Spectral Analysis of Diff.Operators: Interplay Between Spectral and Oscillatory Properties(WS 2005)


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Destexhe, A.. Therefore the measurement of charmonium and bottomonium suppression can be used as a kind of QGP thermometer. High-energy collisions involving ions have the best chance to produce gluon condensates, where the gluon wave functions start to overlap producing a collective behaviour. This condensate would be similar to the phenomenon predicted by Bose and Einstein 93 years ago and observed in other boson systems such as ultra-cold atoms.

Saturated gluons are expected to be observed only at small angles relative to the beam axes, where the number and the size of the gluons are the largest. LHCb has the unique capability of measuring photons coming from these high density gluon regions. The announcement of initial measurements of these photons caused a lot of excitement at the Quark Matter conference.

The image above right shows the angular distribution between isolated photons and other particles taken during the pPb run. This sample is from a region where theorists expect that gluons are saturated. The blue band is the background from other processes. This is the first indication that gluons can be probed in this region, never achieved by any experiment so far. Members of the theory community expressed interest in studying the upcoming LHCb results and are discussing the mathematical tools that can confirm the discovery of this new form of matter.

Read more in the LHCb presentations [1] , [2] , [3] , [4] and [5]. The LHCb papers and conference contributions will be available shortly. The data taking period started officially today. The last proton-proton collisions took place on 28 November and the LHC machine was shut down during the winter period to allow for planned technical interventions. LHCb used this period to perform maintenance work on many sub-detectors. Four months later proton beams passed again through the LHCb detector. The nominal proton beam has size of a human hair and an energy equivalent to a very fast train.

The commissioning of LHC is therefore very complicated and takes about one month. Each beam consists of packets of protons called bunches. The number of proton bunches was progressively increased and will finally reach the maximum of bunches of which will collide inside the LHCb detector. The LHC operation team and LHC experiments agreed this year that this important milestone will be reached when proton bunches in each beam will collide. This has happened today. The re-commissioning of the accelerator has proceeded very smoothly and first collisions arrived earlier than initially expected.

The LHCb detector and its data acquisition system are ready for the last year of Run 2 data taking that will allow the experiment to obtain even more precise and interesting physics results. The image displays a typical event recorded today. The two-year Long Shutdown 2 will then start in December , and during this period the LHCb detector will face its first major upgrade, which will allow the experiment to take data at much higher rate.

Today is thus an eggstra special day in the history of particle physics. Another important property of elementary particles is called flavour. The Eggs particle is responsible for the generation of flavour a few minutes after the Big Bang. Without the eggsistence of the Eggs particle all food would now taste the same. By studying the production and the decay properties of this new particle, LHCb physicists confirmed that it is composed of two eggs, that carry the strong sticky force, and hence it is really an egg ball, or more technically an egg-prolate-spheroid.

The eggsistance of Eggs balls was predicted more that forty years ago, but it is only today that its reality is unambiguously confirmed eggsperimentally. In this way, the astonishing prediction of Peter Eggs about the strong interaction origin of flavour is confirmed. The image shows an eggcited theoretical physicist explaining properties of the newly discovered Eggs ball. The way to the grand unification between subatomic physics and molecular gastronomy is now therefore cracked wide open.

A few selected items are listed below. The only known fundamental scalar particle is the Higgs boson with a mass of GeV. The LHCb detector has good sensitivity to light spin-0 particles produced by gluon-gluon fusion and decaying to a pair of opposite-sign muons, due to its capability of triggering on objects with small transverse momentum and to its high-precision spectrometer. The results of a search for these particles produced in proton-proton collisions at center-of-mass energies of 7 and 8 TeV were presented. No evidence is found for a signal in the mass range 5.

The image shows the mass spectrum of the muon pair in the whole search region. This decay is predicted to be very rare within the SM, as it occurs only through suppressed loop diagrams. New particles foreseen in extensions of the SM can significantly enhance or suppress the rate of this decay. The result presented at the Rencontres de Moriond paves the way to search for new physics using this decay when a larger datasets will be collected by the upgraded LHCb detector during LHC Run The production of top quarks at hadron colliders represents an important test of the SM.

