it is standard to interpret cosmological observations simply with the spatially homogeneous and isotropic friedmann - lemaitre - robertson - walker ( flrw ) models . since the universe is not exactly spatially homogeneous and isotropic , it is not _ a priori _ clear that this is a valid approach .however , it was argued in that flrw light propagation relations should be approximately valid for average observations in spacetimes exhibiting compensation of inhomogeneities along light rays .such considerations have later been studied more thoroughly in , with the conclusion being that average observations should be well described by the volume averaged dynamics and geometry in ( spatially ) statistically homogeneous and isotropic spacetimes with structures evolving slowly compared to the time it takes a light ray to pass through them , _ if _ light samples spacetime fairly . under these requirements , and to the extent that cosmic backreaction ( see _ e.g. _ ) and a possible local inhomogeneity can be neglected , it is then valid to use flrw models to interpret observations . however , the requirement of tracing spacetime fairly is not necessarily fulfilled for observations involving point - like sources such as supernovae ( see _ e.g. _ ) .the requirement _ is _ on the other hand expected to be fulfilled for observations related to sources with large angular extent such as the cmb . as pointed out in _ , analyses of cmb observations depend crucially on the angular diameter distance to the last scattering surface , .it is therefore important to know if there are small systematic shifts in the redshift - distance relation in inhomogeneous spacetimes fulfilling the above listed requirements , compared to the relation obtained using the volume averages of such spacetimes ( which correspond to flrw models when backreaction can be neglected ) .studies involving both perturbation theory , n - body simulations and exact , inhomogeneous solutions to einstein s equations ( _ e.g. _ ) imply that small shifts in the average value of due to inhomogeneities indeed occur .however , as discussed in _ e.g. _ , this shift in is already included in standard analyses of the cmb based on perturbation theory .this conclusion is only valid to the extent that a small average shift in actually is described adequately by perturbation theory .it is in this respect important to acknowledge that perturbation theory is only an approximation scheme and that subtle effects important for precision cosmology may be beyond its reach .this makes it important to test results obtained with perturbation theory to make sure that they are actually true for spacetimes exhibiting , especially nonlinear , structure formation .one tool for performing such tests is swiss cheese models based on the spherically symmetric lemaitre - tolman - bondi ( ltb ) structures glued together in areas where they have reduced to flrw backgrounds " representing the volume averages of the given swiss cheese models .these ( toy-)models give a description of spacetimes with energy - density fluctuations on well - defined average flrw backgrounds and can thus be considered as describing exact perturbations " on flrw backgrounds .this makes swiss cheese models based on ltb structures excellent for testing perturbative results . by using swiss cheese models based on ltb structures, it has been found that there are no large shifts in the average redshift - distance relation except as selection effects arising when only special light rays in a given spacetime are considered ( see _ e.g. _ vs. ) .a noticeable exception is where a shift in the average redshift - distance relation compared to that computed using volume averaged quantities was found at .however , it is not entirely clear to what extent this large shift is related to the occurrence of surface layers in the studied model .results based on swiss cheese models with ltb structures _ should _ be in line with perturbation theory ; it has been shown with direct comparisons that perturbation theory based on ( semi-)nonlinear ltb density fields and the corresponding exact velocity fields reproduces exact light propagation in ltb models very well ( see also for the relation between ltb models and perturbation theory ) .an important factor for the impressive reproductions of light propagation in ltb models with perturbation theory in is the spherical symmetry of the ltb models ; the reproduction is inhibited for anisotropic szekeres structures because the method employed for obtaining velocity fields leads to artificial non - vanishing peculiar velocity fields outside non - spherically symmetric structures .that is , the method leads to non - vanishing peculiar velocity fields in regions where the exact anisotropic szekeres models have reduced to flrw models and hence should not give rise to peculiar velocity fields .this is unfortunate since real structures are not exactly spherically symmetric , making it important to know if spherical symmetry is essential for the agreement between swiss cheese and perturbative results .it is in this respect notable that anisotropic szekeres models exhibit structure formation that deviates significantly from both that of the underlying ltb models and from predictions of perturbation theory .in addition , it is very clear from figure 7a in that light propagation is significantly affected by anisotropy ; in that figure , it is seen that an initially radial light ray following an exact geodesic of an anisotropic szekeres model moves through a significantly different density field than a light ray traced simply by using the born approximation .such a result is not possible for light rays in ltb models since the spherical symmetry dictates that an initially radial light ray remains so .the above considerations imply an importance of studying whether the anisotropy of quasi - spherical szekeres structures affects average observations of _e.g. _ the angular diameter distance to the surface of last scattering .however , while there are ample studies involving light propagation in ltb models , only few studies involve observations in swiss cheese models based on anisotropic szekeres structures . in fact , only a single study of average observations in swiss cheese models based on szekeres models seems to exist , namely .the results from that paper indicate a small average shift in the angular diameter distance compared to the background value .the study in uses an ensemble average which is typical of swiss cheese studies with an important exception being where line - of - sight averages are used .as shown in ( see _ e.g. _ also ) , second order perturbation theory predicts that ensemble averages lead to a positive shift in the angular diameter distance while line - of - sight averages lead to a negative shift ( with the shift defined as such that a negative shift corresponds to image magnification , and a positive to demagnification ) . in partial contradiction to this , positive shifts are obtained in both and , where the shift is , however , not statistically significant in the former .furthermore , it has been shown that the ensemble average of the magnification is equal to 1 while its inverse averages to 1 upon line - of - sight averages .these results hold at least to second order in perturbation theory , insofar that perturbations to the areas of constant - redshift surfaces are negligible ( which is generally assumed to be the case ) .however , the study in finds a low probability of an average equal to while the erratum to describes a good agreement with an average value of of 1 . in summary, the studies of average light propagation in swiss cheese models do not seem to be entirely consistent with neither each other nor with perturbation theory .the swiss cheese models used in and differ from each other in significant ways aside from the anisotropy issue .hence , a comparison between the results in those two papers can not yield much insight into the possible effects anisotropic structure formation has on observations .the purpose with the work presented here is to obtain such information , _i.e. _ the study presented here is concerned with determining if the anisotropy of quasi - spherical szekeres structures affects observations such as and whether or not the anisotropy of the szekeres models influences ( dis-)agreement with perturbative results .the study is conducted by constructing four different swiss cheese models that only differ in the shape of their individual structures .the structures in two of the models are anisotropic szekeres structures , while their underlying ltb models are used for constructing the other two . by studying swiss cheese models with both anisotropic structures and their underlying " spherically symmetric structures, it is possible by direct comparisons to quantify effects of anisotropy . besides computing the average of in each of the models , the density , the distance - redshift relation ,the shear and the expansion rate along individual rays are also studied .this section serves to introduce the szekeres models and specify the particular models used in this study .in addition , a brief description of the construction of swiss cheese models is given .the quasi - spherical szekeres models are exact , inhomogeneous dust solutions to einstein s equations which can easily incorporate a homogeneous component with pressure such as a cosmological constant . in general , the quasi - spherical szekeres models have no killing vectors and hence no symmetries . their line element can be written as : subscripted commas followed by a coordinate or its index indicates partial derivative with respect to that coordinate .the