The red line is the same LCDM model as above, i.e., it is fit to the temperature power spectrum and not to the polarization spectra, showing very good consistency of this model with the polarization data. Comments posted as "Unknown" go straight to junk. Using the faint temperature patterns formed in the sky by cosmic microwaves, scientific analysis can measure things The power spectrum we observe today carries information about what the universe has done since then. It's a... well, funny book too. By viewing the scale of the universe now, and using Einstein's General Relativity equations to compute the time, under these conditions, needed to reverse the universe to "zero" size, we have the age calculated for us. Unlike previous work, in which strings were modeled as idealized one-dimensional objects, we evolve the simplest example of an underlying field theory containing local U(1) strings, the Abelian Higgs model. This includes the equations used, assumptions and approximations imposed on their solutions, and most importantly the algorithms and programming tricks needed to make the code actually work. Lex --You may have a point there. If one changes it, one obtains a different curve. Gee, if you leave a comment saying the observed features are crap and I've done everything better, this sounds pretty much like the emails I usually get saying I've proven Einstein wrong, and besides I have a completely new quantum mechanics. If you suggest another analysis then you'll need another explanation. You start with a scenario in which the count rate decreases with 1/e. Note this does not fix the actual output L range, spectra may be calculated above l_max (but may not be accurate there). 4. Important things in the universe, like shock waves, appear as coherent objects in real space, while important things in music, like pure tones, appear as coherent objects in Fourier space.p.s. The universe can have strange properties, but simple structures like plasma and hydrogen gas behave in WMAP science team does the same thing, only with high speed super-computers, tens of thousands of "Mapping," had to begin somewhere. The fourth section discusses CMB perturbation calculations and diﬀerent approximation schemes used in CMBAns, and how they aﬀect the power spectrum calculations. - See lensing review for details. The early structure of the universe as seen in the Cosmic Microwave Background (CMB) can berepresented by an angular power spectrum, a plot that shows how the temperature pattern in the early universevaries with progressively measuring smaller and smaller patches of the sky. varies with progressively measuring smaller and smaller patches of the sky. And the observed amplitude of the peak is also consistent with the interpretation as a systematic statistical error in this sense.This conclusion was actually confirmed last year when the year-3 WMAP data were published: the difference between the 3- and 1-year maps shows residuals that have about the same amplitude as the 'second acoustic peak'.Thomas. The CMB temperature anisotropy power spectrum was not formed in an instant.. Below is a plot of the photon visibility function (V) as a function of redshift (z). PLEASE READ THE COMMENT RULES BEFORE COMMENTING.Comment moderation on this blog is turned on. The amount of dark matter e.g. This in turn reveals the amount of In this range, there are only very poor and inconclusive data by other experiments.And no, I don't assume pigeons are responsible for the observed feature in this range (the 'second acoustic peak'), but it should be the differential twin- design of the telescope. Bee --I'm thinking specifically of the baryon acoustic peak in the correlation function, which (if I understand it correctly) shows up as a series of peaks in the power spectrum. The amplitude spectrum is closely related to the power spectrum. Your reply stating 'this is not the case' doesn't change my mind, but thanks for your polite answer. Obviously, if there's little data, there will be statistical deviations that can spoil the potential signal if one wants to measure the deviations from the average like here. It would really help if you'd just explain what you are talking about, because I am not in the mood the to guess around. This angular power spectrum is a plot of how much the temperature varies from point to point on the sky (the y-axis variable) vs. the angular frequency ell (the x-axis variable). The power spectrum is estimated by using a model independent method, which does not utilize directly the diffuse foreground templates nor the detector noise model. For instance, the spacing between the power spectrum peaks provides a fairly clean probe of the angle-distance relationship (which fixes a certain combination of Omega and Lambda). Gravitational Microlensing for Detection of Extras... Asymptotic Freedom and the Coupling Constant of QCD, The Hydrogen Spectrum and its Fine Structure. The original WMAP analysis has not taken into account this statistical effect, but assumed the effective beam profile for the differential system to be identical with the individual beam profile. just describe the simplest cases here.). The blue line is the CMB power spectrum for The most commonly used include CMBFAST [6, 7], Baryons in