Information Model of Cosmic Background Radiation

An informat ion based model is used to reanalyze the Cosmic Background Radiat ion (CBR) measured by the Cosmic Background Explorer, COBE, NASA satellite. The advantages of this mathematical model are that a radiation scattering parameter and the temperature at the detector can be calculated. from the data. Three models, the simple Black Body Radiation (BBR) curve, a model by D. J. Fixsen et al. and the Information Based Model (IBM) where compared to the measured data. The best Χ fit was obtained for the IBM. For the monopole radiation one obtains a temperature of 2.72503 K and a scattering parameter o f zero. For the d ipole radiation a temperature of 3.285843 K and a scattering parameter r equal to 2.0705567 is obtained.


Introduction
In a previous paper 1 the properties of the Cosmic Backg round Radiation (CBR) were analy zed using a Black Body Rad iation (BBR) model and a BBR model that included a chemical potential. A lso the author of this paper thanks D. J. Fixsen 1 et. al. for the excellent processing of the measured data.
Here an Info rmation Based Model (IBM) is used to reanalyze all the data. The analysis includes the effect of scattering of the radiation. The same model is used both for the monopole and dipole data discussed by D. J. Fixsen at al 1 . This model does not require the postulating of an addit ional chemical potential. A ll parameters are derived fro m basic principles, though the exact form of one of the functions had to be assumed. The model reverts to the standard BBR theory in the limit of no scattering.
The data was obtained by the Far Infra Red Absolute Spectrometer FIRAS on board the Cosmic Background Exp lorer COBE NASA satellite. Three models, the simp le BBR curve, a model by D. J. Fixsen et al. and the IBM where compared to the measured data. The best Χ 2 fit was obtained for the IBM.
The IBM, describes radiation that traveled through a scattering med iu m fro m a hot body. Using this model the temperature and a scattering parameter can be calculated. The conventional BBR function used in the analysis of thermal systems is derived by the use of Statistical Mechanics. The model used here is derived using Information Theory.
The basis of Informat ion Theory was developed by C. E. Shannon 2 at Bell Laboratories. The information theoretical model facilitates the inclusion of the effect of information propagating through noisy channels. This provides for including the effect of the thermal rad iation being scattered on the way fro m the hot Black Body source to the detector.
The energy of the electromagnetic radiation oscillat ing with any given frequency is divided into energy quanta or photons. The informat ion transmitted at any frequency by the hot body is encoded in the number of photons radiated. Different number of photons radiated represent different informat ion, see Fig. 1. The amount of informat ion in each photon packet is not equal to the number of photons but to a function of the number of photons. This is illustrated in Fig 1. The photon packets are represented by mail bags in Fig. 1. The information in each mail bag is displayed on the tags. The units of informat ion used in this schematic representation is in binary b its. Ho wever, the information used in this article is in Joules per Kelvin. The photons travel through space where some are absorbed or scattered. Maybe, the radiation when passing trough a charged cloud is even amplified by stimulated emission. This can occur by generating additional photons of the same frequency as the incoming radiation and neutralizes some of the charges.
One result of including the effect of scattering is a blue shift of the distribution of photons, see Fig. 2. Unlike in the case of the Doppler effect the wavelength of the indiv idual photons does not change. However, the distribution changes to mo re short wave photons. The total number of photons can, also, change to fewer or mo re photons.

Information Model
The IBM for thermal radiat ion through a scattering med iu m is analyzed below. The concepts used here can be found in a paper by C. E. Shannon 2 and many Probability texts 3,4,5 . The informat ion is encoded by the number n of photons radiated by the hot object, see h n = -k ln P n (1.1) where k is Bo ltzmann's constant and P n is the probability that a signal of n photons is being received. The average detected Shannon informat ion is equal to the average value H of all the information packets.
The propagation of the photons from the hot Black Body source to the detector is modeled by conditional probabilities P(m photons radiated | n Photons received) that m photons are radiated provided n photons were received. Associated with the conditional probabilities having the same condition of n photons being received are conditional entropies H(S | n). Here S is the set of all the different nu mbers m of radiated photons.
Since the details of the scattering model are not known the conditional probabilities can not be specified. However, one The temperature T of the radiat ion at the receiver is known.
where the average energy U of the received photons is given by: Its value, at this point, is not known. Here  is Plan k's constant divided by 2π and ω is the oscillating frequency of the radiation.
The probabilit ies P n are normalized.
The probabilit ies P n can be derived by finding an extremu m value of the informat ion I subject to what is known about the system. In this case the temperature T at the receiver and the fact that the probabilities are normalized are known about the system. However, the equation 1.6 for the temperature, is not in the form of a constraint equation like equations 1.7 and 1.8. Therefore, it can not be used in this process directly. One has to use the average energy U instead, at first. By mult iply ing the two constraint equations, equations 1.7 and 1.8 by convenient constants αk and -βk and adding them to the equation for the informat ion I one obtains: The information I will have an extremu m value when all its derivatives with respect to the probabilities P n are equal to zero. By taking the derivative of the information I with respect to one of the probabilities P n , setting the result equal to zero and solving for the probability P n one obtains: where the exp licit form of the conditional entropy H(S | n) fro m equation 1.3 was used in equation 1.10. The values of the constants α and β are not known at this point. In order to evaluate the constant α one substitutes the probability of equation 1.10 into the first constraint equation, equation 1.8 to obtain for exp -1 -α The constant α can be eliminated by substituting equation 1.11 into equation 1.10.
The constant β has yet to be evaluated. In order to accomplish this one, first, calculates the information -kln P n ( ) of receiving n photons by taking the logarithm of the probability P n and mult iplying the result by minus the Bolt zmann's constant.
By substituting the information associate with receiv ing n photons, equation 1.13, into the average Shannon informat ion of equation 1.2 one obtains: where equation 1.7 was used for the average energy U. By solving equation 1.14 for the average energy U, substituting the resulting expression into equation 1.6 and solving for β one attains the well known expression: The probability P n of receiv ing n photons can now be completely specified by substituting equation 1.15 for the constant β in to equation 1.12 The average energy of a one dimensional radiat ing system where the radiation passes through a scattering med iu m with average scattering parameter ρ is derived by substituting the probability of equation where c is the speed of light in free space. This is the Black Body Radiation law fo r systems radiating through a scattering med iu m. Note that the temperature T is the observed temperature at the receiver. Since the scattering parameter ρ is an Entropy amplitude it must always be positive. This shifts the peak of the distribution to larger values of the normalized frequency Joules m 3 -Hertz -K .
As will be shown in chapter III, RESULTS, for the CBR that the data obtained from a monopole detector exactly matches a standard BBR curve as given by equation 1.20b. However, the data fro m a d ipole detector better fits equation 1.19. The explanation given here of the shift of the photon distribution to higher frequencies is doe to the effect of the scattering process as discussed above. This is a much simp ler explanation than the one given by D. J. Fixsen et al. Thus, by Occam's Razor, perhaps the scattering exp lanation is the correct one?