metric function is given by , where and are continuous but otherwise arbitrary functions of ( with ) .the functions , and represent dipole distortions of the spacetimes of underlying " spherically symmetric ltb models .if they are chosen to be constant , the particular quasi - spherical szekeres model reduces to such an underlying ltb model .in addition , the quasi - spherical szekeres models have the flrw models as their homogeneous and isotropic limit . the flrw limit is achieved by setting , and , with being the scale factor of the flrw model . in the following, quasi - spherical szekeres models will be referred to simply as szekeres models .since ltb models are also strictly speaking szekeres models , szekeres models which are specifically not spherically symmetric are sometimes referred to as anisotropic szekeres models . for a szekeres metric combined with dust and a cosmological constant ,the diagonal components of einstein s equation yield ( denotes newton s constant ) :  the curvature function , , and the integration constant , , both depend on the radial coordinate only while depends on both and .the top equation is also valid for ltb models , while the expression for the density in the ltb limit reduces to because in the limit of spherical symmetry ..specification of anisotropic szekeres models and their underlying ltb models . and specify and hence the ltb models while and specify the functions of the anisotropic szekeres models .[ cols="^,^,^,^,^,^",options="header " , ]light propagation in four swiss cheese models only differing in terms of the shapes of their individual structures was studied . in particular , the redshift - distance relation , shear and expansion rate were studied along light rays initialized equivalently in the four models . despite the anisotropies of the szekeres models , light rays initialized equivalently in the four models were all found to travel through equivalent portions of spacetime . the angular diameter distance along light rays in two of the models , models sz2 and ltb2 , deviate significantly less from the background value than in the other two models , models sz1 and ltb1 .this is consistent with the much more prominent density contrasts in the latter two models . while the angular diameter distance along equivalent light rays in models sz1 and ltb1 are similar , there is a noticeable difference between the angular diameter distances computed along the equivalent light rays of models sz2 and ltb2 .this can presumably be attributed the particular anisotropies in the structures of models sz1 and sz2 .analogous similarities were found for the expansion rate and projected shear along the light rays .in addition , the integrals of the projected shear and of the fluctuations in the expansion rate along the light rays were found to cancel with each other to a high precision .this was also found in and it would be interesting to learn if it is a particular feature of szekers / ltb models or if the feature persists for a larger group of spacetimes .the line - 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Abstract: light propagation in two swiss cheese models based on anisotropic szekeres structures is studied and compared with light propagation in swiss cheese models based on the szekeres models underlying ltb models . the study shows that the anisotropy of the szekeres models has only a small effect on quantities such as redshift - distance relations , projected shear and expansion rate along individual light rays . the average angular diameter distance to the last scattering surface is computed for each model . contrary to earlier studies , the results obtained here are ( mostly ) in agreement with perturbative results . in particular , a small negative shift , , in the angular diameter distance is obtained upon line - of - sight averaging in three of the four models . the results are , however , not statistically significant . in the fourth model , there is a small positive shift which has an especially small statistical significance . the line - of - sight averaged inverse magnification at is consistent with to a high level of confidence for all models , indicating that the area of the surface corresponding to is close to that of the background .


revealing the true nature of dark energy ( de ) has become one of the most important tasks in cosmology .considering the plethora of de models proposed in the literature , a model independent reconstruction of de would be an appealing alternative to testing all models separately .in two recent papers  attempts were made by alam to reconstruct the dark energy equation of state parameter in a model independent manner , using the latest supernova data  . in these two papersa truncated taylor series was used to model the dark energy density .the results indicate an evolution of , a behaviour they call metamorphosis . from the reported analysis, it would seem that this is a significant effect prompting for `` exotic '' models for the de .  since other parameterizations  suggest that the type ia supernova data collected so far are consistent with the simplest de model of all , that of a cosmological constant ( ), it is important to investigate how such different conclusions can be reached starting from the same sets of data . in this paperwe argue that the method of model independent reconstruction proposed by alam suffers from a number of serious shortcomings . for alternative methods of de reconstructionsee e.g. references  .a fundamental requirement on a model independent reconstruction of must be that all de alternatives are treated on an equal footing .although the method of alam at first sight seems to be capable of an exact reproduction of the equation of state parameter of the cosmological constant , it actually favours evolving .the confidence contours , describing the level to which the reconstructed is known , exhibit two related problems .first , contours enclosing regions of high confidence level ( cl ) and high redshift tend to diverge .second , by construction , the low level cl regions shrink for high redshifts .in this section we present the basic formulae needed for confronting the model of alam with supernova data .the ansatz for the dark energy density proposed by alam is a truncated taylor series where . in the following, we will assume that the universe is flat and dominated by matter and de .the hubble parameter in such a universe , with the de parametrized as in equation ( [ eq : ansatz ] ) , is hence given by if the matter density is assumed to be known , only two of the parameters describing the de are independent , and .the third parameter is given by the normalization condition , .the equation of state parameter relates the pressure to the energy density and can be used to characterize an energy species .the dark energy equation of state parameter can be calculated given the hubble parameter and the matter density  by using the parametrization in equation ( [ eq : hubble ] ) , can be expressed in terms of the parameters , and  using the derivative of equation ( [ eq : hubble ] ) in the derivation of equation ( [ eq : de_eos ] ) , is however a questionable step . if the parametrization of is incorrect , is likely to be even more erroneous .the parametrization ( [ eq : de_eos ] ) is capable of reproducing the equation of state parameter for the cosmological constant ( , ) , and topological defects in the form of cosmic strings ( , ) and domain walls ( , ) .models of de are tested against observations by deriving distances to type ia supernovae through their measured brightness .cosmic distances are related through the luminosity distance to the brightness of a supernova measured in magnitudes where will be treated as a `` nuisance '' parameter in the fitting procedure and the absolute magnitude of a supernova is denoted by .the expression for the luminosity distance in a flat universe is observed and theoretically predicted magnitudes can thus be compared by means of the equations ( [ eq : mag ] ) and ( [ eq : dl ] ) and the model of the de enters via equation ( [ eq : hubble ] ) .however , since the integrand in ( [ eq : dl ] ) has to be real , i.e. , only a part of the parameter plane is allowed .as long as this condition is fulfilled , the dark energy density can be negative .in this section we reconstruct the dark energy equation of state parameter by the method proposed by alam .we have used two sets of supernova data , which will be referred to as the _ small _ and the _ large _ data set .the _ small _ data set is a combination of supernovae from tonry  and barris  , and it is the same set as alam used in reference  .the supernovae in the _ small _ set have low extinction and .large _ data set is an extension of the _ small _ set with additional high redshift supernovae from riess  .a maximum likelihood analysis of the data sets was used to estimate the best fit parameters of the dark energy model . the following negative log likelihood function was used where and are observed magnitude and magnitude error of a supernova at redshift .the parameter , related to the hubble constant and the absolute supernova magnitude , was treated as a `` nuisance '' parameter in the fitting procedure described below .first , the negative log likelihood function in equation ( [ eq : l ] ) was computed on a cubic lattice in the parameter space spanned by , and .thereafter a gaussian prior , containing information on the matter density , was applied to is not critical for our argument . ] .the plane containing the minimum of the negative log likelihood function was then identified . finally ,the best fit values and the cl contours in this plane were translated into through equation ( [ eq : de_para ] ) .  