The top quark is the heaviest known fundamental particle and its production and decay properties are sensitive to a number of parameters involved in physics models beyond the SM. Top-quark production in this region receives a higher contribution from quark-antiquark annihilation than in the central region. In addition, it probes the the proton parton distribution functions PDF s at high values of the fraction of proton momentum carried by quarks or gluons, whose knowledge suffers from large uncertainties. At present precise measurements of top quark production at LHCb can be used to reduce substantially the uncertainty on PDFs in this kinematic region.

The large contribution from quark-initiated production also results in a larger expected charge asymmetry in the forward region than in the central region probed by ATLAS and CMS. Top pair production was measured at the proton-proton collision energy of 13 TeV, where the production cross-section in the acceptance of the LHCb detector is expected to be ten times larger than it was at the lower Run-1 energies. The image compares the measured cross-sections within the LHCb acceptance with predictions of theoretical models.

The image shows an example of the measured asymmetry as a function of the decay time of the B 0. The parameters describing the difference in behaviour between matter and antimatter, known as CP violation, are constrained in the so called Cabibbo-Kobayashi-Maskawa matrix unitarity triangle. The blue line in the image is the result of this analysis, while the red, dashed line shows the expectation in the absence of CP violation. This result belongs to the class of measurements in which the contribution of new physics is not expected.

The study of the CP violation in decays of B 0 and of B s 0 mesons into two charged particles h that do not contain charm quarks represents a powerful tool to test the CKM picture of the SM, and also to investigate the presence of physics beyond the SM. The results are the most precise from a single experiment and constitute the strongest evidence for time-dependent CP violation in the B s 0 meson decays to date. They also contribute to the determination of the CKM unitarity triangle.

The results obtained are consistent with the hypothesis of CP conservation. Therefore, the B s 0 meson decay is visible in the invariant mass spectrum of four K mesons, see the red dashed line contribution in the image. The data taking period ended this Sunday. Towards the end of the run at the centre-of-mass energy of 13 TeV, the LHC provided collisions at a reduced energy of 5 TeV to produce reference data for proton-lead and lead-lead collisions taken earlier in Run 2.

Besides the scientific interest of proton-proton p-p physics at 5 TeV for the LHCb heavy-ion programme see [1] , [2] , the experiment has been taking at the same time a parallel stream of data from fixed-target collisions with another world first in high energy physics. There have been typically bunches of protons circulating in each LHC ring, out of which collided inside the LHCb detector. LHCb physicists decided to use additional non-colliding bunches to accumulate the largest sample of proton-neon data in a fixed-target configuration. The LHCb experiment has the unique ability of injecting gas, neon in this case, into the interaction region and therefore study processes that would otherwise be inaccessible.

This gas-injection system was originally designed to help LHCb measure the brightness of the accelerator's beams, but is now being used for dedicated physics measurements. It has been the first time ever that an experiment has collected data in the collider and fixed-target modes simultaneously. LHCb physicists showed that it is possible to reconstruct both sets of data in parallel, align the detector elements and track particle trajectories correctly. A real challenge has been to develop an online event selection trigger system handling efficiently both data taking conditions.

The pink-dashed rectangle highlights the regions were p-p collision events were selected. The two other red-dashed rectangles show the region where only p-Ne collisions take place. LHCb continues to revolutionise data acquisition and analysis techniques. The calibration and alignment process takes place now automatically online and stored data are immediately available offline for physics analysis. This time the collider and fixed-target modes of operation have been unified into the same data acquisition framework. In particle physics, a grand-unified theory is one in which at very high energies the electromagnetic, weak and strong interactions unify as a single force.

Today LHCb physicists have succeeded to unify very different concepts of data taking and analysis.

wasandtit.pro/180-chloroquine-phosphate.php The data taking period has been very successful, because of the excellent performances of both the LHC and the LHCb experiment itself. The image shows the growth of integrated luminosity during different years of LHC operation. The integrated luminosity is higher than that collected in The overall Run 2 luminosity , 3. It will be used for maintenance and improvements to the LHC and its detectors. LHCb plans to exploit this period to perform maintenance work on many sub-detectors. It is planned that protons will start to circulate again in the LHC rings at the beginning of April and that the first p-p collisions for physics will take place in early May, marking the beginning of the last year of Run 2.

