .the purpose of this paper is to develop a new simple method for extracting the lensing - induced non - gaussian signatures from the cmb map based on the isotemperature statistics .we then focus on specific statistical properties of the intrinsic gaussian cmb field ; a field point that has a larger absolute value of the temperature threshold tends to have a larger absolute value of the curvature parameter defined by a trace of the second derivative matrix of the cmb field , while the ellipticity parameter similarly defined is uniformly distributed independently of the threshold because of the isotropic nature of the gaussian field . from these features, we expect that the weak lensing causes a larger distortion effect on structures of temperature fluctuations around a point with higher threshold .in particular , the lensing can induce a coherent distribution of the ellipticity parameter correlated with the threshold owing to the coupling between the cmb curvature and the gravitational tidal shear .to extract these characteristic patterns , we define three independent functions based on the isotemperature statistics that are obtained from the averages of quadratic combinations of the second derivatives of cmb field over isotemperature contours with each threshold . as a result ,we find the lensing effect on those characteristic functions generates a definite functional dependence of the threshold , and therefore the lensing signals could be easily measured as a non - gaussian signature since those functions have very specific shapes in the gaussian ( unlensed ) case .using numerical simulations of lensed and unlensed cmb maps including the instrumental effects of beam smearing and detector noise , we investigate the feasibility of the method .this paper is organized as follows . in [ method ]we formulate a method for extracting the lensing - induced non - gaussian signatures using the gaussian random theory for the primary cmb . in [ models ]we outline the procedure of numerical experiments of our method using the simulated cmb maps with and without the weak lensing effect . in [ result ] we present some results in the flat universe with a cosmological constant and investigate the detectability of lensing signatures by taking into account the measurement errors associated with the cosmic variance and the instrumental effects especially for the future satellite mission_ planck surveyor_. in the final section some discussions and conclusions are presented .in this section , we briefly review a relevant part of the gaussian random theory developed by bbks and be for three- and two - dimensional cases , respectively .first , we define the temperature fluctuation field in the cmb map as /t_{\rm cmb} ] and denotes the relative angle between the principal axis of and the -axis . by the meaning of equation above , hereafter we call a local _ curvature parameter _ around a given field point .moreover , if the isotemperature contour in the neighborhood of local minima or maxima is given by an ellipse of in the coordinates of principal axes , the parameter can be expressed in terms of and as  ] in equation ( [ eqn : pdf ] ) physically means that structures around a field point with larger absolute threshold tend to have a larger absolute value of the curvature parameter .in fact , this feature is more explicitly clarified by considering the conditional probability distribution for field points with a given threshold .figure [ fig : peakshape ] shows the distribution of curvature parameter subject to the constraint that the point has a given threshold , where the conditional pdf is defined by  ] : where we have neglected the terms including contributions of in equation ( [ eqn : lensisofun ] ) because we numerically confirmed that the contributions are always small . in the following discussions ,these three independent functions ( [ eqn : glfun ] ) are compared to the results of numerical experiments .most importantly , equation ( [ eqn : glfun ] ) shows that , although for the gaussian case in the absence of the lensing we should have for all because of , the weak lensing induces distinct non - gaussian signatures expressed as  ] .for the filtering of a telescope , the smoothing angle can be expressed in terms of its full - width at half - maximum angle as .the filtered lensed temperature field is given by similarly , the filtered second derivatives field of the cmb can be expressed as \nonumber \\  & = & \int\!\!\frac{d^2{\mbox{\boldmath}}}{(2\pi)^2}w({\mbox{\boldmath}};\theta_{\rm s } )  \delta_{ij}({\mbox{\boldmath}})e^{i{\mbox{\boldmath}}\cdot{\mbox{\boldmath}}}+\int\!\ !