the best fit parameters to the _ small _ data set ( and ) and the and 1 cl contours in the parameter plane are shown in panel ( a ) in figure [ fig : hzt_w ] .the 1 cl contour is drawn in order to facilitate comparison with reference  and  .note that alam _et al _ report their confidence regions as 1 levels , i.e. as in their maximum likelihood analysis .as two parameters are jointly fitted this corresponds only to a cl .the reconstructed history of the dark energy equation of state parameter , shown in panel ( b ) in figure [ fig : hzt_w ] , is clearly consistent with a cosmological constant at the , but not at the cl .there are however some peculiarities of the reconstructed that should be noted .the and confidence regions drawn in figure [ fig : hzt_w ] diverge at large redshifts .this corresponds to the contour regions crossing the dark shaded region in the plane , where grows very rapidly .the and 1 confidence regions , on the other hand , converge at large redshift .moreover , there is a waist at where all contours are squeezed together .these peculiarities indicate that this reconstruction is in fact not model independent , as will be further explained in the next section .figure [ fig : hzt_16hz ] shows obtained from the _ large _ data set with additional supernovae .this equation of state parameter is consistent with the cosmological constant even at the cl .however , the confidence regions still exhibit the same behaviour as the ones obtained for the _ small _ data set .the unusual behaviour of the results in figure [ fig : hzt_w ] and [ fig : hzt_16hz ] appear since the dark energy equation of state parametrization ( [ eq : de_eos ] ) has a few peculiarities , which we discuss in this section .the dark energy equation of state parameter will obviously diverge whenever the denominator in equation ( [ eq : de_eos ] ) vanishes .regions in the parameter plane where blows up ( ) or diverges somewhere in the redshift interval are shown in dark gray in e.g. panel ( a ) in figure [ fig : hzt_w ] .the point , corresponding to the cosmological constant , lies very close to this region .if the best fit to data is near this point , the error in will blow up .diverging equation of state parameter is the reason why the confidence region in e.g. figure [ fig : hzt_w](b ) blows up .from equation ( [ eq : de_eos ] ) one can easily see that the parametrized equation of state depends on the order of the taylor expansion at the high redshift limit .if , for instance , only the first order is considered , for high redshifts , while for the second order . since the proposed ansatz ( [ eq : ansatz ] ) is a truncated taylor series we can not expect equation ( [ eq : de_eos ] ) to be valid at very high redshifts. however , the effects of these high redshift limits are noticeable already at the moderate redshifts where we try to model de .these limits are responsible for the shrinkage of the non diverging confidence regions at high redshift ( e.g. , see the confidence region in figure [ fig : hzt_w](b ) ) .the dark energy equation of state parameter described by equation ( [ eq : de_eos ] ) can be thought of as a family of curves , where each curve is parametrized by , and . in this sectionwe study the stability of these curves to perturbations in the parameters and .moreover , we will show that some curves in this family become more stable with redshift , with respect to these perturbations , while others become more unstable .the change in a curve , described by the parameters and , due to small perturbations and is given by the curve that corresponds to the cosmological constant ( ) becomes increasingly unstable due to perturbations as redshift increases , a curve that starts close to the line will thus departure from this line with increasing redshift .equation ( [ eq : de_eos ] ) can also make an exact reproduction of two types of topological defects .the equation of state of domain walls ( ) grows stable to perturbations in with , but not to perturbations in  the curve corresponding to cosmic strings ( ) is stable to perturbations in both and at high redshifts the curves corresponding to topological defects are thus very special . all other curves ( except the one corresponding to the cosmological constant ) approach one of these two curves as redshift increases . the effect of perturbations in and depends on the value of these parameters .this is in contrast to perturbations of linear parameterizations of , e.g. , which are independent of parameter values . from the above discussion about the stability of with redshift, we see that the parameterization in equation ( [ eq : de_eos ] ) treats the cosmological constant unfairly in favour of other de alternatives .   the parametrized equation of state parameter ( [ eq : de_eos ] ) will not only favour , but also `` force '' evolution .all curves , describing an evolving equation of state parameter , will cross at a certain redshift . at this redshift , where , the numerator and denominator are equal and the dark energy equation of state assumes the value . the parameters and , obtained by fitting supernova data to the ansatz ( [ eq : ansatz ] ) ,are highly correlated and the ratio is thus nearly constant .the value of this ratio depends on the slope of the cl contours in the plane which in turn is correlated with the assumed prior on the matter density .this implies that the location of the point is more or less independent of the parameters and for a given .the parametrization ( [ eq : de_eos ] ) of the equation of state parameter thus favours de evolution .if the reconstructed equation of state at present is less ( larger ) than minus one it must have been larger ( less ) than minus one in the past .any significant departure from the line will hence force the reconstructed equation of state parameter to cross that line . since most curves will pass through this point , the family of curves will have a waist centered at , where denotes the slope of the cl contours in the plane .the transition redshift in e.g. figure [ fig : hzt_w ] is which corresponds to , in good agreement with the slopes of the ellipses in panel ( a ) .to test the reliability of this way of reconstructing the dark energy we tried to recover a fiducial cosmology from simulated data sets generated using snoc  , a monte carlo simulation package for high z supernova observations .we generated data sets , each one consisting of supernovae with the same distribution of redshifts and magnitude errors as the _ small _ data set used in section [ sec : reconstruction ] .a flat universe with a cosmological constant and was used as fiducial cosmology .the parameters and were fitted to each simulated data set with a fixed value of .  a scatter plot of best fit values to the simulated data setsis shown in figure [ fig : a1a2sim ] . from this plotit is clear that the best fit values of the parameters and are highly correlated , as anticipated in section [ sec : force ] . of the simulated data sets , were found to be consistent with the cosmological constant at the cl .these are indicated by full circles in the plot .data sets not consistent with the cosmological constant at the cl , on the other hand , are indicated by open circles .the best fit to real data obtained in section [ sec : reconstruction ] is indicated with a star in the expanded figure in figure [ fig : a1a2sim ] .the expansion shows a region where simulated data sets have a reconstructed dark energy equation of state parameter similar to the one obtained from real data .the simulated data sets in this region are inconsistent with the fiducial cosmology at the cl .figure [ fig : exmpl_sim ] shows a few examples of reconstructed from simulated data sets .the reconstructed in panel ( b ) is almost indistinguishable from the one obtained from real data ( see figure [ fig : hzt_w ] ) .rapidly evolving equation of state parameters , resembling the behaviour of the result obtained from real data , can thus be obtained from simulated data sets in a cosmology ( ) with the same distributions of redshifts and magnitude errors as the real set .we also simulated data sets with the same distribution of redshifts and magnitude errors as the _ large _ data set used in section [ sec : reconstruction ] .figure [ fig : a1a2sim_16hz ] shows a scatter plot of the best fit values to these simulations . in this case of the simulated data sets were consistent with the fiducial cosmology on the cl .as can be seen from figure [ fig : a1a2sim_16hz ] , the additional high redshift supernovae decrease the spread of the best fit values .  