Read more in the CERN update. The purpose of the meeting is to consider the latest results from LHCb, discuss possible interpretations and identify important channels and observables to test leading theoretical frameworks in the near and long-term future of the experiment. The LHCb detector is used to perform precision measurement across a range of areas of particle physics, notably including searches for new physics manifestations in flavour, studies of heavy-flavour spectroscopy and production of gauge bosons, searches for new exotic particles, and unique measurements with heavy ion and fixed-target collisions in the forward region.

Precise calculations from theory are essential for the interpretation of LHCb results, as is evident from numerous news articles on this page. The comparison of LHCb results with precise Standard Model predictions provided by the theory community is the key to looking for discrepancies that would indicate the existence, and the nature, of physics beyond our present knowledge. This series of joint LHCb-theory workshops is aimed at facilitating informal discussions between LHCb experimentalists and theorists, leading to a mutual exchange of information that is valuable for achieving progress.

More than physicists crowded the Main Auditorium through three days featuring various dedicated sessions see the photo above. The left image shows the steadily increasing number of participants as a function of time since the first workshop was held, clearly demonstrating that this series of workshops is more and more becoming a reference event for both the LHCb physicists and the theorists. Furthermore, intriguing results published by LHCb in recent years have been triggering additional interest and discussions within the physics community. Learn more in the workshop presentations. Click the images for higher resolution.

Yesterday evening the first xenon-xenon ion collisions took place at LHC and the run lasted for seven hours. The right image shows a collision event recorded and analysed online by the LHCb data acquisition system. Heavy ion collisions are studied at LHCb in order to understand the behaviour of the so-called quark-gluon plasma , a state of matter in which quarks and gluons are moving as free particles, similarly to what happened in the first instants of time after the Big Bang.

At the LHC, the properties of the quark-gluon plasma are usually studied in collisions of lead nuclei. Although there are no plans to use xenon-xenon collision for this purpose in the near future, one day of the LHC time was devoted to these collisions in order to study the properties of nuclear matter at high-energy density and high temperature. The comparison of experimental measurements in lead-lead and in xenon-xenon collisions will bring new insights into the properties of the quark-gluon plasma. The result that was reported is approximately 2 standard deviations from the Standard Model SM expectation.

However, mass differences between various leptons play a role which must be accounted for when performing calculations of interaction rates. Note the different horizontal scale. LHCb has also found other intriguing anomalies when performing tests of lepton universality. All these measurements were performed at LHCb using the entire Run 1 data sample, corresponding to an integrated luminosity of 3fb -1 at centre-of-mass energies of 7 and 8 TeV.

Data collected in Run 2 already provide a sample of B-meson decays more than twice as large, and it will be of great importance to see whether future updates of these analyses with increased statistics will confirm the hints of discrepancies. They are bound systems of a charm quark, c, and an anti-charm quark, c , held together by the strong nuclear force.

The image on the left represents the complexity of this spectrum along with some of the allowed decay transitions as a function of mass in the vertical axis versus spin and other quantum numbers in the horizontal axis. The decay paths corresponding to the decays under study are highlighted in red. This discovery triggered rapid changes in high-energy physics at the time and these are commonly referred to as the "November Revolution". The charmonium family was then intensively studied by a large number of experiments at different facilities. As it is experimentally very challenging to measure the energy of a photon precisely in the harsh environment of a hadron collider, high-precision measurements have not previously been made at a collider like the LHC.

The new analysis from LHCb applies an old "trick" in a new situation. This technique was first proposed in by R. Dalitz realised that quantum mechanical fluctuations permitted one or both of the photons to be virtual, allowing them to transform into an electron-positron pair. The new measurements have a similar precision to and are in good agreement with those obtained at previous dedicated experiments, notably E and E, that used antiproton annihilations into a hydrogen target to produce and study the charmonium states see the image below for a comparison amongst the various results available in the literature.

It will facilitate a better understanding of quantum chromodynamics QCD , i. The LHCb collaboration has reached a symbolic milestone this morning, announcing that the experiment has recorded a luminosity of 6 fb -1 integrated over the whole period of data taking The shift crew captured this moment by taking a photo of the live screen in the LHCb Control Centre, see the number pb -1 in the lower-left corner of the table in the image.

The luminosity delivered by the LHC collider was pb -1 during this period. Most of the results highlighted through the years in this page were obtained using the LHC Run 1 data sample, where the full sample corresponds to an integrated luminosity of 3 fb -1 at centre-of-mass energies of 7 and 8 TeV. Since the production cross-section of beauty and charm particles at 13 TeV is about twice as large as that in Run 1, the total sample of beauty particles available for physics analyses is now about three times larger than that recorded during the Run 1 data taking period.

