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The type Ia supernovae and the Hubble's constant

The Hubble's constant is usually surmised to be a constant; but the experiments show a large spread and conflicting estimates. According to the plasma-redshift theory, the Hubble's constant varies with the plasma densities along the line of sight. It varies then slightly with the direction and the distance to a supernova and a galaxy. The relation between the magnitudes of type Ia supernovae and their observed redshifts results in an Hubble's constant with an average value in intergalactic space of 59.44 km per s per Mpc. The standard deviation from this average value is only 0.6 km per s per Mpc, but the standard deviation in a single measurement is about 8.2 km per s per Mpc. These deviations do not include possible absolute calibration errors. The experiments show that the Hubble's constant varies with the intrinsic redshifts of the Milky Way galaxy and the host galaxies for type Ia supernovae, and that it varies with the galactic latitude. These findings support the plasma-redshift theory and contradict the contemporary big-bang theory. Together with the previously reported absence of time dilation in type Ia supernovae measurements, these findings have profound consequences for the standard cosmological theory.


Introduction
We assume that the universe is homogeneous and isotropic on a scale of about R = c/H 0 ≈ 5, 000 Mpc ≈ 1.5 · 10 26 m. Einstein assumed this "cosmological principle" in his first static model of the universe. On smaller scales the universe is inhomogeneous and non-isotropic. Einstein's static model was replaced by a big-bang theory with an explosive event leading to an expansion, which was supposed to supplant Einstein's Λ term. Later, it has been necessary to reintroduce the Λ term in form of a time variable dark energy, and to assume dark matter for explaining the observations [1]. In this article, we usually refer to this as the contemporary big-bang theory. We have shown that plasma redshift, which is based entirely on well proven conventional physics, gives a much simpler explanation of the observations [2,3]. Plasma redshift is an overlooked cross section for energy loss of photons as they penetrate hot sparse plasma. When considering the plasma-redshift crosssection, there is no need for big bang, dark matter, or time variable dark energy. Plasma redshift leads to natural explanation of the Hubble's constant and its observed variations, as we show in the following. Plasma redshift also gives a natural explanation of the highly isotropic cosmic microwave background [2] (see Section 5.9 of that source). It results in a quasi-static and constantly renewing universe without black holes [2] (see Section 6 of that source).
The estimates of the Hubble's constant have standard deviations that are much smaller than the spread, usually between 40 and 100 km s −1 Mpc −1 , as reported by different observers. The measurements of the Hubble's constant are often done using objects (stars, Cepheid variables, galaxies, and supernovae) that are at intermediate distances. The basic distance scales are rooted in parallaxes to nearby stars and Cepheid variables. These distance scales are then used as measuring rods to galaxies and supernovae. In the big-bang theory we assume that the distance R is proportional to the redshift, z, and convert the distance scale to a redshift scale.
According to the plasma-redshift theory, the conversion to redshift scale can lead to errors, because the redshifts per unit distance to the Cepheids in nearby galaxies are relatively large. The distortion is caused by the higher plasma densities in the coronas of the Milky Way and the host galaxy, and in the space between the galaxies in clusters. The average electron densities inside the local clusters have higher electron densities between the galaxies than that in the space between the clusters. The higher electron densities along the lines to nearby galaxies will result in high estimates of the Hubble's constant, H 0 . For example, it may be estimated for the nearby galaxies to be about 80 km s −1 Mpc −1 , when in the space between the clusters it may be less than 60 km s −1 Mpc −1 .
We give in Section 2 the basic equations that determine the Hubble's constant and the magnituderedshift relation. In Section 3, we compare the equations and the data reported by Riess et al. [1].
In Section 4, we analyze the variations in H 0 . In Section 5, we give the conclusions.
2 Relations between distances, electron densities, Hubble's constant, redshifts, and magnitudes Brynjolfsson has shown [2] (see in particular Eq. (47) of that source) that the plasma redshift cross section for interaction of photons with an electron plasma with electron density N e cm −3 leads to where z is the observed plasma redshift, and x and R are in units of cm. When the electron density, N e , from 0 to R is replaced with its average value, (N e ) av , along the stretch, we get that For R = 1 Mpc = 3.0851 · 10 24 cm, the redshift is z = H 0 /c, where the Hubble's constant, H 0 , is in km s −1 Mpc −1 , and the velocity of light, c, is in km s −1 . We get then from Eq. (2) that ln (1 + z) ≈ z = H 0 c = 3.3262 · 10 −25 · 3.0851 · 10 24 (N e ) av , or H 0 = 3.0764 · 10 5 (N e ) av or (N e ) av = 3.2506 · 10 −6 · H 0 where the Hubble's constant, H 0 , is in km s −1 Mpc −1 , and the electron density, (N e ) av , is in cm −3 . From Eqs. (2) and (3), we get when R is in pc, c in km s −1 , and H 0 , is in km s −1 Mpc −1 , that For small z-values, we get the Hubble's law: The reduction in the light intensity from a star, a supernova, or a galaxy with R is given by where the factor exp (−a av R) accounts for the absorption by atoms, molecules and grains along the line of sight, including the absorption in the corona of the Milky Way and the host galaxy. The factor exp (−b (N e ) av ) accounts for the reduction in light intensity caused by the redshift, and exp (−2b (N e ) av ) accounts for the Compton scattering in the narrow beam geometry of intergalactic space. The factor b = 3.3262 · 10 −25 is the cross section for the plasma redshift, and 2b is the cross section for the Compton scattering.
Eq. (6) can be simplified with help of the first relation in Eq. (2). We get The observed magnitude, m, is defined such that I(R) = 10 −0.4 m . The absolute magnitude, M, is defined as the intensity at a distance of 10 pc without any absorbing materials between the observer and the object; that is I 0 /10 2 = 10 −0.4 M . When we insert these values into Eq. (7) and take the logarithm of the expression, we get where the magnitude, A, of absorption is defined as A = −2.5 · log (exp{ −a av R}) .
When we for R insert Eq. (4) into Eq. (8), we get For small redshifts, we can replace the natural logarithm of (1 + z) with z, and get then the Hubble's law. It is of interest to compare this equation with a similar equation in the big-bang theory, and disregard the deceleration and the acceleration (expansion) terms, see Eq. (23) by Sandage [4]: The second term on the right side consists of two parts, 2.5 · log (1 + z) + 2.5 · log (1 + z). The first half accounts for the energy loss caused by redshift, and the second half for the intensity loss caused by time dilation (or slower rate of clocks at high velocities). The two equations (9) and (10), although fundamentally very different, are numerically almost equal. When we subtract Eq. (9) from (10), we get The numerical values of Eq. (11) are listed in Table 1. It is seen that the values of the two is nearly the same. Even at z = 2, the magnitude value of ∆(m − M) = 0.1081 is hardly detectable.  The magnitude-redshift relation can usually not be used to see which theory is right. In the contemporary big-bang theory, it is not reasonable to omit the deceleration and acceleration terms. These terms can be made to fit almost any magnitude-redshift observations. In the magnituderedshift relation defined by the plasma redshift, Eq. (9), there is no such flexibility.