to save computational time ,a simplified fitting procedure was used in the reconstruction of from the simulated data sets , as was kept fixed at the input value .the value of the matter density affects the slope of the confidence contour . by using a more realistic fitting procedure the spread of the best fit values would increase .as we have seen above , equation ( [ eq : de_eos ] ) involves some difficulties associated with the reconstruction of the dark energy equation of state parameter .could an extension of the ansatz overcome these difficulties ?studies of an arbitrary power series describing the dark energy density might help us to answer this question where can be zero or any positive or negative integer .perturbations in the parameters will cause a change in that is independent of these parameters a description of the density of dark energy in terms of a power series will thus not favour any particular model .the equation of state generated by a power series is given by ( [ eq : w(h ) ] ) if only one coefficient in ( [ eq : eos_power ] ) is non zero , assumes a constant value .the power series parametrization of the equation of state parameter is hence capable of reproducing a whole spectrum of constant separated by  the cosmological constant corresponds to .the high redshift limit for a truncated power series with leading term resembles this spectrum the change in due to any perturbation of the parameters in the neighborhood of the cosmological constant ( and ) is given by the effect of perturbations in the parameters depends upon the leading term in the power series . if is positive , the change in the equation of state parameter will increase with redshift .however , if is negative the change will decrease with redshift . in order to discriminate between the cosmological constant and evolving de alternatives, we have to contrast this behaviour with the curves describing evolution .the perturbed equation of state parameter corresponding to non zero at a high redshift ( ) with the leading term is : the effects of perturbations decrease with redshift if is positive , or increase with redshift if is negative .this is the very opposite of the behaviour for the case of a cosmological constant .the instability problem discussed above can thus not be resolved by adding extra terms to the ansatz ( [ eq : ansatz ] ) .the ansatz proposed by alam may be useful for modelling the dark energy density , but its usefulness for revealing the nature of the de seems limited .the parametrization of the dark energy equation of state parameter based on a truncated taylor series involves a number of severe problems .evolution is both favoured and forced by this parametrization .the cosmological constant is thus mistreated by the ansatz proposed in  .not even an extension of the ansatz seems to be able to overcome these difficulties .the equation of state parameter expressed as in equation ( [ eq : de_eos ] ) also diverges in large regions of the parameter space .more data , which focus the solutions to the stable regions , may solve this problem .however , the region close to the point describing a universe with a cosmological constant will be in a disfavoured part of the parameter space.  the dark energy equation of state parameter reconstructed from the data sets presented in reference  and  is inconsistent with the cosmological constant at the confidence level .simulations show that the rapidly changing behaviour of could also be expected with this ansatz for a universe .the scatter of the best fit parameters can be reduced if additional data points at high redshift are added to the simulated data sets .our best fit to real data with additional high redshift supernovae was consistent with the cosmological constant at the confidence level .we concluded that the suggested `` model independent '' method of reconstructing the dark energy equation of state parameter in , is in fact model dependent , and that the seemingly striking results are likely to be due to this deficiency .99 alam u , sahni v , saini t d and starobinsky a a 2003 _ preprint _ astro - ph/0311364 alam u , sahni v and starobinsky a a 2004 _ jcap _ * 0406 * 008 tonry j 2003 _ astrophys .j. _ * 594 * 1 barris b 2004 _ astrophys. j. _ * 602 * 571 riess a 2004 _ astrophys .j. _ * 607 * 665 knop r 2003 _ astrophys . j. _ * 598 * 102 wang y and mukherje p 2004 _ astrophys . j. _ * 606 * 654 wang y and freese k 2004 _ preprint _ astro - ph/0402208 wang y and tegmark m 2004 _ _ * 92 * 241302 huterer d and cooray a 2004 _ preprint _ astro - ph/0404062 daly r and djorgovski s 2004 _ preprint _ astro - ph/0403664 saini t d , raychaudhury s , sahni v and starobinsky a a 2000 _ phys .lett . _ * 85 * 1162 goobar a , mrtsell e , amanullah r , goliath m , bergstrm l and dahln t 2002 _ a & a _ * 392 * 751 ( ` http://www.physto.se/~ariel/snoc ` )

Abstract: recent attempts to fit type ia supernova data by modeling the dark energy density as a truncated taylor series have suggested the possibility of _ metamorphosis _ , i.e. , a rapidly evolving equation of state parameter , . however , we show that fits using that parametrization have significant problems . evolution of is both favoured and in some sense forced , and the equation of state parameter blows up or diverges in large regions of the parameter space . to further elucidate these problems we have simulated sets of supernova data in a to show that the suggested `` evidence '' for metamorphosis is also common for


periodicity analysis of unevenly collected data is a relevant issue in several scientific fields .classical spectral analysis methods are unsatisfactory to solve the problem . in this paperwe present a neural network based estimator system which performs well the frequency extraction from unevenly sampled signals .it uses an unsupervised hebbian nonlinear neural algorithm to extract the principal components of the signal auto - correlation matrix , which , in turn , are used by the music frequency estimator algorithm to extract the frequencies , , .we generalize this method to avoid an interpolation preprocessing step , which generally adds high noise to the signal , and improve the system performance by using a new stop criterion to avoid overfitting problems .the experimental results are obtained comparing our methodology with the others known in literature ( see , , , , , ) .in what follows , we assume to be a physical variable measured at discrete times . can be written as the sum of the signal and random errors : .the problem we are dealing with is how to estimate fundamental frequencies which may be present in the signal  , , .if is measured at uniform time steps ( even sampling ) there are a lot of tools to effectively solve the problem which are based on fourier analysis , , .these methods , however , are usually unreliable for unevenly sampled data . for instance , the typical approach of resampling the data into an evenly sampled sequence , through interpolation , introduces a strong amplification of the noise which affects the effectiveness of all fourier based techniques which are strongly dependent on the noise level . to solve the problem of unevenly sampled data , we consider two classes of spectral estimators :  spectral estimators based on fourier trasform ( least squares methods ) ;  spectral estimators based on the eingevalues and eingevectors of the covariance matrix ( maximum likelihood methods ) .classic periodogram , , , lomb s periodogram , scargle s periodogram , dcdft are the methods of the first class that we use , while music , , and esprit belong to second class .the methods based on the covariance matrix are more recent and have great potentiality .starting by this consideration , we develop a method based on the music estimator .it is compared with classic methods to highlight the results .in the last years several papers dealed with learning in pca neural nets , , , , finding advantages , problems and difficulties of such neural networks . in what follows we shall use a robust hierarchical learning algorithm   where is the weight vector of the output neuron at step , is the corresponding output , is the learning rate and is learning function because it has been experimentally shown that it is the best performing one in our problem , .our neural estimator ( ne ) can be summarized as follows :  * preprocessing : calculate and subtract the average pattern to obtain zero mean process with unity variance .interpolate input data if it is the case .* initialize the weight matrix and the other neural network parameters ; * input the pattern  ] . in the generalization tonon interpolated input data , ^{h} ] .the estimated frequency without interpolation is ( 1/jd ) .a comparison is made with the other methods cited in a previous section and the experimental results are shown in figure 4 and in table 1 .only the lomb s periodogram is in agreement with the right periodicity , but showing some spurious peaks .furthermore , if we enlarge the frequency window for the two best performing methods , while the ne continues to work well , the lomb s periodogram does not work at all as illustrated in figure 5 .in this paper we have illustrated an improved technique based on pca neural networks and music to estimate the frequency of unevenly sampled data .it has been shown that it obtains good results on real data ( here we used the su cygni light curve ) compared with other well - known methods .in fact , it obtains a good estimate of the signal frequency also with few unevenly sampled inputs , it reduces the noise problems related to input data interpolation , it optimizes the convergence by introducing an early stopping criterion , and , finally , it is more resistant to the dimension of the frequency windows .future research lines regard the introduction of genetic algorithms to optimize the weight initialization of the pca neural networks and to use filters to extract and identify one frequency at each time when dealing with multi - frequency signals .m. rasile , l. milano , r. tagliaferri , g. longo , periodicity analysis of unvenly spaced data by means of neural networks , in neural nets wirn vietri 97 , m. marinaro and r. tagliaferri editors , springer verlag , pag.201 - 212 , 1997 .r. roy , t. kailath , esprit- estimation of signal parameters via rotational invariance tecniques , in f.a .grnbaum , j. w. helton and p. khargonear editors , signal processing part ii : control theory and applications , springer verlag , new york , pag.369 - 411 , 1990 .r. tagliaferri , a. ciaramella , l. milano , f. barone , g. longo , spectral analysis of stellar light curves by means of neural networks , accepted for publication , astronomy & astrophysics supplement series , vol.137 , june , 1999 .