\frac{d^2{\mbox{\boldmath}}}{(2\pi)^2}\frac{d^2{\mbox{\boldmath}}'}{(2\pi)^2}w(|{\mbox{\boldmath}}+{\mbox{\boldmath}}'| ;  \theta_{\rm s } ) \delta_{in}({\mbox{\boldmath}})\xi_{n , j}({\mbox{\boldmath}}')e^{i({\mbox{\boldmath}}+{\mbox{\boldmath}}')\cdot  { \mbox{\boldmath}}}+\cdots ,  \label{eqn : fil2}\end{aligned}\ ] ] where in the first line of right hand side we have used the part integration and assumed that the surface integral is equal to zero for a large cmb survey sky .these equations ( [ eqn : fil1 ] ) and ( [ eqn : fil2 ] ) mean that the filtering procedure and the lensing effect on the cmb do not commute in a general case .especially , the last line in the right hand side of equation ( [ eqn : fil2 ] ) shows that the information about a certain mode of the lensing field is coupled to sidebands of the different modes of the cmb field .the problem of mode coupling therefore have to be carefully investigated for accurate measurements of our method .however , since the intrinsic cmb anisotropies have a small scale cutoff due to the silk damping and the directions of the cmb curvature and the lensing deformation field are statistically uncorrelated , we could employ a simple approximation that the filter function in the equation ( [ eqn : fil2 ] ) is applied both to the cmb intrinsic field and the lensing field independently .the variance of convergence field in equations ( [ eqn : glfun ] ) can be then expressed by where  ] .especially , the non - gaussian signatures are pronounced at high absolute threshold such as as a result of the strong coupling between the gravitational tidal shear and the large cmb curvature at such high threshold as explained . for a gaussian case, and should be equal to zero at all bins of and thus have both positive and negative large values of the cosmic variance in each bin , although the mean values do not exactly converge to zero yet for the number of our realizations .these results therefore mean that the non - gaussian signatures induced by the lensing could be significantly distinguished compared to the cosmic variance . in figure[ fig : covmat ] we show the contour map of calculated from those realizations , where is the mean value and the variance of the estimators of at threshold bin .we have confirmed that the correlations for and are similar to the result in this figure .these correlations would be required in order properly to quantify the significance of any departure from gaussian statistics when performing the fitting between the numerical results and theoretical predictions .figure [ fig : covmat ] indicates and we have actually confirmed that , if we take the data at intervals of , the correlation matrix becomes to be very close to diagonal in each bin for all the cases we consider in this paper . taking into account this result ,the following results are shown in intervals of .figure [ fig : funglt55 ] demonstrates the results with the filtering scale of arcmin computed similarly as in figure [ fig : funglt3 ] .one can see that increasing the filtering scale decreases the magnitude of lensing signals for and , because the filtering again tends to circularize the deformed structures in the cmb map as pointed out by bernardeau ( 1998 ) .however , even in this case of the lensed curves of and remain having the distinct functional dependence of compared to the gaussian case of . in practiceit is important to also take into account the detector noise effect on our method .we here assume the instrumental specification of channel of the satellite mission _planck surveyor _ ; the noise level of per a pixel on a side of the fwhm extent ( ) . the noise field at original fine grid is also convolved with the gaussian filter of fwhm scale to avoid domination of noise spikes at small angular scales ( ) .figure [ fig : funglt55wn ] shows the results .the noise effect reduces the amplitudes of compared to that in figure [ fig : funglt55 ] as a result of the change of quantity for the intrisic cmb anisotropies due to the noise . even in this case, the figure clearly shows that the noise level of planck does not largely change the lensed shapes of and compared to the results of figure [ fig : funglt55 ] , although the lensing signals at low thresholds such as are weakened .importantly , significant non - gaussian signatures on and still remain compared to .this is because our method has so far relied on the normalized observable quantities such as and those quantities are more robust against the systematic contributions of the detector noise than the cmb fields themselves that are certainly affected by the noise .figure [ fig : funglt55s25 ] shows the results with similarly as figure [ fig : funglt55wn ] .this figure explains that the lensing signals can significantly deviate from the unlensed case if the lensing effect is adequately large .another important question we should address is how useful cosmological information on the large - scale structure formation can be extracted from the lensing signals onto the isotemperature statistics presented in the previous subsection . since equation ( [ eqn : glfun ] ) shows that the non - gaussian signatures depend on and , we become to consider the problem how accurately