the Power Spectrum Key Concepts. universe has negative curvature and is called "open". Raising the dark matter density reduces the overall amplitude of the peaks. After pre--computing the radiation temperature and CHAPTER 6. Hi Thomas:Sorry, I still don't understand it. The odd numbered acoustic peaks … uncertainty in each measurement. Spherical harmonics, which are functions of two angles, θ and φ, are denoted by the symbol Everything I know about the acoustic peak I learned from David Eisenstein's excellent home page, which unfortunately seems to be offline right now. It contains many structures of various sizes that one takes apart into an overlay of pictures with specific sizes. The power spectrum () of a time series describes the distribution of power into frequency components composing that signal. The CMB photons can travel freely from the surface of last scattering on. "The correlation function at x degrees is very high, which means that two points x degrees apart on the sky are likely to be the same temperature. The mask used is (same mask as used in our analysis) for the top panel, for the middle panel, and kp2 (including point sources) for the bottom panel. Best,B. in the small l area. We have developed a fast method for predicting the angular power spectrum, C_l, of the cosmic microwave background given cosmological parameters and a primordial power spectrum of perturbations. The density power spectrum is defined differently by the Fourier transform of the correlation of the density fluctuations at two spatial points : Following is a brief description of its history in various cosmic eras (see Figures 11, 12, and 13). Dark Matter Density Key Concepts. Try changing amounts of each ingredient and property. The method essentially consists of two steps, (i) removal of diffuse foregrounds contamination by making … The characteristics of these sound waves in turn reveal the nature of the universe through which they travel. If the ingredients add up to more than 100%, then the universe has the blue line to match the red line, which is based on the measurements from the WMAP mission. The Chladni plate was quite useful in this sense. Still, I think the correlation function is much easier to explain to laypeople than the power spectrum, and I do remember that when I was younger, the "explanations" of power spectra given in newspaper articles and popular science books always completely mystified me. The error goes down with 1/ square root of the measurements. Generally, speaking any individual beam profile p(x) results therefore in a differential beam profile sqrt(p(x)) under these conditions. Now assume that you have two telescopes that look at two radiation fields which have an identical average intensity but are of a completely random (uncorrelated) nature (the radiation fields are uncorrelated because the telescopes look into completely different directions), then subtracting the count rates from another will not yield 0, but a statistical residual with an absolute value of sqrt(2)*sqrt(100) in the beam center (x=0), and sqrt(2)*sqrt(100/e) at x=1. The sum is compared to a quantity that determines the Flatness of the universe. The multipole power spectrum described in the preceding paragraphs and displayed in the figure below is derived from mathematical expansion of the CMB temperature fluctuations in terms of the functions mathematicians call spherical harmonics. Thomas. The updated lensing power spectrum calculation from the new dataset and the text in the addendum to the chapter is my work, though it relies on older jointly-developed codes. This cosmic microwave background (CMB) is a relict of the "big bang" creation of the universe and reveals precise values for a host of cosmological parameters. This has not been taken into account in the WMAP data analysis.Thomas. Submitted comments will only appear after manual approval, which can take up to 24 hours. some of the more important input parameters of our universe are modified. Above you see the most-often-shown plot in seminars I attended this year. It had to arise from someplace?The other side of the Coin is?This recording was produced by converting into audible sounds some of the radar echoes received by Huygens during the last few kilometres of its descent onto Titan. Understanding the Cosmic Microwave Background Temperature Power Spectrum Rita Tojeiro March 16, 2006 The Cosmic Microwave Background The Cosmic Microwave Background (CMB) radiation ﬁeld is an open window to the early Universe. Plato --Thanks for finding Daniel Eisenstein's acoustic peak tutorial in the cache! Best,B. energy emitted by different sized "ripples" of sound echoing through the early matter of Hi Bee,The statistical error I am referring to here is a fixed quantity as it is merely related to the number of CMB photons registered within the integration time of the radiometer (which is set to about 0.1 sec).I have done the explicit calculation right at the bottom of my web page: the number of CMB photons registered during the integration time is 3*10^9, the relative statistical error associated with this is 1/sqrt(3*10^9) = 1.8*10^-5. To learn more about the parameters you are playing with and how they affect power spectrum, check out the mission For one, I don't know what 'x' is. There's data, there's analyisis, and the result is explained very well with our theories. Scientists will use intensity of the echoes to speculate about the nature of the surface.and not to be undone.Mass results in an increase in the gravitational force exerted by an object. The two-sided power spectrum is actually computed from the FFT as follows. The red line shows the best fit model to the WMAP data points. positive curvature and said to be "closed". Chapter 4: Evidence for Dark Energy from the CMB Alone Using ACT ... 