Redshift-magnitude relations and Malmquist bias
Let us assume that the type Ia supernovae (SNe Ia) are perfect standard candles with a well-defined maximum luminosity. Given these assumptions, would the researchers measuring the magnitudes and the redshifts be able to see which of Eq. (9) or (10) is right? The answer is no, because both equations give answers that within the accuracy of the measurements fit the expectations. Could we omit the time dilation? Those who assume that the plasma redshift is correct would say: "There is no time dilation." The supernova researchers, who are used to the contemporary big-bang theory would say: "No, we cannot omit time dilation, because if we omit the time dilation the SNe Ia would appear brighter than that observed. The supernova could not be a standard candle, which other indicator show it is." The supernova researchers, who are unaware of plasma-redshift cross-section, find that because the observations fit the magnitude-redshift relation in the contemporary big-bang theory, the time dilation is required. It is thus seen that if the supernovae are perfect standard candles, the observations of the magnitude-redshift relation cannot tell us which theory is right.
We might, however, use other methods to see which theory is more reasonable [3]. One of them uses the fact that the SNe Ia are imperfect standard candles [5].  5)). When using these equations as a rough guide, we should observe a significant increase in brightness with increasing redshift z. The fact that Goldhaber et al. [8] (see in particular Fig. 3b of that source) observe no increase whatsoever in the brightness with increasing z indicates that their assumed time dilation, when making the light-curve correction, reduces the absolute-brightness estimates. This decrease in the estimated brightness is introduced when the supernova researcher adjust the brightness according to the width of the light curve [3]. The same applies to the data reported by Richardson et al. [6].
Brynjolfsson [3] has shown that we get about the right brightness of each SNe Ia, when we correct the absolute magnitude, as determined by the supernova researcher, by the term ∆M = −2.5 log (1 + z). (A slightly better method would be to use the stretch factor, s ′ , obtained by setting the width of the light curve equal to w = s ′ , instead of s, where the measured width is w = s (1 + z). ) We therefore have that the actual absolute magnitude M is where M exp is the absolute magnitude as reported by the supernova researchers, who have assumed that the time dilation applies when they use the light-curve width-correction for estimating the absolute magnitude. The supernova researchers assumed that the contemporary big-bang theory was correct. Therefore, they had to use the time dilation for this correction. The plasma-redshift theory assumes that the universe is quasi-static and has therefore no time dilation, see Eq. (9). The light-curve width-correction, which the supernova researchers applied, resulted in an underestimate of the brightness. When we correct this underestimate by use of Eq. (12), we get an unusually good fit between the observed magnitude-redshift relation and that predicted by the plasma redshift. For this, see Fig. 1.  Although these corrections are small and nearly insignificant, they are included here because this order of magnitude for the coronal electron density is consistent with many observations [2] (see in particular Section 5.7.1 of that source), which are independent of the present estimates that are derived only from the supernova experiments. The average electron density in each corona includes the higher densities (because of lower temperatures) in the transition zone to each corona and the densities in the HII-regions within the galaxies.

Spread in the Hubble's constant
The value of H 0 determined in this way for the nearby 75 SNe Ia gives about the same standard deviation, σ(m − M) = 0.296, from the theoretical curve along the entire curve. It is similar to that about σ(m − M) = 0.29, reported by Riess et al. [1] (see Section 5.2 of that source). In accordance with Eq. (3), we find that the standard deviation in H 0 is caused partially by a electron density along the line of sight to the supernova. When the line of sight passes through the corona of another galaxy, or through the higher densities in the space between galaxies in a cluster, the average density is higher. According to Eq. (3), this higher electron density results in larger value of H 0 . A large Hubble's constant according to Eq. (9) results in a smaller value of (m − M).
The distribution of the magnitude around the theoretical curve is nearly gaussian for the 186 SNe Ia (the silver as well as gold samples, as listed by Riess et al. [1]). 49 samples out of the 186 (or 26.3 %) have |∆(m − M)| ≥ 0.3. If the distribution is gaussian, we should expect about 57.8 samples outside the 0.3 limit. The standard deviation in 57.8 is about 7.6. We find therefore that the 49 samples outside the 0.3-limit is consistent with a gaussian distribution. The supernova researchers select the good samples. It is likely that other concurrent indicators give reason to discard samples with large deviations.
Of It is remarkable to find how consistent the latitude effect is independent of the redshift z. In each case the latitude difference exceeds about ∆b = 10 • . A priori, we expect the electron density integral, R 0 N e dx, to increase slightly with decreasing galactic latitude. As the galactic latitude decreases the integral increases because of the increased path length, mainly through the transition zone to the galactic corona. We expect this to be a small effect for the Milky Way, because of many other factors affecting the deviations from the theoretical curve. It is of interest that this galactic latitude effect is clearly supported by the observations. The latitude effect caused by absorptions A in Eqs. (9) and (10) has been thoroughly documented. The supernova researchers have done a thorough work at evaluating all corrections. We have therefore good reason to expect that Riess et al. [1] have made these corrections correctly. We find therefore that not only the average correction, ∆z = −0.00185, for the average intrinsic redshifts for the Milky Way and the host galaxy for the supernova, but also the above documented latitude effect are consistent with Eq. (3), which in turn affects Eq. (9) in the observed way. These effects, which are consistent with the plasma-redshift theory, are inconsistent with the contemporary big-bang theory.

Conclusions
In reference [3], we found that the supernova data [1, 5, 6, 8, and 10] show that there is no time dilation. This contradicts the contemporary big-bang theory, which has time dilation as a basic premise. The contemporary big-bang theory in addition to big bang and time dilation surmises dark matter and time variable dark energy for explaining the observations. In contrast the plasmaredshift, an overlooked cross section for interaction of photons with hot, sparse plasma, is based on conventional physics, and requires no such adjustable parameters for explaining the observations, as illustrated in Fig. 1. In this paper, we have shown that the experimental data, are consistent with plasma redshift, and indicate an intrinsic redshift for the Milky Way and the host galaxy for each supernova. This intrinsic galactic redshift, ∆z = 0.000925, in the Milky Way corresponds to R 0 N e dx = 2.78 · 10 21 cm −2 . This average column density for the electrons derived from the supernova data is consistent with independent observations of the coronal redshifts analyzed in reference [2] (see in particular Section 5.7.1 of that source). The observations indicate that this intrinsic redshift depends slightly on the galactic latitude. We have thus a series of observations, all of which contradict the contemporary big-bang theory, but are consistent with the plasma-redshift theory [2].
The result from the Hubble Space Telescope Key Project [9] found the Hubble constant from type Ia supernovae experiments to be H 0 = 71 ± 3 random ± 7 systematic km s −1 Mpc −1 . The supernova data with z ≤ 0.97 reported by Riess et al. [10] were used in reference [2] without correcting them for the time dilation (see Fig. 5 of that source). This resulted in H 0 = 65.23. But already then these experimental data indicated a need to back-correct them for the faulty time dilation. When evaluating the experimental data reported by Riess et al. [1], with z ≤ 1.755, it became necessary to correct them for the faulty time dilation. In reference [3], the removal of the time dilation from the experimental determination of the magnitude of the supernovae results in them being brighter than that estimated by Riess et al. [1]. This together with the expanded data in [1] brings the value of the Hubble's constant from about 65.23 to about H 0 ≈ 61.6 km s −1 Mpc −1 . A smaller correction is caused by the intrinsic redshift in the Milky Way and the host galaxy, which reduces the intergalactic value of the Hubble's constant to H 0 ≈ 59.44 km s −1 Mpc −1 . The random uncertainty in the average is small or about ±0.6. To this we must add the uncertainty in the determination of the absolute or systematic value of H 0 , which is based mainly on the uncertainty in the Cepheid data. The evaluation in light of plasma redshift, has already reduced a significant fraction of those systematic errors in H 0 that are caused by intrinsic redshifts of the galaxies and galaxy clusters. It is likely that in the future, we can make better estimates of the electron density along the line of sight and use such estimates for estimating the Hubble's constant, which then can be used to reduce the deviations and uncertainties in the magnitudes of the supernovae, and Cepheid variables.