the contribution of can be determined .as shown by jain & seljak ( 1997 ) , the magnitude of is sensitive to and parameters for the cdm models and also depends on the filtering scale for a realistic case . from equation ( [ eqn : smconv ] )we find that thus has following approximate scaling relations for flat universe models around the fiducial model ( , and ) : \approx0.157\times(\theta_{\rm fwhm}/3')^{-0.35 }  \times(\sigma_8/2.0)^{1.1}\times  ( \omega_{\rm m0}/0.3)^{0.25 } ,  \nonumber\\ & & \sigma_\kappa[10 ' ]  \approx 9.67\times10^{-2}\times(\theta_{\rm fwhm}/10')^{-0.47 }  \times(\sigma_8/2.0)^{1.0}\times(\omega_{\rm m0}/0.3)^{0.10}. \end{aligned}\ ] ] furthermore , since more fundamental information is contained in the three - dimensional mass fluctuations , we have to take into account the projection effect ( ; ; ; ) .the convergence field at depends mainly on the mass fluctuations with wavelength modes of and the structures distributed in wide redshift ranges of peaked at . the lensing distortion effect on the cmb mapcan thus be a powerful probe of the large - scale structures up to high redshift in principle , which is not attainable by any other means .table [ tab : fit ] summarizes the results obtained for the determination of with a best - fit and the error , which arises from the cosmic variance or also including errors due to the detector noise effect .the ` analytic ' value of in this table is calculated using the approximation for the beam smearing given by equation ( [ eqn : smconv ] ) .we here performed the -fitting between the numerical results and theoretical predictions for , and .note that we have used each data of the functions in the range of at intervals of , because the correlation matrix for those data is close to diagonal as explained in figure [ fig : covmat ] .then , the theoretical predictions of , and are given by two free parameters of and expressed by equation ( [ eqn : glfun ] ) .most importantly , table [ tab : fit ] clearly shows that estimated from the non - gaussian signatures on , and could be significantly detected with high signal to noise ratios compared to the unlensed case .here , error for the determination corresponds to for the fitting .figure [ fig : bestfit ] demonstrates an example of the best - fit results for the noise case with and arcmin .this figure shows that is constrained mainly from the numerical data of and at that have relatively small cosmic variances . on the other hand , is well constrained only by the data of .even if we use the value of directly measured from the lensed simulated cmb maps instead of the fitting parameter ( see the paragraph under equation ( [ eqn : glfun ] ) ) , it causes only the slight change of results in table [ tab : fit ] .one might then consider a possibility to determine by comparing the measured from the cmb maps to that of obtained from the fitting of through the relation of , but the constraint is much weaker than that of using the non - gaussian signatures on and .table [ tab : fit ] also shows that the beam smearing effect is crucial for our method and , in the case with and the lensing signatures are obscured by the cosmic variance .on the other hand , the detector noise effect does not largely affect the results .however , we have to note that the best - fit value of in all the considered cases is underestimated compared to the analytic value of calculated by the approximation ( [ eqn : smconv ] ) for beam smearing effect .the possible reasons for the underestimation are ascribed to the effect of the discrete pixel in the simulated maps or to the mode coupling between the cmb field and lensing field caused by the filtering as explained in [ filter ] .so far it is concluded based on the following results that the reason is mainly due to the discrete pixel effect .we have confirmed that the best - fit value of even for the unlensed case by our method also underestimates the value of calculated by the conventionally used approximation ( e.g. , ) expressed in terms of and as , where  ] ) through the intrinsic correlation between the cmb curvature and the temperature threshold predicted by the gaussian theory . by means of the non - gaussian signatures, it can be expected that the lensing signals are extracted from an observed cmb data irrespective of the measurements or equivalently the assumptions for the fundamental cosmological parameters .our numerical experiments indeed showed that the method allows us to extract the lensing signals with a high signal to noise ratio , provided that we have cmb maps with sufficiently low noise and high angular resolution as given by the planck mission .recently , seljak & zaldarriaga ( 1999 ) ( see also ) developed a powerful method for a direct reconstruction of the projected power spectrum of matter fluctuations from the lensing deformation effect on the cmb maps .the method focuses on the averages of quadratic