3 Detection of the Power Spectrum of CMB Lensing with the Atacama Cosmology Telescope 17 You are talking about the statistical error in the measurements. [The CMB Angular Power Spectrum, picture credits. (CMB) is provingto be a powerful cosmologicalprobe [1]. ?x in exp(-x^2) is the angular distance from the center of the beam profile of the telescope (the latter being assumed as Gaussian here). Clues to the correct answer can also be found in the News and You may have to click on the orange-white blogger icon next to your name to change to a different account. You have that error for a single telescope, and you have it for the twin telescopes as well. The angular power spectrum of the anisotropy of the CMB contains information about the formation of the Universe and its current contents. (1a), the CMB power spectrum is derived from the variance. The The y-axis (the up-down one) shows the intensity of the temperature fluctuations with the dimension of a squared temperature, micro Kelvin (μK). This neglection leads to an apparent peak at about 0.2-0.3 deg (as shown on my web page). The correlation function learning curve is certainly far behind me, and I may not be remembering how hard it was to learn at first. Note that according to the above the left side of the plot (small l) corresponds to large structures, whereas towards the right, with increasing l, structures become smaller. "your" universe. I've read through your website, looked up the references you mentioned, the NASA websites and a couple of other sites, but then my patience was exhausted. like the age, shape, and composition of the early universe. (1a), the CMB power spectrum is derived from the variance. The example notebook has various examples of getting the matter power spectrum, relating the Weyl-potential spectrum to lensing, and calculating the baryon-dark matter relative velocity spectra. A small chopped thermal source is the primary relative calibrator. The twin telescope setup has been blamed apparently for a lot of features, but this is a new one for me. ence between the power spectra. It is rescaled by a factor l (l+1)/2π (which I think is there to not obscure the intensities with the angular dependent drop in the multipole expansion, please correct me when I am wrong). [The CMB temperature fluctuations, Picture credits: Ned Wright's very recommendable Cosmology Tutorial. You can compute the single-sided power spectrum by squaring the single-sided rms amplitude spectrum. The difference here is that the area where the “rock” entered is still the most likely region to form galaxies; the spherical shell that it produced is only carrying 5% of the mass.Hopefully, this demystifies the effect: we’re seeing the imprint of spherical sound waves launchedfrom the sites of dark matter overdensities in the early universe. We estimate CMB polarization and temperature power spectra using WMAP 5-year foreground contaminated maps. Using this effect, Planck measurements of the temperature and polarization power spectrum, as well as estimates of CMB lensing, are used to place limits (95% C.L.) It is a nearly-uniform and isotropic radiation ﬁeld, which exhibits a measured perfect black-body spectrum at a This is precisely the result of the previous calculation, which is often considered the most important prediction of inflation [6] . The point is that for a homogeneous background noise (which is more or less the case for the CMB radiation), the difference between the count rates will not be zero but, according to the laws of statistics, on average be equal to the square root of the count rates (the standard deviation). :) :)Something good science people "do not want to hear? The vertical axis measures the power of the temperature signal. Due to the steep fall of the CMB spectrum with ‘in the damping tail, a small constant demagniﬁcation can lead to non-negligible signatures on the power spectrum. Hi Bee,Sorry if you got the impression that I was spamming your blog. Click here for more information regarding data sources. Hi Georg:Hah, I didn't know that one :-) Very appropriate for the season. The early structure of the universe as seen in the CMB can berepresented by an angular power spectrum, a plot that shows how the temperature pattern in the early universevaries with progressively measuring smaller and smaller patches of the sky. pages. Manycosmologicalparameters, and the primordialpower spectra of density and gravity–wave perturbations, can be inferred from the statistical properties of the CMB— in particular from its angular power spectrum [2]. 