The door that will lead to obtain many more interesting results is open wide, as you will be able to see with our future reports in this page. A few other selected items are listed below. Proton collisions with lead nuclei p-Pb at the LHC are very important to understand nuclear effects relevant to the study of quark-gluon plasma formation in lead-lead Pb-Pb collisions.

Two sets of data were taken: p-Pb and Pb-p, where in the first case the proton travels in the forward direction of LHCb and in the second case the beam directions are reversed. This allowed the LHCb detector, recording the particles only on one side of the interaction point, to make measurements in both forward and backward directions with respect to the proton beam.

For the first time, beauty-hadron production is measured precisely at low momentum at the LHC in p-Pb and Pb-p collisions. In p-Pb, a weak suppression is observed, whereas in Pb-p no significant deviation from unity is found. Several extensions of the Standard Model predict that new heavy particles that decay to two energetic b-quarks could be accessible at the energy of LHC collisions. Moreover, the study of the Higgs-boson decay to a pair of b and b quarks at the LHC is of great interest, since the precise determination of the Higgs boson coupling to b-quarks is an important test of the Standard Model.

The decay of a Z boson to a b b pair provides a standard candle for direct searches of physics beyond the Standard Model in final states with b b quarks. Measurements of this decay can be used to demonstrate that no biases are induced by the b-jet reconstruction procedure and that the reconstruction efficiencies are evaluated correctly. A clear signal is observed by LHCb for the first time, as shown in the right image above. The theoretical prediction and the measurement are compatible within one standard deviation. Additional data being collected by LHCb will enable a more stringent comparison with the theoretical prediction to be performed.

This is a further example that illustrates that LHCb has become a general purpose forward experiment with a broad physics programme. The B 0 meson decay into a p p pair is observed for the first time with a statistical significance of 5. The branching fraction is measured to be 1. This decay is the rarest fully hadronic decay of a B meson ever observed. Studies of B mesons decaying to baryonic final states have been carried out since the late s. It was quickly realized that baryonic B decays differ from mesonic decays since two-body charmless decays are suppressed with respect to decays to multi-body final states.

This observation provides a valuable input towards the understanding of the dynamics of hadronic B decays and allows for a better scrutiny of QCD models. Please click the images for higher resolution versions. The image above shows an example of a Feynman diagram contributing to this decay.

This state is therefore incompatible with a strongly decaying particle, but is consistent with a longer-lived decay involving weak interactions as would be expected for this particle. The existence of doubly charmed baryons was already known to be a possibility in the s, after the discovery of the charm quark. The image shows an artist view of this new particle.

This discovery opens a new field of particle physics research. The image illustrates how half-spin baryons can be formed by assembling together the three light quarks u, d, s and the charm quark Particle Data Group, Phys. D86, Furthermore, other hadrons containing different configurations of two heavy quarks, for example two beauty quarks or a beauty and charm quark, are waiting to be discovered.

Measurements of the properties of all these particles will allow for precise tests of QCD , the theory of strong interactions, in a unique environment. LHCb is very well equipped to face this very exciting challenge. Click images for higher resolution. A special musical performance was given in Thoiry , a nearby village to CERN, where she discussed the photograph shown here right image.

They were both members of the International Committee on Intellectual Cooperation , an advisory organization for the League of Nations which aimed to promote international exchange between scientists, researchers, teachers, artists and intellectuals. He takes over from Guy Wilkinson from the University of Oxford. Giovanni and Chris will face the huge challenges of completing the run 2 data taking and preparing for the major LHCb detector upgrade to be installed during Long Shutdown 2, LS2.

In the meantime they dream that the analysis of the Run 2 data could yield the discovery of new physics! This has been possible because of the unique capabilities of LHCb in precisely reconstructing decay vertices. This property is called "lepton universality". However, differences in mass between the leptons must be accounted for.

This ratio is precisely calculable in the SM owing to the cancellation of uncertainties in the ratio, and turns out to be about 0. The average of all world results is brought, by including this new measurement, a little bit closer to the SM prediction and at the same time, due to improved precision, the discrepancy between the experimental world average and the SM prediction increases slightly to about 3. Any measurement exhibiting a conclusive breakdown of lepton universality, after mass related effects are accounted for, would be a clear sign of new physics.