Results a) Anal ysis method
In order to determining the values of the temperature T at the receiver and the average scattering parameter ρ fo r the radiation of the CBR the data listed in Tables 1 or 2 is compared to data calculated from equation 1.19. To accomplish this comparison the observed CBR data is normalized by divid ing the second column of the data by its maximu m value. The data calcu lated fro m equation 1.19 is similarly normalized by div iding it by its maximu m value.
Next the Xi squared Χ 2 value is calculated as follo ws:  The monopole data was measured using the FIRAS wh ich measures the difference between the free sky radiation and an internal ideal BBR source. The data underwent substantial processing to eliminate various instrument errors. Depending on the error eliminating technique D. J. Fixsen at al 1 . calculate a CBR temperature of between 2.723 ± 0.001 o K and 2.7255 ± 0.0009 o K .
The data in Table 1 was copied fro m "on line" NA SA COBE/ FIRAS Monopole data. Colu mn 1 is the spatial frequency in units of cm -1 . Colu mn 2 is the FIRAS monopole spectrum co mputed as the sum of a 2.725 o K BBR spectrum and the residual in column 3. The units are MJy/sr (Mega Jansky per sterradian). Colu mn 3 is the residual monopole spectrum fro m Tab le 4 of Fixsen et al 1 in units of kJy/sr. Colu mn 4 is the one standard deviation spectrum uncertainty fro m Table 4 o f Fixes et al, in un its of kJy/sr, and co lu mn 5 is the modeled Galaxy spectrum at the Galactic poles fro m Table 4 of Fixsen et al, also in units of kJy/sr.
By matching the data in colu mns 1 and 2 of Table 1 to equation 1.19 one obtains a best fit for a temperature of 2.725(030813) o K, and a scattering parameter ρ of zero for a Χ 2 value of 1.50643015 x 10 -6 . This agrees well with the values calculated by D. J. Fixsen et al. Since this result was calculated fro m the data in Table 1 it was subject to the same error correct ion method used by D. J. Fixsen et al. It assumes that since the scattering parameter is an Entropy amplitude it must be positive c) Anal ysis of di pole data.
The best fit of the CBR dipole data listed in colu mns 1 and 2 of Table 2 to a standard BBR curve, equation 2.20b is for a temperature of 3.45306053 o K, corresponding to a Χ 2 value of 0.10854900837.
The best fit of the Cosmic Background Rad iation dipole data listed in colu mns 1 and 2 of Table 2   The IBM has the lowest value of X 2 and is thus the best fit to the dipole data of the three models discussed here. Recall that the three models were the simple BBR curve, Fixsen's BBR curve with chemical potential and the IBM . A comparison of models is listed in Table 3  The interpretation of the dipole result differs fro m the one given by D. J. Fixsen et al. The interpretation here is that the dipole co mponent of the radiation, unlike the radiation measured with a monopole instrument, is scattered before it reaches the detector.. The effect of the uncertainty in the data in Table 2 on the best fit temperature and scattering parameter are listed in Table 4.  Table 2 is shown in Fig.  4. The data in Fig. 4 is plotted as a function of the wave vector 2π λ in m-1.
The explanation given here of the shift of the photon distribution to higher frequencies is doe to the effect of the scattering process as discussed above. This is a much simp ler explanation than the one given by D. J. Fixsen et al. Thus by Occam's Razor, perhaps the scattering exp lanation is the correct one?.   Table 2  The reason that the dipole rad iation measured at the detector appears to be subject to scattering while the scattering parameter o f the monopole radiat ion is equal to zero as well as the meaning of the value of the temperature of the dipole rad iation source is left to wiser researchers than the author. Data plus uncertainty Data minus uncertainty