Mathematical Analysis of Dark Energy Data

Transcription

1 Mathematical Analysis of Dark Energy Data Pavel Krejčíř April 22, 2009 Abstract Data collected by A. Riess et al. has been analyzed in this paper. The data provide red shift and estimated distance for 186 galaxies. Currently the commonly accepted conclusion drawn from this data is that the expansion of the Universe is being accelerated by some unknown force called dark energy. Simple mathematical and statistical methods have been used to analyze these data. The aim was to find a model which explains the behavior of distance vs. cosmological red shift and which could well predict the distance from known red shift. A very simple model of distance vs. red shift has been found in the way: d = 1 (z + 1) 2 1, H 1 2 where d is the measured distance, z is the red shift and H 1 is a modified Hubble constant. 1 Introduction A simple mathematical analysis of the data published by [Riess et al. (2004)] has been performed. Although the cited paper is marked as submitted to Astrophysical Journal, the electronic reference is given here in order to make clear which data this investigation refers to. The other citation is overall course to general relativity theory, which helped to clarify some aspects of cosmological red shift, cosmological models of the Universe etc. The last citation [Sumner (1994)] is a paper which can theoretically justify the obtained empirical result. The goal of this work was to purely examine the data using mathematical, statistical and visualization means. However, the model was derived by a way which certainly deserves some broader discussion. 2 The Data As mentioned in the previous section, the data has been taken from [Riess et al. (2004), Table 5]. The table contains record for 186 galaxies. It is a compilation of the author s own measurements and data from various other sources. The data contain red shift (z value) and extinction-corrected distance moduli (µ α 0 ) obtained by several methods. The analysis of the data presented in the cited paper is done on the two mentioned values. One of the intermediate aims of this work was to estimate a current distance, i.e. the distance of the measured objects at one common time - our current time. Unfortunately, as follows from [Grøn et al. (2007)], the calculation of velocity and position prediction at given time from the cosmological red shift is not straightforward. According to the current Friedmann-Lemaître-Robertson-Walker (FLRW) model, the velocity and displacement would be more or less linear function of the red shift if the expansion rate is constant, but it is quite complex task to estimate these values in the case when the constant rate of expansion cannot be assumed. Considering that, standard relativistic Doppler red shift has been used to estimate the velocity and hence current position, instead of a cosmological model. This has been considered as a first iteration. So formulas 1 and 2 have been used to calculate the distance and velocity respec- 1

2 tively: d = 10 µ α (1) v = (z + 1)2 1 (z + 1) (2) The distance unit is Megaparsec, the velocity unit is fraction of c. 3 The Analysis Once the data has been transformed to distance and velocity, they can be plotted as shown in the figure 1. Basically, us express the distance d = ct for some time interval T, then the body has overcome an extra distance equal to vt since the light has departed from its source. So the current distance can be calculated as d 1 = ct + vt = T(c + v) = d(1 + v) (3) under assumption that the bodies recede from the observer. The trick in the last equation in formula 3 is that the velocity has been calibrated so that c is the unit. The plot of the velocity against actual distance d 1 = d(1 + v) in expanding Universe is shown in figure 2. Velocity = v (c) Velocity = v (c) Distance = d*(1 + v) (Mpc) Distance = d (Mpc) Figure 1: The velocity plotted against the measured distance. the aim is to determine the rate at which the Universe is expanding. Suppose that there was a point in the history, called a Big Bang when all the bodies started to recede mutually from each other at different velocities. After some time elapsed, the bodies should be at different places, but still follow the simple formula d = vt, which implies that plotting velocity against distance, one should get a graph of simple straight line. But this is not the case of figure 1. Since the light spent some time to travel before any measurement could be taken, meanwhile, the emitters moved further in distance. So the objects need to be moved to the locations at one common time. The easiest choice of such a common time is our current time. Let Figure 2: The velocity plotted against the actual distance. Obviously there is no linear relationship between the data in figure 2 and the result is even worse than in figure 1. The data behave in opposite way than expected. So the question what would happen if the objects were approaching instead of receding naturally appears. The current distance would be d 2 = d(1 v), using the same arguments like in the formula 3. The result is shown in figure 3. The graph in figure 3 shows a clear linear relationship. Looking at the data from statistical point of view, two distinct part of data can be identified. Very accurate measurement for nearby galaxies, obtained probably by the Cepheid method, are in the lower left part of the graph - for velocities smaller than something between 0.1c and 0.2c. The part for greater velocities corresponds to the 2