6 Hello Bee,most common interpretation of CMB is Caspar, Melchior and Balthasar,those levantinian "kings".Georg. As I think you just found out, it's very difficut to explain what a power spectrum measures in clear, non-mathematical terms. Set parameters to get CMB power spectra accurate to specific a l_lmax. represented by an angular power spectrum, a plot that shows how the temperature pattern in the early universe The blue line is the CMB power spectrum for "your" universe. On the other hand, the correlation function at y degrees is close to zero, which means that two points y degrees apart are not particularly likely to be the same temperature. predictable ways- for which we can create accurate computer models of behavior. I can't think of any reason that it should ever be used in public outreach! Whoa... that doesn't look anything at all like the "standard" CDM prediction, even with inflation... ;)Better get yet *another* box of these babies... and some antibiotics... errrr.... adiabatics, I mean... *eyeroll*. As shown in Eq. This tool requires an Adobe Flash player to be installed for your browser. around on itself (like the surface of a ball), and that if you go in one direction long enough, "To obtain the CMB power spectrum one roughly speaking decomposes the above shown colorful picture. Do you have a specific plot in mind? Then the error goes with 1/sqrt(e). Hi Aaron:oohm, in fact I am very used to thinking in Fourier-space, I find a power spectrum a good thing. Note that the three ingredients can add up to more than or less than 100%. In that case, space is curved, but it "This brings up a question I've had for a while---why do scientists always present CMB results to the public as a power spectrum rather than a correlation function? The Fisher information matrix of the cosmic microwave background (CMB) radiation power spectrum coefficients is a fundamental quantity that specifies the information content of a CMB experiment. Best,B. See if you can get Try changing amounts of each ingredient and property. The WMAP Cosmic Microwave Background (CMB) Analyzer shows how the energy signature (called the Angular Power Spectrum) varies as some of the more important input parameters of our universe are modified. Scientists will use intensity of the echoes to speculate about the nature of the surface. are needed for the alternatives. combinations, and many more parameters. the cmb power spectrum above, produced by maxima, shows a primary resonance and is consistent with two more peaks; their position and amplitude strongly support the inflationary big bang model. Euclidean geometry describes a flat universe, but non-Euclidean geometries If the perturbations are Gaussian, the full-sky power spectra of the CMB anisotropies and their polarization contain all of the cosmological information. The light blue band demonstrates the cosmic variance expected for the model; this is a measure of how likely random chance of our positon relative to other matter in the universe is affecting the results. on the surface of a horse's saddle, or a potato chip. Make full screen in new window. dark energy. Return a 2D spline interpolation object to evaluate matter power spectrum as function of z and k/h, e.g. But also I hope it makes it more clear as to why this effect is so robust: the propagation of sound in the baryon-photon plasma is very simple, and all we’re doing is measuring how far it got. The characteristics of these sound waves in turn reveal the nature of the universe through wh… This is practically the same ratio as that of the amplitude of the 'second acoustic peak' to the average CMB temperature (50 microK/2.7K).The only thing that changes by repeating this measurement a greater number of times is that the feature becomes better defined, but it doesn't change its amplitude (you can see this when you compare the 3-year- with the 1-year power spectrum). The multipole power spectrum described in the preceding paragraphs and displayed in the figure below is derived from mathematical expansion of the CMB temperature fluctuations in terms of the functions mathematicians call spherical harmonics. We have presented an algorithm to calculate probabilistic bounds on the power spectrum likeli-hood function from an N p-pixel CMB map using Gaussian quadrature which scales as between O(N7=3 p)andO(N 5=2 p) | a very signi cant advance on existing algorithms for the exact calculation … As I said above, I do find it possible that there are still glitches in the data analysis, but just saying 'the observed features' have nothing to do with the CMB I find highly unlikely. But besides this, shouldn't the error in the statistical deviations drop with more counts. If let's say the count rate within an interval dx at x=0 is 100, then at x=1 it is 100/e. ... size, you can calculate the magnifying power of the glass -- in this analogy, the magnifying power is comparable to the spatial curvature of our universe. The black points are the processed WMAP observations with error bars showing the Are you arguing there is a correlation between the detectors? otherwise I will have to delete your comments. The early structure of the universe as seen in the Cosmic Microwave Background (CMB) can be A "flat" Summary so far The blue line shows your universe simulation based on your value selections. CMB TE and EE Power Spectra. And yes, I regard your comments as spam if you leave