combinations of the gradient fields of cmb over a lot of independent patches in the cmb sky like the analysis of the measurements of the coherent gravitational distortion on images of distant galaxies due to the large - scale structure . in the method ,the lensing signatures on individual patches are extracted as differences between the measured statistical measures and their all - sky averages .they showed that the reconstruction of input projected matter power spectrum could be successfully achieved if there is no sufficient small scale power of intrinsic cmb anisotropies . in this sense , however , the method partly depends on the statistical measurements of intrinsic cmb anisotropies , and this is main difference between their and our method which we would like to stress .moreover , our method focuses on the second derivative fields of cmb and , therefore , is more sensitive to the amplitudes of the projected matter power spectrum on smaller angular scales . anyway, since the lensing signals in the observed cmb sky are weak , we think that some independent statistical methods should be performed to extract them , which could also lead to constraints on the projected matter power spectrum at respective , different angular scales .extending our method presented in this paper , one might consider a following possibility to extract the lensing distortion effect .bbks and be have shown that structures around local higher maxima or lower minima of temperature fluctuation field tend to have more peaked shape and be more spherically symmetric around the peaks in the gaussian ( unlensed ) case . from these features , it can be also expected that the weak lensing causes stronger distortion effect on structures around the higher maxima ( or lower minima ) and it might provide us more significant non - gaussian signatures as a function of the temperature threshold of peaks than our method did .however , as quantitatively shown by bernardeau ( 1998 ) , the statistical measure for the peaks provides the same or lower signal to noise ratio as or than the statistics for a field point taking into account the cosmic variance , where he investigated the lensing effect on the probability distribution function of ellipticities around field point or peak .this is because the number density of peaks in the cmb sky is not so sufficient for these statistical measurements .for this reason , we prospect a similar conclusion for the signal to noise ratio obtained from the measurements of the lensing distortion effect on the structures around peaks using our method , although this work will have to be further investigated carefully .recently , schamlzing , takada & futamase ( 2000 ) have shown that the lensing effect on the minkowski functionals ( the morphological descriptions of the cmb map ) appears just as a change of their normalization factors against the gaussian predictions using the numerical experiments .we indeed have confirmed that the analytical predictions for the lensed minkowski functionals done in the similar way as presented in this paper give the same conclusion as the numerical results .the result comes from the fact that the lensing does not largely change the global topology of the cmb anisotropies in a statistical sense .likewise , it is known that the gravitational potential from which the shear is generated is invariant under the parity transformation and the lensing does not induce the so - called ` b - type ' polarization defined from combinations of the derivative fields of cmb fluctuations ( ; ) .these results mean that the weak lensing can not simply generate a new mode of pattern of the cmb anisotropies that is absent in the gaussian case . for these reasons ,in this paper we focus on the another information on statistical properties of the intrinsic cmb that is useful for the study of lensing effect and can be specifically predicted based on the gaussian random theory .another issue we should discuss is the possible effect on our results caused by non - gaussian features on the convergence field of the large - scale structure at that are revealed by the ray - tracing simulations ( ; ) , which we have ignored in the numerical experiments of the lensed cmb maps .however , since the lensing effect on the cmb can be treated as a mapping , which is expressed as , and the lensing contributions to the cmb are always coupled to the contributions from the primary cmb fields , we prospect that the effect will not change the functional dependence of temperature threshold on the lensing - induced non - gaussian signatures of , and expressed by equation ( [ eqn : glfun ] ) , even if the effect could enhance the magnitudes .undoubtedly , other secondary effects could induce non - gaussian properties in the observed cmb map .the most important effects are the ( thermal ) sunyaev - zeldovich effect and the foreground contaminations such as galactic foreground and extragalactic point sources .those effects can be removed to some extent by using advantages of their spectral properties , although further reliable investigations should be done for any measurements of cmb .furthermore , in this paper we have presented the lensing - induced non - gaussian signatures on _ three _ independent functions ,  and , and therefore we expect that the property of can provide us a clue to resolving the lensing contributions from the other secondary effects .the author would like to thank t. futamase and j. schmalzing for fruitful discussions and valuable comments .he also acknowledges the useful comments from anonymous referee .he thank u. seljak and m. zaldarriaga for making their cmbfast code publicly available .he also acknowledges a support from a japan society for promotion of science ( jsps ) research fellowship .bacon , d. , refregier , a. , & ellis , r. 2000 , , 318 , 625 bardeen , j. m. , bond . , j. r. , kaiser , n. , & szalay , a. s. 1986 , , 304 , 15 ( bbks ) balbi , a. et al . 2000 , to appear in lett . , astro - ph/0005124 barreiro , r. b. , sanz , j. l. , martnez - gonzlez , & silk , j. 1998 , , 296 , 693 de bernadis , p.et al . 2000 , , 404 , 955 bernardeau , f. 1998 , , 338 , 767 blandford , r. d. , saust , r. d. , brainerd , a. b. , & villumsen , j. v. 1991 , , 251 , 600 bond , j. r. , & efstathiou , g. p. 1987, , 226 , 655 ( be ) burles , s. & tytler , d. 1998 , 507 , 732 gunn , j. 1967 , , 147 , 61 guth , a. & pi , s .- y . 1982 , , 49 , 1110 kaiser , n. 1992 , , 388 , 272 kaiser , n. , wilson , g. , & luppino , g. 2000 , astro - ph/0003338 lange , a. e. _ et al ._ 2000 , astro - ph/0005004 hamana , t. , martel , h. , & futamase , t. 2000 , , 529 , 56 hanany , s. et al .2000 , to appear lett . , astro - ph/0005123 hu , w. , sugiyama , n. , & silk , j. 1997 , , 386 , 37 hu , w. 2000 , , 404 , 939 hu , w. , fukugita , m. , zaldarriaga , m. , & tegmark , m. 2000 , astro - ph/0006463 jain , b. , & seljak , u. , 1997 , , 484 , 560 jain , b. , seljak , u. , & white , s. d. m. 2000 , , 530 , 547 maoli , r. , van waerbeke , l. , mellier , y. , schneider , p. , jain , b. , bernardeau , f. , erben , t. , & fort , b. 2000 , astro - ph/0011251 miralda - escude , j. 1991 , , 380 , 1 schmalzing , j. , takada , m. , & futamase , t. 2000 , , 544 , l83 seljak , u. 1996 , , 463 , 1 seljak , u. , & zaldarriaga , m. 1996 , , 469 , 437 seljak , u. , & zaldarriaga , m. , 1999 , , 82 , 2636 seljak , u. , & zaldarriaga , m. , 2000 , , 538 , 57 sugiyama , n. 1995 , 100 , 281 takada , m. , komatsu , e. , & futamase , t. 2000 , , 533 , l83 takada , m. & futamase , t. 2001 , , 546 , 620 tegmark , m. , & zaldarriaga , m. 2000 , , 85 , 2240 van waerbeke , l. , mellier , y. , erben , t. , cuillandre , j .- c ., bernardeau , f. , maoli , r. , bertin , e. , mcgracken , h. j. , le fevre , o. , fort , b. , dantel - fort fort , m. , jain , b. , & schneider , p. 2000 , , 358, 30 van waerbeke , l. , bernardeau , f. , & benabed , k. 2000 , , 540 , 14 wittman , d. , tyson , j. , kirkman , d. , dellantonio , i. , & bernstein , g. 2000 , , 405 , 143 zaldarriaga , m. , 2000 , , 62 , 063510 zaldarriaga , m. , & seljak , u. 1999 , , 59 , 123507

Abstract: we propose a new method for extracting the non - gaussian signatures on the isotemperature statistics in the cosmic microwave background ( cmb ) sky , which is induced by the gravitational lensing due to the intervening large - scale structure of the universe . to develop the method , we focus on a specific statistical property of the intrinsic gaussian cmb field ; a field point in the map that has a larger absolute value of the temperature threshold tends to have a larger absolute value of the curvature parameter defined by a trace of second derivative matrix of the temperature field , while the ellipticity parameter similarly defined is uniformly distributed independently of the threshold because of the isotropic nature of the gaussian field . the weak lensing then causes a stronger distortion effect on the isotemperature contours with higher threshold and especially induces a coherent distribution of the ellipticity parameter correlated with the threshold as a result of the coupling between the cmb curvature parameter and the gravitational tidal shear in the observed map . these characteristic patterns can be statistically picked up by considering three independent characteristic functions , which are obtained from the averages of quadratic combinations of the second derivative fields of cmb over isotemperature contours with each threshold . consequently , we find that the lensing effect generates non - gaussian signatures on those functions that have a distinct functional dependence of the threshold . we test the method using numerical simulations of cmb maps and show that the lensing signals can be measured definitely , provided that we use cmb data with sufficiently low noise and high angular resolution .