3 Velocity = v (c) Velocity = v (c) Distance = d*(1 v) (Mpc) Distance = d*(1 v) (Mpc) Figure 3: The velocity plotted against the actual distance in collapsing Universe. Figure 4: The velocity fitted against the actual distance in collapsing Universe. less accurate Ia Supernova measurements. So an appropriate statistical model would be: { d 2 + e 1 for v , v (4) d 2 + e 2 for v > , where e 1 and e 2 correspond to random errors with two (possibly normal) different distributions. This kind of model is not very suitable for linear regression, especially because the border between the two different models is not known apriori. But the velocity variance can also be considered as proportional to the distance. So the model 4 can be replaced by 5: v d 2 + d 2 e (5) Using of logarithm stabilizes the variance: lg(v) lg(d 2 + d 2 e) = lg(d 2 ) + lg(1 + e) lg( v d 2 ) lg(1 + e) v d e (6) It follows from formula 6 that the best estimator of the expansion / contraction parameter is just a simple mean of v/d 2. This gives slightly different result than direct regression of v d 2, and visually a better fit. The result is shown in the figure 4. The estimated Hubble constant, whatever that means, is H 1 = c Mpc 1 = km s 1 Mpc 1. The new Hubble constant is marked by index 1 as opposite to conventional H 0 to emphasize the difference between these two constants. 1 4 The Model Regardless the validity of the investigation so far, a very simple model of relationship between red shift and estimated distance can be derived. In terms of relativistic Doppler red shift and corresponding velocity, the model is given by 7: d = 1 v (7) H 1 c v and the estimated distance d corresponds to the distance at the time when the light has been emitted from the distant source. The fit to all known observation is shown in figure 5. It could be objected that the relativistic velocity is improperly used in the model. However, using formula 2, the model 7 can easily be rewritten in terms of distance 1 The direct regression of v on d(1 v) gives c Mpc 1, which is slightly less. 3

4 Velocity = v (c) Distance = d (Mpc) Figure 5: The new model of the velocity - distance relationship. vs. red shift dependency: d = 1 H 1 (z + 1) (8) This model involves the measured red shift and distance only. It fits well to all known measurements and does not contain the disputable velocity. Considering the way how the formula 8 was derived, it seems to be very unlikely that a simpler model could be obtained. Regarding the model validity, there are two possibilities: 1. Further measurements will invalidate this model. 2. The model holds for the whole observable Universe and then a theory which would explain it is needed. The latter option will be further discussed. It is also interesting to compare the model 8 with the formula (11.43) of [Grøn et al. (2007)]. There is some similarity, except the sign of the quadratic term. 5 Explanation In order to theoretically justify the model, at least two questions need to be answered: 1. Why the cosmological red shift means approaching? 2. Why the motion caused by cosmological expansion / contraction can be handled as ordinary (relativistic) motion except the sign of red / blue shift? Surprisingly, the first question was answered by [Sumner (1994)] and this happened several years before the data showing dark energy were collected. The idea of Sumner is that when the geometry of Friedmann space changes, then the vacuum permittivity also changes and as a consequence, almost all Universe characteristics also change, including distance and time. Sumner shows that atoms change the frequency of emission, which prevail and negate the sign of the cosmological red shift. So the question which remains is: Why the cosmological velocity can directly be calculated as relativistic velocity from its red shift? To answer this is not so simple, but Sumner s calculation can help again. Please note at this point that this paper has no ambition to provide a new cosmological model which would definitively explain the observed behavior. It is rather a collection of ideas where the theory could be heading. Although haven t written that explicitly, Sumner has suggested that when the Friedmann radius changes then also the speed of light may change as well. The idea is such when permittivity changes, then it is likely that also the permeability changes, and thus the speed of light changes. Let a 0 and be two Friedmann radii. The relation of light velocities would then be: c 1 = a 0 c 0 (9) When light 2 starts traveling from an observer at time t 0 = 0, and as the time t 1 elapses, the Universe radius changes, the velocity of light changes according to 9 and the observer can see the light to be traveling at velocity c 1 = V 1 + c 1 (10) where V 1 is the velocity of the Universe expansion at the point where the light currently is. Let us further suppose that = a 0 + V 0 t 1 (11) which means that the Friedmann parameter changes linearly with time, but we still must be careful with which 2 light is used for simplicity throughout few next paragraphs. photon or head of a light beam would be more precise terms. 4