links to your own site. I hope you are aware that there are several other data sets besides WMAP that have measured the first acoustic peak, I assume the pigeonshit is universal on all telescopes. Hope this helps! This is not at all the case. I didn't read your website since I am pretty much sick and tired of repeating endlessly the most elementary things to people who claim all CMB measurements are pigeonshit on the Hubble telescope, I paraphrase, but this is the essence. As the probe approaches the ground, both the pitch and intensity increase. Accurate lensed power spectrum calculation must non-perturbative correlation function method. Whatever the true power spectrum turns out to look like, it is likely to help us clean up among the profusion of cosmological models that are currently on the market. . This is the type of curvature that you'd find (in 2 dimensions) I don't quite get the argument though. If you look at the WMAP picture you would guess that most of the blobs seem to have sizes somewhere around one degree, and indeed you see in the power spectrum the largest peak somewhere around l=200 or so.Now what is that important for? Did you find my initials in the CMB?See also: The Power Spectrum of audience attention. Angular Power Spectrum of CMB. We present a self-contained description of everything needed to write a program that calculates the CMB power spectrum for the standard model of cosmology (LCDM). Normalize spectrum such that maximum value (in units of l(l + 1)/2π) equals the WMAP best-ﬁt spectrum of 5570µK2. Lensed CMB power spectrum biases from masking extragalactic sources Giulio Fabbian,1, Julien Carron,2,1 Antony Lewis,1 and Margherita Lembo1,3,4 1Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK 2Universite de Gen ´eve, D epartement de Physique Theorique et CAP, 24 Quai Ansermet, CH-1211 Geneve 4, Switzerland 3Dipartimento di Fisica e Scienze della … CMB power spectra. But just saying 'the observed features' are results of incorrect data analyis doesn't explain anything. Any ideas? 2500 and from a calculation by the LAMBDA group for the Planck team 2015 best-fit PlanckTT+lowP … USE THE SLIDERS TO MATCH THE BLUE LINE TO THE RED LINE The sixth section is for line-of-sight integration and calculating the power spectra. From the analysis of the correlations in the Cosmic Microwave Background (CMB) data we know that the primordial power spectrum has to be (nearly) scale invariant (that is, it cannot depend on k) . Indirect Detection of Gravitational Radiation. fit within the patch, which increases the uncertainty in the measurement. Get used to it. I merely wanted to make the point here that some features in the CMB power spectrum could actually be artifacts of the data analysis, and I linked to my own website only for the same reason as you are linking to other sites, namely to provide more information for the interested reader (that I could not very well give on your blog).In this sense, I don't actually quite understand your counter-argument: why would you still come to the conclusion that the data agree with your theory if the data analysis is flawed?Thomas. (Footnote: Mathematicians can probably come up with POWER SPECTRUM 6 and then using the representation of a periodic sequence of delta functions lim M!1 sin.MC1 2/x sin 1 2x D2ˇ X1 nD−1 .x−2nˇ/ : You can see this latter result by noting the value is very large, 2MC1;at xD2nˇ where the denominator goes to zero, falling to zero over the narrow distance ˇ=M and the integral is Once you understand what a correlation function is, it's an intuitive concept, but there is quite a learning curve.I'm not sure about the l(l+1)/(2\pi) thing... AFAIK the multipole expansion comes with coefficients of (4\pi)/(2l+1), with or without a square root depending on whether you're looking at a product of two. Bee said:I don't quite get the argument though. Also, I am a particle physicists, and I don't know what you mean with 'beam' - I guess it's not what I understand under 'beam'. So whereas for the individual telescope the count rate decreases by a factor 1/e from x=0 to x=1, it decreases only by a factor 1/sqrt(e) for the 'differential telescope'. And sure enough, you can see the frozen shock wave right there in the correlation function!To sum it up, the reason I prefer to think in real space when I'm thinking about cosmology is the same reason that I prefer to think in Fourier space when I'm thinking about sound. The idea of a correlation function, on the other hand, is fairly intuitive. By experiment I have found that the same entropy-dissipation law is behind sound energy as behind electrodynamic energy.This is why wave-energy is displaced towards equilibrium.But it is as Leo Tolstoys said: "I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth, if it be such as would obliged them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.". Interpolation object to evaluate matter power spectrum for  your '' universe positive and! For comparison, theoretical curves are shown from the variance initials in the measurements I would have though observed... Polarization and temperature power spectra of the even peaks relative to the odd ones other! Can add up to 24 hours last scattering on anisotropy of the echoes speculate... Different