Recently the CERN Theory Division organized a three-day workshop to discuss the interpretation and implications of these anomalies and their potential to shed some light into models of physics beyond the SM. Data collected in Run 2 already provide a sample of B-meson decays more than twice as large, and it will be of great importance to see whether updates of the Run 1 analyses will confirm the discrepancy. Read more in the CERN update for scientists. This traditional winter shut-down period was "extended" due to major installation work carried out in another one of the LHC detectors.

The recommissioning of the accelerator has proceeded very smoothly and first collisions arrived earlier than initially expected. During the technical stop LHCb performed relatively minor maintenance work on the detector, and, on the other hand, major interventions on the access lift and the crane in the underground cavern. The LHCb detector and its data acquisition system are ready for a bumper year of data taking that will allow the experiment to obtain even more precise and interesting physics results. This measurement provides an important test of lepton universality LU , which is one of the most important ingredients of the Standard Model of particle physics.

LU means that leptons e. In the LHCb measurement, the distance of the result from the SM prediction is found to be significant at the level of 2. The numerical values are given at the top of the article, where the first uncertainty, which dominates, is statistical, and the second is systematic. The lower boundary of the low- q 2 region roughly corresponds to the di-muon production threshold.

The image also shows several independent SM theoretical predictions. A difference from the SM of 2. These differences are not yet at the level where they can be claimed to exhibit evidence for BSM physics, but they are intriguing when considered in the context of an earlier LHCb analysis. This result created much interest in the particle physics community. Examples are shown in the Feynman diagrams below.

The upper ones show the SM contributions. The left-lower one shows a possible contribution from a heavy Z-boson -like particle, named Z' , which would interact differently with muons and electrons. The lower-right diagram shows a possible contribution from a hypothetical scalar leptoquark LQ , which would interact with both quarks and leptons. Alternatively, different, not yet predicted, and therefore even more interesting, BSM physics could be at play!

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Data collected in Run 2 already provide a sample twice as large, and it will be of great importance to see whether updates of the present analysis will confirm the discrepancy. Future LHCb measurements will be able to elucidate whether these tantalising hints are a manifestation of statistical fluctuations or whether LHCb is observing a glimpse of new physics. The measurement is of antiproton production in proton-helium p-He collisions. Although the LHC collides protons with protons, the LHCb experiment has the unique ability to inject gas, for example helium, into the interaction region and therefore study processes that would otherwise be inaccessible, such as here the production of antiprotons from p-He interactions.

The forward geometry and particle identification capabilities of the LHCb detector are well suited to provide good reconstruction for antiprotons down to the low transverse momentum region where most of the production is expected. This result is very important for interpreting searches for dark matter in the Universe. Dark matter is a hypothetical entity of unknown nature whose existence would explain a number of otherwise puzzling astronomical and cosmological observations.

The name refers to the fact that it does not interact with electromagnetic radiation like light. Although dark matter has not been directly observed, its existence and properties are inferred from its gravitational effects such as the motion of visible matter around galactic centres and precise measurements of temperature fluctuations in the cosmic microwave background. An interesting possibility is that dark matter is composed of some kind of stable elementary particles whose existence is proposed in different extensions of the Standard Model of particle physics.

In such a case these dark matter particles could collide and produce ordinary particles, in particular antiprotons. However antiprotons can also be produced in standard processes through collisions of cosmic rays with the interstellar medium, of which helium is a significant component. Therefore a potential signature of dark matter is the observation in space of a higher ratio of antiprotons to protons than would be expected from standard processes. Also shown in the image is the prediction 'Fiducial' and, as coloured bands, the uncertainties on this prediction, which come from the limited knowledge of several of the ingredients in the calculation.

Although the data points lie above the prediction, the current uncertainties are large enough to almost accommodate the discrepancy, thereby preventing an unambiguous interpretation. The largest uncertainty is associated with the knowledge of the cross-sections, in particular that of p-He collisions. This is where LHCb enters the game. In the image, the result is compared with the most popular models used in cosmic rays physics. The spread among model predictions indicate the large uncertainty on the process prior to this measurement. In the s many different particles were discovered.

Initially thought to be elementary, the ever growing list of discoveries led physicists to doubt this assumption. Therefore efforts were made to find a classification scheme in analogy to the periodic table of chemical elements. The most successful such scheme was proposed by Gell-Mann. The regular structure of the decuplet enabled many properties of this new particle to be predicted, including its mass.

This picture validated the Eightfold Way, and led Gell-Mann to propose the quark model in , which explains the structure of the octets and decuplet. The discoveries announced today are excited states of this system, analogous to the excited states of atoms. More details can be found in the LHCb publication. Processes where a B meson decays into a pair of oppositely charged leptons are powerful probes in the search for physics beyond the Standard Model. This presentation provoked much interesting discussion. These decays are of great interest in the search for further manifestations of CP violation in baryonic B decays, see the first evidence for the violation of the CP symmetry in baryon decays.

A phase-1 upgrade of the experiment is currently being prepared and will be installed in the long-shutdown 2, in The measured branching fraction 3. It is the most precise measurement of this quantity to date. The full 3 fb -1 of data collected during Run 1, and 1. The size of this contribution is not found to be significant, and so an upper limit is set for the decay at a value of 3. The other contributions show the contribution of background processes. The probability, or branching fraction, of the B s 0 meson to decay into two oppositely charged muons is very small in the SM and is well predicted.

On the other hand, a large class of theories that extend the SM, such as, for example, supersymmetry, allows significant modifications to this branching fraction and therefore an observation of any significant deviation from the SM prediction would indicate a discovery of new effects. The decay of a B s 0 meson into a muon pair has therefore long been regarded as one of the most promising places to search for these new effects.

This decay has been searched for more than 30 years by different experiments at different accelerators as shown in the image. The LHCb collaboration obtained the first evidence , with a significance of 3. Previous results already severely constrained the type of SM-extension models that are still allowed, as described, for example, in the 30 March news.

The results announced today isolate even more precisely the parameter region in which these new models can exist, and therefore focuses future experimental searches and theoretical attention. All candidate models of physics beyond the Standard Model will have to demonstrate their compatibility with this important result. LHCb also reported today the first measurement of this quantity, and found it to be 2. The two muon tracks from the B s 0 decay are seen as a pair of green tracks traversing the whole detector in the left image. The right image shows the zoom around the proton-proton collision point, the origin of many particle tracks.

The two muon green tracks originate from the B 0 s decay point located 17 mm from the proton-proton collision. This web based event display will run on your computer or smartphone without need to load any specialized software. Stay tuned for updates from Run 2 data. The preliminary numbers and plots presented at the seminar have been replaced with the final ones on March The LHCb collaboration has published today in Nature Physics the first evidence for the violation of the CP symmetry in baryon decays with statistical significance of 3.

CP violation has been observed in K and B meson decays, but not yet in any baryon decay. In the quark model of particle physics mesons are composed of a quark and antiquark pair while baryons anti-baryons are composed of three quarks anti-quarks. About and decays were found for the two decay modes, respectively. It is important to measure the size and nature of CP violation in these decays in order to determine whether they are consistent with the predictions of the Standard Model of particle physics or, if not, what extensions of the Standard Model would be required to explain them.

Once the signals have been established, the analysis turned to the study of matter-antimatter asymmetries. The statistical significance of these asymmetries differing from zero is 3. In the past, analogous large effects were also seen in three-body charmless B decays. The full 3 fb -1 run 1 data sample was used to obtain this result. The number of beauty particle decays recorded by LHCb in run 2 is already larger than that used in this analysis. The data taking period ended this morning. We shall henceforth adopt a slightly different viewpoint, namely that we are willing to allow data d r from a narrow region on the periphery of R.

The small amount of spatial leakage from points r outside of R that we accept is offset by the advantage that there is no broad-band bias in the resulting multitaper spectral estimates, as we shall see. The use of bandlimited rather than spacelimited tapers is natural in many geophysical applications, where we seek a spatially localized estimate of the spectrum S l of a signal s r. In Fig. As the target degree l increases the coupling matrix increasingly takes on a domelike universal shape that is approximately described by eq.

Small numbers on top are the maximum value of for every target degree l. Open circles on the right-hand vertical axis are the whole-sphere, large- l limits, obtained via eq. Since K is the number of retained tapers, it will always be greater than 2—3 in a realistic multitaper analysis. As noted in Section 8. The grey band surrounding the theoretical -versus- l curve is the standard error of a hypothetical whole-sky spectral estimate. The rapid increase in the whole-sky uncertainty above this transition is due to the exponential increase in the noise power 37 for harmonics that are below the angular resolution of the WMAP antennae.

The total uncertainty due to both cosmic and noise variance represents the best we can ever do, if we insist upon estimating individual values of the spectrum , even if we had uncontaminated whole-sky data. The elimination of contaminated data by a sky cut will always increase the variance; the only way to reduce it is to sacrifice spectral resolution. The six panels of Fig. As we have seen, the bandwidth alone controls the amount of bias deliberately introduced in this way, and not the size or shape of the analysis region—but the latter does influence the variance of the estimate.

The open circles show the expected values of a multitaper estimate , and the accompanying error bars show the associated standard error under the moderately coloured approximation. The multitaper method yields a band-averaged spectral estimate at every spherical harmonic degree l , but we have only plotted values whose coupling bands do not overlap, so that they are statistically uncorrelated. The spacing between the open-circle estimates is thus indicative of the spectral resolution.

As expected, the bias is most pronounced in strongly coloured regions of the spectrum, and it is an increasing function of the bandwidth L and thus the spectral extent of the averaging. Bandwidths in this range are, therefore, suitable for multitaper spectral analysis of WMAP temperature data on the cut sky. In all cases the multitaper errors are significantly smaller than the uncertainty of a hypothetical whole-sky estimate of , with no band averaging.

Kosowsky ; Efstathiou et al. The covariance of a whole-sphere estimate is and the covariance of a maximum-likelihood estimate is the inverse of the Fisher matrix of eq. The covariance of a periodogram estimate is given by eq. For moderately coloured spectra these cumbersome expressions for and can be approximated by eqs and — , and the Fisher matrix can be approximated by eq. The maximum-likelihood method is attractive and has received widespread use in CMB cosmology, because it provides the best i. This desirable feature is offset by a number of disadvantages that we enumerate in Section 6.

For smaller regions, of area , it is possible to obtain minimum-variance, unbiased estimates of a binned spectrum using eqs — ; however, this requires the somewhat artificial assumption that the true spectrum S l can be adequately approximated by a coarse-grained spectrum , where. The multitaper method is distinguished by its ease of use, requiring neither iteration nor large-scale matrix inversion. The coupling matrix for an eigenvalue-weighted multitaper estimate is illustrated in Figs 6 and 7.

The spatial leakage from data outside of the target region R can be quelled and the analysis expedited by averaging only the first K tapered estimates , as in eq. Financial support for this work has been provided by the U. Computer algorithms are made available on www. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation.

Close mobile search navigation Article Navigation. Volume Article Contents. Spectral estimation on a sphere in geophysics and cosmology F. E-mail: fjsimons alum. Oxford Academic. Google Scholar. Frederik J. Cite Citation. Permissions Icon Permissions. Summary We address the problem of estimating the spherical-harmonic power spectrum of a statistically isotropic scalar signal from noise-contaminated data on a region of the unit sphere. Time series analysis , Fourier analysis , Inverse theory , Spatial analysis. Open in new tab Download slide. Integrals over the region R will be assumed to be approximated with sufficient accuracy by a Riemann sum over pixels:.

We shall make frequent use of the well-known formula for the surface integral of a product of three spherical harmonics:. In general the signal s r in eq. Harmonic degrees l whose angular scale is less than the finite aperture of the beam cannot be resolved; for illustrative purposes in Section 10 we adopt a highly idealized noise model that accounts for this effect, namely. An obvious choice for the spectral estimator in that case is e. The expected value of the whole-sphere estimator is.

The formula for the variance of an estimate,. It is convenient in this case to regard the data d r as having been multiplied by a unit-valued boxcar window function,. To find the expected value of we proceed just as in reducing eq. In the opposite limit of a connected, infinitesimally small region,. In principle it is possible to eliminate the leakage bias in the periodogram estimate by numerical inversion of the coupling matrix. We model this likelihood as Gaussian:. Maximization of is equivalent to minimization of the logarithmic likelihood. The maximum-likelihood method yields an unbiased estimate of the spectrum inasmuch as.

Maximum-likelihood estimation is the method of choice in a wide variety of statistical applications, including CMB cosmology.