Abstract: in this paper we present a neural network based estimator system which performs well the frequency extraction from unevenly sampled signals . it uses an unsupervised hebbian nonlinear neural algorithm to extract the principal components which , in turn , are used by the music frequency estimator algorithm to extract the frequencies .  we generalize this method to avoid an interpolation preprocessing step and to improve the performance by using a new stop criterion to avoid overfitting .   the experimental results are obtained comparing our methodology with the others known in literature .


i am going to talk about processes that form the basis of life and evolution .the hypothesis that living organisms have adapted to their environment , and have exploited the available material resources and the physical laws governing them to the best of their capability , is the legacy of charles darwin survival of the fittest .this is an optimisation problem , but it is not easy to quantify it in mathematical terms. often we can explain various observed features of living organisms .the explanation becomes more and more believable , as more and more of its ingredients are verified experimentally .yet even when definite predictions exist , an explanation is an explanation and not a proof ; there is no way we can ask evolution to repeat itself and observe it like many common scientific experiments .  with this attitude ,let us look at life .living organisms try to perpetuate themselves .the disturbances from the environment , and the damage they cause , make it impossible for a particular structure to survive forever .so the perpetuation is carried out through the process of replication .one generation of organisms produces the next generation , which is essentially a copy of itself .the self - similarity is maintained by the hereditary information the genetic code that is passed on from one generation to the next .the long chains of dna molecules residing in the nuclei of the cells form the repository of the genetic information .these dna molecules control life in two ways : ( 1 ) their own highly faithful replication , which passes on the information to the next generation ( each life begins as a single cell , and each cell in a complex living organism contains identical dna molecules ) , and ( 2 ) the synthesis of proteins which govern all the processes of a living organism ( haemoglobin , insulin , immunoglobulin etc .are well - known examples of proteins ) .computation is nothing but processing of information .so we can study what dna does from the view - point of computer science . in the process of designing and building modern computers, we have learnt the importance of various software and hardware features .let us look at some of them .the first is the process of digitisation . instead of handling a single variable covering a large range ,it is easier to handle several variables each spanning a smaller range .any desired accuracy can be maintained by putting together as many as necessary of the smaller range variables , while the instruction set required to manipulate each variable is substantially simplified .this simplification means that only a limited number of processes have to be physically implemented , leading to high speed computation .discretisation also makes it possible to correct small errors arising from local fluctuations .there are disadvantages of digitisation in terms of increase in the depth of calculation and power consumption , but the advantages are so great that digital computers have pushed away analogue computers to obscurity . even before the discovery of dna, erwin schrdinger had emphasised the fact that an aperiodic chain of building blocks carries information , just like our systems of writing numbers and sentences .the structure of dna and protein reveals that life has indeed taken the route of digitising its information .dna and rna chains use an alphabet of 4 nucleotide bases , while proteins use an alphabet of 20 amino acids .the second is the packing of the information .when there are repetitive structures or correlations amongst different sections of a message , that reduces its capacity to convey new information part of the variables are wasted in repeating what is already conveyed .claude shannon showed that the information content of a fixed length message is maximised when all the correlations are eliminated and each of the variables is made as random as possible .our languages are not that efficient ; we can immediately notice that consonants and vowels roughly alternate in their structure .when we easily compress our text files on a computer , we remove such correlations without losing information .detailed analyses of dna sequences have found little correlation amongst the letters of its alphabet , and we have to marvel at the fact that life has achieved the close to maximum entropy structure of coding its information .  the third is the selection of the letters of the alphabet .this clearly depends on the task to be accomplished and the choices available as symbols .a practical criterion for fast error - free information processing is that various symbols should be easily distinguishable from each - other .we use the decimal system of numbers because we , at least in india , learnt to count with our fingers .there is no way to prove this , but it is a better explanation than anything else . we can offer a better justification for why the computers we have designed use the binary system of numbers . 2 is the smallest base available for a number system , and that leads to maximal simplification of the elementary instruction set .( the difference is obvious when we compare the mathematical tables we learnt in primary schools to the corresponding operations in binary arithmetic . ) 2 is also the maximum number of items that can be distinguished with a single yes / no question .( detecting on / off in an electrical circuit is much easier than identifying more values of voltages and currents . )thus it is worth investigating what life optimised in selecting the letters of its alphabet .the computational task involved in dna replication is assembly .the desired components already exist ( they are floating around in a random ensemble ) ; they are just picked up one by one and arranged in the required order . to pick up the desired component, one must be able to identify it uniquely .this is a variant of the unsorted database search problem , unsorted because prior to their selection the components are not arranged in any particular order .( it is important to note that this replication is not a copy process .copy means writing a specific symbol in a blank location , and unlike assembly , it is forbidden by the linearity of quantum mechanics . )the optimisation criterion for this task is now clear one must distinguish the maximum number of items with a minimum number of identifying questions .i have already pointed out that in a classical search , a single yes / no question can distinguish two items .the interesting point is that a quantum search can do better .let the database contain distinct objects arranged in a random order .a certain object has to be located in the database by asking a set of questions .each query is a yes / no question based on a property of the desired object ( e.g. is this the object that i want or not ? ) . in the search process , the same query is repeated using different input states until the desired object is found .let be the number of queries required to locate the desired object in the database .using classical probability analysis , it can be easily seen that ( a ) when all objects are available with equal probability for each query ( i.e. each query has a success probability of ) , and ( b ) when the objects which have been rejected earlier in the search process are not picked up again for a query . herethe angular brackets represent the average expectation values .option ( b ) is available only when the system possesses memory to recognise what has already been tried before . in the random cellular environment, the rejected object is thrown back into the database , and only option ( a ) is available to a classical assembly operation .lov grover discovered a quantum database search algorithm that locates the desired object using fewer queries . quantum algorithms work with amplitudes , which evolve in time by unitary transformations . at any stage , the observation probability of a state is the absolute value square of the corresponding amplitude .the quantum database is represented as an hilbert space , with the distinct objects as its orthonormal basis vectors .the quantum query can be applied not only to the basis vectors , but also to their all possible superpositions ( i.e. to any state in the hilbert space ) .let be the desired state and be the symmetric superposition state . and be the reflection operators corresponding to these states .the operator distinguishes between the desired state and the rest .it flips the sign of the amplitude in the desired state , and is the query or the quantum oracle .the operator treats all objects on an equal footing .it implements the reflection about the average operation .grover s algorithm starts with the input state , and at each step applies the combination to it .each step just rotates the state vector by a fixed angle ( determined by ) in the plane formed by and . applications of rotate the state vector all the way to , at which stage the desired state is located and the algorithm is terminated . this relation is readily solved , since the state vector rotates at a constant rate , giving over the last few years , this algorithm has been studied in detail .i just summarise some of the important features :  * for a given , the solution for satisfying eq.(3 ) may not be an integer .this means that the algorithm will have to stop without the final state being exactly on the r.h.s . of eq.(2 ) .there will remain a small admixture of other states in the output , implying an error in the search process .the size of this admixture is determined by how close one can gets to on the r.h.s . of eq.(3 ) . apart from this ,the algorithm is fully deterministic . *the algorithm is known to be optimal , going from to along a geodesic .no other algorithm , classical or quantum , can locate the desired object in an unsorted database with a fewer number of queries . *the iterative steps of the algorithm can be viewed as the discretised evolution of the state vector in the hilbert space , governed by a hamiltonian containing two terms , and .the former represents a potential energy attracting the state towards , while the latter represents a kinetic energy diffusing the state throughout the hilbert space .the alteration between and in the discretised steps is reminiscent of trotter s formula used in construction of the transfer matrix from a discretised feynman s path integral . *asymptotically , .the best that the classical algorithms can do is to random walk through all the possibilities , and that produces as mentioned above . with the use of superposition of all possibilities at the start , the quantum algorithm performs a directed walk to the final result and achieves the square - root speed - up .* the result in eq.(3 ) depends only on ; the phases of various components of can be arbitrary , i.e. they can have the symmetry of bosons , fermions or even anyons .  to come back to the genetic code ,let us look at two of the solutions of eq.(3 ) for small .the only exact integral solution is , .base - pairing during dna replication can be looked upon as a yes / no query , either the pairing takes place through molecular bond formations or it does not , and its task is to distinguish between 4 possibilities .the other interesting solution is , .the well - known triplet code of dna has 3 consecutive nucleotide bases carrying 21 signals , 20 for the amino acids plus a stop .3 base - pairings between t - rna and m - rna transfer this code to the amino acid chain .these solutions are highly provocative .this is the first time they have come out of an algorithm that performs the actual task accomplished by dna .it is fascinating that they are the optimal solutions .indeed it is imperative to investigate whether dna has the quantum hardware necessary to implement the quantum search algorithm .over the last fifty years , molecular biologists have learnt a lot about the structure and function of dna by careful experiments ( they have also been rewarded with many nobel prizes ) .let us quickly go through some of the facts they have unravelled .* dna has the structure of a double helix .it can be schematically represented as a ladder , as in fig.1 .the sides of the ladder have a periodic structure with alternating sugar and phosphate groups .the nucleotide base pairs form the rungs of the ladder , and the genetic information is encoded in the order of these base pairs .* dna contains 4 nucleotide bases a , t , c , g which are closely related to each - other in chemical structure .the bases are always paired as a - t and c - g along the rungs of the ladder by hydrogen bonds .this base - pairing makes the two dna strands complementary in character . * the sugar and phosphate groups along the side of the ladderare held together by covalent bonds .their bonding is completely insensitive to the bases attached to them , and takes place in the presence of dna polymerase enzymes . * during replication , the helicase enzyme separates the two strands of dna by breaking the hydrogen bonds , much like opening a zipper .the unpaired bases along each of the separated strands find their partners from the surrounding molecules , producing two copies of the original dna .* the sides of the ladder have asymmetric ends .the replication process is directed , always proceeding from the end to the end ( these numbers label the position of the carbon atoms in the sugar rings ) of the strand being constructed .the dna polymerase enzyme slides along the intact strand , adding one base at a time to the growing strand . during this process ,base - pairing and sugar - phosphate bonding alternate .* rna molecules carry the nucleotide bases a , u , c , g with u very similar in chemical structure to t. a - u pairing is as strong as a - t pairing . messenger rna ( m - rna ) has a single strand structure . in the first step of protein synthesis, the rna polymerase enzyme separates the paired dna strands and constructs an m - rna strand on the dna template by base - pairing ( the process is the same as in dna replication ) .the m - rna strand grows as the rna polymerase enzyme slides along the dna from the promoter to the terminator base sequence .finally the rna polymerase enzyme detaches itself from the dna , the fully constructed m - rna strand floats away to the ribosomes in the cytoplasm of the cell , and the separated dna strands pair up again .* transfer rna ( t - rna ) molecules have 3 rna bases at one end and an amino acid at the other , with a many - to - one mapping between the two . inside cellular structures called ribosomes ,3 t - rna bases line up against the matching bases of m - rna , aligning the amino acids at the other end .the aligned amino acids then split off from the t - rna molecules and bind themselves into a chain .the process again proceeds monotonically from the end to the end of the m - rna . after the amino acids split off ,the remnant t - rna molecules are recycled .this completes the transfer of the genetic code from dna to proteins .* enzymes play a crucial role in many of the above steps .in addition to facilitating various processes by their catalytic action , they store energy needed for various processes , ensure that dna keeps out u and rna keeps out t , and perform error correction by their exonuclease action ( i.e. reversing the assembly process to remove a mismatched base pair ) . * hereditary dnais accurately assembled , with an error rate of per base pair , after the proof - reading exonuclease action .proteins are assembled less accurately , with an error rate of per amino acid .let us also recollect some useful facts about various chemical bonds .* ionic bonds are strong , can form at any angle , and can be explained in terms of electrostatic forces .ions often separate in solutions .* covalent bonds are strong , form at specific angles , and can be explained in terms of coulomb forces between electrons and nuclei and the exclusion principle .* van der waals bonds are weak , not very angle dependent , and explainable in terms of interactions between virtual electric dipoles .they play an important role in transitions between solid , liquid and gas phases , as well as in folding and linking of polymers .* hydrogen bonds are weak , highly angle dependent , and explainable in terms of a proton ( ) tunneling between two attractive energy minima .the situation is a genuine illustration of a particle in a double well potential , e.g. .high sensitivity of the tunneling amplitude to the shape of the energy barrier make hydrogen bonds extremely sensitive to the distances and angles involved .they are the most quantum of all bonds ; water is a well - known example .* delocalisation of electrons and protons over distances of the order of a few angstroms greatly helps in molecular bond formation .it is important to note that these distances are much bigger than the compton wavelengths of the particles , yet delocalisation is common and maintains quantum coherence . in case of electrons ,the phenomena are called resonance and hybridisation , e.g. the benzene ring . in case of protons ,the different configurations are called tautomers , e.g. amino imino and keto enol fluctuations of the nucleotide bases .  with all this information , the quantum search algorithmic requirements from the dna structure are clear .it is convenient to take the distinct nucleotide bases as the quantum basis states in the hilbert space. then ( 1 ) the quantum query transformation must be found in the base - pairing with hydrogen bonds .( 2 ) the symmetric transformation must be found in the base - independent processes occurring along the sides of the ladder .( 3 ) an environment with good quantum coherence must exist .thermal noise is inevitable at inside the cells , so the transformations must be stable against such fluctuations .figuratively , the best that can be achieved is ~~.\ ] ] thus we need quantum features that smoothly cross over to the classical regime , i.e. features that are reasonably stable against small decoherent fluctuations .examples are : ( a ) geometric and topological phases , and ( b ) projection / measurement operators .during dna replication , the intact strand of dna acts as a template on which the growing strand is assembled . at each step, the base on the intact strand decides which one of the four possible bases can pair with it .this is exactly the yes / no query ( also called the oracle ) used in the database search algorithm . to connect this oracle to the quantum transformation , we have to look at how molecular bond formations transform a quantum state .the generic quantum evolution operator is , with being the total hamiltonian of the system .global conservation of energy means that the overall phase will completely factor out of the evolution and will not affect the final probabilities .what matters is only the relative phase between pairing and non - pairing bases . during the pairing process ,the bases come together in an initial scattering state , discover that there is a lower energy binding state available , and decay to that state releasing the extra energy as a quantum .the interaction hamiltonian for the bond formation process can be represented as where are the transition operators between the excited and ground states of the reactants , and are the transition operators between zero and one quantum states of energy released .both the terms in eq.(5 ) are necessary for the hamiltonian to be hermitian .the phase change during the bond formation satisfies with only two states involved ,eq.(6 ) is easily solved by diagonalising . in the two dimensional space ,let and eigenvalues .eq.(6 ) then reduces to with the solution .this is the geometric phase well - known in quantum optics .a complete rabi cycle in a two - level system gives a phase change of , and the transition process corresponds to half the cycle .the phase does not depend on specific values of or , but only on the fact that the transition takes place .quantum mechanics does not specify how will be divided between the bound state and the released energy quantum . in quantum optics , thisbreak - up is determined by the phase of the laser .here i assume that the process of decoherence is such that the energy quantum does not carry away any phase information .  at this stage , we discover a pleasant surprise that the base - pairing takes place not with a single hydrogen bond but with multiple hydrogen bonds ( two for a - t and a - u , and three for c - g ) , as shown in fig.2 .multiple hydrogen bonds are necessary for the mechanical stability of the helix .but they are also of different length , making it likely that they form asynchronously . assuming a two - step deexcitation process during base - pairing , the geometric phase change becomes , just what is needed to implement energy of a single hydrogen bond is , giving .this roughly explains the observed error rate in dna replication .the tunneling amplitude for bond formation is related to both and , which determines the time scale of the base - pairing ,   next step is to look for the transformation in the processes occurring along the sides of the dna - ladder . during these processes ,quantum evolution produces various phases as molecular bonds get formed and broken . but these bonds treat all the nucleotide bases in the same manner , so the phases completely factor out and have no effect on the final probabilities .thus i leave the phases out .suppose that is the equilibrium state of the physical system , favoured by the processes that occur along the sides of the dna - ladder .this means that any other initial state will gradually relax towards , with the damping provided by the environment .let be the time scale for this relaxation process .now is a superposition state of nucleotide bases , and it can be created only if the cellular environment provides transition matrix elements between its various components .( in free space , transition matrix elements between nucleotide bases of different chemical composition vanish . )the magnitude of these transition matrix elements decides how quickly cycles through its various components .let be the time scale for these oscillations .now let us look at the dna replication process , when the above defined time scales satisfy the hierarchy   1 .  in the initial stage ,the randomly floating around nucleotide bases come into contact with the growing dna strand , and relax to the state .2 .  when the nucleotide base finds its proper orientation , hydrogen bond formation suddenly takes place , changing the state to .this state is entangled between the nucleotide bases and the energy quanta , $ ] .3 .  after this sudden change , the relaxation process again tries to bring the system back to the state . with the timescales obeying eq.(10 ) , this relaxation occurs as damped oscillations , much like what happens when one gives a sudden jerk to a damped pendulum .the opposite end of the damped oscillation is .when the system evolves to this opposite end , it discovers that it is no longer entangled between the nucleotide bases and the energy quanta , for . at this point ,the energy quanta are free to wander off with minimal disturbance to the quantum coherence .the departure of the energy quanta confirms the base - pairing , providing a projective measurement of the system .( i take the measurement time scale to be much smaller than . )the energy quanta that have wandered off are unlikely to return , making the process irreversible .the replication then continues to add the next base onto the growing strand .these steps are schematically shown in fig.3 .they provide a highly tuned yet a robust algorithm .there are no fine - tuned parameters ; the hierarchy of eq.(10 ) has enough room to take care of substantial variation in individual time scales .life could nt be simpler !  to illustrate the possibility that processes with time scales obeying the hierarchy of eq.(10 )do physically occur , let us look at a quantum example .consider the processes involved in the functioning of the maser . is a molecule with two equivalent configurations , corresponding to the nitrogen atom being above or below the triangle of hydrogen atoms .these two configurations can be distinguished by the direction of their electric dipole moments .quantum tunneling of the nitrogen atom through the triangle of hydrogen atoms mixes these two configurations , and the ground state is the symmetric superposition of the two . in gas ,any initial state decays towards this equilibrium ground state ( molecular collisions help in this relaxation ) .if a electric field is applied to the gas for a short duration , it favours one of the two configurations and kicks the molecules out of their equilibrium state . after the removal of the electric field , the kicked molecules oscillate from one configuration to the other , till the oscillations are damped out by the decay process . shining the molecules with a radio - frequency pulse resonant with removes the extra energy quickly by stimulated emission and produces a coherent maser.  the relaxation time scale depends on temperature as well as on various molecular concentrations , and governs the overall replication rate . under normal circumstances ,dna replication is observed to occur at the rate of 1000 base - pairings / sec , constraining to be smaller than sec . without any knowledge of the transition matrix elements, i do not have any estimate of .strictly speaking , i should not talk about quantum states in processes that involve damping ; the proper mathematical formulation must be in the language of density matrices .but when the damping is small , as is the case here , it is easier to talk about states .the steps above can be easily transcribed in the language of density matrices , without changing their outcomes .now we can proceed to the remaining pieces needed to complete the scenario : a mechanism that favours the state as the equilibrium state , and an environment that permits an almost coherent quantum evolution .for that i have to appeal to the ingredients ignored so far the enzymes .enzymes play a very important role in many biochemical reactions , and some of the things they do were mentioned in section iii .the rates of various biochemical reactions , when estimated with the standard thermodynamical analysis ( probability distributions , diffusion processes , kinetic theory , etc . ) , fall too short of the observed rates , often by orders of magnitudes .one is left with no choice but to admit that these reactions are catalysed .enzymes are the objects which catalyse biochemical reactions .they are large complicated molecules , much larger than the reactants they help , made of several peptide chains .their shapes play an important part in catalysis , and often they completely surround the reaction region .they do not bind to either the reactants or the products , just help them along the way .the standard explanation is that enzymes lower reaction barriers .just how this lowering of reaction barriers occurs is not clearly understood , and is an active area of research ( for example , enzymes can form weak bonds with the transition state or suck out solvent molecules from in between the reactants ) . ultimately it must be explained in terms of some underlying physical laws .i put forward two specific hypotheses about what enzymes accomplish in the processing of genetic information .  *_ enzymes provide a shielded environment where quantum coherence of the reactants is maintained ._ this is a rather passive task , consistent with the properties of enzymes mentioned above , and it is also plausible . for instance , diamagnetic electrons do an extraordinarily good job of shielding the nuclear spins from the environment the coherence time observed in nmr is sec , much longer than the thermal environment relaxation time ( sec ) and the molecular collision time ( sec ) , and still neighbouring nuclear spins couple through the electron cloud .a few orders of magnitude increase in coherence time is sufficient in many reactions for faster quantum algorithms to take over from their classical counterparts , and provide the catalytic speed - up . *_ enzymes are able to create superposed states of chemically distinct molecules ._ this is an active task .various nucleotide bases differ from each other only in terms of small chemical groups , containing less than 10 atoms , at their hydrogen bonding end .to convert one base into another , enzymes have to be repositories of these chemical groups which differentiate between various nucleotide bases .enzymes are known to do cut - and - paste jobs with such chemical groups ( e.g. one of the simplest substitution processes is methylation , replacing by , which converts u to t ) .given such transition matrix elements , quantum dynamics automatically produces a superposition state as the lowest energy equilibrium state .( note that the cut - and - paste job in a classical environment would produce a mixture , but in a quantum environment it produces superposition . )it is mandatory that the enzymes do the cut - and - paste job only on the growing strand and not on the intact strand .perhaps this is ensured by other molecular bonds .these hypotheses are powerful enough to explain many observed properties of enzymes from a new perspective . for example ,  * it is obvious why dna replication always takes place in the presence of enzymes . if base - pairing were to occur by chance collisions, it would occur anywhere along the exposed unpaired strand . *since enzymes control the transition matrix elements , they can keep u out of dna and t out of rna . *quantum processes can tunnel through the reaction barriers , instead of climbing over them .* as long as quantum coherence is maintained , the replication process is reversible .this can easily explain the error - correcting exonuclease action of the polymerase enzymes .more importantly , these hypotheses are experimentally testable , provided one can observe the replication process at its intermediate stages .that is within the grasp of modern techniques such as x - ray diffraction analysis , electron microscopy , nmr spectroscopy , radioactive tagging of atoms in chemical reactions , and femtosecond photography .i have proposed a quantum algorithmic mechanism for dna replication and protein synthesis .this genetic information processing takes place at the molecular level , where quantum physics is indeed the dominant dynamics ( classical physics effects appear as decoherence and are subdominant ) .it is reasonable to expect that if there was something to be gained from quantum computation , life would have taken advantage of that at this physical scale . for dna replication ,the quantum search algorithm provides a factor of two speed - up over classical search , but it is still an advantage. quantum algorithms can provide a bigger advantage for more complicated processes involving many steps .implementation of quantum database search does not require a general purpose quantum computer .a system that can implement the quantum oracle and quantum state reflections is sufficient .i have described the physical analogues of these operations in genetic information processing , and it is not unusual for living organisms to find the correct ingredients without bothering about generalities .in fact various pieces of the scenario have fitted together so nicely ( database search paradigm , optimal values and , quantum transformations implementing and ) , that with the courage of conviction , i have made bold hypotheses to fill the remaining gaps . the role that enzymes play , as described in sectionvi , is both plausible and experimentally verifiable .the other assumptions i have made ( i.e. two - step cascade deexcitation in base - pairing , energy quanta not carrying away any phase information , energy quanta departing only when they cause minimal decoherence to the system ) concern how decoherence modifies pure quantum evolution , and are also experimentally testable .such experimental tests would decide the future of this proposal .it is clear that if experiments verify the quantum scenario for genetic information processing presented here , there will be a significant overhaul of conventional molecular biology .there is nothing inappropriate in that the subject of molecular biology was born out of quantum physics , and quantum physics has not yet had its last word on it .i also want to acknowledge that molecular biology is a far more complicated subject than the simplest features i have explored in this work .if quantum physics provides better explanations for other more complex phenomena of life , it would be a wonderful development .i am grateful to the audience for their many comments and questions .i summarise below some of the important concepts they brought out .a : in case of dna replication , enzymes separate only a limited region of paired strands as they slide along replicating the information .complete separation of the strands would break too many hydrogen bonds and cost too much in energy .thus the replication remains a local process . in case of protein synthesis , random pairing of t - rna and m - rna at several different locations would lead to a lot of mismatches and errors , since the triplet code needs precise starting and ending points .parallel processing does take place though : several ribosomes work simultaneously on a single m - rna strand , each one traversing the full length of m - rna from one end to the other and constructing identical amino acid chains .a : when the dna assembly takes place , with each base as a separate unit , the number of possibilities explored is indeed .but this is not what happens in the matching between t - rna and m - rna .the three bases come as a single group , without any possibility of their rearrangement .the whole group has to be accepted or rejected as a single entity .in such a situation , the number of objects that can be distinguished is smaller , as given by eq.(3 ) .a : with only 21 signals embedded amongst 64 possibilities , the amino acid code is indeed degenerate .moreover , all the amino acids are not present in living organisms with equal frequency , quite unlike the nucleotide bases .ribosomes are also much bigger and more complicated molecules than the polymerase enzymes , and so capable of carrying out more complex tasks .all this makes protein synthesis a more difficult process to study than dna replication .i have not analysed it in detail , and i am unable to provide any understanding of the degeneracy of the triplet code .q : the energy quanta do not have to be released exactly at the opposite end of the oscillation .oscillations spend more time near their extrema anyway , and so even random emission will give high probability for correct base - pairing .a : classical algorithms can do everything that quantum algorithms do , albeit slower . in the absence of enzymes, various steps will take place only by random chance , and so the start - up will be slow .but once processes get going , as in a living cell , enzymes will be manufactured along the way and there will be no turning back to slower algorithms .   a : in my proposal , the polymerase enzymes substitute one chemical group for another . with the dna base sequence being random , the chemical groups are recycled with high probability .so with a reasonable initial stock an enzyme can perform its task for a long time .of course if an enzyme runs out of its stock , it has no choice but to quit and replenish its stock .a : dna replication can take place without cellular environment . with proper enzymes in the solution, the polymerase chain reaction can rapidly multiply dna .this is used in dna fingerprinting from dead cells .a : in my view , catalysts convert random walk processes into directed walks , providing a speed - up in the number of steps .quantum superposition is one way to achieve this , but it does not have to be the only way . there may be even classical mechanisms which can do the same job .a : understanding the basic processes of life will always lead to new applications .for example , molecular biologists have been working on accelerating synthesis of desired proteins and inhibiting growth of cancer and harmful viruses .i am not inclined to speculate more on this issue right now .

Abstract: replication of dna and synthesis of proteins are studied from the view - point of quantum database search . identification of a base - pairing with a quantum query gives a natural ( and first ever ! ) explanation of why living organisms have 4 nucleotide bases and 20 amino acids . it is amazing that these numbers arise as solutions to an optimisation problem . components of the dna structure which implement grover s algorithm are identified , and a physical scenario is presented for the execution of the quantum algorithm . it is proposed that enzymes play a crucial role in maintaining quantum coherence of the process . experimental tests that can verify this scenario are pointed out .