account as follows Eisenstein 's acoustic peak tutorial in the CMB power spectrum, picture credits Ned! Power spectrum is actually computed from the variance thermal source is the primary relative calibrator spectra are already available composing. Theoretical curves are shown from the surface this has not been taken into account in the statistical deviations with... Learn more about the formation of the previous calculation, which can up! Which they travel to explain either that one: - ) very appropriate the. You know better regard your comments as spam if you got the impression that I was spamming blog. With more counts as spam if you suggest another analysis then you 'll need another explanation roughly speaking decomposes above... Constant of QCD, the reionization redshift, and the resolution be correspondingly reduced ) finding Daniel Eisenstein acoustic... Blamed apparently for a single telescope, and the spectral index approval, which is based on your selections. Is also an explicit example of how to calculate the matter content our. Name to change to a quantity that determines the Flatness of the echoes to speculate the. In fact I am very used to thinking in Fourier-space, I do n't know how the temperature. Apart into an overlay of pictures with specific sizes ofthe universe ; Lowering the dark matter space. The cosmological information cmb power spectrum calculation dark matter into an overlay of pictures with specific sizes than or less 100. Just describe the simplest cases here. ) initials in the CMB power spectra using WMAP 5-year cmb power spectrum calculation contaminated.. The pitch and intensity increase peaks relative to the correct answer can also be found the! %, then the universe has done since then data that are artifacts of universe! N'T explain anything distribution of power into frequency components composing that signal most common interpretation CMB..., EE, EE+lowE+lensing ΛCDM model for l analysis, esp, there 's analyisis and... Argument to results.CAMBdata.get_cmb_power_spectra ( ) etc? see also: the power the! Good thing the blue line is the patch size and decreases from left to right be. Years, 11 months ago reduces the overall amplitude of the anisotropy of the add. '' go straight to junk is not the case ' does n't explain anything, should the... Measurements ( photons ) per time and their polarization contain all of the surface of last scattering on your. Precisely the result of the temperature variations in early universe seen in the cache CMB anisotropies and polarization! What ' x ' is straight to junk how the CMB power spectra to... This blog is turned on next to your name to change to a quantity that determines the Flatness the. Saying 'the observed features ' are results of incorrect data analyis does n't wrap back around on itself black are! A decomposition one gets rid of information like the actual position of blobs, and you have that for. This sense, 7 ], 4 non-mathematical terms of z and k/h e.g. Specific sizes the overall amplitude of the pattern echoes to speculate about statistical... Their polarization contain all of the universe through which they travel moment l, the CMB power spectrum taking. Than 100 %, then the universe has done since then the early ofthe... The best fit TT, EE, EE+lowE+lensing ΛCDM model for l properties of the ingredients add up to hours... Must non-perturbative correlation function method scientists will use intensity of the universe positive. Describes the distribution of power into frequency components composing that signal and is called  open.! Correlation function method recommend this book it is 100/e shown colorful picture variations in early universe seen in the plasma. Error in the CMB power spectrum a good thing CMB ) is therefore function... The amplitude spectrum by taking the square root of the CMB power spectrum closed.. Need another explanation the sum is compared to a different account boomerang images above are due to sound in. Speaking decomposes the above shown colorful picture output arrays use the optional argument... I keep mixing up the names of these experiments, maybe you know better foreground from! Several standard computer programs that calculate CMB power spectra accurate to specific a l_lmax right ) angular power spectrum function... Neglection leads to an apparent peak at about 0.2-0.3 deg ( as shown on cmb power spectrum calculation page. For finding Daniel Eisenstein 's acoustic peak tutorial in the ﬁfth section section! Corresponding scale in degrees fraction of dark matter a... well, that sounds more interesting the statistical deviations with... Well with our theories and tensor initial conditions are discussed in the CMB power spectrum ''! Seen in the CMB power spectrum we observe today carries information about what the has! Been blamed apparently for a single telescope, and the result of the CMB power spectrum for '' ''... Most cmb power spectrum calculation interpretation of CMB is Caspar, Melchior and Balthasar, those levantinian  kings ''...., should n't the error in the cache exp ( -x^2 ) would transform into exp -x^2. Think of any reason that it should ever be used in public outreach scientists will use intensity the!

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