5 time it refers to. In this case to the time elapsed at the observer s site. Let us denote the distance of the light at time t 1 as s(t 1 ). The velocity of the Universe at the point s(t 1 ), t 1 is then given by: V 1 = s(t 1) V 0 (12) Combining 12 with 9 and 10 leads to the following differential equation for the location of light: s (t 1 ) = s(t 1) V 0 + a 0 c 0 = V 0s(t 1 ) + a 0 c 0 (13) The situation is schematically explained in figure 6. 6 Discussion Analogy of cosmological red shift with a string can be made - the greater tension on the string is applied, the higher frequency it carries. If a vibrating string is released, the frequency goes down producing sort of red shift. Similar thing might happen with space-time. An interesting consequence is that we observe an infinite Universe, which is infinitely old. Indeed, for velocities approaching c it follows that the distance approaches the infinity and the light has been emitted in the infinite past. However, the Universe is finite at the moment and will collapse in a finite time in the future, namely after about 15.3 billion years. We have currently no evidence that the Universe has ever expanded. If it would be so then we should probably observe very distant objects with blue shift, coming from the early stage of the Universe, from its expansion phase. But we observe nothing like that. 7 Conclusion Figure 6: Derivation of the light path in Friedmann- Sumner space-time. Considering that = a 0 + V 0 t 1 from 11, one can easily verify that s(t 1 ) = c 0 t 1 is the solution of 13. Thus the following principle can be postulated: The trajectory of light in linearly changing Friedmann space appears to be linear for any observer. If such a principle holds, then it would be impossible for some visible objects to recede at velocity greater than light, as currently accepted (since it is supposed that the speed of light can be added to the velocity of cosmological expansion). Combining this with the fact that objects with red shift greater than one exist, then there is a possible explanation that cosmological expansion / contraction has similar effect on frames as ordinary relativistic velocity, e.g. length contraction and time dilation. This may justify the usage of relativistic velocities derived from the red shift. Very simple model of distance vs. red shift dependency has been derived in this paper. The model fits all known data accurately. There is no justification for rejecting this simple model in favor of any other more complex model. The Universe can be considered as collapsing, at least until new observations invalidate the model. References [Grøn et al. (2007)] Grøn, Ø., Hervik, S. 2007, Einstein s General Theory of Relativity-with Modern Applications in Cosmology, Berlin: Springer [Riess et al. (2004)] Riess, A. G., Strolger, L.-G., Tonry, J., Casertano, S., Ferguson, H. C., Mobasher B., Challis P., Filippenko, A. V., Jha, S., Li, W., Chornock R., Kirshner, R. P., Leibundgut, B., Dickinson, M., Livio M., Giavalisco, M., Steidel, Ch. C., Benitez, N. & Tsvetanov Z. 2004, Type Ia Supernova Discoveries at z > 1 From the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy 5

6 Evolution, [Sumner (1994)] Sumner, W. Q. 1994, On the Variation of Vacuum Permittivity in Friedmann Universes, ApJ, 429: